%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT179^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:45:02 AM UTC 2026
% Result : Theorem 9.74s 2.75s
% Output : Refutation 9.74s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT179^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n020.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 06:10:57 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.42 ---- Original DTF file ---
% 0.19/0.42 thf(spec,logic,$$dhol).
% 0.19/0.42 %------------------------------------------------------------------------------
% 0.19/0.42 % File : DAT179^1 : TPTP v9.2.1. Released v7.0.0.
% 0.19/0.42 % Domain : Data Structures
% 0.19/0.42 % Problem : Lazy lists II 596
% 0.19/0.42 % Version : [Bla16] axioms : Especial.
% 0.19/0.42 % English :
% 0.19/0.42
% 0.19/0.42 % Refs : [Fri04] Friedrich (2004), Lazy Lists II
% 0.19/0.42 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.19/0.42 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.19/0.42 % Source : [Bla16]
% 0.19/0.42 % Names : llist2__596.p [Bla16]
% 0.19/0.42
% 0.19/0.42 % Status : Theorem
% 0.19/0.42 % Rating : 0.67 v8.1.0, 0.50 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% 0.19/0.42 % Syntax : Number of formulae : 360 ( 141 unt; 66 typ; 0 def)
% 0.19/0.42 % Number of atoms : 701 ( 292 equ; 0 cnn)
% 0.19/0.42 % Maximal formula atoms : 13 ( 2 avg)
% 0.19/0.42 % Number of connectives : 2979 ( 69 ~; 11 |; 36 &;2544 @)
% 0.19/0.42 % ( 0 <=>; 319 =>; 0 <=; 0 <~>)
% 0.19/0.42 % Maximal formula depth : 19 ( 6 avg)
% 0.19/0.42 % Number of types : 5 ( 4 usr)
% 0.19/0.42 % Number of type conns : 219 ( 219 >; 0 *; 0 +; 0 <<)
% 0.19/0.42 % Number of symbols : 65 ( 62 usr; 7 con; 0-5 aty)
% 0.19/0.42 % Number of variables : 866 ( 29 ^; 760 !; 21 ?; 866 :)
% 0.19/0.42 % ( 56 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.42 % SPC : TH1_THM_EQU_NAR
% 0.19/0.42
% 0.19/0.42 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.19/0.42 % 2016-07-13 14:47:34.314
% 0.19/0.42 %------------------------------------------------------------------------------
% 0.19/0.42 %----Could-be-implicit typings (8)
% 0.19/0.42 thf(ty_t_Coinductive__List_Ollist,type,
% 0.19/0.42 coinductive_llist: $tType > $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_t_Option_Ooption,type,
% 0.19/0.42 option: $tType > $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_t_Set_Oset,type,
% 0.19/0.42 set: $tType > $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_t_Num_Onum,type,
% 0.19/0.42 num: $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_t_Nat_Onat,type,
% 0.19/0.42 nat: $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_t_Int_Oint,type,
% 0.19/0.42 int: $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_t_itself,type,
% 0.19/0.42 itself: $tType > $tType ).
% 0.19/0.42
% 0.19/0.42 thf(ty_tf_a,type,
% 0.19/0.42 a: $tType ).
% 0.19/0.42
% 0.19/0.42 %----Explicit typings (58)
% 0.19/0.42 thf(sy_cl_HOL_Otype,type,
% 0.19/0.42 type:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.42 ord:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.42 order:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Rings_Osemiring__1,type,
% 0.19/0.42 semiring_1:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.42 linorder:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.42 preorder:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.19/0.42 semiring_char_0:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.19/0.42 linordered_semidom:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.19/0.42 linord219039673up_add:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 0.19/0.42 archim1727834104eiling:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.19/0.42 comple187826305attice:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.19/0.42 archim1804426504_field:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.42 canoni770627133id_add:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.19/0.42 ordere236663937imp_le:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.19/0.42 ordere516151231imp_le:
% 0.19/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Archimedean__Field_Oround,type,
% 0.19/0.42 archimedean_round:
% 0.19/0.42 !>[A: $tType] : ( A > int ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.19/0.42 coinductive_LNil:
% 0.19/0.42 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.19/0.42 plus_plus:
% 0.19/0.42 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.42 zero_zero:
% 0.19/0.42 !>[A: $tType] : A ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 0.19/0.42 hilbert_GreatestM:
% 0.19/0.42 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 0.19/0.42 hilbert_LeastM:
% 0.19/0.42 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_If,type,
% 0.19/0.42 if:
% 0.19/0.42 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Int_Onat,type,
% 0.19/0.42 nat2: int > nat ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
% 0.19/0.42 lList2435255213lllsts:
% 0.19/0.42 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
% 0.19/0.42 lList22096119349pslsts:
% 0.19/0.42 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
% 0.19/0.42 lList21612149805nflsts:
% 0.19/0.42 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Olbutlast,type,
% 0.19/0.42 lList2370560421utlast:
% 0.19/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oll2f,type,
% 0.19/0.42 lList2228235758e_ll2f:
% 0.19/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( option @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollength,type,
% 0.19/0.42 lList21232602520length:
% 0.19/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oltake,type,
% 0.19/0.42 lList22119844313_ltake:
% 0.19/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
% 0.19/0.42 lList21148268032oslsts:
% 0.19/0.42 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Nat_OSuc,type,
% 0.19/0.42 suc: nat > nat ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.19/0.42 semiring_1_of_nat:
% 0.19/0.42 !>[A: $tType] : ( nat > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 0.19/0.42 semiri532925092at_aux:
% 0.19/0.42 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.19/0.42 size_size:
% 0.19/0.42 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
% 0.19/0.42 nat_tr1645093318rphism:
% 0.19/0.42 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Nat__Transfer_Otsub,type,
% 0.19/0.42 nat_tsub: int > int > int ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.19/0.42 numeral_numeral:
% 0.19/0.42 !>[A: $tType] : ( num > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Obind,type,
% 0.19/0.42 bind:
% 0.19/0.42 !>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ocombine__options,type,
% 0.19/0.42 combine_options:
% 0.19/0.42 !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ois__none,type,
% 0.19/0.42 is_none:
% 0.19/0.42 !>[A: $tType] : ( ( option @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_ONone,type,
% 0.19/0.42 none:
% 0.19/0.42 !>[A: $tType] : ( option @ A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_OSome,type,
% 0.19/0.42 some:
% 0.19/0.42 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_Ocase__option,type,
% 0.19/0.42 case_option:
% 0.19/0.42 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_Opred__option,type,
% 0.19/0.42 pred_option:
% 0.19/0.42 !>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_Orec__option,type,
% 0.19/0.42 rec_option:
% 0.19/0.42 !>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_Oset__option,type,
% 0.19/0.42 set_option:
% 0.19/0.42 !>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_Osize__option,type,
% 0.19/0.42 size_option:
% 0.19/0.42 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Option_Ooption_Othe,type,
% 0.19/0.42 the:
% 0.19/0.42 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.42 ord_less_eq:
% 0.19/0.42 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Pure_Otype,type,
% 0.19/0.42 type2:
% 0.19/0.42 !>[A: $tType] : ( itself @ A ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_Set_OCollect,type,
% 0.19/0.42 collect:
% 0.19/0.42 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_c_member,type,
% 0.19/0.42 member:
% 0.19/0.42 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(sy_v_Q,type,
% 0.19/0.42 q: $o ).
% 0.19/0.42
% 0.19/0.42 thf(sy_v_i,type,
% 0.19/0.42 i: nat ).
% 0.19/0.42
% 0.19/0.42 thf(sy_v_j,type,
% 0.19/0.42 j: nat ).
% 0.19/0.42
% 0.19/0.42 thf(sy_v_t,type,
% 0.19/0.42 t: coinductive_llist @ a ).
% 0.19/0.42
% 0.19/0.42 thf(sy_v_x,type,
% 0.19/0.42 x: a ).
% 0.19/0.42
% 0.19/0.42 %----Relevant facts (253)
% 0.19/0.42 thf(fact_0_H,axiom,
% 0.19/0.42 ! [Y: a] :
% 0.19/0.42 ( ( ( lList2228235758e_ll2f @ a @ t @ j )
% 0.19/0.42 = ( some @ a @ Y ) )
% 0.19/0.42 => q ) ).
% 0.19/0.42
% 0.19/0.42 % H
% 0.19/0.42 thf(fact_1_jlei,axiom,
% 0.19/0.42 ord_less_eq @ nat @ j @ i ).
% 0.19/0.42
% 0.19/0.42 % jlei
% 0.19/0.42 thf(fact_2_tisome,axiom,
% 0.19/0.42 ( ( lList2228235758e_ll2f @ a @ t @ i )
% 0.19/0.42 = ( some @ a @ x ) ) ).
% 0.19/0.42
% 0.19/0.42 % tisome
% 0.19/0.42 thf(fact_3_option_Oinject,axiom,
% 0.19/0.42 ! [A: $tType,X2: A,Y2: A] :
% 0.19/0.42 ( ( ( some @ A @ X2 )
% 0.19/0.42 = ( some @ A @ Y2 ) )
% 0.19/0.42 = ( X2 = Y2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.inject
% 0.19/0.42 thf(fact_4_order__refl,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % order_refl
% 0.19/0.42 thf(fact_5_le__refl,axiom,
% 0.19/0.42 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.19/0.42
% 0.19/0.42 % le_refl
% 0.19/0.42 thf(fact_6_le__trans,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ J @ K )
% 0.19/0.42 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_trans
% 0.19/0.42 thf(fact_7_eq__imp__le,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( M = N )
% 0.19/0.42 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % eq_imp_le
% 0.19/0.42 thf(fact_8_le__antisym,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ N @ M )
% 0.19/0.42 => ( M = N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_antisym
% 0.19/0.42 thf(fact_9_nat__le__linear,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.42 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_le_linear
% 0.19/0.42 thf(fact_10_ex__has__least__nat,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.19/0.42 ( ( P @ K )
% 0.19/0.42 => ? [X3: A] :
% 0.19/0.42 ( ( P @ X3 )
% 0.19/0.42 & ! [Y3: A] :
% 0.19/0.42 ( ( P @ Y3 )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ex_has_least_nat
% 0.19/0.42 thf(fact_11_le__funD,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 => ! [F: A > B,G: A > B,X: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.19/0.42 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_funD
% 0.19/0.42 thf(fact_12_le__funE,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 => ! [F: A > B,G: A > B,X: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.19/0.42 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_funE
% 0.19/0.42 thf(fact_13_le__funI,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 => ! [F: A > B,G: A > B] :
% 0.19/0.42 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.19/0.42 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_funI
% 0.19/0.42 thf(fact_14_le__fun__def,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 => ( ( ord_less_eq @ ( A > B ) )
% 0.19/0.42 = ( ^ [F2: A > B,G2: A > B] :
% 0.19/0.42 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_fun_def
% 0.19/0.42 thf(fact_15_dual__order_Oantisym,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [B2: A,A2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.42 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dual_order.antisym
% 0.19/0.42 thf(fact_16_dual__order_Otrans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [B2: A,A2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dual_order.trans
% 0.19/0.42 thf(fact_17_linorder__wlog,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.19/0.42 ( ! [A3: A,B3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.19/0.42 => ( P @ A3 @ B3 ) )
% 0.19/0.42 => ( ! [A3: A,B3: A] :
% 0.19/0.42 ( ( P @ B3 @ A3 )
% 0.19/0.42 => ( P @ A3 @ B3 ) )
% 0.19/0.42 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % linorder_wlog
% 0.19/0.42 thf(fact_18_dual__order_Orefl,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % dual_order.refl
% 0.19/0.42 thf(fact_19_order__trans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A,Z: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.19/0.42 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % order_trans
% 0.19/0.42 thf(fact_20_order__class_Oorder_Oantisym,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.42 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % order_class.order.antisym
% 0.19/0.42 thf(fact_21_ord__le__eq__trans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ord @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.42 => ( ( B2 = C2 )
% 0.19/0.42 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ord_le_eq_trans
% 0.19/0.42 thf(fact_22_ord__eq__le__trans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ord @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.42 ( ( A2 = B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.42 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ord_eq_le_trans
% 0.19/0.42 thf(fact_23_antisym__conv,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [Y: A,X: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ Y @ X )
% 0.19/0.42 => ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 = ( X = Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % antisym_conv
% 0.19/0.42 thf(fact_24_le__cases3,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A,Z: A] :
% 0.19/0.42 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.19/0.42 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.19/0.42 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.19/0.42 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.19/0.42 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.19/0.42 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_cases3
% 0.19/0.42 thf(fact_25_order_Otrans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.42 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % order.trans
% 0.19/0.42 thf(fact_26_le__cases,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_cases
% 0.19/0.42 thf(fact_27_eq__refl,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % eq_refl
% 0.19/0.42 thf(fact_28_linear,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % linear
% 0.19/0.42 thf(fact_29_antisym,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( ( ord_less_eq @ A @ Y @ X )
% 0.19/0.42 => ( X = Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % antisym
% 0.19/0.42 thf(fact_30_eq__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.19/0.42 = ( ^ [X4: A,Y5: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.19/0.42 & ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % eq_iff
% 0.19/0.42 thf(fact_31_ord__le__eq__subst,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 & ( ord @ A @ ( type2 @ A ) ) )
% 0.19/0.42 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.42 => ( ( ( F @ B2 )
% 0.19/0.42 = C2 )
% 0.19/0.42 => ( ! [X3: A,Y6: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X3 @ Y6 )
% 0.19/0.42 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 0.19/0.42 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ord_le_eq_subst
% 0.19/0.42 thf(fact_32_ord__eq__le__subst,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 & ( ord @ A @ ( type2 @ A ) ) )
% 0.19/0.42 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.42 ( ( A2
% 0.19/0.42 = ( F @ B2 ) )
% 0.19/0.42 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.19/0.42 => ( ! [X3: B,Y6: B] :
% 0.19/0.42 ( ( ord_less_eq @ B @ X3 @ Y6 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 0.19/0.42 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ord_eq_le_subst
% 0.19/0.42 thf(fact_33_order__subst2,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType] :
% 0.19/0.42 ( ( ( order @ C @ ( type2 @ C ) )
% 0.19/0.42 & ( order @ A @ ( type2 @ A ) ) )
% 0.19/0.42 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.42 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.19/0.42 => ( ! [X3: A,Y6: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X3 @ Y6 )
% 0.19/0.42 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 0.19/0.42 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % order_subst2
% 0.19/0.42 thf(fact_34_order__subst1,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( ( order @ B @ ( type2 @ B ) )
% 0.19/0.42 & ( order @ A @ ( type2 @ A ) ) )
% 0.19/0.42 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.19/0.42 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.19/0.42 => ( ! [X3: B,Y6: B] :
% 0.19/0.42 ( ( ord_less_eq @ B @ X3 @ Y6 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 0.19/0.42 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % order_subst1
% 0.19/0.42 thf(fact_35_LeastMI2,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 => ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ! [Y6: A] :
% 0.19/0.42 ( ( P @ Y6 )
% 0.19/0.42 => ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y6 ) ) )
% 0.19/0.42 => ( ! [X3: A] :
% 0.19/0.42 ( ( P @ X3 )
% 0.19/0.42 => ( ! [Y3: A] :
% 0.19/0.42 ( ( P @ Y3 )
% 0.19/0.42 => ( ord_less_eq @ B @ ( M @ X3 ) @ ( M @ Y3 ) ) )
% 0.19/0.42 => ( Q @ X3 ) ) )
% 0.19/0.42 => ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % LeastMI2
% 0.19/0.42 thf(fact_36_lfp_Oleq__antisym,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( ( ord_less_eq @ A @ Y @ X )
% 0.19/0.42 => ( X = Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lfp.leq_antisym
% 0.19/0.42 thf(fact_37_gfp_Oleq__antisym,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [Y: A,X: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ Y @ X )
% 0.19/0.42 => ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( X = Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % gfp.leq_antisym
% 0.19/0.42 thf(fact_38_lfp_Oleq__trans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A,Z: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.19/0.42 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lfp.leq_trans
% 0.19/0.42 thf(fact_39_gfp_Oleq__trans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [Y: A,X: A,Z: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ Y @ X )
% 0.19/0.42 => ( ( ord_less_eq @ A @ Z @ Y )
% 0.19/0.42 => ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % gfp.leq_trans
% 0.19/0.42 thf(fact_40_lfp_Oleq__refl,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % lfp.leq_refl
% 0.19/0.42 thf(fact_41_wlog__linorder__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.19/0.42 ( ! [A3: A,B3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.19/0.42 => ( P @ A3 @ B3 ) )
% 0.19/0.42 => ( ( ( P @ B2 @ A2 )
% 0.19/0.42 => ( P @ A2 @ B2 ) )
% 0.19/0.42 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % wlog_linorder_le
% 0.19/0.42 thf(fact_42_GreatestMI2,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.42 => ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ! [Y6: A] :
% 0.19/0.42 ( ( P @ Y6 )
% 0.19/0.42 => ( ord_less_eq @ B @ ( M @ Y6 ) @ ( M @ X ) ) )
% 0.19/0.42 => ( ! [X3: A] :
% 0.19/0.42 ( ( P @ X3 )
% 0.19/0.42 => ( ! [Y3: A] :
% 0.19/0.42 ( ( P @ Y3 )
% 0.19/0.42 => ( ord_less_eq @ B @ ( M @ Y3 ) @ ( M @ X3 ) ) )
% 0.19/0.42 => ( Q @ X3 ) ) )
% 0.19/0.42 => ( Q @ ( hilbert_GreatestM @ A @ B @ M @ P ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % GreatestMI2
% 0.19/0.42 thf(fact_43_infinite__idx__SomeE,axiom,
% 0.19/0.42 ! [A: $tType,T: coinductive_llist @ A,A4: set @ A,I: nat] :
% 0.19/0.42 ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A4 ) )
% 0.19/0.42 => ~ ! [X3: A] :
% 0.19/0.42 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 0.19/0.42 != ( some @ A @ X3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % infinite_idx_SomeE
% 0.19/0.42 thf(fact_44_infinite__Some,axiom,
% 0.19/0.42 ! [A: $tType,T: coinductive_llist @ A,A4: set @ A,I: nat] :
% 0.19/0.42 ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A4 ) )
% 0.19/0.42 => ? [A3: A] :
% 0.19/0.42 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 0.19/0.42 = ( some @ A @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % infinite_Some
% 0.19/0.42 thf(fact_45_mem__Collect__eq,axiom,
% 0.19/0.42 ! [A: $tType,A2: A,P: A > $o] :
% 0.19/0.42 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.19/0.42 = ( P @ A2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % mem_Collect_eq
% 0.19/0.42 thf(fact_46_Collect__mem__eq,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A] :
% 0.19/0.42 ( ( collect @ A
% 0.19/0.42 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 0.19/0.42 = A4 ) ).
% 0.19/0.42
% 0.19/0.42 % Collect_mem_eq
% 0.19/0.42 thf(fact_47_Collect__cong,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.42 ( ! [X3: A] :
% 0.19/0.42 ( ( P @ X3 )
% 0.19/0.42 = ( Q @ X3 ) )
% 0.19/0.42 => ( ( collect @ A @ P )
% 0.19/0.42 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Collect_cong
% 0.19/0.42 thf(fact_48_ext,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.19/0.42 ( ! [X3: A] :
% 0.19/0.42 ( ( F @ X3 )
% 0.19/0.42 = ( G @ X3 ) )
% 0.19/0.42 => ( F = G ) ) ).
% 0.19/0.42
% 0.19/0.42 % ext
% 0.19/0.42 thf(fact_49_inflsts__mono,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21612149805nflsts @ A @ A4 ) @ ( lList21612149805nflsts @ A @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % inflsts_mono
% 0.19/0.42 thf(fact_50_LeastM__nat__le,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % LeastM_nat_le
% 0.19/0.42 thf(fact_51_LeastM__nat__lemma,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.19/0.42 ( ( P @ K )
% 0.19/0.42 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) )
% 0.19/0.42 & ! [Y3: A] :
% 0.19/0.42 ( ( P @ Y3 )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ Y3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % LeastM_nat_lemma
% 0.19/0.42 thf(fact_52_option_Osimps_I7_J,axiom,
% 0.19/0.42 ! [C: $tType,A: $tType,F1: C,F22: A > C,X2: A] :
% 0.19/0.42 ( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
% 0.19/0.42 = ( F22 @ X2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(7)
% 0.19/0.42 thf(fact_53_combine__options__simps_I3_J,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.19/0.42 ( ( combine_options @ A @ F @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
% 0.19/0.42 = ( some @ A @ ( F @ A2 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_simps(3)
% 0.19/0.42 thf(fact_54_ll2f__lem1,axiom,
% 0.19/0.42 ! [A: $tType,T: coinductive_llist @ A,I: nat,X: A] :
% 0.19/0.42 ( ( ( lList2228235758e_ll2f @ A @ T @ ( suc @ I ) )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 => ? [Y6: A] :
% 0.19/0.42 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 0.19/0.42 = ( some @ A @ Y6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ll2f_lem1
% 0.19/0.42 thf(fact_55_ll2f__Suc__Some,axiom,
% 0.19/0.42 ! [A: $tType,T: coinductive_llist @ A,I: nat,X: A] :
% 0.19/0.42 ( ( ( lList2228235758e_ll2f @ A @ T @ ( suc @ I ) )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 => ~ ! [X3: A] :
% 0.19/0.42 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 0.19/0.42 != ( some @ A @ X3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ll2f_Suc_Some
% 0.19/0.42 thf(fact_56_ll2f__None__le,axiom,
% 0.19/0.42 ! [A: $tType,T: coinductive_llist @ A,J: nat,I: nat] :
% 0.19/0.42 ( ( ( lList2228235758e_ll2f @ A @ T @ J )
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ J @ I )
% 0.19/0.42 => ( ( lList2228235758e_ll2f @ A @ T @ I )
% 0.19/0.42 = ( none @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ll2f_None_le
% 0.19/0.42 thf(fact_57_of__nat__le__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_le_iff
% 0.19/0.42 thf(fact_58_is__none__code_I2_J,axiom,
% 0.19/0.42 ! [B: $tType,X: B] :
% 0.19/0.42 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_none_code(2)
% 0.19/0.42 thf(fact_59_option_Opred__inject_I2_J,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,A2: A] :
% 0.19/0.42 ( ( pred_option @ A @ P @ ( some @ A @ A2 ) )
% 0.19/0.42 = ( P @ A2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.pred_inject(2)
% 0.19/0.42 thf(fact_60_nat_Oinject,axiom,
% 0.19/0.42 ! [X2: nat,Y2: nat] :
% 0.19/0.42 ( ( ( suc @ X2 )
% 0.19/0.42 = ( suc @ Y2 ) )
% 0.19/0.42 = ( X2 = Y2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat.inject
% 0.19/0.42 thf(fact_61_old_Onat_Oinject,axiom,
% 0.19/0.42 ! [Nat: nat,Nat2: nat] :
% 0.19/0.42 ( ( ( suc @ Nat )
% 0.19/0.42 = ( suc @ Nat2 ) )
% 0.19/0.42 = ( Nat = Nat2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.nat.inject
% 0.19/0.42 thf(fact_62_of__nat__eq__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ( semiring_1_of_nat @ A @ M )
% 0.19/0.42 = ( semiring_1_of_nat @ A @ N ) )
% 0.19/0.42 = ( M = N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_eq_iff
% 0.19/0.42 thf(fact_63_Suc__le__mono,axiom,
% 0.19/0.42 ! [N: nat,M: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.19/0.42
% 0.19/0.42 % Suc_le_mono
% 0.19/0.42 thf(fact_64_not__Some__eq,axiom,
% 0.19/0.42 ! [A: $tType,X: option @ A] :
% 0.19/0.42 ( ( ! [Y5: A] :
% 0.19/0.42 ( X
% 0.19/0.42 != ( some @ A @ Y5 ) ) )
% 0.19/0.42 = ( X
% 0.19/0.42 = ( none @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_Some_eq
% 0.19/0.42 thf(fact_65_not__None__eq,axiom,
% 0.19/0.42 ! [A: $tType,X: option @ A] :
% 0.19/0.42 ( ( X
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 = ( ? [Y5: A] :
% 0.19/0.42 ( X
% 0.19/0.42 = ( some @ A @ Y5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_None_eq
% 0.19/0.42 thf(fact_66_is__none__code_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_none_code(1)
% 0.19/0.42 thf(fact_67_combine__options__simps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,X: option @ A] :
% 0.19/0.42 ( ( combine_options @ A @ F @ X @ ( none @ A ) )
% 0.19/0.42 = X ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_simps(2)
% 0.19/0.42 thf(fact_68_combine__options__simps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Y: option @ A] :
% 0.19/0.42 ( ( combine_options @ A @ F @ ( none @ A ) @ Y )
% 0.19/0.42 = Y ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_simps(1)
% 0.19/0.42 thf(fact_69_LeastM__natI,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.19/0.42 ( ( P @ K )
% 0.19/0.42 => ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % LeastM_natI
% 0.19/0.42 thf(fact_70_ll2f__None__Suc,axiom,
% 0.19/0.42 ! [A: $tType,T: coinductive_llist @ A,I: nat] :
% 0.19/0.42 ( ( ( lList2228235758e_ll2f @ A @ T @ I )
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 => ( ( lList2228235758e_ll2f @ A @ T @ ( suc @ I ) )
% 0.19/0.42 = ( none @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ll2f_None_Suc
% 0.19/0.42 thf(fact_71_option_Osimps_I6_J,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType,F1: C,F22: A > C] :
% 0.19/0.42 ( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
% 0.19/0.42 = F1 ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(6)
% 0.19/0.42 thf(fact_72_is__none__simps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_none_simps(1)
% 0.19/0.42 thf(fact_73_option_Opred__inject_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.pred_inject(1)
% 0.19/0.42 thf(fact_74_Suc__inject,axiom,
% 0.19/0.42 ! [X: nat,Y: nat] :
% 0.19/0.42 ( ( ( suc @ X )
% 0.19/0.42 = ( suc @ Y ) )
% 0.19/0.42 => ( X = Y ) ) ).
% 0.19/0.42
% 0.19/0.42 % Suc_inject
% 0.19/0.42 thf(fact_75_n__not__Suc__n,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( N
% 0.19/0.42 != ( suc @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % n_not_Suc_n
% 0.19/0.42 thf(fact_76_Option_Ois__none__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( is_none @ A )
% 0.19/0.42 = ( ^ [X4: option @ A] :
% 0.19/0.42 ( X4
% 0.19/0.42 = ( none @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Option.is_none_def
% 0.19/0.42 thf(fact_77_combine__options__assoc,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A,Z: option @ A] :
% 0.19/0.42 ( ! [X3: A,Y6: A,Z3: A] :
% 0.19/0.42 ( ( F @ ( F @ X3 @ Y6 ) @ Z3 )
% 0.19/0.42 = ( F @ X3 @ ( F @ Y6 @ Z3 ) ) )
% 0.19/0.42 => ( ( combine_options @ A @ F @ ( combine_options @ A @ F @ X @ Y ) @ Z )
% 0.19/0.42 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_assoc
% 0.19/0.42 thf(fact_78_combine__options__commute,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A] :
% 0.19/0.42 ( ! [X3: A,Y6: A] :
% 0.19/0.42 ( ( F @ X3 @ Y6 )
% 0.19/0.42 = ( F @ Y6 @ X3 ) )
% 0.19/0.42 => ( ( combine_options @ A @ F @ X @ Y )
% 0.19/0.42 = ( combine_options @ A @ F @ Y @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_commute
% 0.19/0.42 thf(fact_79_combine__options__left__commute,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Y: option @ A,X: option @ A,Z: option @ A] :
% 0.19/0.42 ( ! [X3: A,Y6: A] :
% 0.19/0.42 ( ( F @ X3 @ Y6 )
% 0.19/0.42 = ( F @ Y6 @ X3 ) )
% 0.19/0.42 => ( ! [X3: A,Y6: A,Z3: A] :
% 0.19/0.42 ( ( F @ ( F @ X3 @ Y6 ) @ Z3 )
% 0.19/0.42 = ( F @ X3 @ ( F @ Y6 @ Z3 ) ) )
% 0.19/0.42 => ( ( combine_options @ A @ F @ Y @ ( combine_options @ A @ F @ X @ Z ) )
% 0.19/0.42 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_left_commute
% 0.19/0.42 thf(fact_80_full__nat__induct,axiom,
% 0.19/0.42 ! [P: nat > $o,N: nat] :
% 0.19/0.42 ( ! [N2: nat] :
% 0.19/0.42 ( ! [M2: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
% 0.19/0.42 => ( P @ M2 ) )
% 0.19/0.42 => ( P @ N2 ) )
% 0.19/0.42 => ( P @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % full_nat_induct
% 0.19/0.42 thf(fact_81_not__less__eq__eq,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_less_eq_eq
% 0.19/0.42 thf(fact_82_Suc__n__not__le__n,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.19/0.42
% 0.19/0.42 % Suc_n_not_le_n
% 0.19/0.42 thf(fact_83_le__Suc__eq,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.19/0.42 = ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.42 | ( M
% 0.19/0.42 = ( suc @ N ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_Suc_eq
% 0.19/0.42 thf(fact_84_Suc__le__D,axiom,
% 0.19/0.42 ! [N: nat,M3: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M3 )
% 0.19/0.42 => ? [M4: nat] :
% 0.19/0.42 ( M3
% 0.19/0.42 = ( suc @ M4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Suc_le_D
% 0.19/0.42 thf(fact_85_le__SucI,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.42 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_SucI
% 0.19/0.42 thf(fact_86_le__SucE,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.19/0.42 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.19/0.42 => ( M
% 0.19/0.42 = ( suc @ N ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_SucE
% 0.19/0.42 thf(fact_87_Suc__leD,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.19/0.42 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % Suc_leD
% 0.19/0.42 thf(fact_88_combine__options__cases,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
% 0.19/0.42 ( ( ( X
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 => ( P @ X @ Y ) )
% 0.19/0.42 => ( ( ( Y
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 => ( P @ X @ Y ) )
% 0.19/0.42 => ( ! [A3: A,B3: B] :
% 0.19/0.42 ( ( X
% 0.19/0.42 = ( some @ A @ A3 ) )
% 0.19/0.42 => ( ( Y
% 0.19/0.42 = ( some @ B @ B3 ) )
% 0.19/0.42 => ( P @ X @ Y ) ) )
% 0.19/0.42 => ( P @ X @ Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_cases
% 0.19/0.42 thf(fact_89_split__option__all,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ^ [P2: ( option @ A ) > $o] :
% 0.19/0.42 ! [X5: option @ A] : ( P2 @ X5 ) )
% 0.19/0.42 = ( ^ [P3: ( option @ A ) > $o] :
% 0.19/0.42 ( ( P3 @ ( none @ A ) )
% 0.19/0.42 & ! [X4: A] : ( P3 @ ( some @ A @ X4 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % split_option_all
% 0.19/0.42 thf(fact_90_split__option__ex,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ^ [P2: ( option @ A ) > $o] :
% 0.19/0.42 ? [X5: option @ A] : ( P2 @ X5 ) )
% 0.19/0.42 = ( ^ [P3: ( option @ A ) > $o] :
% 0.19/0.42 ( ( P3 @ ( none @ A ) )
% 0.19/0.42 | ? [X4: A] : ( P3 @ ( some @ A @ X4 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % split_option_ex
% 0.19/0.42 thf(fact_91_option_Oinducts,axiom,
% 0.19/0.42 ! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
% 0.19/0.42 ( ( P @ ( none @ A ) )
% 0.19/0.42 => ( ! [X3: A] : ( P @ ( some @ A @ X3 ) )
% 0.19/0.42 => ( P @ Option ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.inducts
% 0.19/0.42 thf(fact_92_option_Oexhaust,axiom,
% 0.19/0.42 ! [A: $tType,Y: option @ A] :
% 0.19/0.42 ( ( Y
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ~ ! [X22: A] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( some @ A @ X22 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.exhaust
% 0.19/0.42 thf(fact_93_option_OdiscI,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A,X2: A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 = ( some @ A @ X2 ) )
% 0.19/0.42 => ( Option
% 0.19/0.42 != ( none @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.discI
% 0.19/0.42 thf(fact_94_option_Odistinct_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,X2: A] :
% 0.19/0.42 ( ( none @ A )
% 0.19/0.42 != ( some @ A @ X2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.distinct(1)
% 0.19/0.42 thf(fact_95_is__none__simps_I2_J,axiom,
% 0.19/0.42 ! [B: $tType,X: B] :
% 0.19/0.42 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_none_simps(2)
% 0.19/0.42 thf(fact_96_lift__Suc__antimono__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [F: nat > A,N: nat,N3: nat] :
% 0.19/0.42 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ N @ N3 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lift_Suc_antimono_le
% 0.19/0.42 thf(fact_97_lift__Suc__mono__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [F: nat > A,N: nat,N3: nat] :
% 0.19/0.42 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ N @ N3 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lift_Suc_mono_le
% 0.19/0.42 thf(fact_98_subset__antisym,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.19/0.42 => ( A4 = B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % subset_antisym
% 0.19/0.42 thf(fact_99_subsetI,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ! [X3: A] :
% 0.19/0.42 ( ( member @ A @ X3 @ A4 )
% 0.19/0.42 => ( member @ A @ X3 @ B4 ) )
% 0.19/0.42 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.19/0.42
% 0.19/0.42 % subsetI
% 0.19/0.42 thf(fact_100_real__arch__simple,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % real_arch_simple
% 0.19/0.42 thf(fact_101_poslsts__mono,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21148268032oslsts @ A @ A4 ) @ ( lList21148268032oslsts @ A @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % poslsts_mono
% 0.19/0.42 thf(fact_102_fpslsts__mono,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList22096119349pslsts @ A @ A4 ) @ ( lList22096119349pslsts @ A @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fpslsts_mono
% 0.19/0.42 thf(fact_103_zle__int,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % zle_int
% 0.19/0.42 thf(fact_104_transfer__int__nat__relations_I3_J,axiom,
% 0.19/0.42 ! [X: nat,Y: nat] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ X @ Y ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_int_nat_relations(3)
% 0.19/0.42 thf(fact_105_int__if__cong,axiom,
% 0.19/0.42 ! [P: $o,X: nat,Y: nat] :
% 0.19/0.42 ( ( P
% 0.19/0.42 => ( ( semiring_1_of_nat @ int @ X )
% 0.19/0.42 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y ) ) ) )
% 0.19/0.42 & ( ~ P
% 0.19/0.42 => ( ( semiring_1_of_nat @ int @ Y )
% 0.19/0.42 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % int_if_cong
% 0.19/0.42 thf(fact_106_int__int__eq,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ( semiring_1_of_nat @ int @ M )
% 0.19/0.42 = ( semiring_1_of_nat @ int @ N ) )
% 0.19/0.42 = ( M = N ) ) ).
% 0.19/0.42
% 0.19/0.42 % int_int_eq
% 0.19/0.42 thf(fact_107_transfer__int__nat__relations_I1_J,axiom,
% 0.19/0.42 ! [X: nat,Y: nat] :
% 0.19/0.42 ( ( ( semiring_1_of_nat @ int @ X )
% 0.19/0.42 = ( semiring_1_of_nat @ int @ Y ) )
% 0.19/0.42 = ( X = Y ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_int_nat_relations(1)
% 0.19/0.42 thf(fact_108_set__mp,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ( member @ A @ X @ A4 )
% 0.19/0.42 => ( member @ A @ X @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % set_mp
% 0.19/0.42 thf(fact_109_in__mono,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ( member @ A @ X @ A4 )
% 0.19/0.42 => ( member @ A @ X @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % in_mono
% 0.19/0.42 thf(fact_110_subsetD,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ( member @ A @ C2 @ A4 )
% 0.19/0.42 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % subsetD
% 0.19/0.42 thf(fact_111_subsetCE,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ( member @ A @ C2 @ A4 )
% 0.19/0.42 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % subsetCE
% 0.19/0.42 thf(fact_112_equalityE,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( A4 = B4 )
% 0.19/0.42 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % equalityE
% 0.19/0.42 thf(fact_113_subset__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.42 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.19/0.42 ! [X4: A] :
% 0.19/0.42 ( ( member @ A @ X4 @ A5 )
% 0.19/0.42 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % subset_eq
% 0.19/0.42 thf(fact_114_equalityD1,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( A4 = B4 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.19/0.42
% 0.19/0.42 % equalityD1
% 0.19/0.42 thf(fact_115_equalityD2,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( A4 = B4 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.19/0.42
% 0.19/0.42 % equalityD2
% 0.19/0.42 thf(fact_116_set__rev__mp,axiom,
% 0.19/0.42 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( member @ A @ X @ A4 )
% 0.19/0.42 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( member @ A @ X @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % set_rev_mp
% 0.19/0.42 thf(fact_117_subset__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.42 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.19/0.42 ! [T2: A] :
% 0.19/0.42 ( ( member @ A @ T2 @ A5 )
% 0.19/0.42 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % subset_iff
% 0.19/0.42 thf(fact_118_rev__subsetD,axiom,
% 0.19/0.42 ! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( member @ A @ C2 @ A4 )
% 0.19/0.42 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % rev_subsetD
% 0.19/0.42 thf(fact_119_subset__refl,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.19/0.42
% 0.19/0.42 % subset_refl
% 0.19/0.42 thf(fact_120_Collect__mono,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.42 ( ! [X3: A] :
% 0.19/0.42 ( ( P @ X3 )
% 0.19/0.42 => ( Q @ X3 ) )
% 0.19/0.42 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Collect_mono
% 0.19/0.42 thf(fact_121_subset__trans,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % subset_trans
% 0.19/0.42 thf(fact_122_set__eq__subset,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 0.19/0.42 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.42 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % set_eq_subset
% 0.19/0.42 thf(fact_123_contra__subsetD,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ~ ( member @ A @ C2 @ B4 )
% 0.19/0.42 => ~ ( member @ A @ C2 @ A4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % contra_subsetD
% 0.19/0.42 thf(fact_124_Collect__mono__iff,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.19/0.42 = ( ! [X4: A] :
% 0.19/0.42 ( ( P @ X4 )
% 0.19/0.42 => ( Q @ X4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Collect_mono_iff
% 0.19/0.42 thf(fact_125_round__of__nat,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: nat] :
% 0.19/0.42 ( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.19/0.42 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % round_of_nat
% 0.19/0.42 thf(fact_126_option_Ocollapse,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( some @ A @ ( the @ A @ Option ) )
% 0.19/0.42 = Option ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.collapse
% 0.19/0.42 thf(fact_127_transfer__morphismI,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: A > B,A4: A > $o] : ( nat_tr1645093318rphism @ A @ B @ F @ A4 ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_morphismI
% 0.19/0.42 thf(fact_128_of__nat__aux_Osimps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [Inc: A > A,N: nat,I: A] :
% 0.19/0.42 ( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N ) @ I )
% 0.19/0.42 = ( semiri532925092at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_aux.simps(2)
% 0.19/0.42 thf(fact_129_round__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ Y )
% 0.19/0.42 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % round_mono
% 0.19/0.42 thf(fact_130_transfer__morphism__def,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( nat_tr1645093318rphism @ B @ A )
% 0.19/0.42 = ( ^ [F2: B > A,A5: B > $o] : $true ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_morphism_def
% 0.19/0.42 thf(fact_131_option_Oexpand,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 0.19/0.42 ( ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( Option2
% 0.19/0.42 = ( none @ A ) ) )
% 0.19/0.42 => ( ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( Option2
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( the @ A @ Option )
% 0.19/0.42 = ( the @ A @ Option2 ) ) ) )
% 0.19/0.42 => ( Option = Option2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.expand
% 0.19/0.42 thf(fact_132_option_Osel,axiom,
% 0.19/0.42 ! [A: $tType,X2: A] :
% 0.19/0.42 ( ( the @ A @ ( some @ A @ X2 ) )
% 0.19/0.42 = X2 ) ).
% 0.19/0.42
% 0.19/0.42 % option.sel
% 0.19/0.42 thf(fact_133_option_Oexhaust__sel,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( Option
% 0.19/0.42 = ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.exhaust_sel
% 0.19/0.42 thf(fact_134_option_Osplit__sel__asm,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 0.19/0.42 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 0.19/0.42 = ( ~ ( ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 & ~ ( P @ F1 ) )
% 0.19/0.42 | ( ( Option
% 0.19/0.42 = ( some @ A @ ( the @ A @ Option ) ) )
% 0.19/0.42 & ~ ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.split_sel_asm
% 0.19/0.42 thf(fact_135_option_Osplit__sel,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 0.19/0.42 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 0.19/0.42 = ( ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 => ( P @ F1 ) )
% 0.19/0.42 & ( ( Option
% 0.19/0.42 = ( some @ A @ ( the @ A @ Option ) ) )
% 0.19/0.42 => ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.split_sel
% 0.19/0.42 thf(fact_136_option_Ocase__eq__if,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( case_option @ B @ A )
% 0.19/0.42 = ( ^ [F12: B,F23: A > B,Option3: option @ A] :
% 0.19/0.42 ( if @ B
% 0.19/0.42 @ ( Option3
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 @ F12
% 0.19/0.42 @ ( F23 @ ( the @ A @ Option3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.case_eq_if
% 0.19/0.42 thf(fact_137_is__none__bind,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A )] :
% 0.19/0.42 ( ( is_none @ A @ ( bind @ B @ A @ F @ G ) )
% 0.19/0.42 = ( ( is_none @ B @ F )
% 0.19/0.42 | ( is_none @ A @ ( G @ ( the @ B @ F ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_none_bind
% 0.19/0.42 thf(fact_138_bind__runit,axiom,
% 0.19/0.42 ! [A: $tType,X: option @ A] :
% 0.19/0.42 ( ( bind @ A @ A @ X @ ( some @ A ) )
% 0.19/0.42 = X ) ).
% 0.19/0.42
% 0.19/0.42 % bind_runit
% 0.19/0.42 thf(fact_139_bind_Obind__lzero,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: A > ( option @ B )] :
% 0.19/0.42 ( ( bind @ A @ B @ ( none @ A ) @ F )
% 0.19/0.42 = ( none @ B ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind.bind_lzero
% 0.19/0.42 thf(fact_140_bind__eq__Some__conv,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A ),X: A] :
% 0.19/0.42 ( ( ( bind @ B @ A @ F @ G )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 = ( ? [Y5: B] :
% 0.19/0.42 ( ( F
% 0.19/0.42 = ( some @ B @ Y5 ) )
% 0.19/0.42 & ( ( G @ Y5 )
% 0.19/0.42 = ( some @ A @ X ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_eq_Some_conv
% 0.19/0.42 thf(fact_141_bind_Obind__lunit,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: A,F: A > ( option @ B )] :
% 0.19/0.42 ( ( bind @ A @ B @ ( some @ A @ X ) @ F )
% 0.19/0.42 = ( F @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind.bind_lunit
% 0.19/0.42 thf(fact_142_Option_Obind__cong,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ! [A3: A] :
% 0.19/0.42 ( ( Y
% 0.19/0.42 = ( some @ A @ A3 ) )
% 0.19/0.42 => ( ( F @ A3 )
% 0.19/0.42 = ( G @ A3 ) ) )
% 0.19/0.42 => ( ( bind @ A @ B @ X @ F )
% 0.19/0.42 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Option.bind_cong
% 0.19/0.42 thf(fact_143_bind__option__cong__code,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B )] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ( bind @ A @ B @ X @ F )
% 0.19/0.42 = ( bind @ A @ B @ Y @ F ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_option_cong_code
% 0.19/0.42 thf(fact_144_option_Osimps_I4_J,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F1: B,F22: A > B] :
% 0.19/0.42 ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
% 0.19/0.42 = F1 ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(4)
% 0.19/0.42 thf(fact_145_option_Osimps_I5_J,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F1: B,F22: A > B,X2: A] :
% 0.19/0.42 ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
% 0.19/0.42 = ( F22 @ X2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(5)
% 0.19/0.42 thf(fact_146_bind__split,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F: B > ( option @ A )] :
% 0.19/0.42 ( ( P @ ( bind @ B @ A @ M @ F ) )
% 0.19/0.42 = ( ( ( M
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 => ( P @ ( none @ A ) ) )
% 0.19/0.42 & ! [V: B] :
% 0.19/0.42 ( ( M
% 0.19/0.42 = ( some @ B @ V ) )
% 0.19/0.42 => ( P @ ( F @ V ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_split
% 0.19/0.42 thf(fact_147_bind__split__asm,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F: B > ( option @ A )] :
% 0.19/0.42 ( ( P @ ( bind @ B @ A @ M @ F ) )
% 0.19/0.42 = ( ~ ( ( ( M
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 & ~ ( P @ ( none @ A ) ) )
% 0.19/0.42 | ? [X4: B] :
% 0.19/0.42 ( ( M
% 0.19/0.42 = ( some @ B @ X4 ) )
% 0.19/0.42 & ~ ( P @ ( F @ X4 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_split_asm
% 0.19/0.42 thf(fact_148_bind__eq__None__conv,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,A2: option @ B,F: B > ( option @ A )] :
% 0.19/0.42 ( ( ( bind @ B @ A @ A2 @ F )
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( ( A2
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 | ( ( F @ ( the @ B @ A2 ) )
% 0.19/0.42 = ( none @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_eq_None_conv
% 0.19/0.42 thf(fact_149_alllsts__mono,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.19/0.42 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A4 ) @ ( lList2435255213lllsts @ A @ B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % alllsts_mono
% 0.19/0.42 thf(fact_150_nat__le__iff,axiom,
% 0.19/0.42 ! [X: int,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 0.19/0.42 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_le_iff
% 0.19/0.42 thf(fact_151_ll2f__LNil,axiom,
% 0.19/0.42 ! [A: $tType,X: nat] :
% 0.19/0.42 ( ( lList2228235758e_ll2f @ A @ ( coinductive_LNil @ A ) @ X )
% 0.19/0.42 = ( none @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % ll2f_LNil
% 0.19/0.42 thf(fact_152_option_Oset__sel,axiom,
% 0.19/0.42 ! [A: $tType,A2: option @ A] :
% 0.19/0.42 ( ( A2
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( member @ A @ ( the @ A @ A2 ) @ ( set_option @ A @ A2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.set_sel
% 0.19/0.42 thf(fact_153_nat__int,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 0.19/0.42 = N ) ).
% 0.19/0.42
% 0.19/0.42 % nat_int
% 0.19/0.42 thf(fact_154_elem__set,axiom,
% 0.19/0.42 ! [A: $tType,X: A,Xo: option @ A] :
% 0.19/0.42 ( ( member @ A @ X @ ( set_option @ A @ Xo ) )
% 0.19/0.42 = ( Xo
% 0.19/0.42 = ( some @ A @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % elem_set
% 0.19/0.42 thf(fact_155_poslsts__iff,axiom,
% 0.19/0.42 ! [A: $tType,S: coinductive_llist @ A,A4: set @ A] :
% 0.19/0.42 ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ A4 ) )
% 0.19/0.42 = ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A4 ) )
% 0.19/0.42 & ( S
% 0.19/0.42 != ( coinductive_LNil @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % poslsts_iff
% 0.19/0.42 thf(fact_156_ospec,axiom,
% 0.19/0.42 ! [A: $tType,A4: option @ A,P: A > $o,X: A] :
% 0.19/0.42 ( ! [X3: A] :
% 0.19/0.42 ( ( member @ A @ X3 @ ( set_option @ A @ A4 ) )
% 0.19/0.42 => ( P @ X3 ) )
% 0.19/0.42 => ( ( A4
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 => ( P @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ospec
% 0.19/0.42 thf(fact_157_option_Oset__cases,axiom,
% 0.19/0.42 ! [A: $tType,E: A,A2: option @ A] :
% 0.19/0.42 ( ( member @ A @ E @ ( set_option @ A @ A2 ) )
% 0.19/0.42 => ( A2
% 0.19/0.42 = ( some @ A @ E ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.set_cases
% 0.19/0.42 thf(fact_158_option_Oset__intros,axiom,
% 0.19/0.42 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( set_option @ A @ ( some @ A @ A2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.set_intros
% 0.19/0.42 thf(fact_159_nat__if__cong,axiom,
% 0.19/0.42 ! [P: $o,X: int,Y: int] :
% 0.19/0.42 ( ( P
% 0.19/0.42 => ( ( nat2 @ X )
% 0.19/0.42 = ( nat2 @ ( if @ int @ P @ X @ Y ) ) ) )
% 0.19/0.42 & ( ~ P
% 0.19/0.42 => ( ( nat2 @ Y )
% 0.19/0.42 = ( nat2 @ ( if @ int @ P @ X @ Y ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_if_cong
% 0.19/0.42 thf(fact_160_alllsts_OLNil__all,axiom,
% 0.19/0.42 ! [A: $tType,A4: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A4 ) ) ).
% 0.19/0.42
% 0.19/0.42 % alllsts.LNil_all
% 0.19/0.42 thf(fact_161_infsubsetall,axiom,
% 0.19/0.42 ! [A: $tType,X: coinductive_llist @ A,A4: set @ A] :
% 0.19/0.42 ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A4 ) )
% 0.19/0.42 => ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % infsubsetall
% 0.19/0.42 thf(fact_162_bind__option__cong,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ! [Z3: A] :
% 0.19/0.42 ( ( member @ A @ Z3 @ ( set_option @ A @ Y ) )
% 0.19/0.42 => ( ( F @ Z3 )
% 0.19/0.42 = ( G @ Z3 ) ) )
% 0.19/0.42 => ( ( bind @ A @ B @ X @ F )
% 0.19/0.42 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_option_cong
% 0.19/0.42 thf(fact_163_option_Opred__cong,axiom,
% 0.19/0.42 ! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
% 0.19/0.42 ( ( X = Ya )
% 0.19/0.42 => ( ! [Z3: A] :
% 0.19/0.42 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 0.19/0.42 => ( ( P @ Z3 )
% 0.19/0.42 = ( Pa @ Z3 ) ) )
% 0.19/0.42 => ( ( pred_option @ A @ P @ X )
% 0.19/0.42 = ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.pred_cong
% 0.19/0.42 thf(fact_164_option_Opred__mono__strong,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
% 0.19/0.42 ( ( pred_option @ A @ P @ X )
% 0.19/0.42 => ( ! [Z3: A] :
% 0.19/0.42 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 0.19/0.42 => ( ( P @ Z3 )
% 0.19/0.42 => ( Pa @ Z3 ) ) )
% 0.19/0.42 => ( pred_option @ A @ Pa @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.pred_mono_strong
% 0.19/0.42 thf(fact_165_nat__mono,axiom,
% 0.19/0.42 ! [X: int,Y: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ X @ Y )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_mono
% 0.19/0.42 thf(fact_166_lbutlast__LNil,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
% 0.19/0.42 = ( coinductive_LNil @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % lbutlast_LNil
% 0.19/0.42 thf(fact_167_le__nat__iff,axiom,
% 0.19/0.42 ! [K: int,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 0.19/0.42 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_nat_iff
% 0.19/0.42 thf(fact_168_int__nat__eq,axiom,
% 0.19/0.42 ! [Z: int] :
% 0.19/0.42 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.19/0.42 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.19/0.42 = Z ) )
% 0.19/0.42 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.19/0.42 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.19/0.42 = ( zero_zero @ int ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % int_nat_eq
% 0.19/0.42 thf(fact_169_of__nat__eq__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [M: nat] :
% 0.19/0.42 ( ( ( semiring_1_of_nat @ A @ M )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( M
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_eq_0_iff
% 0.19/0.42 thf(fact_170_of__nat__0__eq__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: nat] :
% 0.19/0.42 ( ( ( zero_zero @ A )
% 0.19/0.42 = ( semiring_1_of_nat @ A @ N ) )
% 0.19/0.42 = ( ( zero_zero @ nat )
% 0.19/0.42 = N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_0_eq_iff
% 0.19/0.42 thf(fact_171_of__nat__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_0
% 0.19/0.42 thf(fact_172_of__nat__le__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [M: nat] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( M
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_le_0_iff
% 0.19/0.42 thf(fact_173_round__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.19/0.42 => ( ( archimedean_round @ A @ ( zero_zero @ A ) )
% 0.19/0.42 = ( zero_zero @ int ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % round_0
% 0.19/0.42 thf(fact_174_less__eq__int__code_I1_J,axiom,
% 0.19/0.42 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_int_code(1)
% 0.19/0.42 thf(fact_175_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 0.19/0.42 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.19/0.42
% 0.19/0.42 % Nat_Transfer.transfer_nat_int_function_closures(5)
% 0.19/0.42 thf(fact_176_of__nat__0__le__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_0_le_iff
% 0.19/0.42 thf(fact_177_of__nat__neq__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: nat] :
% 0.19/0.42 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 0.19/0.42 != ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_neq_0
% 0.19/0.42 thf(fact_178_transfer__morphism__nat__int,axiom,
% 0.19/0.42 nat_tr1645093318rphism @ int @ nat @ nat2 @ ( ord_less_eq @ int @ ( zero_zero @ int ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_morphism_nat_int
% 0.19/0.42 thf(fact_179_transfer__nat__int__relations_I1_J,axiom,
% 0.19/0.42 ! [X: int,Y: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.19/0.42 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.19/0.42 => ( ( ( nat2 @ X )
% 0.19/0.42 = ( nat2 @ Y ) )
% 0.19/0.42 = ( X = Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_nat_int_relations(1)
% 0.19/0.42 thf(fact_180_eq__nat__nat__iff,axiom,
% 0.19/0.42 ! [Z: int,Z4: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.19/0.42 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
% 0.19/0.42 => ( ( ( nat2 @ Z )
% 0.19/0.42 = ( nat2 @ Z4 ) )
% 0.19/0.42 = ( Z = Z4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % eq_nat_nat_iff
% 0.19/0.42 thf(fact_181_ex__nat,axiom,
% 0.19/0.42 ( ( ^ [P2: nat > $o] :
% 0.19/0.42 ? [X5: nat] : ( P2 @ X5 ) )
% 0.19/0.42 = ( ^ [P3: nat > $o] :
% 0.19/0.42 ? [X4: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.19/0.42 & ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ex_nat
% 0.19/0.42 thf(fact_182_all__nat,axiom,
% 0.19/0.42 ( ( ^ [P2: nat > $o] :
% 0.19/0.42 ! [X5: nat] : ( P2 @ X5 ) )
% 0.19/0.42 = ( ^ [P3: nat > $o] :
% 0.19/0.42 ! [X4: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.19/0.42 => ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % all_nat
% 0.19/0.42 thf(fact_183_zero__le__imp__eq__int,axiom,
% 0.19/0.42 ! [K: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.42 => ? [N2: nat] :
% 0.19/0.42 ( K
% 0.19/0.42 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_le_imp_eq_int
% 0.19/0.42 thf(fact_184_nonneg__int__cases,axiom,
% 0.19/0.42 ! [K: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.42 => ~ ! [N2: nat] :
% 0.19/0.42 ( K
% 0.19/0.42 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nonneg_int_cases
% 0.19/0.42 thf(fact_185_nonneg__eq__int,axiom,
% 0.19/0.42 ! [Z: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.19/0.42 => ~ ! [M4: nat] :
% 0.19/0.42 ( Z
% 0.19/0.42 != ( semiring_1_of_nat @ int @ M4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nonneg_eq_int
% 0.19/0.42 thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 0.19/0.42 ! [Z: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z ) ) ).
% 0.19/0.42
% 0.19/0.42 % Nat_Transfer.transfer_nat_int_function_closures(9)
% 0.19/0.42 thf(fact_187_transfer__int__nat__quantifiers_I1_J,axiom,
% 0.19/0.42 ! [P: int > $o] :
% 0.19/0.42 ( ( ! [X4: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.19/0.42 => ( P @ X4 ) ) )
% 0.19/0.42 = ( ! [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_int_nat_quantifiers(1)
% 0.19/0.42 thf(fact_188_transfer__int__nat__quantifiers_I2_J,axiom,
% 0.19/0.42 ! [P: int > $o] :
% 0.19/0.42 ( ( ? [X4: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.19/0.42 & ( P @ X4 ) ) )
% 0.19/0.42 = ( ? [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_int_nat_quantifiers(2)
% 0.19/0.42 thf(fact_189_transfer__nat__int__relations_I3_J,axiom,
% 0.19/0.42 ! [X: int,Y: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.19/0.42 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
% 0.19/0.42 = ( ord_less_eq @ int @ X @ Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_nat_int_relations(3)
% 0.19/0.42 thf(fact_190_nat__0__le,axiom,
% 0.19/0.42 ! [Z: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.19/0.42 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 0.19/0.42 = Z ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_0_le
% 0.19/0.42 thf(fact_191_int__eq__iff,axiom,
% 0.19/0.42 ! [M: nat,Z: int] :
% 0.19/0.42 ( ( ( semiring_1_of_nat @ int @ M )
% 0.19/0.42 = Z )
% 0.19/0.42 = ( ( M
% 0.19/0.42 = ( nat2 @ Z ) )
% 0.19/0.42 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % int_eq_iff
% 0.19/0.42 thf(fact_192_le__zero__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.42 = ( N
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_zero_eq
% 0.19/0.42 thf(fact_193_imp__le__cong,axiom,
% 0.19/0.42 ! [X: int,X6: int,P: $o,P4: $o] :
% 0.19/0.42 ( ( X = X6 )
% 0.19/0.42 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 0.19/0.42 => ( P = P4 ) )
% 0.19/0.42 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.19/0.42 => P )
% 0.19/0.42 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 0.19/0.42 => P4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % imp_le_cong
% 0.19/0.42 thf(fact_194_conj__le__cong,axiom,
% 0.19/0.42 ! [X: int,X6: int,P: $o,P4: $o] :
% 0.19/0.42 ( ( X = X6 )
% 0.19/0.42 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 0.19/0.42 => ( P = P4 ) )
% 0.19/0.42 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.19/0.42 & P )
% 0.19/0.42 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 0.19/0.42 & P4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % conj_le_cong
% 0.19/0.42 thf(fact_195_le0,axiom,
% 0.19/0.42 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.19/0.42
% 0.19/0.42 % le0
% 0.19/0.42 thf(fact_196_nat__0,axiom,
% 0.19/0.42 ( ( nat2 @ ( zero_zero @ int ) )
% 0.19/0.42 = ( zero_zero @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_0
% 0.19/0.42 thf(fact_197_nat__le__0,axiom,
% 0.19/0.42 ! [Z: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
% 0.19/0.42 => ( ( nat2 @ Z )
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_le_0
% 0.19/0.42 thf(fact_198_nat__0__iff,axiom,
% 0.19/0.42 ! [I: int] :
% 0.19/0.42 ( ( ( nat2 @ I )
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_0_iff
% 0.19/0.42 thf(fact_199_nat_Odistinct_I1_J,axiom,
% 0.19/0.42 ! [X2: nat] :
% 0.19/0.42 ( ( zero_zero @ nat )
% 0.19/0.42 != ( suc @ X2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat.distinct(1)
% 0.19/0.42 thf(fact_200_old_Onat_Odistinct_I2_J,axiom,
% 0.19/0.42 ! [Nat3: nat] :
% 0.19/0.42 ( ( suc @ Nat3 )
% 0.19/0.42 != ( zero_zero @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.nat.distinct(2)
% 0.19/0.42 thf(fact_201_old_Onat_Odistinct_I1_J,axiom,
% 0.19/0.42 ! [Nat2: nat] :
% 0.19/0.42 ( ( zero_zero @ nat )
% 0.19/0.42 != ( suc @ Nat2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.nat.distinct(1)
% 0.19/0.42 thf(fact_202_nat_OdiscI,axiom,
% 0.19/0.42 ! [Nat: nat,X2: nat] :
% 0.19/0.42 ( ( Nat
% 0.19/0.42 = ( suc @ X2 ) )
% 0.19/0.42 => ( Nat
% 0.19/0.42 != ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat.discI
% 0.19/0.42 thf(fact_203_nat__induct,axiom,
% 0.19/0.42 ! [P: nat > $o,N: nat] :
% 0.19/0.42 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.42 => ( ! [N2: nat] :
% 0.19/0.42 ( ( P @ N2 )
% 0.19/0.42 => ( P @ ( suc @ N2 ) ) )
% 0.19/0.42 => ( P @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_induct
% 0.19/0.42 thf(fact_204_diff__induct,axiom,
% 0.19/0.42 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.19/0.42 ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
% 0.19/0.42 => ( ! [Y6: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y6 ) )
% 0.19/0.42 => ( ! [X3: nat,Y6: nat] :
% 0.19/0.42 ( ( P @ X3 @ Y6 )
% 0.19/0.42 => ( P @ ( suc @ X3 ) @ ( suc @ Y6 ) ) )
% 0.19/0.42 => ( P @ M @ N ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % diff_induct
% 0.19/0.42 thf(fact_205_zero__induct,axiom,
% 0.19/0.42 ! [P: nat > $o,K: nat] :
% 0.19/0.42 ( ( P @ K )
% 0.19/0.42 => ( ! [N2: nat] :
% 0.19/0.42 ( ( P @ ( suc @ N2 ) )
% 0.19/0.42 => ( P @ N2 ) )
% 0.19/0.42 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_induct
% 0.19/0.42 thf(fact_206_Suc__neq__Zero,axiom,
% 0.19/0.42 ! [M: nat] :
% 0.19/0.42 ( ( suc @ M )
% 0.19/0.42 != ( zero_zero @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % Suc_neq_Zero
% 0.19/0.42 thf(fact_207_Zero__neq__Suc,axiom,
% 0.19/0.42 ! [M: nat] :
% 0.19/0.42 ( ( zero_zero @ nat )
% 0.19/0.42 != ( suc @ M ) ) ).
% 0.19/0.42
% 0.19/0.42 % Zero_neq_Suc
% 0.19/0.42 thf(fact_208_Zero__not__Suc,axiom,
% 0.19/0.42 ! [M: nat] :
% 0.19/0.42 ( ( zero_zero @ nat )
% 0.19/0.42 != ( suc @ M ) ) ).
% 0.19/0.42
% 0.19/0.42 % Zero_not_Suc
% 0.19/0.42 thf(fact_209_old_Onat_Oexhaust,axiom,
% 0.19/0.42 ! [Y: nat] :
% 0.19/0.42 ( ( Y
% 0.19/0.42 != ( zero_zero @ nat ) )
% 0.19/0.42 => ~ ! [Nat4: nat] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( suc @ Nat4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.nat.exhaust
% 0.19/0.42 thf(fact_210_old_Onat_Oinducts,axiom,
% 0.19/0.42 ! [P: nat > $o,Nat: nat] :
% 0.19/0.42 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.42 => ( ! [Nat4: nat] :
% 0.19/0.42 ( ( P @ Nat4 )
% 0.19/0.42 => ( P @ ( suc @ Nat4 ) ) )
% 0.19/0.42 => ( P @ Nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.nat.inducts
% 0.19/0.42 thf(fact_211_not0__implies__Suc,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( ( N
% 0.19/0.42 != ( zero_zero @ nat ) )
% 0.19/0.42 => ? [M4: nat] :
% 0.19/0.42 ( N
% 0.19/0.42 = ( suc @ M4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not0_implies_Suc
% 0.19/0.42 thf(fact_212_dependent__nat__choice,axiom,
% 0.19/0.42 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 0.19/0.42 ( ? [X1: A] : ( P @ ( zero_zero @ nat ) @ X1 )
% 0.19/0.42 => ( ! [X3: A,N2: nat] :
% 0.19/0.42 ( ( P @ N2 @ X3 )
% 0.19/0.42 => ? [Y3: A] :
% 0.19/0.42 ( ( P @ ( suc @ N2 ) @ Y3 )
% 0.19/0.42 & ( Q @ N2 @ X3 @ Y3 ) ) )
% 0.19/0.42 => ? [F3: nat > A] :
% 0.19/0.42 ! [N4: nat] :
% 0.19/0.42 ( ( P @ N4 @ ( F3 @ N4 ) )
% 0.19/0.42 & ( Q @ N4 @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dependent_nat_choice
% 0.19/0.42 thf(fact_213_exists__least__lemma,axiom,
% 0.19/0.42 ! [P: nat > $o] :
% 0.19/0.42 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.19/0.42 => ( ? [X1: nat] : ( P @ X1 )
% 0.19/0.42 => ? [N2: nat] :
% 0.19/0.42 ( ~ ( P @ N2 )
% 0.19/0.42 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % exists_least_lemma
% 0.19/0.42 thf(fact_214_le__0__eq,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.19/0.42 = ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_0_eq
% 0.19/0.42 thf(fact_215_less__eq__nat_Osimps_I1_J,axiom,
% 0.19/0.42 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_nat.simps(1)
% 0.19/0.42 thf(fact_216_of__nat__aux_Osimps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [Inc: A > A,I: A] :
% 0.19/0.42 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 0.19/0.42 = I ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_aux.simps(1)
% 0.19/0.42 thf(fact_217_transfer__int__nat__numerals_I1_J,axiom,
% 0.19/0.42 ( ( zero_zero @ int )
% 0.19/0.42 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % transfer_int_nat_numerals(1)
% 0.19/0.42 thf(fact_218_nat__eq__iff,axiom,
% 0.19/0.42 ! [W: int,M: nat] :
% 0.19/0.42 ( ( ( nat2 @ W )
% 0.19/0.42 = M )
% 0.19/0.42 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.19/0.42 => ( W
% 0.19/0.42 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.19/0.42 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.19/0.42 => ( M
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_eq_iff
% 0.19/0.42 thf(fact_219_nat__eq__iff2,axiom,
% 0.19/0.42 ! [M: nat,W: int] :
% 0.19/0.42 ( ( M
% 0.19/0.42 = ( nat2 @ W ) )
% 0.19/0.42 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.19/0.42 => ( W
% 0.19/0.42 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.19/0.42 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.19/0.42 => ( M
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_eq_iff2
% 0.19/0.42 thf(fact_220_zero__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_le
% 0.19/0.42 thf(fact_221_option_Osize__gen_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,X: A > nat] :
% 0.19/0.42 ( ( size_option @ A @ X @ ( none @ A ) )
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.size_gen(1)
% 0.19/0.42 thf(fact_222_LList2__Mirabelle__hamjzmohle_Ollength__LNil,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( lList21232602520length @ A @ ( coinductive_LNil @ A ) )
% 0.19/0.42 = ( zero_zero @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % LList2_Mirabelle_hamjzmohle.llength_LNil
% 0.19/0.42 thf(fact_223_option_Osize_I4_J,axiom,
% 0.19/0.42 ! [A: $tType,X2: A] :
% 0.19/0.42 ( ( size_size @ ( option @ A ) @ ( some @ A @ X2 ) )
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.size(4)
% 0.19/0.42 thf(fact_224_option_Osize_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( size_size @ ( option @ A ) @ ( none @ A ) )
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.size(3)
% 0.19/0.42 thf(fact_225_option_Osize__neq,axiom,
% 0.19/0.42 ! [A: $tType,X: option @ A] :
% 0.19/0.42 ( ( size_size @ ( option @ A ) @ X )
% 0.19/0.42 != ( zero_zero @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.size_neq
% 0.19/0.42 thf(fact_226_LList2__Mirabelle__hamjzmohle_Oltake_Osimps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,L: coinductive_llist @ A] :
% 0.19/0.42 ( ( lList22119844313_ltake @ A @ L @ ( zero_zero @ nat ) )
% 0.19/0.42 = ( coinductive_LNil @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % LList2_Mirabelle_hamjzmohle.ltake.simps(1)
% 0.19/0.42 thf(fact_227_list__decode_Ocases,axiom,
% 0.19/0.42 ! [X: nat] :
% 0.19/0.42 ( ( X
% 0.19/0.42 != ( zero_zero @ nat ) )
% 0.19/0.42 => ~ ! [N2: nat] :
% 0.19/0.42 ( X
% 0.19/0.42 != ( suc @ N2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % list_decode.cases
% 0.19/0.42 thf(fact_228_le__numeral__extra_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_numeral_extra(3)
% 0.19/0.42 thf(fact_229_int__eq__iff__numeral,axiom,
% 0.19/0.42 ! [M: nat,V2: num] :
% 0.19/0.42 ( ( ( semiring_1_of_nat @ int @ M )
% 0.19/0.42 = ( numeral_numeral @ int @ V2 ) )
% 0.19/0.42 = ( ( M
% 0.19/0.42 = ( nat2 @ ( numeral_numeral @ int @ V2 ) ) )
% 0.19/0.42 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % int_eq_iff_numeral
% 0.19/0.42 thf(fact_230_numeral__le__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [M: num,N: num] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.42 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % numeral_le_iff
% 0.19/0.42 thf(fact_231_of__nat__numeral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: num] :
% 0.19/0.42 ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
% 0.19/0.42 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_numeral
% 0.19/0.42 thf(fact_232_round__numeral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: num] :
% 0.19/0.42 ( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
% 0.19/0.42 = ( numeral_numeral @ int @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % round_numeral
% 0.19/0.42 thf(fact_233_not__numeral__le__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: num] :
% 0.19/0.42 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_numeral_le_zero
% 0.19/0.42 thf(fact_234_zero__le__numeral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_le_numeral
% 0.19/0.42 thf(fact_235_option_Osize__gen_I2_J,axiom,
% 0.19/0.42 ! [A: $tType,X: A > nat,X2: A] :
% 0.19/0.42 ( ( size_option @ A @ X @ ( some @ A @ X2 ) )
% 0.19/0.42 = ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.size_gen(2)
% 0.19/0.42 thf(fact_236_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
% 0.19/0.42 ! [X: int,Y: int] :
% 0.19/0.42 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.19/0.42 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.19/0.42 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( nat_tsub @ X @ Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Nat_Transfer.transfer_nat_int_function_closures(3)
% 0.19/0.42 thf(fact_237_add__le__cancel__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [C2: A,A2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_cancel_left
% 0.19/0.42 thf(fact_238_add__le__cancel__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,C2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_cancel_right
% 0.19/0.42 thf(fact_239_Nat_Oadd__0__right,axiom,
% 0.19/0.42 ! [M: nat] :
% 0.19/0.42 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.19/0.42 = M ) ).
% 0.19/0.42
% 0.19/0.42 % Nat.add_0_right
% 0.19/0.42 thf(fact_240_add__is__0,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ( plus_plus @ nat @ M @ N )
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 = ( ( M
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_is_0
% 0.19/0.42 thf(fact_241_add__Suc__right,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( plus_plus @ nat @ M @ ( suc @ N ) )
% 0.19/0.42 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_Suc_right
% 0.19/0.42 thf(fact_242_nat__add__left__cancel__le,axiom,
% 0.19/0.42 ! [K: nat,M: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_add_left_cancel_le
% 0.19/0.42 thf(fact_243_add__le__same__cancel1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [B2: A,A2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 0.19/0.42 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_same_cancel1
% 0.19/0.42 thf(fact_244_add__le__same__cancel2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.19/0.42 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_same_cancel2
% 0.19/0.42 thf(fact_245_le__add__same__cancel1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_add_same_cancel1
% 0.19/0.42 thf(fact_246_le__add__same__cancel2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_add_same_cancel2
% 0.19/0.42 thf(fact_247_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % double_add_le_zero_iff_single_add_le_zero
% 0.19/0.42 thf(fact_248_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [A2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_le_double_add_iff_zero_le_single_add
% 0.19/0.42 thf(fact_249_of__nat__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.19/0.42 => ! [M: nat,N: nat] :
% 0.19/0.42 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
% 0.19/0.42 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % of_nat_add
% 0.19/0.42 thf(fact_250_nat__numeral,axiom,
% 0.19/0.42 ! [K: num] :
% 0.19/0.42 ( ( nat2 @ ( numeral_numeral @ int @ K ) )
% 0.19/0.42 = ( numeral_numeral @ nat @ K ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_numeral
% 0.19/0.42 thf(fact_251_add__is__1,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ( plus_plus @ nat @ M @ N )
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.42 = ( ( ( M
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) )
% 0.19/0.42 | ( ( M
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_is_1
% 0.19/0.42 thf(fact_252_one__is__add,axiom,
% 0.19/0.42 ! [M: nat,N: nat] :
% 0.19/0.42 ( ( ( suc @ ( zero_zero @ nat ) )
% 0.19/0.42 = ( plus_plus @ nat @ M @ N ) )
% 0.19/0.42 = ( ( ( M
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) )
% 0.19/0.42 | ( ( M
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % one_is_add
% 0.19/0.42
% 0.19/0.42 %----Type constructors (37)
% 0.19/0.42 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.19/0.42 ! [A6: $tType,A7: $tType] :
% 0.19/0.42 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 0.19/0.42 => ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.19/0.42 ! [A6: $tType,A7: $tType] :
% 0.19/0.42 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.19/0.42 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.19/0.42 ! [A6: $tType,A7: $tType] :
% 0.19/0.42 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.19/0.42 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_fun___Orderings_Oord,axiom,
% 0.19/0.42 ! [A6: $tType,A7: $tType] :
% 0.19/0.42 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.19/0.42 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.19/0.42 ordere516151231imp_le @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.19/0.42 ordere236663937imp_le @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.19/0.42 linord219039673up_add @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.19/0.42 linordered_semidom @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.19/0.42 semiring_char_0 @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.19/0.42 preorder @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.19/0.42 linorder @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.19/0.42 semiring_1 @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.19/0.42 order @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.19/0.42 ord @ int @ ( type2 @ int ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4,axiom,
% 0.19/0.42 ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_5,axiom,
% 0.19/0.42 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.19/0.42 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 0.19/0.42 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7,axiom,
% 0.19/0.42 semiring_char_0 @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Orderings_Opreorder_8,axiom,
% 0.19/0.42 preorder @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Orderings_Olinorder_9,axiom,
% 0.19/0.42 linorder @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Rings_Osemiring__1_10,axiom,
% 0.19/0.42 semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Orderings_Oorder_11,axiom,
% 0.19/0.42 order @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Nat_Onat___Orderings_Oord_12,axiom,
% 0.19/0.42 ord @ nat @ ( type2 @ nat ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Num_Onum___Orderings_Opreorder_13,axiom,
% 0.19/0.42 preorder @ num @ ( type2 @ num ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Num_Onum___Orderings_Olinorder_14,axiom,
% 0.19/0.42 linorder @ num @ ( type2 @ num ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Num_Onum___Orderings_Oorder_15,axiom,
% 0.19/0.42 order @ num @ ( type2 @ num ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Num_Onum___Orderings_Oord_16,axiom,
% 0.19/0.42 ord @ num @ ( type2 @ num ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_17,axiom,
% 0.19/0.42 ! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Set_Oset___Orderings_Opreorder_18,axiom,
% 0.19/0.42 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Set_Oset___Orderings_Oorder_19,axiom,
% 0.19/0.42 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_Set_Oset___Orderings_Oord_20,axiom,
% 0.19/0.42 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21,axiom,
% 0.19/0.42 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_HOL_Obool___Orderings_Opreorder_22,axiom,
% 0.19/0.42 preorder @ $o @ ( type2 @ $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_HOL_Obool___Orderings_Olinorder_23,axiom,
% 0.19/0.42 linorder @ $o @ ( type2 @ $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 0.19/0.42 order @ $o @ ( type2 @ $o ) ).
% 0.19/0.42
% 0.19/0.42 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 0.19/0.42 ord @ $o @ ( type2 @ $o ) ).
% 0.19/0.42
% 0.19/0.42 %----Helper facts (3)
% 0.19/0.42 thf(help_If_3_1_T,axiom,
% 0.19/0.42 ! [P: $o] :
% 0.19/0.42 ( ( P = $true )
% 0.19/0.42 | ( P = $false ) ) ).
% 0.19/0.42
% 0.19/0.42 thf(help_If_2_1_T,axiom,
% 0.19/0.42 ! [A: $tType,X: A,Y: A] :
% 0.19/0.42 ( ( if @ A @ $false @ X @ Y )
% 0.19/0.42 = Y ) ).
% 0.19/0.42
% 0.19/0.42 thf(help_If_1_1_T,axiom,
% 0.19/0.42 ! [A: $tType,X: A,Y: A] :
% 0.19/0.42 ( ( if @ A @ $true @ X @ Y )
% 0.19/0.42 = X ) ).
% 0.19/0.42
% 0.19/0.42 %----Conjectures (1)
% 0.19/0.42 thf(conj_0,conjecture,
% 0.19/0.42 q ).
% 0.19/0.42
% 0.19/0.42 %------------------------------------------------------------------------------
% 0.19/0.42 ------------------------
% 2.43/1.42 ---- Embedded in THF ---
% 2.43/1.42 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.43/1.42 %%% Generated on Tue Feb 24 06:10:58 EST 2026
% 2.43/1.42 %%% using '$$dhol' embedding, version 1.3.0.
% 2.43/1.42 %%% Logic specification used:
% 2.43/1.42 %%% thf(spec, logic, $$dhol).
% 2.43/1.42
% 2.43/1.42 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2DTF12325.p
% 2.43/1.42 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.43/1.42 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 2.43/1.42 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.43/1.42 thf(ty_t_Num_Onum, type, num: $tType).
% 2.43/1.42 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.43/1.42 thf(ty_t_Int_Oint, type, int: $tType).
% 2.43/1.42 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.43/1.42 thf(ty_tf_a, type, a: $tType).
% 2.43/1.42 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim1727834104eiling: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.43/1.42 thf(sy_c_Archimedean__Field_Oround, type, archimedean_round: (!> [A:$tType]: ((A > int)))).
% 2.43/1.42 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 2.43/1.42 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.43/1.42 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.43/1.42 thf(sy_c_Hilbert__Choice_OGreatestM, type, hilbert_GreatestM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.43/1.42 thf(sy_c_Hilbert__Choice_OLeastM, type, hilbert_LeastM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.43/1.42 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.43/1.42 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts, type, lList2435255213lllsts: (!> [A:$tType]: (((set @ A) > (set @ (coinductive_llist @ A)))))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts, type, lList22096119349pslsts: (!> [A:$tType]: (((set @ A) > (set @ (coinductive_llist @ A)))))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts, type, lList21612149805nflsts: (!> [A:$tType]: (((set @ A) > (set @ (coinductive_llist @ A)))))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Olbutlast, type, lList2370560421utlast: (!> [A:$tType]: (((coinductive_llist @ A) > (coinductive_llist @ A))))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oll2f, type, lList2228235758e_ll2f: (!> [A:$tType]: (((coinductive_llist @ A) > (nat > (option @ A)))))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollength, type, lList21232602520length: (!> [A:$tType]: (((coinductive_llist @ A) > nat)))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oltake, type, lList22119844313_ltake: (!> [A:$tType]: (((coinductive_llist @ A) > (nat > (coinductive_llist @ A)))))).
% 2.43/1.42 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts, type, lList21148268032oslsts: (!> [A:$tType]: (((set @ A) > (set @ (coinductive_llist @ A)))))).
% 2.43/1.42 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.43/1.42 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.43/1.42 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri532925092at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))).
% 2.43/1.42 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.43/1.42 thf(sy_c_Nat__Transfer_Otransfer__morphism, type, nat_tr1645093318rphism: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > $o))))).
% 2.43/1.42 thf(sy_c_Nat__Transfer_Otsub, type, nat_tsub: (int > (int > int))).
% 2.43/1.42 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 2.43/1.42 thf(sy_c_Option_Obind, type, bind: (!> [A:$tType,B:$tType]: (((option @ A) > ((A > (option @ B)) > (option @ B)))))).
% 2.43/1.42 thf(sy_c_Option_Ocombine__options, type, combine_options: (!> [A:$tType]: (((A > (A > A)) > ((option @ A) > ((option @ A) > (option @ A))))))).
% 2.43/1.42 thf(sy_c_Option_Ois__none, type, is_none: (!> [A:$tType]: (((option @ A) > $o)))).
% 2.43/1.42 thf(sy_c_Option_Ooption_ONone, type, none: (!> [A:$tType]: ((option @ A)))).
% 2.43/1.42 thf(sy_c_Option_Ooption_OSome, type, some: (!> [A:$tType]: ((A > (option @ A))))).
% 2.43/1.42 thf(sy_c_Option_Ooption_Ocase__option, type, case_option: (!> [B:$tType,A:$tType]: ((B > ((A > B) > ((option @ A) > B)))))).
% 2.43/1.42 thf(sy_c_Option_Ooption_Opred__option, type, pred_option: (!> [A:$tType]: (((A > $o) > ((option @ A) > $o))))).
% 2.43/1.42 thf(sy_c_Option_Ooption_Orec__option, type, rec_option: (!> [C:$tType,A:$tType]: ((C > ((A > C) > ((option @ A) > C)))))).
% 2.43/1.42 thf(sy_c_Option_Ooption_Oset__option, type, set_option: (!> [A:$tType]: (((option @ A) > (set @ A))))).
% 2.43/1.42 thf(sy_c_Option_Ooption_Osize__option, type, size_option: (!> [A:$tType]: (((A > nat) > ((option @ A) > nat))))).
% 2.43/1.42 thf(sy_c_Option_Ooption_Othe, type, the: (!> [A:$tType]: (((option @ A) > A)))).
% 2.43/1.42 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.43/1.42 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.43/1.42 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.43/1.42 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.43/1.42 thf(sy_v_Q, type, q: $o).
% 2.43/1.42 thf(sy_v_i, type, i: nat).
% 2.43/1.42 thf(sy_v_j, type, j: nat).
% 2.43/1.42 thf(sy_v_t, type, t: (coinductive_llist @ a)).
% 2.43/1.42 thf(sy_v_x, type, x: a).
% 2.43/1.42 thf(fact_0_H, axiom, (! [Y:a]: ((((((lList2228235758e_ll2f @ a) @ t) @ j) = ((some @ a) @ Y)) => q)))).
% 2.43/1.42 thf(fact_1_jlei, axiom, (((ord_less_eq @ nat) @ j) @ i)).
% 2.43/1.42 thf(fact_2_tisome, axiom, ((((lList2228235758e_ll2f @ a) @ t) @ i) = ((some @ a) @ x))).
% 2.43/1.42 thf(fact_3_option_Oinject, axiom, (! [A:$tType,X2:A,Y2:A]: (((((some @ A) @ X2) = ((some @ A) @ Y2)) = (X2 = Y2))))).
% 2.43/1.42 thf(fact_4_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.43/1.42 thf(fact_5_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.43/1.42 thf(fact_6_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.43/1.42 thf(fact_7_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.43/1.42 thf(fact_8_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.43/1.42 thf(fact_9_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.43/1.42 thf(fact_10_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (? [X3:A]: (((P @ X3) & (! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ nat) @ (M @ X3)) @ (M @ Y3)))))))))))).
% 2.43/1.42 thf(fact_11_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.43/1.42 thf(fact_12_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.43/1.42 thf(fact_13_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.43/1.42 thf(fact_14_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F2 @ X4)) @ (G2 @ X4))))))))))).
% 2.43/1.42 thf(fact_15_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.43/1.42 thf(fact_16_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.43/1.42 thf(fact_17_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((! [A3:A,B3:A]: ((((P @ B3) @ A3) => ((P @ A3) @ B3)))) => ((P @ A2) @ B2))))))))).
% 2.43/1.42 thf(fact_18_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.43/1.42 thf(fact_19_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.43/1.42 thf(fact_20_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.43/1.42 thf(fact_21_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.43/1.42 thf(fact_22_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.43/1.42 thf(fact_23_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.43/1.42 thf(fact_24_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.43/1.42 thf(fact_25_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.43/1.42 thf(fact_26_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.43/1.42 thf(fact_27_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.43/1.42 thf(fact_28_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.43/1.42 thf(fact_29_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.43/1.42 thf(fact_30_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X4))))))))).
% 2.43/1.42 thf(fact_31_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X3:A,Y6:A]: (((((ord_less_eq @ A) @ X3) @ Y6) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y6))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.43/1.42 thf(fact_32_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y6:B]: (((((ord_less_eq @ B) @ X3) @ Y6) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y6))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.43/1.42 thf(fact_33_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y6:A]: (((((ord_less_eq @ A) @ X3) @ Y6) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y6))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.43/1.42 thf(fact_34_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y6:B]: (((((ord_less_eq @ B) @ X3) @ Y6) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y6))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.43/1.42 thf(fact_35_LeastMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ B) @ (M @ X)) @ (M @ Y6))))) => ((! [X3:A]: (((P @ X3) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ B) @ (M @ X3)) @ (M @ Y3))))) => (Q @ X3))))) => (Q @ ((((hilbert_LeastM @ A) @ B) @ M) @ P))))))))))).
% 2.43/1.42 thf(fact_36_lfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.43/1.42 thf(fact_37_gfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) => (X = Y))))))))).
% 2.43/1.42 thf(fact_38_lfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.43/1.42 thf(fact_39_gfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X:A,Z:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z) @ Y) => (((ord_less_eq @ A) @ Z) @ X))))))))).
% 2.43/1.42 thf(fact_40_lfp_Oleq__refl, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.43/1.42 thf(fact_41_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 2.43/1.42 thf(fact_42_GreatestMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ B) @ (M @ Y6)) @ (M @ X))))) => ((! [X3:A]: (((P @ X3) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ B) @ (M @ Y3)) @ (M @ X3))))) => (Q @ X3))))) => (Q @ ((((hilbert_GreatestM @ A) @ B) @ M) @ P))))))))))).
% 2.43/1.42 thf(fact_43_infinite__idx__SomeE, axiom, (! [A:$tType,T:(coinductive_llist @ A),A4:(set @ A),I:nat]: (((((member @ (coinductive_llist @ A)) @ T) @ ((lList21612149805nflsts @ A) @ A4)) => (~ ((! [X3:A]: (((((lList2228235758e_ll2f @ A) @ T) @ I) != ((some @ A) @ X3)))))))))).
% 2.43/1.42 thf(fact_44_infinite__Some, axiom, (! [A:$tType,T:(coinductive_llist @ A),A4:(set @ A),I:nat]: (((((member @ (coinductive_llist @ A)) @ T) @ ((lList21612149805nflsts @ A) @ A4)) => (? [A3:A]: (((((lList2228235758e_ll2f @ A) @ T) @ I) = ((some @ A) @ A3)))))))).
% 2.43/1.42 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.43/1.42 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A4)))) = A4)))).
% 2.43/1.42 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.43/1.42 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.43/1.42 thf(fact_49_inflsts__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ (coinductive_llist @ A))) @ ((lList21612149805nflsts @ A) @ A4)) @ ((lList21612149805nflsts @ A) @ B4)))))).
% 2.43/1.42 thf(fact_50_LeastM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M:(A > nat)]: (((P @ X) => (((ord_less_eq @ nat) @ (M @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P))) @ (M @ X)))))).
% 2.43/1.42 thf(fact_51_LeastM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => ((P @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P)) & (! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ nat) @ (M @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P))) @ (M @ Y3)))))))))).
% 2.43/1.42 thf(fact_52_option_Osimps_I7_J, axiom, (! [C:$tType,A:$tType,F1:C,F22:(A > C),X2:A]: (((((((rec_option @ C) @ A) @ F1) @ F22) @ ((some @ A) @ X2)) = (F22 @ X2))))).
% 2.43/1.42 thf(fact_53_combine__options__simps_I3_J, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((((combine_options @ A) @ F) @ ((some @ A) @ A2)) @ ((some @ A) @ B2)) = ((some @ A) @ ((F @ A2) @ B2)))))).
% 2.43/1.42 thf(fact_54_ll2f__lem1, axiom, (! [A:$tType,T:(coinductive_llist @ A),I:nat,X:A]: ((((((lList2228235758e_ll2f @ A) @ T) @ (suc @ I)) = ((some @ A) @ X)) => (? [Y6:A]: (((((lList2228235758e_ll2f @ A) @ T) @ I) = ((some @ A) @ Y6)))))))).
% 2.43/1.42 thf(fact_55_ll2f__Suc__Some, axiom, (! [A:$tType,T:(coinductive_llist @ A),I:nat,X:A]: ((((((lList2228235758e_ll2f @ A) @ T) @ (suc @ I)) = ((some @ A) @ X)) => (~ ((! [X3:A]: (((((lList2228235758e_ll2f @ A) @ T) @ I) != ((some @ A) @ X3)))))))))).
% 2.43/1.42 thf(fact_56_ll2f__None__le, axiom, (! [A:$tType,T:(coinductive_llist @ A),J:nat,I:nat]: ((((((lList2228235758e_ll2f @ A) @ T) @ J) = (none @ A)) => ((((ord_less_eq @ nat) @ J) @ I) => ((((lList2228235758e_ll2f @ A) @ T) @ I) = (none @ A))))))).
% 2.43/1.42 thf(fact_57_of__nat__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 2.43/1.42 thf(fact_58_is__none__code_I2_J, axiom, (! [B:$tType,X:B]: ((~ (((is_none @ B) @ ((some @ B) @ X))))))).
% 2.43/1.42 thf(fact_59_option_Opred__inject_I2_J, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((((pred_option @ A) @ P) @ ((some @ A) @ A2)) = (P @ A2))))).
% 2.43/1.42 thf(fact_60_nat_Oinject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))).
% 2.43/1.42 thf(fact_61_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.43/1.42 thf(fact_62_of__nat__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.43/1.42 thf(fact_63_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 2.43/1.42 thf(fact_64_not__Some__eq, axiom, (! [A:$tType,X:(option @ A)]: (((! [Y5:A]: ((X != ((some @ A) @ Y5)))) = (X = (none @ A)))))).
% 2.43/1.42 thf(fact_65_not__None__eq, axiom, (! [A:$tType,X:(option @ A)]: (((X != (none @ A)) = (? [Y5:A]: ((X = ((some @ A) @ Y5)))))))).
% 2.43/1.42 thf(fact_66_is__none__code_I1_J, axiom, (! [A:$tType]: (((is_none @ A) @ (none @ A))))).
% 2.43/1.42 thf(fact_67_combine__options__simps_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),X:(option @ A)]: ((((((combine_options @ A) @ F) @ X) @ (none @ A)) = X)))).
% 2.43/1.42 thf(fact_68_combine__options__simps_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Y:(option @ A)]: ((((((combine_options @ A) @ F) @ (none @ A)) @ Y) = Y)))).
% 2.43/1.42 thf(fact_69_LeastM__natI, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (P @ ((((hilbert_LeastM @ A) @ nat) @ M) @ P)))))).
% 2.43/1.42 thf(fact_70_ll2f__None__Suc, axiom, (! [A:$tType,T:(coinductive_llist @ A),I:nat]: ((((((lList2228235758e_ll2f @ A) @ T) @ I) = (none @ A)) => ((((lList2228235758e_ll2f @ A) @ T) @ (suc @ I)) = (none @ A)))))).
% 2.43/1.42 thf(fact_71_option_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,F1:C,F22:(A > C)]: (((((((rec_option @ C) @ A) @ F1) @ F22) @ (none @ A)) = F1)))).
% 2.43/1.42 thf(fact_72_is__none__simps_I1_J, axiom, (! [A:$tType]: (((is_none @ A) @ (none @ A))))).
% 2.43/1.42 thf(fact_73_option_Opred__inject_I1_J, axiom, (! [A:$tType,P:(A > $o)]: ((((pred_option @ A) @ P) @ (none @ A))))).
% 2.43/1.42 thf(fact_74_Suc__inject, axiom, (! [X:nat,Y:nat]: ((((suc @ X) = (suc @ Y)) => (X = Y))))).
% 2.43/1.42 thf(fact_75_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.43/1.42 thf(fact_76_Option_Ois__none__def, axiom, (! [A:$tType]: (((is_none @ A) = (^ [X4:(option @ A)]: ((X4 = (none @ A)))))))).
% 2.43/1.42 thf(fact_77_combine__options__assoc, axiom, (! [A:$tType,F:(A > (A > A)),X:(option @ A),Y:(option @ A),Z:(option @ A)]: (((! [X3:A,Y6:A,Z3:A]: ((((F @ ((F @ X3) @ Y6)) @ Z3) = ((F @ X3) @ ((F @ Y6) @ Z3))))) => (((((combine_options @ A) @ F) @ ((((combine_options @ A) @ F) @ X) @ Y)) @ Z) = ((((combine_options @ A) @ F) @ X) @ ((((combine_options @ A) @ F) @ Y) @ Z))))))).
% 2.43/1.42 thf(fact_78_combine__options__commute, axiom, (! [A:$tType,F:(A > (A > A)),X:(option @ A),Y:(option @ A)]: (((! [X3:A,Y6:A]: ((((F @ X3) @ Y6) = ((F @ Y6) @ X3)))) => (((((combine_options @ A) @ F) @ X) @ Y) = ((((combine_options @ A) @ F) @ Y) @ X)))))).
% 2.43/1.42 thf(fact_79_combine__options__left__commute, axiom, (! [A:$tType,F:(A > (A > A)),Y:(option @ A),X:(option @ A),Z:(option @ A)]: (((! [X3:A,Y6:A]: ((((F @ X3) @ Y6) = ((F @ Y6) @ X3)))) => ((! [X3:A,Y6:A,Z3:A]: ((((F @ ((F @ X3) @ Y6)) @ Z3) = ((F @ X3) @ ((F @ Y6) @ Z3))))) => (((((combine_options @ A) @ F) @ Y) @ ((((combine_options @ A) @ F) @ X) @ Z)) = ((((combine_options @ A) @ F) @ X) @ ((((combine_options @ A) @ F) @ Y) @ Z)))))))).
% 2.43/1.42 thf(fact_80_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less_eq @ nat) @ (suc @ M2)) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.43/1.42 thf(fact_81_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 2.43/1.42 thf(fact_82_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.43/1.42 thf(fact_83_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 2.43/1.42 thf(fact_84_Suc__le__D, axiom, (! [N:nat,M3:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M3) => (? [M4:nat]: ((M3 = (suc @ M4)))))))).
% 2.43/1.42 thf(fact_85_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 2.43/1.42 thf(fact_86_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 2.43/1.42 thf(fact_87_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.43/1.42 thf(fact_88_combine__options__cases, axiom, (! [A:$tType,B:$tType,X:(option @ A),P:((option @ A) > ((option @ B) > $o)),Y:(option @ B)]: ((((X = (none @ A)) => ((P @ X) @ Y)) => (((Y = (none @ B)) => ((P @ X) @ Y)) => ((! [A3:A,B3:B]: (((X = ((some @ A) @ A3)) => ((Y = ((some @ B) @ B3)) => ((P @ X) @ Y))))) => ((P @ X) @ Y))))))).
% 2.43/1.42 thf(fact_89_split__option__all, axiom, (! [A:$tType]: (((^ [P2:((option @ A) > $o)]: ((! [X5:(option @ A)]: ((P2 @ X5))))) = (^ [P3:((option @ A) > $o)]: (((P3 @ (none @ A)) & (! [X4:A]: ((P3 @ ((some @ A) @ X4))))))))))).
% 2.43/1.42 thf(fact_90_split__option__ex, axiom, (! [A:$tType]: (((^ [P2:((option @ A) > $o)]: ((? [X5:(option @ A)]: ((P2 @ X5))))) = (^ [P3:((option @ A) > $o)]: (((P3 @ (none @ A)) | (? [X4:A]: ((P3 @ ((some @ A) @ X4))))))))))).
% 2.43/1.42 thf(fact_91_option_Oinducts, axiom, (! [A:$tType,P:((option @ A) > $o),Option:(option @ A)]: (((P @ (none @ A)) => ((! [X3:A]: ((P @ ((some @ A) @ X3)))) => (P @ Option)))))).
% 2.43/1.42 thf(fact_92_option_Oexhaust, axiom, (! [A:$tType,Y:(option @ A)]: (((Y != (none @ A)) => (~ ((! [X22:A]: ((Y != ((some @ A) @ X22)))))))))).
% 2.43/1.42 thf(fact_93_option_OdiscI, axiom, (! [A:$tType,Option:(option @ A),X2:A]: (((Option = ((some @ A) @ X2)) => (Option != (none @ A)))))).
% 2.43/1.42 thf(fact_94_option_Odistinct_I1_J, axiom, (! [A:$tType,X2:A]: (((none @ A) != ((some @ A) @ X2))))).
% 2.43/1.42 thf(fact_95_is__none__simps_I2_J, axiom, (! [B:$tType,X:B]: ((~ (((is_none @ B) @ ((some @ B) @ X))))))).
% 2.43/1.42 thf(fact_96_lift__Suc__antimono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N2))) @ (F @ N2)))) => ((((ord_less_eq @ nat) @ N) @ N3) => (((ord_less_eq @ A) @ (F @ N3)) @ (F @ N)))))))))).
% 2.43/1.42 thf(fact_97_lift__Suc__mono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less_eq @ nat) @ N) @ N3) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N3)))))))))).
% 2.43/1.42 thf(fact_98_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 2.43/1.42 thf(fact_99_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.43/1.42 thf(fact_100_real__arch__simple, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [N2:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.43/1.42 thf(fact_101_poslsts__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ (coinductive_llist @ A))) @ ((lList21148268032oslsts @ A) @ A4)) @ ((lList21148268032oslsts @ A) @ B4)))))).
% 2.43/1.42 thf(fact_102_fpslsts__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ (coinductive_llist @ A))) @ ((lList22096119349pslsts @ A) @ A4)) @ ((lList22096119349pslsts @ A) @ B4)))))).
% 2.43/1.42 thf(fact_103_zle__int, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.43/1.42 thf(fact_104_transfer__int__nat__relations_I3_J, axiom, (! [X:nat,Y:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y)) = (((ord_less_eq @ nat) @ X) @ Y))))).
% 2.43/1.42 thf(fact_105_int__if__cong, axiom, (! [P:$o,X:nat,Y:nat]: (((P => (((semiring_1_of_nat @ int) @ X) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X) @ Y)))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ Y) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X) @ Y)))))))).
% 2.43/1.42 thf(fact_106_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.43/1.42 thf(fact_107_transfer__int__nat__relations_I1_J, axiom, (! [X:nat,Y:nat]: (((((semiring_1_of_nat @ int) @ X) = ((semiring_1_of_nat @ int) @ Y)) = (X = Y))))).
% 2.43/1.42 thf(fact_108_set__mp, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 2.43/1.42 thf(fact_109_in__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 2.43/1.42 thf(fact_110_subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B4)))))).
% 2.43/1.42 thf(fact_111_subsetCE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C2) @ A4) => (((member @ A) @ C2) @ B4)))))).
% 2.43/1.42 thf(fact_112_equalityE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))))).
% 2.43/1.42 thf(fact_113_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B5)))))))))).
% 2.43/1.42 thf(fact_114_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.43/1.42 thf(fact_115_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.43/1.42 thf(fact_116_set__rev__mp, axiom, (! [A:$tType,X:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ X) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ A) @ X) @ B4)))))).
% 2.43/1.42 thf(fact_117_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A5) => (((member @ A) @ T2) @ B5)))))))))).
% 2.43/1.42 thf(fact_118_rev__subsetD, axiom, (! [A:$tType,C2:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C2) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ A) @ C2) @ B4)))))).
% 2.43/1.42 thf(fact_119_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 2.43/1.42 thf(fact_120_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.43/1.42 thf(fact_121_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less_eq @ (set @ A)) @ A4) @ C3)))))).
% 2.43/1.42 thf(fact_122_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z2:(set @ A)]: ((Y4 = Z2))) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))).
% 2.43/1.42 thf(fact_123_contra__subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((~ ((((member @ A) @ C2) @ B4))) => (~ ((((member @ A) @ C2) @ A4)))))))).
% 2.43/1.42 thf(fact_124_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 2.43/1.42 thf(fact_125_round__of__nat, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [N:nat]: ((((archimedean_round @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 2.43/1.42 thf(fact_126_option_Ocollapse, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (((some @ A) @ ((the @ A) @ Option)) = Option))))).
% 2.43/1.42 thf(fact_127_transfer__morphismI, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(A > $o)]: (((((nat_tr1645093318rphism @ A) @ B) @ F) @ A4)))).
% 2.43/1.42 thf(fact_128_of__nat__aux_Osimps_I2_J, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [Inc:(A > A),N:nat,I:A]: ((((((semiri532925092at_aux @ A) @ Inc) @ (suc @ N)) @ I) = ((((semiri532925092at_aux @ A) @ Inc) @ N) @ (Inc @ I))))))))).
% 2.43/1.42 thf(fact_129_round__mono, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ int) @ ((archimedean_round @ A) @ X)) @ ((archimedean_round @ A) @ Y))))))))).
% 2.43/1.42 thf(fact_130_transfer__morphism__def, axiom, (! [A:$tType,B:$tType]: ((((nat_tr1645093318rphism @ B) @ A) = (^ [F2:(B > A),A5:(B > $o)]: ($true)))))).
% 2.43/1.42 thf(fact_131_option_Oexpand, axiom, (! [A:$tType,Option:(option @ A),Option2:(option @ A)]: ((((Option = (none @ A)) = (Option2 = (none @ A))) => (((Option != (none @ A)) => ((Option2 != (none @ A)) => (((the @ A) @ Option) = ((the @ A) @ Option2)))) => (Option = Option2)))))).
% 2.43/1.42 thf(fact_132_option_Osel, axiom, (! [A:$tType,X2:A]: ((((the @ A) @ ((some @ A) @ X2)) = X2)))).
% 2.43/1.42 thf(fact_133_option_Oexhaust__sel, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (Option = ((some @ A) @ ((the @ A) @ Option))))))).
% 2.43/1.42 thf(fact_134_option_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (~ ((((Option = (none @ A)) & (~ ((P @ F1)))) | ((Option = ((some @ A) @ ((the @ A) @ Option))) & (~ ((P @ (F22 @ ((the @ A) @ Option))))))))))))).
% 2.43/1.42 thf(fact_135_option_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (((Option = (none @ A)) => (P @ F1)) & ((Option = ((some @ A) @ ((the @ A) @ Option))) => (P @ (F22 @ ((the @ A) @ Option))))))))).
% 2.43/1.42 thf(fact_136_option_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((case_option @ B) @ A) = (^ [F12:B,F23:(A > B),Option3:(option @ A)]: (((((if @ B) @ (Option3 = (none @ A))) @ F12) @ (F23 @ ((the @ A) @ Option3))))))))).
% 2.43/1.42 thf(fact_137_is__none__bind, axiom, (! [A:$tType,B:$tType,F:(option @ B),G:(B > (option @ A))]: ((((is_none @ A) @ ((((bind @ B) @ A) @ F) @ G)) = (((is_none @ B) @ F) | ((is_none @ A) @ (G @ ((the @ B) @ F)))))))).
% 2.43/1.42 thf(fact_138_bind__runit, axiom, (! [A:$tType,X:(option @ A)]: ((((((bind @ A) @ A) @ X) @ (some @ A)) = X)))).
% 2.43/1.42 thf(fact_139_bind_Obind__lzero, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B))]: ((((((bind @ A) @ B) @ (none @ A)) @ F) = (none @ B))))).
% 2.43/1.42 thf(fact_140_bind__eq__Some__conv, axiom, (! [A:$tType,B:$tType,F:(option @ B),G:(B > (option @ A)),X:A]: (((((((bind @ B) @ A) @ F) @ G) = ((some @ A) @ X)) = (? [Y5:B]: (((F = ((some @ B) @ Y5)) & ((G @ Y5) = ((some @ A) @ X))))))))).
% 2.43/1.42 thf(fact_141_bind_Obind__lunit, axiom, (! [B:$tType,A:$tType,X:A,F:(A > (option @ B))]: ((((((bind @ A) @ B) @ ((some @ A) @ X)) @ F) = (F @ X))))).
% 2.43/1.42 thf(fact_142_Option_Obind__cong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > (option @ B)),G:(A > (option @ B))]: (((X = Y) => ((! [A3:A]: (((Y = ((some @ A) @ A3)) => ((F @ A3) = (G @ A3))))) => (((((bind @ A) @ B) @ X) @ F) = ((((bind @ A) @ B) @ Y) @ G))))))).
% 2.43/1.42 thf(fact_143_bind__option__cong__code, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > (option @ B))]: (((X = Y) => (((((bind @ A) @ B) @ X) @ F) = ((((bind @ A) @ B) @ Y) @ F)))))).
% 2.43/1.42 thf(fact_144_option_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F22:(A > B)]: (((((((case_option @ B) @ A) @ F1) @ F22) @ (none @ A)) = F1)))).
% 2.43/1.42 thf(fact_145_option_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F22:(A > B),X2:A]: (((((((case_option @ B) @ A) @ F1) @ F22) @ ((some @ A) @ X2)) = (F22 @ X2))))).
% 2.43/1.42 thf(fact_146_bind__split, axiom, (! [A:$tType,B:$tType,P:((option @ A) > $o),M:(option @ B),F:(B > (option @ A))]: (((P @ ((((bind @ B) @ A) @ M) @ F)) = (((M = (none @ B)) => (P @ (none @ A))) & (! [V:B]: (((M = ((some @ B) @ V)) => (P @ (F @ V)))))))))).
% 2.43/1.42 thf(fact_147_bind__split__asm, axiom, (! [A:$tType,B:$tType,P:((option @ A) > $o),M:(option @ B),F:(B > (option @ A))]: (((P @ ((((bind @ B) @ A) @ M) @ F)) = (~ ((((M = (none @ B)) & (~ ((P @ (none @ A))))) | (? [X4:B]: (((M = ((some @ B) @ X4)) & (~ ((P @ (F @ X4)))))))))))))).
% 2.43/1.42 thf(fact_148_bind__eq__None__conv, axiom, (! [B:$tType,A:$tType,A2:(option @ B),F:(B > (option @ A))]: (((((((bind @ B) @ A) @ A2) @ F) = (none @ A)) = ((A2 = (none @ B)) | ((F @ ((the @ B) @ A2)) = (none @ A))))))).
% 2.43/1.42 thf(fact_149_alllsts__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((ord_less_eq @ (set @ (coinductive_llist @ A))) @ ((lList2435255213lllsts @ A) @ A4)) @ ((lList2435255213lllsts @ A) @ B4)))))).
% 2.43/1.42 thf(fact_150_nat__le__iff, axiom, (! [X:int,N:nat]: (((((ord_less_eq @ nat) @ (nat2 @ X)) @ N) = (((ord_less_eq @ int) @ X) @ ((semiring_1_of_nat @ int) @ N)))))).
% 2.43/1.42 thf(fact_151_ll2f__LNil, axiom, (! [A:$tType,X:nat]: (((((lList2228235758e_ll2f @ A) @ (coinductive_LNil @ A)) @ X) = (none @ A))))).
% 2.43/1.42 thf(fact_152_option_Oset__sel, axiom, (! [A:$tType,A2:(option @ A)]: (((A2 != (none @ A)) => (((member @ A) @ ((the @ A) @ A2)) @ ((set_option @ A) @ A2)))))).
% 2.43/1.42 thf(fact_153_nat__int, axiom, (! [N:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N)) = N)))).
% 2.43/1.42 thf(fact_154_elem__set, axiom, (! [A:$tType,X:A,Xo:(option @ A)]: (((((member @ A) @ X) @ ((set_option @ A) @ Xo)) = (Xo = ((some @ A) @ X)))))).
% 2.43/1.42 thf(fact_155_poslsts__iff, axiom, (! [A:$tType,S:(coinductive_llist @ A),A4:(set @ A)]: (((((member @ (coinductive_llist @ A)) @ S) @ ((lList21148268032oslsts @ A) @ A4)) = ((((member @ (coinductive_llist @ A)) @ S) @ ((lList2435255213lllsts @ A) @ A4)) & (S != (coinductive_LNil @ A))))))).
% 2.43/1.42 thf(fact_156_ospec, axiom, (! [A:$tType,A4:(option @ A),P:(A > $o),X:A]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set_option @ A) @ A4)) => (P @ X3)))) => ((A4 = ((some @ A) @ X)) => (P @ X)))))).
% 2.43/1.42 thf(fact_157_option_Oset__cases, axiom, (! [A:$tType,E:A,A2:(option @ A)]: (((((member @ A) @ E) @ ((set_option @ A) @ A2)) => (A2 = ((some @ A) @ E)))))).
% 2.43/1.42 thf(fact_158_option_Oset__intros, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ ((set_option @ A) @ ((some @ A) @ A2)))))).
% 2.43/1.42 thf(fact_159_nat__if__cong, axiom, (! [P:$o,X:int,Y:int]: (((P => ((nat2 @ X) = (nat2 @ ((((if @ int) @ P) @ X) @ Y)))) & ((~ (P)) => ((nat2 @ Y) = (nat2 @ ((((if @ int) @ P) @ X) @ Y)))))))).
% 2.43/1.42 thf(fact_160_alllsts_OLNil__all, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ (coinductive_llist @ A)) @ (coinductive_LNil @ A)) @ ((lList2435255213lllsts @ A) @ A4))))).
% 2.43/1.42 thf(fact_161_infsubsetall, axiom, (! [A:$tType,X:(coinductive_llist @ A),A4:(set @ A)]: (((((member @ (coinductive_llist @ A)) @ X) @ ((lList21612149805nflsts @ A) @ A4)) => (((member @ (coinductive_llist @ A)) @ X) @ ((lList2435255213lllsts @ A) @ A4)))))).
% 2.43/1.42 thf(fact_162_bind__option__cong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > (option @ B)),G:(A > (option @ B))]: (((X = Y) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ Y)) => ((F @ Z3) = (G @ Z3))))) => (((((bind @ A) @ B) @ X) @ F) = ((((bind @ A) @ B) @ Y) @ G))))))).
% 2.43/1.42 thf(fact_163_option_Opred__cong, axiom, (! [A:$tType,X:(option @ A),Ya:(option @ A),P:(A > $o),Pa:(A > $o)]: (((X = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ Ya)) => ((P @ Z3) = (Pa @ Z3))))) => ((((pred_option @ A) @ P) @ X) = (((pred_option @ A) @ Pa) @ Ya))))))).
% 2.43/1.42 thf(fact_164_option_Opred__mono__strong, axiom, (! [A:$tType,P:(A > $o),X:(option @ A),Pa:(A > $o)]: (((((pred_option @ A) @ P) @ X) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ X)) => ((P @ Z3) => (Pa @ Z3))))) => (((pred_option @ A) @ Pa) @ X)))))).
% 2.43/1.42 thf(fact_165_nat__mono, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ X) @ Y) => (((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y)))))).
% 2.43/1.42 thf(fact_166_lbutlast__LNil, axiom, (! [A:$tType]: ((((lList2370560421utlast @ A) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.43/1.42 thf(fact_167_le__nat__iff, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ nat) @ N) @ (nat2 @ K)) = (((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K)))))).
% 2.43/1.42 thf(fact_168_int__nat__eq, axiom, (! [Z:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = Z)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = (zero_zero @ int))))))).
% 2.43/1.42 thf(fact_169_of__nat__eq__0__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.43/1.42 thf(fact_170_of__nat__0__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.43/1.42 thf(fact_171_of__nat__0, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.43/1.42 thf(fact_172_of__nat__le__0__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.43/1.42 thf(fact_173_round__0, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (((archimedean_round @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 2.43/1.42 thf(fact_174_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.43/1.42 thf(fact_175_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.43/1.42 thf(fact_176_of__nat__0__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 2.43/1.42 thf(fact_177_of__nat__neq__0, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ (suc @ N)) != (zero_zero @ A)))))))).
% 2.43/1.42 thf(fact_178_transfer__morphism__nat__int, axiom, ((((nat_tr1645093318rphism @ int) @ nat) @ nat2) @ ((ord_less_eq @ int) @ (zero_zero @ int)))).
% 2.43/1.42 thf(fact_179_transfer__nat__int__relations_I1_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (((nat2 @ X) = (nat2 @ Y)) = (X = Y))))))).
% 2.43/1.42 thf(fact_180_eq__nat__nat__iff, axiom, (! [Z:int,Z4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z4) => (((nat2 @ Z) = (nat2 @ Z4)) = (Z = Z4))))))).
% 2.43/1.42 thf(fact_181_ex__nat, axiom, ((^ [P2:(nat > $o)]: ((? [X5:nat]: ((P2 @ X5))))) = (^ [P3:(nat > $o)]: ((? [X4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) & (P3 @ (nat2 @ X4))))))))).
% 2.43/1.42 thf(fact_182_all__nat, axiom, ((^ [P2:(nat > $o)]: ((! [X5:nat]: ((P2 @ X5))))) = (^ [P3:(nat > $o)]: ((! [X4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) => (P3 @ (nat2 @ X4))))))))).
% 2.43/1.42 thf(fact_183_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: ((K = ((semiring_1_of_nat @ int) @ N2)))))))).
% 2.43/1.42 thf(fact_184_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: ((K != ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.43/1.42 thf(fact_185_nonneg__eq__int, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (~ ((! [M4:nat]: ((Z != ((semiring_1_of_nat @ int) @ M4)))))))))).
% 2.43/1.42 thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J, axiom, (! [Z:nat]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((semiring_1_of_nat @ int) @ Z))))).
% 2.43/1.42 thf(fact_187_transfer__int__nat__quantifiers_I1_J, axiom, (! [P:(int > $o)]: (((! [X4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) => (P @ X4)))) = (! [X4:nat]: ((P @ ((semiring_1_of_nat @ int) @ X4)))))))).
% 2.43/1.42 thf(fact_188_transfer__int__nat__quantifiers_I2_J, axiom, (! [P:(int > $o)]: (((? [X4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) & (P @ X4)))) = (? [X4:nat]: ((P @ ((semiring_1_of_nat @ int) @ X4)))))))).
% 2.43/1.42 thf(fact_189_transfer__nat__int__relations_I3_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => ((((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y)) = (((ord_less_eq @ int) @ X) @ Y))))))).
% 2.43/1.42 thf(fact_190_nat__0__le, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((semiring_1_of_nat @ int) @ (nat2 @ Z)) = Z))))).
% 2.43/1.42 thf(fact_191_int__eq__iff, axiom, (! [M:nat,Z:int]: (((((semiring_1_of_nat @ int) @ M) = Z) = ((M = (nat2 @ Z)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z)))))).
% 2.43/1.42 thf(fact_192_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.43/1.42 thf(fact_193_imp__le__cong, axiom, (! [X:int,X6:int,P:$o,P4:$o]: (((X = X6) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X6) => (P = P4)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X6) => P4))))))).
% 2.43/1.42 thf(fact_194_conj__le__cong, axiom, (! [X:int,X6:int,P:$o,P4:$o]: (((X = X6) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X6) => (P = P4)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) & P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X6) & P4))))))).
% 2.43/1.42 thf(fact_195_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.43/1.42 thf(fact_196_nat__0, axiom, ((nat2 @ (zero_zero @ int)) = (zero_zero @ nat))).
% 2.43/1.42 thf(fact_197_nat__le__0, axiom, (! [Z:int]: (((((ord_less_eq @ int) @ Z) @ (zero_zero @ int)) => ((nat2 @ Z) = (zero_zero @ nat)))))).
% 2.43/1.42 thf(fact_198_nat__0__iff, axiom, (! [I:int]: ((((nat2 @ I) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I) @ (zero_zero @ int)))))).
% 2.43/1.42 thf(fact_199_nat_Odistinct_I1_J, axiom, (! [X2:nat]: (((zero_zero @ nat) != (suc @ X2))))).
% 2.43/1.42 thf(fact_200_old_Onat_Odistinct_I2_J, axiom, (! [Nat3:nat]: (((suc @ Nat3) != (zero_zero @ nat))))).
% 2.43/1.42 thf(fact_201_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 2.43/1.42 thf(fact_202_nat_OdiscI, axiom, (! [Nat:nat,X2:nat]: (((Nat = (suc @ X2)) => (Nat != (zero_zero @ nat)))))).
% 2.43/1.42 thf(fact_203_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((P @ N2) => (P @ (suc @ N2))))) => (P @ N)))))).
% 2.43/1.42 thf(fact_204_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [X3:nat]: (((P @ X3) @ (zero_zero @ nat)))) => ((! [Y6:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y6)))) => ((! [X3:nat,Y6:nat]: ((((P @ X3) @ Y6) => ((P @ (suc @ X3)) @ (suc @ Y6))))) => ((P @ M) @ N))))))).
% 2.43/1.42 thf(fact_205_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N2:nat]: (((P @ (suc @ N2)) => (P @ N2)))) => (P @ (zero_zero @ nat))))))).
% 2.43/1.42 thf(fact_206_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 2.43/1.42 thf(fact_207_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.43/1.42 thf(fact_208_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.43/1.42 thf(fact_209_old_Onat_Oexhaust, axiom, (! [Y:nat]: (((Y != (zero_zero @ nat)) => (~ ((! [Nat4:nat]: ((Y != (suc @ Nat4)))))))))).
% 2.43/1.42 thf(fact_210_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat4:nat]: (((P @ Nat4) => (P @ (suc @ Nat4))))) => (P @ Nat)))))).
% 2.43/1.42 thf(fact_211_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M4:nat]: ((N = (suc @ M4)))))))).
% 2.43/1.42 thf(fact_212_dependent__nat__choice, axiom, (! [A:$tType,P:(nat > (A > $o)),Q:(nat > (A > (A > $o)))]: (((? [X1:A]: (((P @ (zero_zero @ nat)) @ X1))) => ((! [X3:A,N2:nat]: ((((P @ N2) @ X3) => (? [Y3:A]: ((((P @ (suc @ N2)) @ Y3) & (((Q @ N2) @ X3) @ Y3))))))) => (? [F3:(nat > A)]: ((! [N4:nat]: ((((P @ N4) @ (F3 @ N4)) & (((Q @ N4) @ (F3 @ N4)) @ (F3 @ (suc @ N4))))))))))))).
% 2.43/1.42 thf(fact_213_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X1:nat]: ((P @ X1))) => (? [N2:nat]: (((~ ((P @ N2))) & (P @ (suc @ N2)))))))))).
% 2.43/1.42 thf(fact_214_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.43/1.42 thf(fact_215_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.43/1.42 thf(fact_216_of__nat__aux_Osimps_I1_J, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [Inc:(A > A),I:A]: ((((((semiri532925092at_aux @ A) @ Inc) @ (zero_zero @ nat)) @ I) = I))))))).
% 2.43/1.42 thf(fact_217_transfer__int__nat__numerals_I1_J, axiom, ((zero_zero @ int) = ((semiring_1_of_nat @ int) @ (zero_zero @ nat)))).
% 2.43/1.42 thf(fact_218_nat__eq__iff, axiom, (! [W:int,M:nat]: ((((nat2 @ W) = M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M = (zero_zero @ nat)))))))).
% 2.43/1.42 thf(fact_219_nat__eq__iff2, axiom, (! [M:nat,W:int]: (((M = (nat2 @ W)) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M = (zero_zero @ nat)))))))).
% 2.43/1.42 thf(fact_220_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.43/1.42 thf(fact_221_option_Osize__gen_I1_J, axiom, (! [A:$tType,X:(A > nat)]: (((((size_option @ A) @ X) @ (none @ A)) = (suc @ (zero_zero @ nat)))))).
% 2.43/1.42 thf(fact_222_LList2__Mirabelle__hamjzmohle_Ollength__LNil, axiom, (! [A:$tType]: ((((lList21232602520length @ A) @ (coinductive_LNil @ A)) = (zero_zero @ nat))))).
% 2.43/1.42 thf(fact_223_option_Osize_I4_J, axiom, (! [A:$tType,X2:A]: ((((size_size @ (option @ A)) @ ((some @ A) @ X2)) = (suc @ (zero_zero @ nat)))))).
% 2.43/1.42 thf(fact_224_option_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (option @ A)) @ (none @ A)) = (suc @ (zero_zero @ nat)))))).
% 2.43/1.42 thf(fact_225_option_Osize__neq, axiom, (! [A:$tType,X:(option @ A)]: ((((size_size @ (option @ A)) @ X) != (zero_zero @ nat))))).
% 2.43/1.42 thf(fact_226_LList2__Mirabelle__hamjzmohle_Oltake_Osimps_I1_J, axiom, (! [A:$tType,L:(coinductive_llist @ A)]: (((((lList22119844313_ltake @ A) @ L) @ (zero_zero @ nat)) = (coinductive_LNil @ A))))).
% 2.43/1.42 thf(fact_227_list__decode_Ocases, axiom, (! [X:nat]: (((X != (zero_zero @ nat)) => (~ ((! [N2:nat]: ((X != (suc @ N2)))))))))).
% 2.43/1.42 thf(fact_228_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.43/1.42 thf(fact_229_int__eq__iff__numeral, axiom, (! [M:nat,V2:num]: (((((semiring_1_of_nat @ int) @ M) = ((numeral_numeral @ int) @ V2)) = ((M = (nat2 @ ((numeral_numeral @ int) @ V2))) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ ((numeral_numeral @ int) @ V2))))))).
% 2.43/1.42 thf(fact_230_numeral__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 2.43/1.42 thf(fact_231_of__nat__numeral, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [N:num]: ((((semiring_1_of_nat @ A) @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ A) @ N)))))))).
% 2.43/1.42 thf(fact_232_round__numeral, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [N:num]: ((((archimedean_round @ A) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ int) @ N)))))))).
% 2.43/1.42 thf(fact_233_not__numeral__le__zero, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 2.43/1.42 thf(fact_234_zero__le__numeral, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 2.43/1.42 thf(fact_235_option_Osize__gen_I2_J, axiom, (! [A:$tType,X:(A > nat),X2:A]: (((((size_option @ A) @ X) @ ((some @ A) @ X2)) = (((plus_plus @ nat) @ (X @ X2)) @ (suc @ (zero_zero @ nat))))))).
% 2.43/1.42 thf(fact_236_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ ((nat_tsub @ X) @ Y))))))).
% 2.43/1.42 thf(fact_237_add__le__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.43/1.42 thf(fact_238_add__le__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.43/1.42 thf(fact_239_Nat_Oadd__0__right, axiom, (! [M:nat]: (((((plus_plus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 2.43/1.42 thf(fact_240_add__is__0, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 2.43/1.42 thf(fact_241_add__Suc__right, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ M) @ (suc @ N)) = (suc @ (((plus_plus @ nat) @ M) @ N)))))).
% 2.43/1.42 thf(fact_242_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.43/1.42 thf(fact_243_add__le__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.43/1.42 thf(fact_244_add__le__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.43/1.42 thf(fact_245_le__add__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.43/1.42 thf(fact_246_le__add__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.43/1.42 thf(fact_247_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.43/1.42 thf(fact_248_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.43/1.42 thf(fact_249_of__nat__add, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M) @ N)) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.43/1.42 thf(fact_250_nat__numeral, axiom, (! [K:num]: (((nat2 @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ nat) @ K))))).
% 2.43/1.42 thf(fact_251_add__is__1, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (suc @ (zero_zero @ nat))) = (((M = (suc @ (zero_zero @ nat))) & (N = (zero_zero @ nat))) | ((M = (zero_zero @ nat)) & (N = (suc @ (zero_zero @ nat))))))))).
% 2.43/1.42 thf(fact_252_one__is__add, axiom, (! [M:nat,N:nat]: ((((suc @ (zero_zero @ nat)) = (((plus_plus @ nat) @ M) @ N)) = (((M = (suc @ (zero_zero @ nat))) & (N = (zero_zero @ nat))) | ((M = (zero_zero @ nat)) & (N = (suc @ (zero_zero @ nat))))))))).
% 2.43/1.42 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A6:$tType,A7:$tType]: ((((comple187826305attice @ A7) @ (type2 @ A7)) => ((comple187826305attice @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.43/1.42 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.43/1.42 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.43/1.42 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.43/1.42 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, ((ordere516151231imp_le @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, ((ordere236663937imp_le @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, ((linord219039673up_add @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, ((semiring_char_0 @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, ((preorder @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, ((linorder @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, ((semiring_1 @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, ((order @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, ((ord @ int) @ (type2 @ int))).
% 2.43/1.42 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4, axiom, ((ordere516151231imp_le @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_5, axiom, ((ordere236663937imp_le @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7, axiom, ((semiring_char_0 @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Orderings_Opreorder_8, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Orderings_Olinorder_9, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Rings_Osemiring__1_10, axiom, ((semiring_1 @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Orderings_Oorder_11, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Nat_Onat___Orderings_Oord_12, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.43/1.42 thf(tcon_Num_Onum___Orderings_Opreorder_13, axiom, ((preorder @ num) @ (type2 @ num))).
% 2.43/1.42 thf(tcon_Num_Onum___Orderings_Olinorder_14, axiom, ((linorder @ num) @ (type2 @ num))).
% 2.43/1.42 thf(tcon_Num_Onum___Orderings_Oorder_15, axiom, ((order @ num) @ (type2 @ num))).
% 2.43/1.42 thf(tcon_Num_Onum___Orderings_Oord_16, axiom, ((ord @ num) @ (type2 @ num))).
% 2.43/1.42 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_17, axiom, (! [A6:$tType]: (((comple187826305attice @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.43/1.42 thf(tcon_Set_Oset___Orderings_Opreorder_18, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.43/1.42 thf(tcon_Set_Oset___Orderings_Oorder_19, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.43/1.42 thf(tcon_Set_Oset___Orderings_Oord_20, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.43/1.42 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 2.43/1.42 thf(tcon_HOL_Obool___Orderings_Opreorder_22, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.43/1.42 thf(tcon_HOL_Obool___Orderings_Olinorder_23, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.43/1.42 thf(tcon_HOL_Obool___Orderings_Oorder_24, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.43/1.42 thf(tcon_HOL_Obool___Orderings_Oord_25, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.43/1.42 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.43/1.42 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.43/1.42 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.43/1.42 thf(conj_0, conjecture, q).
% 2.43/1.42 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2DTF12325.p
% 2.43/1.42 ------------------------
% 2.43/1.46 ---- Cleaned THF ---
% 2.43/1.46 thf(ty_t_Coinductive__List_Ollist,type,
% 2.43/1.46 coinductive_llist: $tType > $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_t_Option_Ooption,type,
% 2.43/1.46 option: $tType > $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_t_Set_Oset,type,
% 2.43/1.46 set: $tType > $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_t_Num_Onum,type,
% 2.43/1.46 num: $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_t_Nat_Onat,type,
% 2.43/1.46 nat: $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_t_Int_Oint,type,
% 2.43/1.46 int: $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_t_itself,type,
% 2.43/1.46 itself: $tType > $tType ).
% 2.43/1.46
% 2.43/1.46 thf(ty_tf_a,type,
% 2.43/1.46 a: $tType ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_HOL_Otype,type,
% 2.43/1.46 type:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Orderings_Oord,type,
% 2.43/1.46 ord:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Orderings_Oorder,type,
% 2.43/1.46 order:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Rings_Osemiring__1,type,
% 2.43/1.46 semiring_1:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Orderings_Olinorder,type,
% 2.43/1.46 linorder:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Orderings_Opreorder,type,
% 2.43/1.46 preorder:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.43/1.46 semiring_char_0:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.43/1.46 linordered_semidom:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.43/1.46 linord219039673up_add:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 2.43/1.46 archim1727834104eiling:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.43/1.46 comple187826305attice:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.43/1.46 archim1804426504_field:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.43/1.46 canoni770627133id_add:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.43/1.46 ordere236663937imp_le:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.43/1.46 ordere516151231imp_le:
% 2.43/1.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Archimedean__Field_Oround,type,
% 2.43/1.46 archimedean_round:
% 2.43/1.46 !>[A: $tType] : ( A > int ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 2.43/1.46 coinductive_LNil:
% 2.43/1.46 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.43/1.46 plus_plus:
% 2.43/1.46 !>[A: $tType] : ( A > A > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.43/1.46 zero_zero:
% 2.43/1.46 !>[A: $tType] : A ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 2.43/1.46 hilbert_GreatestM:
% 2.43/1.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 2.43/1.46 hilbert_LeastM:
% 2.43/1.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_If,type,
% 2.43/1.46 if:
% 2.43/1.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Int_Onat,type,
% 2.43/1.46 nat2: int > nat ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
% 2.43/1.46 lList2435255213lllsts:
% 2.43/1.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
% 2.43/1.46 lList22096119349pslsts:
% 2.43/1.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
% 2.43/1.46 lList21612149805nflsts:
% 2.43/1.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Olbutlast,type,
% 2.43/1.46 lList2370560421utlast:
% 2.43/1.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oll2f,type,
% 2.43/1.46 lList2228235758e_ll2f:
% 2.43/1.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( option @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollength,type,
% 2.43/1.46 lList21232602520length:
% 2.43/1.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oltake,type,
% 2.43/1.46 lList22119844313_ltake:
% 2.43/1.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
% 2.43/1.46 lList21148268032oslsts:
% 2.43/1.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Nat_OSuc,type,
% 2.43/1.46 suc: nat > nat ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.43/1.46 semiring_1_of_nat:
% 2.43/1.46 !>[A: $tType] : ( nat > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 2.43/1.46 semiri532925092at_aux:
% 2.43/1.46 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.43/1.46 size_size:
% 2.43/1.46 !>[A: $tType] : ( A > nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
% 2.43/1.46 nat_tr1645093318rphism:
% 2.43/1.46 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Nat__Transfer_Otsub,type,
% 2.43/1.46 nat_tsub: int > int > int ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 2.43/1.46 numeral_numeral:
% 2.43/1.46 !>[A: $tType] : ( num > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Obind,type,
% 2.43/1.46 bind:
% 2.43/1.46 !>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ocombine__options,type,
% 2.43/1.46 combine_options:
% 2.43/1.46 !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ois__none,type,
% 2.43/1.46 is_none:
% 2.43/1.46 !>[A: $tType] : ( ( option @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_ONone,type,
% 2.43/1.46 none:
% 2.43/1.46 !>[A: $tType] : ( option @ A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_OSome,type,
% 2.43/1.46 some:
% 2.43/1.46 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_Ocase__option,type,
% 2.43/1.46 case_option:
% 2.43/1.46 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_Opred__option,type,
% 2.43/1.46 pred_option:
% 2.43/1.46 !>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_Orec__option,type,
% 2.43/1.46 rec_option:
% 2.43/1.46 !>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_Oset__option,type,
% 2.43/1.46 set_option:
% 2.43/1.46 !>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_Osize__option,type,
% 2.43/1.46 size_option:
% 2.43/1.46 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Option_Ooption_Othe,type,
% 2.43/1.46 the:
% 2.43/1.46 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.43/1.46 ord_less_eq:
% 2.43/1.46 !>[A: $tType] : ( A > A > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Pure_Otype,type,
% 2.43/1.46 type2:
% 2.43/1.46 !>[A: $tType] : ( itself @ A ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_Set_OCollect,type,
% 2.43/1.46 collect:
% 2.43/1.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_c_member,type,
% 2.43/1.46 member:
% 2.43/1.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(sy_v_Q,type,
% 2.43/1.46 q: $o ).
% 2.43/1.46
% 2.43/1.46 thf(sy_v_i,type,
% 2.43/1.46 i: nat ).
% 2.43/1.46
% 2.43/1.46 thf(sy_v_j,type,
% 2.43/1.46 j: nat ).
% 2.43/1.46
% 2.43/1.46 thf(sy_v_t,type,
% 2.43/1.46 t: coinductive_llist @ a ).
% 2.43/1.46
% 2.43/1.46 thf(sy_v_x,type,
% 2.43/1.46 x: a ).
% 2.43/1.46
% 2.43/1.46 thf(fact_0_H,axiom,
% 2.43/1.46 ! [Y: a] :
% 2.43/1.46 ( ( ( lList2228235758e_ll2f @ a @ t @ j )
% 2.43/1.46 = ( some @ a @ Y ) )
% 2.43/1.46 => q ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_1_jlei,axiom,
% 2.43/1.46 ord_less_eq @ nat @ j @ i ).
% 2.43/1.46
% 2.43/1.46 thf(fact_2_tisome,axiom,
% 2.43/1.46 ( ( lList2228235758e_ll2f @ a @ t @ i )
% 2.43/1.46 = ( some @ a @ x ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_3_option_Oinject,axiom,
% 2.43/1.46 ! [A: $tType,X2: A,Y2: A] :
% 2.43/1.46 ( ( ( some @ A @ X2 )
% 2.43/1.46 = ( some @ A @ Y2 ) )
% 2.43/1.46 = ( X2 = Y2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_4_order__refl,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( preorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_5_le__refl,axiom,
% 2.43/1.46 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_6_le__trans,axiom,
% 2.43/1.46 ! [I: nat,J: nat,K: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ I @ J )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ J @ K )
% 2.43/1.46 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_7_eq__imp__le,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( M = N )
% 2.43/1.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_8_le__antisym,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ N @ M )
% 2.43/1.46 => ( M = N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_9_nat__le__linear,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.43/1.46 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_10_ex__has__least__nat,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.43/1.46 ( ( P @ K )
% 2.43/1.46 => ? [X3: A] :
% 2.43/1.46 ( ( P @ X3 )
% 2.43/1.46 & ! [Y3: A] :
% 2.43/1.46 ( ( P @ Y3 )
% 2.43/1.46 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y3 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_11_le__funD,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType] :
% 2.43/1.46 ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 => ! [F: A > B,G: A > B,X: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.43/1.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_12_le__funE,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType] :
% 2.43/1.46 ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 => ! [F: A > B,G: A > B,X: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.43/1.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_13_le__funI,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType] :
% 2.43/1.46 ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 => ! [F: A > B,G: A > B] :
% 2.43/1.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.43/1.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_14_le__fun__def,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType] :
% 2.43/1.46 ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 => ( ( ord_less_eq @ ( A > B ) )
% 2.43/1.46 = ( ^ [F2: A > B,G2: A > B] :
% 2.43/1.46 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_15_dual__order_Oantisym,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [B2: A,A2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.43/1.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.43/1.46 => ( A2 = B2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_16_dual__order_Otrans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [B2: A,A2: A,C2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.43/1.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.43/1.46 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_17_linorder__wlog,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.43/1.46 ( ! [A3: A,B3: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.43/1.46 => ( P @ A3 @ B3 ) )
% 2.43/1.46 => ( ! [A3: A,B3: A] :
% 2.43/1.46 ( ( P @ B3 @ A3 )
% 2.43/1.46 => ( P @ A3 @ B3 ) )
% 2.43/1.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_18_dual__order_Orefl,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_19_order__trans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( preorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A,Z: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.43/1.46 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_20_order__class_Oorder_Oantisym,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.43/1.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.43/1.46 => ( A2 = B2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_21_ord__le__eq__trans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ord @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A,C2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.43/1.46 => ( ( B2 = C2 )
% 2.43/1.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_22_ord__eq__le__trans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ord @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A,C2: A] :
% 2.43/1.46 ( ( A2 = B2 )
% 2.43/1.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.43/1.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_23_antisym__conv,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [Y: A,X: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ Y @ X )
% 2.43/1.46 => ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 = ( X = Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_24_le__cases3,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A,Z: A] :
% 2.43/1.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.43/1.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.43/1.46 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.43/1.46 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.43/1.46 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.43/1.46 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.43/1.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.43/1.46 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.43/1.46 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.43/1.46 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.43/1.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_25_order_Otrans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A,C2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.43/1.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.43/1.46 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_26_le__cases,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A] :
% 2.43/1.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_27_eq__refl,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( preorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A] :
% 2.43/1.46 ( ( X = Y )
% 2.43/1.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_28_linear,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_29_antisym,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( ( ord_less_eq @ A @ Y @ X )
% 2.43/1.46 => ( X = Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_30_eq__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.43/1.46 = ( ^ [X4: A,Y5: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.43/1.46 & ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_31_ord__le__eq__subst,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType] :
% 2.43/1.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 & ( ord @ A @ ( type2 @ A ) ) )
% 2.43/1.46 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.43/1.46 => ( ( ( F @ B2 )
% 2.43/1.46 = C2 )
% 2.43/1.46 => ( ! [X3: A,Y6: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X3 @ Y6 )
% 2.43/1.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 2.43/1.46 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_32_ord__eq__le__subst,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType] :
% 2.43/1.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 & ( ord @ A @ ( type2 @ A ) ) )
% 2.43/1.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.43/1.46 ( ( A2
% 2.43/1.46 = ( F @ B2 ) )
% 2.43/1.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.43/1.46 => ( ! [X3: B,Y6: B] :
% 2.43/1.46 ( ( ord_less_eq @ B @ X3 @ Y6 )
% 2.43/1.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 2.43/1.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_33_order__subst2,axiom,
% 2.43/1.46 ! [A: $tType,C: $tType] :
% 2.43/1.46 ( ( ( order @ C @ ( type2 @ C ) )
% 2.43/1.46 & ( order @ A @ ( type2 @ A ) ) )
% 2.43/1.46 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.43/1.46 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.43/1.46 => ( ! [X3: A,Y6: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X3 @ Y6 )
% 2.43/1.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 2.43/1.46 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_34_order__subst1,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType] :
% 2.43/1.46 ( ( ( order @ B @ ( type2 @ B ) )
% 2.43/1.46 & ( order @ A @ ( type2 @ A ) ) )
% 2.43/1.46 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.43/1.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.43/1.46 => ( ! [X3: B,Y6: B] :
% 2.43/1.46 ( ( ord_less_eq @ B @ X3 @ Y6 )
% 2.43/1.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y6 ) ) )
% 2.43/1.46 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_35_LeastMI2,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType] :
% 2.43/1.46 ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 => ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
% 2.43/1.46 ( ( P @ X )
% 2.43/1.46 => ( ! [Y6: A] :
% 2.43/1.46 ( ( P @ Y6 )
% 2.43/1.46 => ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y6 ) ) )
% 2.43/1.46 => ( ! [X3: A] :
% 2.43/1.46 ( ( P @ X3 )
% 2.43/1.46 => ( ! [Y3: A] :
% 2.43/1.46 ( ( P @ Y3 )
% 2.43/1.46 => ( ord_less_eq @ B @ ( M @ X3 ) @ ( M @ Y3 ) ) )
% 2.43/1.46 => ( Q @ X3 ) ) )
% 2.43/1.46 => ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_36_lfp_Oleq__antisym,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( ( ord_less_eq @ A @ Y @ X )
% 2.43/1.46 => ( X = Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_37_gfp_Oleq__antisym,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [Y: A,X: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ Y @ X )
% 2.43/1.46 => ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( X = Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_38_lfp_Oleq__trans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A,Z: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.43/1.46 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_39_gfp_Oleq__trans,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [Y: A,X: A,Z: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ Y @ X )
% 2.43/1.46 => ( ( ord_less_eq @ A @ Z @ Y )
% 2.43/1.46 => ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_40_lfp_Oleq__refl,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_41_wlog__linorder__le,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linorder @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [P: A > A > $o,B2: A,A2: A] :
% 2.43/1.46 ( ! [A3: A,B3: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.43/1.46 => ( P @ A3 @ B3 ) )
% 2.43/1.46 => ( ( ( P @ B2 @ A2 )
% 2.43/1.46 => ( P @ A2 @ B2 ) )
% 2.43/1.46 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_42_GreatestMI2,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType] :
% 2.43/1.46 ( ( ord @ B @ ( type2 @ B ) )
% 2.43/1.46 => ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
% 2.43/1.46 ( ( P @ X )
% 2.43/1.46 => ( ! [Y6: A] :
% 2.43/1.46 ( ( P @ Y6 )
% 2.43/1.46 => ( ord_less_eq @ B @ ( M @ Y6 ) @ ( M @ X ) ) )
% 2.43/1.46 => ( ! [X3: A] :
% 2.43/1.46 ( ( P @ X3 )
% 2.43/1.46 => ( ! [Y3: A] :
% 2.43/1.46 ( ( P @ Y3 )
% 2.43/1.46 => ( ord_less_eq @ B @ ( M @ Y3 ) @ ( M @ X3 ) ) )
% 2.43/1.46 => ( Q @ X3 ) ) )
% 2.43/1.46 => ( Q @ ( hilbert_GreatestM @ A @ B @ M @ P ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_43_infinite__idx__SomeE,axiom,
% 2.43/1.46 ! [A: $tType,T: coinductive_llist @ A,A4: set @ A,I: nat] :
% 2.43/1.46 ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A4 ) )
% 2.43/1.46 => ~ ! [X3: A] :
% 2.43/1.46 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 2.43/1.46 != ( some @ A @ X3 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_44_infinite__Some,axiom,
% 2.43/1.46 ! [A: $tType,T: coinductive_llist @ A,A4: set @ A,I: nat] :
% 2.43/1.46 ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A4 ) )
% 2.43/1.46 => ? [A3: A] :
% 2.43/1.46 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 2.43/1.46 = ( some @ A @ A3 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_45_mem__Collect__eq,axiom,
% 2.43/1.46 ! [A: $tType,A2: A,P: A > $o] :
% 2.43/1.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.43/1.46 = ( P @ A2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_46_Collect__mem__eq,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A] :
% 2.43/1.46 ( ( collect @ A
% 2.43/1.46 @ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
% 2.43/1.46 = A4 ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_47_Collect__cong,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.43/1.46 ( ! [X3: A] :
% 2.43/1.46 ( ( P @ X3 )
% 2.43/1.46 = ( Q @ X3 ) )
% 2.43/1.46 => ( ( collect @ A @ P )
% 2.43/1.46 = ( collect @ A @ Q ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_48_ext,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.43/1.46 ( ! [X3: A] :
% 2.43/1.46 ( ( F @ X3 )
% 2.43/1.46 = ( G @ X3 ) )
% 2.43/1.46 => ( F = G ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_49_inflsts__mono,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21612149805nflsts @ A @ A4 ) @ ( lList21612149805nflsts @ A @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_50_LeastM__nat__le,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 2.43/1.46 ( ( P @ X )
% 2.43/1.46 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_51_LeastM__nat__lemma,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.43/1.46 ( ( P @ K )
% 2.43/1.46 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) )
% 2.43/1.46 & ! [Y3: A] :
% 2.43/1.46 ( ( P @ Y3 )
% 2.43/1.46 => ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ Y3 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_52_option_Osimps_I7_J,axiom,
% 2.43/1.46 ! [C: $tType,A: $tType,F1: C,F22: A > C,X2: A] :
% 2.43/1.46 ( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
% 2.43/1.46 = ( F22 @ X2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_53_combine__options__simps_I3_J,axiom,
% 2.43/1.46 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.43/1.46 ( ( combine_options @ A @ F @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
% 2.43/1.46 = ( some @ A @ ( F @ A2 @ B2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_54_ll2f__lem1,axiom,
% 2.43/1.46 ! [A: $tType,T: coinductive_llist @ A,I: nat,X: A] :
% 2.43/1.46 ( ( ( lList2228235758e_ll2f @ A @ T @ ( suc @ I ) )
% 2.43/1.46 = ( some @ A @ X ) )
% 2.43/1.46 => ? [Y6: A] :
% 2.43/1.46 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 2.43/1.46 = ( some @ A @ Y6 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_55_ll2f__Suc__Some,axiom,
% 2.43/1.46 ! [A: $tType,T: coinductive_llist @ A,I: nat,X: A] :
% 2.43/1.46 ( ( ( lList2228235758e_ll2f @ A @ T @ ( suc @ I ) )
% 2.43/1.46 = ( some @ A @ X ) )
% 2.43/1.46 => ~ ! [X3: A] :
% 2.43/1.46 ( ( lList2228235758e_ll2f @ A @ T @ I )
% 2.43/1.46 != ( some @ A @ X3 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_56_ll2f__None__le,axiom,
% 2.43/1.46 ! [A: $tType,T: coinductive_llist @ A,J: nat,I: nat] :
% 2.43/1.46 ( ( ( lList2228235758e_ll2f @ A @ T @ J )
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ J @ I )
% 2.43/1.46 => ( ( lList2228235758e_ll2f @ A @ T @ I )
% 2.43/1.46 = ( none @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_57_of__nat__le__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.43/1.46 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_58_is__none__code_I2_J,axiom,
% 2.43/1.46 ! [B: $tType,X: B] :
% 2.43/1.46 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_59_option_Opred__inject_I2_J,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,A2: A] :
% 2.43/1.46 ( ( pred_option @ A @ P @ ( some @ A @ A2 ) )
% 2.43/1.46 = ( P @ A2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_60_nat_Oinject,axiom,
% 2.43/1.46 ! [X2: nat,Y2: nat] :
% 2.43/1.46 ( ( ( suc @ X2 )
% 2.43/1.46 = ( suc @ Y2 ) )
% 2.43/1.46 = ( X2 = Y2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_61_old_Onat_Oinject,axiom,
% 2.43/1.46 ! [Nat: nat,Nat2: nat] :
% 2.43/1.46 ( ( ( suc @ Nat )
% 2.43/1.46 = ( suc @ Nat2 ) )
% 2.43/1.46 = ( Nat = Nat2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_62_of__nat__eq__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ( semiring_1_of_nat @ A @ M )
% 2.43/1.46 = ( semiring_1_of_nat @ A @ N ) )
% 2.43/1.46 = ( M = N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_63_Suc__le__mono,axiom,
% 2.43/1.46 ! [N: nat,M: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.43/1.46 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_64_not__Some__eq,axiom,
% 2.43/1.46 ! [A: $tType,X: option @ A] :
% 2.43/1.46 ( ( ! [Y5: A] :
% 2.43/1.46 ( X
% 2.43/1.46 != ( some @ A @ Y5 ) ) )
% 2.43/1.46 = ( X
% 2.43/1.46 = ( none @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_65_not__None__eq,axiom,
% 2.43/1.46 ! [A: $tType,X: option @ A] :
% 2.43/1.46 ( ( X
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 = ( ? [Y5: A] :
% 2.43/1.46 ( X
% 2.43/1.46 = ( some @ A @ Y5 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_66_is__none__code_I1_J,axiom,
% 2.43/1.46 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_67_combine__options__simps_I2_J,axiom,
% 2.43/1.46 ! [A: $tType,F: A > A > A,X: option @ A] :
% 2.43/1.46 ( ( combine_options @ A @ F @ X @ ( none @ A ) )
% 2.43/1.46 = X ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_68_combine__options__simps_I1_J,axiom,
% 2.43/1.46 ! [A: $tType,F: A > A > A,Y: option @ A] :
% 2.43/1.46 ( ( combine_options @ A @ F @ ( none @ A ) @ Y )
% 2.43/1.46 = Y ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_69_LeastM__natI,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 2.43/1.46 ( ( P @ K )
% 2.43/1.46 => ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_70_ll2f__None__Suc,axiom,
% 2.43/1.46 ! [A: $tType,T: coinductive_llist @ A,I: nat] :
% 2.43/1.46 ( ( ( lList2228235758e_ll2f @ A @ T @ I )
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 => ( ( lList2228235758e_ll2f @ A @ T @ ( suc @ I ) )
% 2.43/1.46 = ( none @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_71_option_Osimps_I6_J,axiom,
% 2.43/1.46 ! [A: $tType,C: $tType,F1: C,F22: A > C] :
% 2.43/1.46 ( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
% 2.43/1.46 = F1 ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_72_is__none__simps_I1_J,axiom,
% 2.43/1.46 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_73_option_Opred__inject_I1_J,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_74_Suc__inject,axiom,
% 2.43/1.46 ! [X: nat,Y: nat] :
% 2.43/1.46 ( ( ( suc @ X )
% 2.43/1.46 = ( suc @ Y ) )
% 2.43/1.46 => ( X = Y ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_75_n__not__Suc__n,axiom,
% 2.43/1.46 ! [N: nat] :
% 2.43/1.46 ( N
% 2.43/1.46 != ( suc @ N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_76_Option_Ois__none__def,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( is_none @ A )
% 2.43/1.46 = ( ^ [X4: option @ A] :
% 2.43/1.46 ( X4
% 2.43/1.46 = ( none @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_77_combine__options__assoc,axiom,
% 2.43/1.46 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A,Z: option @ A] :
% 2.43/1.46 ( ! [X3: A,Y6: A,Z3: A] :
% 2.43/1.46 ( ( F @ ( F @ X3 @ Y6 ) @ Z3 )
% 2.43/1.46 = ( F @ X3 @ ( F @ Y6 @ Z3 ) ) )
% 2.43/1.46 => ( ( combine_options @ A @ F @ ( combine_options @ A @ F @ X @ Y ) @ Z )
% 2.43/1.46 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_78_combine__options__commute,axiom,
% 2.43/1.46 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A] :
% 2.43/1.46 ( ! [X3: A,Y6: A] :
% 2.43/1.46 ( ( F @ X3 @ Y6 )
% 2.43/1.46 = ( F @ Y6 @ X3 ) )
% 2.43/1.46 => ( ( combine_options @ A @ F @ X @ Y )
% 2.43/1.46 = ( combine_options @ A @ F @ Y @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_79_combine__options__left__commute,axiom,
% 2.43/1.46 ! [A: $tType,F: A > A > A,Y: option @ A,X: option @ A,Z: option @ A] :
% 2.43/1.46 ( ! [X3: A,Y6: A] :
% 2.43/1.46 ( ( F @ X3 @ Y6 )
% 2.43/1.46 = ( F @ Y6 @ X3 ) )
% 2.43/1.46 => ( ! [X3: A,Y6: A,Z3: A] :
% 2.43/1.46 ( ( F @ ( F @ X3 @ Y6 ) @ Z3 )
% 2.43/1.46 = ( F @ X3 @ ( F @ Y6 @ Z3 ) ) )
% 2.43/1.46 => ( ( combine_options @ A @ F @ Y @ ( combine_options @ A @ F @ X @ Z ) )
% 2.43/1.46 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_80_full__nat__induct,axiom,
% 2.43/1.46 ! [P: nat > $o,N: nat] :
% 2.43/1.46 ( ! [N2: nat] :
% 2.43/1.46 ( ! [M2: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
% 2.43/1.46 => ( P @ M2 ) )
% 2.43/1.46 => ( P @ N2 ) )
% 2.43/1.46 => ( P @ N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_81_not__less__eq__eq,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 2.43/1.46 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_82_Suc__n__not__le__n,axiom,
% 2.43/1.46 ! [N: nat] :
% 2.43/1.46 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_83_le__Suc__eq,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.43/1.46 = ( ( ord_less_eq @ nat @ M @ N )
% 2.43/1.46 | ( M
% 2.43/1.46 = ( suc @ N ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_84_Suc__le__D,axiom,
% 2.43/1.46 ! [N: nat,M3: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M3 )
% 2.43/1.46 => ? [M4: nat] :
% 2.43/1.46 ( M3
% 2.43/1.46 = ( suc @ M4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_85_le__SucI,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.43/1.46 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_86_le__SucE,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.43/1.46 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 2.43/1.46 => ( M
% 2.43/1.46 = ( suc @ N ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_87_Suc__leD,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.43/1.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_88_combine__options__cases,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
% 2.43/1.46 ( ( ( X
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 => ( P @ X @ Y ) )
% 2.43/1.46 => ( ( ( Y
% 2.43/1.46 = ( none @ B ) )
% 2.43/1.46 => ( P @ X @ Y ) )
% 2.43/1.46 => ( ! [A3: A,B3: B] :
% 2.43/1.46 ( ( X
% 2.43/1.46 = ( some @ A @ A3 ) )
% 2.43/1.46 => ( ( Y
% 2.43/1.46 = ( some @ B @ B3 ) )
% 2.43/1.46 => ( P @ X @ Y ) ) )
% 2.43/1.46 => ( P @ X @ Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_89_split__option__all,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ^ [P2: ( option @ A ) > $o] :
% 2.43/1.46 ! [X5: option @ A] : ( P2 @ X5 ) )
% 2.43/1.46 = ( ^ [P3: ( option @ A ) > $o] :
% 2.43/1.46 ( ( P3 @ ( none @ A ) )
% 2.43/1.46 & ! [X4: A] : ( P3 @ ( some @ A @ X4 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_90_split__option__ex,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ^ [P2: ( option @ A ) > $o] :
% 2.43/1.46 ? [X5: option @ A] : ( P2 @ X5 ) )
% 2.43/1.46 = ( ^ [P3: ( option @ A ) > $o] :
% 2.43/1.46 ( ( P3 @ ( none @ A ) )
% 2.43/1.46 | ? [X4: A] : ( P3 @ ( some @ A @ X4 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_91_option_Oinducts,axiom,
% 2.43/1.46 ! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
% 2.43/1.46 ( ( P @ ( none @ A ) )
% 2.43/1.46 => ( ! [X3: A] : ( P @ ( some @ A @ X3 ) )
% 2.43/1.46 => ( P @ Option ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_92_option_Oexhaust,axiom,
% 2.43/1.46 ! [A: $tType,Y: option @ A] :
% 2.43/1.46 ( ( Y
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 => ~ ! [X22: A] :
% 2.43/1.46 ( Y
% 2.43/1.46 != ( some @ A @ X22 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_93_option_OdiscI,axiom,
% 2.43/1.46 ! [A: $tType,Option: option @ A,X2: A] :
% 2.43/1.46 ( ( Option
% 2.43/1.46 = ( some @ A @ X2 ) )
% 2.43/1.46 => ( Option
% 2.43/1.46 != ( none @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_94_option_Odistinct_I1_J,axiom,
% 2.43/1.46 ! [A: $tType,X2: A] :
% 2.43/1.46 ( ( none @ A )
% 2.43/1.46 != ( some @ A @ X2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_95_is__none__simps_I2_J,axiom,
% 2.43/1.46 ! [B: $tType,X: B] :
% 2.43/1.46 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_96_lift__Suc__antimono__le,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [F: nat > A,N: nat,N3: nat] :
% 2.43/1.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ N @ N3 )
% 2.43/1.46 => ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_97_lift__Suc__mono__le,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( order @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [F: nat > A,N: nat,N3: nat] :
% 2.43/1.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ N @ N3 )
% 2.43/1.46 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_98_subset__antisym,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.43/1.46 => ( A4 = B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_99_subsetI,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ! [X3: A] :
% 2.43/1.46 ( ( member @ A @ X3 @ A4 )
% 2.43/1.46 => ( member @ A @ X3 @ B4 ) )
% 2.43/1.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_100_real__arch__simple,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A] :
% 2.43/1.46 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_101_poslsts__mono,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21148268032oslsts @ A @ A4 ) @ ( lList21148268032oslsts @ A @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_102_fpslsts__mono,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList22096119349pslsts @ A @ A4 ) @ ( lList22096119349pslsts @ A @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_103_zle__int,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 2.43/1.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_104_transfer__int__nat__relations_I3_J,axiom,
% 2.43/1.46 ! [X: nat,Y: nat] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) )
% 2.43/1.46 = ( ord_less_eq @ nat @ X @ Y ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_105_int__if__cong,axiom,
% 2.43/1.46 ! [P: $o,X: nat,Y: nat] :
% 2.43/1.46 ( ( P
% 2.43/1.46 => ( ( semiring_1_of_nat @ int @ X )
% 2.43/1.46 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y ) ) ) )
% 2.43/1.46 & ( ~ P
% 2.43/1.46 => ( ( semiring_1_of_nat @ int @ Y )
% 2.43/1.46 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_106_int__int__eq,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ( semiring_1_of_nat @ int @ M )
% 2.43/1.46 = ( semiring_1_of_nat @ int @ N ) )
% 2.43/1.46 = ( M = N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_107_transfer__int__nat__relations_I1_J,axiom,
% 2.43/1.46 ! [X: nat,Y: nat] :
% 2.43/1.46 ( ( ( semiring_1_of_nat @ int @ X )
% 2.43/1.46 = ( semiring_1_of_nat @ int @ Y ) )
% 2.43/1.46 = ( X = Y ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_108_set__mp,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ( member @ A @ X @ A4 )
% 2.43/1.46 => ( member @ A @ X @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_109_in__mono,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ( member @ A @ X @ A4 )
% 2.43/1.46 => ( member @ A @ X @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_110_subsetD,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ( member @ A @ C2 @ A4 )
% 2.43/1.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_111_subsetCE,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ( member @ A @ C2 @ A4 )
% 2.43/1.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_112_equalityE,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( A4 = B4 )
% 2.43/1.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_113_subset__eq,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) )
% 2.43/1.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.43/1.46 ! [X4: A] :
% 2.43/1.46 ( ( member @ A @ X4 @ A5 )
% 2.43/1.46 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_114_equalityD1,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( A4 = B4 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_115_equalityD2,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( A4 = B4 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_116_set__rev__mp,axiom,
% 2.43/1.46 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( member @ A @ X @ A4 )
% 2.43/1.46 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( member @ A @ X @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_117_subset__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) )
% 2.43/1.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.43/1.46 ! [T2: A] :
% 2.43/1.46 ( ( member @ A @ T2 @ A5 )
% 2.43/1.46 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_118_rev__subsetD,axiom,
% 2.43/1.46 ! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( member @ A @ C2 @ A4 )
% 2.43/1.46 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_119_subset__refl,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_120_Collect__mono,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.43/1.46 ( ! [X3: A] :
% 2.43/1.46 ( ( P @ X3 )
% 2.43/1.46 => ( Q @ X3 ) )
% 2.43/1.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_121_subset__trans,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_122_set__eq__subset,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ^ [Y4: set @ A,Z2: set @ A] : ( Y4 = Z2 ) )
% 2.43/1.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.43/1.46 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_123_contra__subsetD,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ~ ( member @ A @ C2 @ B4 )
% 2.43/1.46 => ~ ( member @ A @ C2 @ A4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_124_Collect__mono__iff,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.43/1.46 = ( ! [X4: A] :
% 2.43/1.46 ( ( P @ X4 )
% 2.43/1.46 => ( Q @ X4 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_125_round__of__nat,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: nat] :
% 2.43/1.46 ( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
% 2.43/1.46 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_126_option_Ocollapse,axiom,
% 2.43/1.46 ! [A: $tType,Option: option @ A] :
% 2.43/1.46 ( ( Option
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 => ( ( some @ A @ ( the @ A @ Option ) )
% 2.43/1.46 = Option ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_127_transfer__morphismI,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,F: A > B,A4: A > $o] : ( nat_tr1645093318rphism @ A @ B @ F @ A4 ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_128_of__nat__aux_Osimps_I2_J,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [Inc: A > A,N: nat,I: A] :
% 2.43/1.46 ( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N ) @ I )
% 2.43/1.46 = ( semiri532925092at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_129_round__mono,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A,Y: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ X @ Y )
% 2.43/1.46 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_130_transfer__morphism__def,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType] :
% 2.43/1.46 ( ( nat_tr1645093318rphism @ B @ A )
% 2.43/1.46 = ( ^ [F2: B > A,A5: B > $o] : $true ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_131_option_Oexpand,axiom,
% 2.43/1.46 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 2.43/1.46 ( ( ( Option
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 = ( Option2
% 2.43/1.46 = ( none @ A ) ) )
% 2.43/1.46 => ( ( ( Option
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 => ( ( Option2
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 => ( ( the @ A @ Option )
% 2.43/1.46 = ( the @ A @ Option2 ) ) ) )
% 2.43/1.46 => ( Option = Option2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_132_option_Osel,axiom,
% 2.43/1.46 ! [A: $tType,X2: A] :
% 2.43/1.46 ( ( the @ A @ ( some @ A @ X2 ) )
% 2.43/1.46 = X2 ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_133_option_Oexhaust__sel,axiom,
% 2.43/1.46 ! [A: $tType,Option: option @ A] :
% 2.43/1.46 ( ( Option
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 => ( Option
% 2.43/1.46 = ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_134_option_Osplit__sel__asm,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 2.43/1.46 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 2.43/1.46 = ( ~ ( ( ( Option
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 & ~ ( P @ F1 ) )
% 2.43/1.46 | ( ( Option
% 2.43/1.46 = ( some @ A @ ( the @ A @ Option ) ) )
% 2.43/1.46 & ~ ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_135_option_Osplit__sel,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 2.43/1.46 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 2.43/1.46 = ( ( ( Option
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 => ( P @ F1 ) )
% 2.43/1.46 & ( ( Option
% 2.43/1.46 = ( some @ A @ ( the @ A @ Option ) ) )
% 2.43/1.46 => ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_136_option_Ocase__eq__if,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType] :
% 2.43/1.46 ( ( case_option @ B @ A )
% 2.43/1.46 = ( ^ [F12: B,F23: A > B,Option3: option @ A] :
% 2.43/1.46 ( if @ B
% 2.43/1.46 @ ( Option3
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 @ F12
% 2.43/1.46 @ ( F23 @ ( the @ A @ Option3 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_137_is__none__bind,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A )] :
% 2.43/1.46 ( ( is_none @ A @ ( bind @ B @ A @ F @ G ) )
% 2.43/1.46 = ( ( is_none @ B @ F )
% 2.43/1.46 | ( is_none @ A @ ( G @ ( the @ B @ F ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_138_bind__runit,axiom,
% 2.43/1.46 ! [A: $tType,X: option @ A] :
% 2.43/1.46 ( ( bind @ A @ A @ X @ ( some @ A ) )
% 2.43/1.46 = X ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_139_bind_Obind__lzero,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,F: A > ( option @ B )] :
% 2.43/1.46 ( ( bind @ A @ B @ ( none @ A ) @ F )
% 2.43/1.46 = ( none @ B ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_140_bind__eq__Some__conv,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A ),X: A] :
% 2.43/1.46 ( ( ( bind @ B @ A @ F @ G )
% 2.43/1.46 = ( some @ A @ X ) )
% 2.43/1.46 = ( ? [Y5: B] :
% 2.43/1.46 ( ( F
% 2.43/1.46 = ( some @ B @ Y5 ) )
% 2.43/1.46 & ( ( G @ Y5 )
% 2.43/1.46 = ( some @ A @ X ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_141_bind_Obind__lunit,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,X: A,F: A > ( option @ B )] :
% 2.43/1.46 ( ( bind @ A @ B @ ( some @ A @ X ) @ F )
% 2.43/1.46 = ( F @ X ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_142_Option_Obind__cong,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 2.43/1.46 ( ( X = Y )
% 2.43/1.46 => ( ! [A3: A] :
% 2.43/1.46 ( ( Y
% 2.43/1.46 = ( some @ A @ A3 ) )
% 2.43/1.46 => ( ( F @ A3 )
% 2.43/1.46 = ( G @ A3 ) ) )
% 2.43/1.46 => ( ( bind @ A @ B @ X @ F )
% 2.43/1.46 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_143_bind__option__cong__code,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B )] :
% 2.43/1.46 ( ( X = Y )
% 2.43/1.46 => ( ( bind @ A @ B @ X @ F )
% 2.43/1.46 = ( bind @ A @ B @ Y @ F ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_144_option_Osimps_I4_J,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,F1: B,F22: A > B] :
% 2.43/1.46 ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
% 2.43/1.46 = F1 ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_145_option_Osimps_I5_J,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,F1: B,F22: A > B,X2: A] :
% 2.43/1.46 ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
% 2.43/1.46 = ( F22 @ X2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_146_bind__split,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F: B > ( option @ A )] :
% 2.43/1.46 ( ( P @ ( bind @ B @ A @ M @ F ) )
% 2.43/1.46 = ( ( ( M
% 2.43/1.46 = ( none @ B ) )
% 2.43/1.46 => ( P @ ( none @ A ) ) )
% 2.43/1.46 & ! [V: B] :
% 2.43/1.46 ( ( M
% 2.43/1.46 = ( some @ B @ V ) )
% 2.43/1.46 => ( P @ ( F @ V ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_147_bind__split__asm,axiom,
% 2.43/1.46 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F: B > ( option @ A )] :
% 2.43/1.46 ( ( P @ ( bind @ B @ A @ M @ F ) )
% 2.43/1.46 = ( ~ ( ( ( M
% 2.43/1.46 = ( none @ B ) )
% 2.43/1.46 & ~ ( P @ ( none @ A ) ) )
% 2.43/1.46 | ? [X4: B] :
% 2.43/1.46 ( ( M
% 2.43/1.46 = ( some @ B @ X4 ) )
% 2.43/1.46 & ~ ( P @ ( F @ X4 ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_148_bind__eq__None__conv,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,A2: option @ B,F: B > ( option @ A )] :
% 2.43/1.46 ( ( ( bind @ B @ A @ A2 @ F )
% 2.43/1.46 = ( none @ A ) )
% 2.43/1.46 = ( ( A2
% 2.43/1.46 = ( none @ B ) )
% 2.43/1.46 | ( ( F @ ( the @ B @ A2 ) )
% 2.43/1.46 = ( none @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_149_alllsts__mono,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.43/1.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.43/1.46 => ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A4 ) @ ( lList2435255213lllsts @ A @ B4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_150_nat__le__iff,axiom,
% 2.43/1.46 ! [X: int,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 2.43/1.46 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_151_ll2f__LNil,axiom,
% 2.43/1.46 ! [A: $tType,X: nat] :
% 2.43/1.46 ( ( lList2228235758e_ll2f @ A @ ( coinductive_LNil @ A ) @ X )
% 2.43/1.46 = ( none @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_152_option_Oset__sel,axiom,
% 2.43/1.46 ! [A: $tType,A2: option @ A] :
% 2.43/1.46 ( ( A2
% 2.43/1.46 != ( none @ A ) )
% 2.43/1.46 => ( member @ A @ ( the @ A @ A2 ) @ ( set_option @ A @ A2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_153_nat__int,axiom,
% 2.43/1.46 ! [N: nat] :
% 2.43/1.46 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 2.43/1.46 = N ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_154_elem__set,axiom,
% 2.43/1.46 ! [A: $tType,X: A,Xo: option @ A] :
% 2.43/1.46 ( ( member @ A @ X @ ( set_option @ A @ Xo ) )
% 2.43/1.46 = ( Xo
% 2.43/1.46 = ( some @ A @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_155_poslsts__iff,axiom,
% 2.43/1.46 ! [A: $tType,S: coinductive_llist @ A,A4: set @ A] :
% 2.43/1.46 ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ A4 ) )
% 2.43/1.46 = ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A4 ) )
% 2.43/1.46 & ( S
% 2.43/1.46 != ( coinductive_LNil @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_156_ospec,axiom,
% 2.43/1.46 ! [A: $tType,A4: option @ A,P: A > $o,X: A] :
% 2.43/1.46 ( ! [X3: A] :
% 2.43/1.46 ( ( member @ A @ X3 @ ( set_option @ A @ A4 ) )
% 2.43/1.46 => ( P @ X3 ) )
% 2.43/1.46 => ( ( A4
% 2.43/1.46 = ( some @ A @ X ) )
% 2.43/1.46 => ( P @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_157_option_Oset__cases,axiom,
% 2.43/1.46 ! [A: $tType,E: A,A2: option @ A] :
% 2.43/1.46 ( ( member @ A @ E @ ( set_option @ A @ A2 ) )
% 2.43/1.46 => ( A2
% 2.43/1.46 = ( some @ A @ E ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_158_option_Oset__intros,axiom,
% 2.43/1.46 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( set_option @ A @ ( some @ A @ A2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_159_nat__if__cong,axiom,
% 2.43/1.46 ! [P: $o,X: int,Y: int] :
% 2.43/1.46 ( ( P
% 2.43/1.46 => ( ( nat2 @ X )
% 2.43/1.46 = ( nat2 @ ( if @ int @ P @ X @ Y ) ) ) )
% 2.43/1.46 & ( ~ P
% 2.43/1.46 => ( ( nat2 @ Y )
% 2.43/1.46 = ( nat2 @ ( if @ int @ P @ X @ Y ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_160_alllsts_OLNil__all,axiom,
% 2.43/1.46 ! [A: $tType,A4: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A4 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_161_infsubsetall,axiom,
% 2.43/1.46 ! [A: $tType,X: coinductive_llist @ A,A4: set @ A] :
% 2.43/1.46 ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A4 ) )
% 2.43/1.46 => ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_162_bind__option__cong,axiom,
% 2.43/1.46 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 2.43/1.46 ( ( X = Y )
% 2.43/1.46 => ( ! [Z3: A] :
% 2.43/1.46 ( ( member @ A @ Z3 @ ( set_option @ A @ Y ) )
% 2.43/1.46 => ( ( F @ Z3 )
% 2.43/1.46 = ( G @ Z3 ) ) )
% 2.43/1.46 => ( ( bind @ A @ B @ X @ F )
% 2.43/1.46 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_163_option_Opred__cong,axiom,
% 2.43/1.46 ! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
% 2.43/1.46 ( ( X = Ya )
% 2.43/1.46 => ( ! [Z3: A] :
% 2.43/1.46 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 2.43/1.46 => ( ( P @ Z3 )
% 2.43/1.46 = ( Pa @ Z3 ) ) )
% 2.43/1.46 => ( ( pred_option @ A @ P @ X )
% 2.43/1.46 = ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_164_option_Opred__mono__strong,axiom,
% 2.43/1.46 ! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
% 2.43/1.46 ( ( pred_option @ A @ P @ X )
% 2.43/1.46 => ( ! [Z3: A] :
% 2.43/1.46 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 2.43/1.46 => ( ( P @ Z3 )
% 2.43/1.46 => ( Pa @ Z3 ) ) )
% 2.43/1.46 => ( pred_option @ A @ Pa @ X ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_165_nat__mono,axiom,
% 2.43/1.46 ! [X: int,Y: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ X @ Y )
% 2.43/1.46 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_166_lbutlast__LNil,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
% 2.43/1.46 = ( coinductive_LNil @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_167_le__nat__iff,axiom,
% 2.43/1.46 ! [K: int,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 2.43/1.46 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_168_int__nat__eq,axiom,
% 2.43/1.46 ! [Z: int] :
% 2.43/1.46 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.43/1.46 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.43/1.46 = Z ) )
% 2.43/1.46 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.43/1.46 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.43/1.46 = ( zero_zero @ int ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_169_of__nat__eq__0__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [M: nat] :
% 2.43/1.46 ( ( ( semiring_1_of_nat @ A @ M )
% 2.43/1.46 = ( zero_zero @ A ) )
% 2.43/1.46 = ( M
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_170_of__nat__0__eq__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: nat] :
% 2.43/1.46 ( ( ( zero_zero @ A )
% 2.43/1.46 = ( semiring_1_of_nat @ A @ N ) )
% 2.43/1.46 = ( ( zero_zero @ nat )
% 2.43/1.46 = N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_171_of__nat__0,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.43/1.46 = ( zero_zero @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_172_of__nat__le__0__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [M: nat] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.43/1.46 = ( M
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_173_round__0,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.43/1.46 => ( ( archimedean_round @ A @ ( zero_zero @ A ) )
% 2.43/1.46 = ( zero_zero @ int ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_174_less__eq__int__code_I1_J,axiom,
% 2.43/1.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_175_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 2.43/1.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_176_of__nat__0__le__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_177_of__nat__neq__0,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: nat] :
% 2.43/1.46 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 2.43/1.46 != ( zero_zero @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_178_transfer__morphism__nat__int,axiom,
% 2.43/1.46 nat_tr1645093318rphism @ int @ nat @ nat2 @ ( ord_less_eq @ int @ ( zero_zero @ int ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_179_transfer__nat__int__relations_I1_J,axiom,
% 2.43/1.46 ! [X: int,Y: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.43/1.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.43/1.46 => ( ( ( nat2 @ X )
% 2.43/1.46 = ( nat2 @ Y ) )
% 2.43/1.46 = ( X = Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_180_eq__nat__nat__iff,axiom,
% 2.43/1.46 ! [Z: int,Z4: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.43/1.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
% 2.43/1.46 => ( ( ( nat2 @ Z )
% 2.43/1.46 = ( nat2 @ Z4 ) )
% 2.43/1.46 = ( Z = Z4 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_181_ex__nat,axiom,
% 2.43/1.46 ( ( ^ [P2: nat > $o] :
% 2.43/1.46 ? [X5: nat] : ( P2 @ X5 ) )
% 2.43/1.46 = ( ^ [P3: nat > $o] :
% 2.43/1.46 ? [X4: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.43/1.46 & ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_182_all__nat,axiom,
% 2.43/1.46 ( ( ^ [P2: nat > $o] :
% 2.43/1.46 ! [X5: nat] : ( P2 @ X5 ) )
% 2.43/1.46 = ( ^ [P3: nat > $o] :
% 2.43/1.46 ! [X4: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.43/1.46 => ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_183_zero__le__imp__eq__int,axiom,
% 2.43/1.46 ! [K: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.43/1.46 => ? [N2: nat] :
% 2.43/1.46 ( K
% 2.43/1.46 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_184_nonneg__int__cases,axiom,
% 2.43/1.46 ! [K: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.43/1.46 => ~ ! [N2: nat] :
% 2.43/1.46 ( K
% 2.43/1.46 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_185_nonneg__eq__int,axiom,
% 2.43/1.46 ! [Z: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.43/1.46 => ~ ! [M4: nat] :
% 2.43/1.46 ( Z
% 2.43/1.46 != ( semiring_1_of_nat @ int @ M4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 2.43/1.46 ! [Z: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_187_transfer__int__nat__quantifiers_I1_J,axiom,
% 2.43/1.46 ! [P: int > $o] :
% 2.43/1.46 ( ( ! [X4: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.43/1.46 => ( P @ X4 ) ) )
% 2.43/1.46 = ( ! [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_188_transfer__int__nat__quantifiers_I2_J,axiom,
% 2.43/1.46 ! [P: int > $o] :
% 2.43/1.46 ( ( ? [X4: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.43/1.46 & ( P @ X4 ) ) )
% 2.43/1.46 = ( ? [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_189_transfer__nat__int__relations_I3_J,axiom,
% 2.43/1.46 ! [X: int,Y: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.43/1.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.43/1.46 => ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
% 2.43/1.46 = ( ord_less_eq @ int @ X @ Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_190_nat__0__le,axiom,
% 2.43/1.46 ! [Z: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.43/1.46 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
% 2.43/1.46 = Z ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_191_int__eq__iff,axiom,
% 2.43/1.46 ! [M: nat,Z: int] :
% 2.43/1.46 ( ( ( semiring_1_of_nat @ int @ M )
% 2.43/1.46 = Z )
% 2.43/1.46 = ( ( M
% 2.43/1.46 = ( nat2 @ Z ) )
% 2.43/1.46 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_192_le__zero__eq,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.43/1.46 = ( N
% 2.43/1.46 = ( zero_zero @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_193_imp__le__cong,axiom,
% 2.43/1.46 ! [X: int,X6: int,P: $o,P4: $o] :
% 2.43/1.46 ( ( X = X6 )
% 2.43/1.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 2.43/1.46 => ( P = P4 ) )
% 2.43/1.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.43/1.46 => P )
% 2.43/1.46 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 2.43/1.46 => P4 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_194_conj__le__cong,axiom,
% 2.43/1.46 ! [X: int,X6: int,P: $o,P4: $o] :
% 2.43/1.46 ( ( X = X6 )
% 2.43/1.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 2.43/1.46 => ( P = P4 ) )
% 2.43/1.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.43/1.46 & P )
% 2.43/1.46 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
% 2.43/1.46 & P4 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_195_le0,axiom,
% 2.43/1.46 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_196_nat__0,axiom,
% 2.43/1.46 ( ( nat2 @ ( zero_zero @ int ) )
% 2.43/1.46 = ( zero_zero @ nat ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_197_nat__le__0,axiom,
% 2.43/1.46 ! [Z: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
% 2.43/1.46 => ( ( nat2 @ Z )
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_198_nat__0__iff,axiom,
% 2.43/1.46 ! [I: int] :
% 2.43/1.46 ( ( ( nat2 @ I )
% 2.43/1.46 = ( zero_zero @ nat ) )
% 2.43/1.46 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_199_nat_Odistinct_I1_J,axiom,
% 2.43/1.46 ! [X2: nat] :
% 2.43/1.46 ( ( zero_zero @ nat )
% 2.43/1.46 != ( suc @ X2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_200_old_Onat_Odistinct_I2_J,axiom,
% 2.43/1.46 ! [Nat3: nat] :
% 2.43/1.46 ( ( suc @ Nat3 )
% 2.43/1.46 != ( zero_zero @ nat ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_201_old_Onat_Odistinct_I1_J,axiom,
% 2.43/1.46 ! [Nat2: nat] :
% 2.43/1.46 ( ( zero_zero @ nat )
% 2.43/1.46 != ( suc @ Nat2 ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_202_nat_OdiscI,axiom,
% 2.43/1.46 ! [Nat: nat,X2: nat] :
% 2.43/1.46 ( ( Nat
% 2.43/1.46 = ( suc @ X2 ) )
% 2.43/1.46 => ( Nat
% 2.43/1.46 != ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_203_nat__induct,axiom,
% 2.43/1.46 ! [P: nat > $o,N: nat] :
% 2.43/1.46 ( ( P @ ( zero_zero @ nat ) )
% 2.43/1.46 => ( ! [N2: nat] :
% 2.43/1.46 ( ( P @ N2 )
% 2.43/1.46 => ( P @ ( suc @ N2 ) ) )
% 2.43/1.46 => ( P @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_204_diff__induct,axiom,
% 2.43/1.46 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.43/1.46 ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
% 2.43/1.46 => ( ! [Y6: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y6 ) )
% 2.43/1.46 => ( ! [X3: nat,Y6: nat] :
% 2.43/1.46 ( ( P @ X3 @ Y6 )
% 2.43/1.46 => ( P @ ( suc @ X3 ) @ ( suc @ Y6 ) ) )
% 2.43/1.46 => ( P @ M @ N ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_205_zero__induct,axiom,
% 2.43/1.46 ! [P: nat > $o,K: nat] :
% 2.43/1.46 ( ( P @ K )
% 2.43/1.46 => ( ! [N2: nat] :
% 2.43/1.46 ( ( P @ ( suc @ N2 ) )
% 2.43/1.46 => ( P @ N2 ) )
% 2.43/1.46 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_206_Suc__neq__Zero,axiom,
% 2.43/1.46 ! [M: nat] :
% 2.43/1.46 ( ( suc @ M )
% 2.43/1.46 != ( zero_zero @ nat ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_207_Zero__neq__Suc,axiom,
% 2.43/1.46 ! [M: nat] :
% 2.43/1.46 ( ( zero_zero @ nat )
% 2.43/1.46 != ( suc @ M ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_208_Zero__not__Suc,axiom,
% 2.43/1.46 ! [M: nat] :
% 2.43/1.46 ( ( zero_zero @ nat )
% 2.43/1.46 != ( suc @ M ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_209_old_Onat_Oexhaust,axiom,
% 2.43/1.46 ! [Y: nat] :
% 2.43/1.46 ( ( Y
% 2.43/1.46 != ( zero_zero @ nat ) )
% 2.43/1.46 => ~ ! [Nat4: nat] :
% 2.43/1.46 ( Y
% 2.43/1.46 != ( suc @ Nat4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_210_old_Onat_Oinducts,axiom,
% 2.43/1.46 ! [P: nat > $o,Nat: nat] :
% 2.43/1.46 ( ( P @ ( zero_zero @ nat ) )
% 2.43/1.46 => ( ! [Nat4: nat] :
% 2.43/1.46 ( ( P @ Nat4 )
% 2.43/1.46 => ( P @ ( suc @ Nat4 ) ) )
% 2.43/1.46 => ( P @ Nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_211_not0__implies__Suc,axiom,
% 2.43/1.46 ! [N: nat] :
% 2.43/1.46 ( ( N
% 2.43/1.46 != ( zero_zero @ nat ) )
% 2.43/1.46 => ? [M4: nat] :
% 2.43/1.46 ( N
% 2.43/1.46 = ( suc @ M4 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_212_dependent__nat__choice,axiom,
% 2.43/1.46 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 2.43/1.46 ( ? [X1: A] : ( P @ ( zero_zero @ nat ) @ X1 )
% 2.43/1.46 => ( ! [X3: A,N2: nat] :
% 2.43/1.46 ( ( P @ N2 @ X3 )
% 2.43/1.46 => ? [Y3: A] :
% 2.43/1.46 ( ( P @ ( suc @ N2 ) @ Y3 )
% 2.43/1.46 & ( Q @ N2 @ X3 @ Y3 ) ) )
% 2.43/1.46 => ? [F3: nat > A] :
% 2.43/1.46 ! [N4: nat] :
% 2.43/1.46 ( ( P @ N4 @ ( F3 @ N4 ) )
% 2.43/1.46 & ( Q @ N4 @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_213_exists__least__lemma,axiom,
% 2.43/1.46 ! [P: nat > $o] :
% 2.43/1.46 ( ~ ( P @ ( zero_zero @ nat ) )
% 2.43/1.46 => ( ? [X1: nat] : ( P @ X1 )
% 2.43/1.46 => ? [N2: nat] :
% 2.43/1.46 ( ~ ( P @ N2 )
% 2.43/1.46 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_214_le__0__eq,axiom,
% 2.43/1.46 ! [N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.43/1.46 = ( N
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_215_less__eq__nat_Osimps_I1_J,axiom,
% 2.43/1.46 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_216_of__nat__aux_Osimps_I1_J,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [Inc: A > A,I: A] :
% 2.43/1.46 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 2.43/1.46 = I ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_217_transfer__int__nat__numerals_I1_J,axiom,
% 2.43/1.46 ( ( zero_zero @ int )
% 2.43/1.46 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_218_nat__eq__iff,axiom,
% 2.43/1.46 ! [W: int,M: nat] :
% 2.43/1.46 ( ( ( nat2 @ W )
% 2.43/1.46 = M )
% 2.43/1.46 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.43/1.46 => ( W
% 2.43/1.46 = ( semiring_1_of_nat @ int @ M ) ) )
% 2.43/1.46 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.43/1.46 => ( M
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_219_nat__eq__iff2,axiom,
% 2.43/1.46 ! [M: nat,W: int] :
% 2.43/1.46 ( ( M
% 2.43/1.46 = ( nat2 @ W ) )
% 2.43/1.46 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.43/1.46 => ( W
% 2.43/1.46 = ( semiring_1_of_nat @ int @ M ) ) )
% 2.43/1.46 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.43/1.46 => ( M
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_220_zero__le,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_221_option_Osize__gen_I1_J,axiom,
% 2.43/1.46 ! [A: $tType,X: A > nat] :
% 2.43/1.46 ( ( size_option @ A @ X @ ( none @ A ) )
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_222_LList2__Mirabelle__hamjzmohle_Ollength__LNil,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( lList21232602520length @ A @ ( coinductive_LNil @ A ) )
% 2.43/1.46 = ( zero_zero @ nat ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_223_option_Osize_I4_J,axiom,
% 2.43/1.46 ! [A: $tType,X2: A] :
% 2.43/1.46 ( ( size_size @ ( option @ A ) @ ( some @ A @ X2 ) )
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_224_option_Osize_I3_J,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( size_size @ ( option @ A ) @ ( none @ A ) )
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_225_option_Osize__neq,axiom,
% 2.43/1.46 ! [A: $tType,X: option @ A] :
% 2.43/1.46 ( ( size_size @ ( option @ A ) @ X )
% 2.43/1.46 != ( zero_zero @ nat ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_226_LList2__Mirabelle__hamjzmohle_Oltake_Osimps_I1_J,axiom,
% 2.43/1.46 ! [A: $tType,L: coinductive_llist @ A] :
% 2.43/1.46 ( ( lList22119844313_ltake @ A @ L @ ( zero_zero @ nat ) )
% 2.43/1.46 = ( coinductive_LNil @ A ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_227_list__decode_Ocases,axiom,
% 2.43/1.46 ! [X: nat] :
% 2.43/1.46 ( ( X
% 2.43/1.46 != ( zero_zero @ nat ) )
% 2.43/1.46 => ~ ! [N2: nat] :
% 2.43/1.46 ( X
% 2.43/1.46 != ( suc @ N2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_228_le__numeral__extra_I3_J,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_229_int__eq__iff__numeral,axiom,
% 2.43/1.46 ! [M: nat,V2: num] :
% 2.43/1.46 ( ( ( semiring_1_of_nat @ int @ M )
% 2.43/1.46 = ( numeral_numeral @ int @ V2 ) )
% 2.43/1.46 = ( ( M
% 2.43/1.46 = ( nat2 @ ( numeral_numeral @ int @ V2 ) ) )
% 2.43/1.46 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_230_numeral__le__iff,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [M: num,N: num] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 2.43/1.46 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_231_of__nat__numeral,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: num] :
% 2.43/1.46 ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
% 2.43/1.46 = ( numeral_numeral @ A @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_232_round__numeral,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: num] :
% 2.43/1.46 ( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
% 2.43/1.46 = ( numeral_numeral @ int @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_233_not__numeral__le__zero,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: num] :
% 2.43/1.46 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_234_zero__le__numeral,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_235_option_Osize__gen_I2_J,axiom,
% 2.43/1.46 ! [A: $tType,X: A > nat,X2: A] :
% 2.43/1.46 ( ( size_option @ A @ X @ ( some @ A @ X2 ) )
% 2.43/1.46 = ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_236_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
% 2.43/1.46 ! [X: int,Y: int] :
% 2.43/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.43/1.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.43/1.46 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( nat_tsub @ X @ Y ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_237_add__le__cancel__left,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [C2: A,A2: A,B2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 2.43/1.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_238_add__le__cancel__right,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,C2: A,B2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 2.43/1.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_239_Nat_Oadd__0__right,axiom,
% 2.43/1.46 ! [M: nat] :
% 2.43/1.46 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 2.43/1.46 = M ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_240_add__is__0,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ( plus_plus @ nat @ M @ N )
% 2.43/1.46 = ( zero_zero @ nat ) )
% 2.43/1.46 = ( ( M
% 2.43/1.46 = ( zero_zero @ nat ) )
% 2.43/1.46 & ( N
% 2.43/1.46 = ( zero_zero @ nat ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_241_add__Suc__right,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( plus_plus @ nat @ M @ ( suc @ N ) )
% 2.43/1.46 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_242_nat__add__left__cancel__le,axiom,
% 2.43/1.46 ! [K: nat,M: nat,N: nat] :
% 2.43/1.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.43/1.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_243_add__le__same__cancel1,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [B2: A,A2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 2.43/1.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_244_add__le__same__cancel2,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 2.43/1.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_245_le__add__same__cancel1,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 2.43/1.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_246_le__add__same__cancel2,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A,B2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 2.43/1.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_247_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.43/1.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_248_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [A2: A] :
% 2.43/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.43/1.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_249_of__nat__add,axiom,
% 2.43/1.46 ! [A: $tType] :
% 2.43/1.46 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.43/1.46 => ! [M: nat,N: nat] :
% 2.43/1.46 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
% 2.43/1.46 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_250_nat__numeral,axiom,
% 2.43/1.46 ! [K: num] :
% 2.43/1.46 ( ( nat2 @ ( numeral_numeral @ int @ K ) )
% 2.43/1.46 = ( numeral_numeral @ nat @ K ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_251_add__is__1,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ( plus_plus @ nat @ M @ N )
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) )
% 2.43/1.46 = ( ( ( M
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) )
% 2.43/1.46 & ( N
% 2.43/1.46 = ( zero_zero @ nat ) ) )
% 2.43/1.46 | ( ( M
% 2.43/1.46 = ( zero_zero @ nat ) )
% 2.43/1.46 & ( N
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(fact_252_one__is__add,axiom,
% 2.43/1.46 ! [M: nat,N: nat] :
% 2.43/1.46 ( ( ( suc @ ( zero_zero @ nat ) )
% 2.43/1.46 = ( plus_plus @ nat @ M @ N ) )
% 2.43/1.46 = ( ( ( M
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) )
% 2.43/1.46 & ( N
% 2.43/1.46 = ( zero_zero @ nat ) ) )
% 2.43/1.46 | ( ( M
% 2.43/1.46 = ( zero_zero @ nat ) )
% 2.43/1.46 & ( N
% 2.43/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.43/1.46 ! [A6: $tType,A7: $tType] :
% 2.43/1.46 ( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
% 2.43/1.46 => ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.43/1.46 ! [A6: $tType,A7: $tType] :
% 2.43/1.46 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 2.43/1.46 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.43/1.46 ! [A6: $tType,A7: $tType] :
% 2.43/1.46 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.43/1.46 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_fun___Orderings_Oord,axiom,
% 2.43/1.46 ! [A6: $tType,A7: $tType] :
% 2.43/1.46 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 2.43/1.46 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.43/1.46 ordere516151231imp_le @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.43/1.46 ordere236663937imp_le @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 2.43/1.46 linord219039673up_add @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.43/1.46 linordered_semidom @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.43/1.46 semiring_char_0 @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.43/1.46 preorder @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.43/1.46 linorder @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.43/1.46 semiring_1 @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.43/1.46 order @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.43/1.46 ord @ int @ ( type2 @ int ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4,axiom,
% 2.43/1.46 ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_5,axiom,
% 2.43/1.46 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.43/1.46 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 2.43/1.46 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7,axiom,
% 2.43/1.46 semiring_char_0 @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Orderings_Opreorder_8,axiom,
% 2.43/1.46 preorder @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Orderings_Olinorder_9,axiom,
% 2.43/1.46 linorder @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Rings_Osemiring__1_10,axiom,
% 2.43/1.46 semiring_1 @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Orderings_Oorder_11,axiom,
% 2.43/1.46 order @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Nat_Onat___Orderings_Oord_12,axiom,
% 2.43/1.46 ord @ nat @ ( type2 @ nat ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Num_Onum___Orderings_Opreorder_13,axiom,
% 2.43/1.46 preorder @ num @ ( type2 @ num ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Num_Onum___Orderings_Olinorder_14,axiom,
% 2.43/1.46 linorder @ num @ ( type2 @ num ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Num_Onum___Orderings_Oorder_15,axiom,
% 2.43/1.46 order @ num @ ( type2 @ num ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Num_Onum___Orderings_Oord_16,axiom,
% 2.43/1.46 ord @ num @ ( type2 @ num ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_17,axiom,
% 2.43/1.46 ! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Set_Oset___Orderings_Opreorder_18,axiom,
% 2.43/1.46 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Set_Oset___Orderings_Oorder_19,axiom,
% 2.43/1.46 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_Set_Oset___Orderings_Oord_20,axiom,
% 2.43/1.46 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21,axiom,
% 2.43/1.46 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_HOL_Obool___Orderings_Opreorder_22,axiom,
% 2.43/1.46 preorder @ $o @ ( type2 @ $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_HOL_Obool___Orderings_Olinorder_23,axiom,
% 2.43/1.46 linorder @ $o @ ( type2 @ $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 2.43/1.46 order @ $o @ ( type2 @ $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 2.43/1.46 ord @ $o @ ( type2 @ $o ) ).
% 2.43/1.46
% 2.43/1.46 thf(help_If_3_1_T,axiom,
% 2.43/1.46 ! [P: $o] :
% 2.43/1.46 ( ( P = $true )
% 2.43/1.46 | ( P = $false ) ) ).
% 2.43/1.46
% 2.43/1.46 thf(help_If_2_1_T,axiom,
% 2.43/1.46 ! [A: $tType,X: A,Y: A] :
% 2.43/1.46 ( ( if @ A @ $false @ X @ Y )
% 2.43/1.46 = Y ) ).
% 2.43/1.46
% 2.43/1.46 thf(help_If_1_1_T,axiom,
% 2.43/1.46 ! [A: $tType,X: A,Y: A] :
% 2.43/1.46 ( ( if @ A @ $true @ X @ Y )
% 2.43/1.46 = X ) ).
% 2.43/1.46
% 2.43/1.46 thf(conj_0,conjecture,
% 2.43/1.46 q ).
% 2.43/1.46 ------------------------
% 2.43/1.48 % (12430)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/183Mi)
% 2.43/1.49 % (12432)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_12325 for (2999ds/27Mi)
% 2.43/1.49 % (12431)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_12325 for (2999ds/4Mi)
% 2.43/1.49 % (12434)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_12325 for (2999ds/2Mi)
% 2.43/1.49 % (12433)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/2Mi)
% 2.43/1.49 % (12435)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_12325 for (2999ds/275Mi)
% 2.43/1.49 % (12436)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/18Mi)
% 2.43/1.49 % (12433)Instruction limit reached!
% 2.43/1.49 % (12433)------------------------------
% 2.43/1.49 % (12433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.49 % (12433)Termination reason: Unknown
% 2.43/1.49 % (12434)Instruction limit reached!
% 2.43/1.49 % (12434)------------------------------
% 2.43/1.49 % (12434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.49 % (12434)Termination reason: Unknown
% 2.43/1.49 % (12434)Termination phase: shuffling
% 2.43/1.49
% 2.43/1.49 % (12434)Memory used [KB]: 1407
% 2.43/1.49 % (12434)Time elapsed: 0.003 s
% 2.43/1.49 % (12434)Instructions burned: 3 (million)
% 2.43/1.49 % (12434)------------------------------
% 2.43/1.49 % (12434)------------------------------
% 2.43/1.49 % (12433)Termination phase: shuffling
% 2.43/1.49
% 2.43/1.49 % (12433)Memory used [KB]: 1407
% 2.43/1.49 % (12433)Time elapsed: 0.003 s
% 2.43/1.49 % (12433)Instructions burned: 3 (million)
% 2.43/1.49 % (12433)------------------------------
% 2.43/1.49 % (12433)------------------------------
% 2.43/1.49 % (12431)Instruction limit reached!
% 2.43/1.49 % (12431)------------------------------
% 2.43/1.49 % (12431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.49 % (12431)Termination reason: Unknown
% 2.43/1.49 % (12431)Termination phase: shuffling
% 2.43/1.49
% 2.43/1.49 % (12431)Memory used [KB]: 1407
% 2.43/1.49 % (12431)Time elapsed: 0.004 s
% 2.43/1.49 % (12431)Instructions burned: 5 (million)
% 2.43/1.49 % (12431)------------------------------
% 2.43/1.49 % (12431)------------------------------
% 2.43/1.50 % (12436)Instruction limit reached!
% 2.43/1.50 % (12436)------------------------------
% 2.43/1.50 % (12436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.50 % (12436)Termination reason: Unknown
% 2.43/1.50 % (12436)Termination phase: Property scanning
% 2.43/1.50
% 2.43/1.50 % (12436)Memory used [KB]: 1663
% 2.43/1.50 % (12436)Time elapsed: 0.011 s
% 2.43/1.50 % (12436)Instructions burned: 18 (million)
% 2.43/1.50 % (12436)------------------------------
% 2.43/1.50 % (12436)------------------------------
% 2.52/1.50 % (12432)Instruction limit reached!
% 2.52/1.50 % (12432)------------------------------
% 2.52/1.50 % (12432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (12432)Termination reason: Unknown
% 2.52/1.50 % (12432)Termination phase: Preprocessing 1
% 2.52/1.50
% 2.52/1.50 % (12432)Memory used [KB]: 1663
% 2.52/1.50 % (12432)Time elapsed: 0.015 s
% 2.52/1.50 % (12432)Instructions burned: 28 (million)
% 2.52/1.50 % (12432)------------------------------
% 2.52/1.50 % (12432)------------------------------
% 2.52/1.50 % (12435)Refutation not found, incomplete strategy
% 2.52/1.50 % (12435)------------------------------
% 2.52/1.50 % (12435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (12435)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.50
% 2.52/1.50
% 2.52/1.50 % (12435)Memory used [KB]: 6140
% 2.52/1.50 % (12435)Time elapsed: 0.017 s
% 2.52/1.50 % (12435)Instructions burned: 30 (million)
% 2.52/1.50 % (12435)------------------------------
% 2.52/1.50 % (12435)------------------------------
% 2.52/1.50 % (12437)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/3Mi)
% 2.52/1.50 % (12438)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_12325 for (2999ds/37Mi)
% 2.52/1.50 % (12439)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_12325 for (2999ds/15Mi)
% 2.52/1.50 % (12437)Instruction limit reached!
% 2.52/1.50 % (12437)------------------------------
% 2.52/1.50 % (12437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.50 % (12437)Termination reason: Unknown
% 2.52/1.50 % (12437)Termination phase: shuffling
% 2.52/1.50
% 2.52/1.50 % (12437)Memory used [KB]: 1407
% 2.52/1.50 % (12437)Time elapsed: 0.003 s
% 2.52/1.50 % (12437)Instructions burned: 4 (million)
% 2.52/1.50 % (12437)------------------------------
% 2.52/1.50 % (12437)------------------------------
% 2.52/1.51 % (12440)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_12325 for (2999ds/3Mi)
% 2.52/1.51 % (12439)Instruction limit reached!
% 2.52/1.51 % (12439)------------------------------
% 2.52/1.51 % (12439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.51 % (12439)Termination reason: Unknown
% 2.52/1.51 % (12439)Termination phase: shuffling
% 2.52/1.51
% 2.52/1.51 % (12439)Memory used [KB]: 1663
% 2.52/1.51 % (12439)Time elapsed: 0.009 s
% 2.52/1.51 % (12439)Instructions burned: 15 (million)
% 2.52/1.51 % (12439)------------------------------
% 2.52/1.51 % (12439)------------------------------
% 2.52/1.51 % (12440)Instruction limit reached!
% 2.52/1.51 % (12440)------------------------------
% 2.52/1.51 % (12440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.51 % (12440)Termination reason: Unknown
% 2.52/1.51 % (12440)Termination phase: shuffling
% 2.52/1.51
% 2.52/1.51 % (12440)Memory used [KB]: 1407
% 2.52/1.51 % (12440)Time elapsed: 0.003 s
% 2.52/1.51 % (12440)Instructions burned: 3 (million)
% 2.52/1.51 % (12440)------------------------------
% 2.52/1.51 % (12440)------------------------------
% 2.52/1.52 % (12442)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/7Mi)
% 2.52/1.52 % (12441)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_12325 for (2999ds/1041Mi)
% 2.52/1.52 % (12438)Instruction limit reached!
% 2.52/1.52 % (12438)------------------------------
% 2.52/1.52 % (12438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.52 % (12438)Termination reason: Unknown
% 2.52/1.52 % (12438)Termination phase: Property scanning
% 2.52/1.52
% 2.52/1.52 % (12438)Memory used [KB]: 2046
% 2.52/1.52 % (12438)Time elapsed: 0.019 s
% 2.52/1.52 % (12438)Instructions burned: 38 (million)
% 2.52/1.52 % (12438)------------------------------
% 2.52/1.52 % (12438)------------------------------
% 2.52/1.52 % (12442)Instruction limit reached!
% 2.52/1.52 % (12442)------------------------------
% 2.52/1.52 % (12442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.52 % (12442)Termination reason: Unknown
% 2.52/1.52 % (12442)Termination phase: shuffling
% 2.52/1.52
% 2.52/1.52 % (12442)Memory used [KB]: 1407
% 2.52/1.52 % (12442)Time elapsed: 0.005 s
% 2.52/1.52 % (12442)Instructions burned: 7 (million)
% 2.52/1.52 % (12442)------------------------------
% 2.52/1.52 % (12442)------------------------------
% 2.52/1.52 % (12443)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/16Mi)
% 2.52/1.53 % (12444)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_12325 for (2999ds/3Mi)
% 2.52/1.53 % (12445)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_12325 for (2999ds/3Mi)
% 2.52/1.53 % (12444)Instruction limit reached!
% 2.52/1.53 % (12444)------------------------------
% 2.52/1.53 % (12444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (12444)Termination reason: Unknown
% 2.52/1.53 % (12444)Termination phase: shuffling
% 2.52/1.53
% 2.52/1.53 % (12444)Memory used [KB]: 1407
% 2.52/1.53 % (12444)Time elapsed: 0.003 s
% 2.52/1.53 % (12444)Instructions burned: 3 (million)
% 2.52/1.53 % (12444)------------------------------
% 2.52/1.53 % (12444)------------------------------
% 2.52/1.53 % (12445)Instruction limit reached!
% 2.52/1.53 % (12445)------------------------------
% 2.52/1.53 % (12445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (12445)Termination reason: Unknown
% 2.52/1.53 % (12445)Termination phase: shuffling
% 2.52/1.53
% 2.52/1.53 % (12445)Memory used [KB]: 1407
% 2.52/1.53 % (12445)Time elapsed: 0.003 s
% 2.52/1.53 % (12445)Instructions burned: 3 (million)
% 2.52/1.53 % (12445)------------------------------
% 2.52/1.53 % (12445)------------------------------
% 2.52/1.53 % (12443)Instruction limit reached!
% 2.52/1.53 % (12443)------------------------------
% 2.52/1.53 % (12443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.53 % (12443)Termination reason: Unknown
% 2.52/1.53 % (12443)Termination phase: Property scanning
% 2.52/1.53
% 2.52/1.53 % (12443)Memory used [KB]: 1663
% 2.52/1.53 % (12443)Time elapsed: 0.010 s
% 2.52/1.53 % (12443)Instructions burned: 17 (million)
% 2.52/1.53 % (12443)------------------------------
% 2.52/1.53 % (12443)------------------------------
% 2.52/1.54 % (12446)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/7Mi)
% 2.52/1.54 % (12430)Instruction limit reached!
% 2.52/1.54 % (12430)------------------------------
% 2.52/1.54 % (12430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (12430)Termination reason: Unknown
% 2.52/1.54 % (12430)Termination phase: Saturation
% 2.52/1.54
% 2.52/1.54 % (12430)Memory used [KB]: 7164
% 2.52/1.54 % (12430)Time elapsed: 0.059 s
% 2.52/1.54 % (12430)Instructions burned: 184 (million)
% 2.52/1.54 % (12430)------------------------------
% 2.52/1.54 % (12430)------------------------------
% 2.52/1.54 % (12447)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/3Mi)
% 2.52/1.54 % (12447)Instruction limit reached!
% 2.52/1.54 % (12447)------------------------------
% 2.52/1.54 % (12447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (12447)Termination reason: Unknown
% 2.52/1.54 % (12447)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (12447)Memory used [KB]: 1407
% 2.52/1.54 % (12447)Time elapsed: 0.003 s
% 2.52/1.54 % (12447)Instructions burned: 4 (million)
% 2.52/1.54 % (12447)------------------------------
% 2.52/1.54 % (12447)------------------------------
% 2.52/1.54 % (12446)Instruction limit reached!
% 2.52/1.54 % (12446)------------------------------
% 2.52/1.54 % (12446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.54 % (12446)Termination reason: Unknown
% 2.52/1.54 % (12446)Termination phase: shuffling
% 2.52/1.54
% 2.52/1.54 % (12446)Memory used [KB]: 1407
% 2.52/1.54 % (12446)Time elapsed: 0.005 s
% 2.52/1.54 % (12446)Instructions burned: 7 (million)
% 2.52/1.54 % (12446)------------------------------
% 2.52/1.54 % (12446)------------------------------
% 2.52/1.55 % (12448)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/4Mi)
% 2.52/1.55 % (12449)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/18Mi)
% 2.52/1.55 % (12451)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/6Mi)
% 2.52/1.55 % (12450)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/710Mi)
% 2.52/1.55 % (12448)Instruction limit reached!
% 2.52/1.55 % (12448)------------------------------
% 2.52/1.55 % (12448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (12448)Termination reason: Unknown
% 2.52/1.55 % (12448)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (12448)Memory used [KB]: 1407
% 2.52/1.55 % (12448)Time elapsed: 0.004 s
% 2.52/1.55 % (12448)Instructions burned: 5 (million)
% 2.52/1.55 % (12448)------------------------------
% 2.52/1.55 % (12448)------------------------------
% 2.52/1.55 % (12451)Instruction limit reached!
% 2.52/1.55 % (12451)------------------------------
% 2.52/1.55 % (12451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.55 % (12451)Termination reason: Unknown
% 2.52/1.55 % (12451)Termination phase: shuffling
% 2.52/1.55
% 2.52/1.55 % (12451)Memory used [KB]: 1407
% 2.52/1.55 % (12451)Time elapsed: 0.003 s
% 2.52/1.55 % (12451)Instructions burned: 6 (million)
% 2.52/1.55 % (12451)------------------------------
% 2.52/1.55 % (12451)------------------------------
% 2.52/1.56 % (12452)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_12325 for (2999ds/902Mi)
% 2.52/1.56 % (12449)Instruction limit reached!
% 2.52/1.56 % (12449)------------------------------
% 2.52/1.56 % (12449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (12449)Termination reason: Unknown
% 2.52/1.56 % (12449)Termination phase: Property scanning
% 2.52/1.56
% 2.52/1.56 % (12449)Memory used [KB]: 1663
% 2.52/1.56 % (12449)Time elapsed: 0.011 s
% 2.52/1.56 % (12449)Instructions burned: 19 (million)
% 2.52/1.56 % (12449)------------------------------
% 2.52/1.56 % (12449)------------------------------
% 2.52/1.56 % (12453)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_12325 for (2999ds/21Mi)
% 2.52/1.56 % (12455)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_12325 for (2999ds/6Mi)
% 2.52/1.56 % (12455)Instruction limit reached!
% 2.52/1.56 % (12455)------------------------------
% 2.52/1.56 % (12455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (12455)Termination reason: Unknown
% 2.52/1.56 % (12455)Termination phase: shuffling
% 2.52/1.56
% 2.52/1.56 % (12455)Memory used [KB]: 1407
% 2.52/1.56 % (12455)Time elapsed: 0.004 s
% 2.52/1.56 % (12450)Refutation not found, incomplete strategy
% 2.52/1.56 % (12450)------------------------------
% 2.52/1.56 % (12450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (12450)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.56
% 2.52/1.56
% 2.52/1.56 % (12450)Memory used [KB]: 6140
% 2.52/1.56 % (12450)Time elapsed: 0.015 s
% 2.52/1.56 % (12450)Instructions burned: 28 (million)
% 2.52/1.56 % (12450)------------------------------
% 2.52/1.56 % (12450)------------------------------
% 2.52/1.56 % (12455)Instructions burned: 7 (million)
% 2.52/1.56 % (12455)------------------------------
% 2.52/1.56 % (12455)------------------------------
% 2.52/1.56 % (12453)Instruction limit reached!
% 2.52/1.56 % (12453)------------------------------
% 2.52/1.56 % (12453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.56 % (12453)Termination reason: Unknown
% 2.52/1.56 % (12453)Termination phase: Property scanning
% 2.52/1.56
% 2.52/1.56 % (12453)Memory used [KB]: 1791
% 2.52/1.56 % (12453)Time elapsed: 0.009 s
% 2.52/1.56 % (12453)Instructions burned: 21 (million)
% 2.52/1.56 % (12453)------------------------------
% 2.52/1.56 % (12453)------------------------------
% 2.52/1.56 % (12454)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/5Mi)
% 2.52/1.57 % (12454)Instruction limit reached!
% 2.52/1.57 % (12454)------------------------------
% 2.52/1.57 % (12454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.57 % (12454)Termination reason: Unknown
% 2.52/1.57 % (12454)Termination phase: shuffling
% 2.52/1.57
% 2.52/1.57 % (12454)Memory used [KB]: 1407
% 2.52/1.57 % (12454)Time elapsed: 0.004 s
% 2.52/1.57 % (12454)Instructions burned: 6 (million)
% 2.52/1.57 % (12454)------------------------------
% 2.52/1.57 % (12454)------------------------------
% 2.52/1.57 % (12456)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/377Mi)
% 2.52/1.57 % (12459)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/879Mi)
% 2.52/1.58 % (12458)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_12325 for (2999ds/19Mi)
% 2.52/1.58 % (12457)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_12325 for (2999ds/779Mi)
% 2.52/1.58 % (12460)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_12325 for (2999ds/17Mi)
% 2.52/1.58 % (12459)Refutation not found, incomplete strategy
% 2.52/1.58 % (12459)------------------------------
% 2.52/1.58 % (12459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.58 % (12459)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.58
% 2.52/1.58
% 2.52/1.58 % (12459)Memory used [KB]: 6140
% 2.52/1.58 % (12459)Time elapsed: 0.011 s
% 2.52/1.58 % (12459)Instructions burned: 33 (million)
% 2.52/1.58 % (12459)------------------------------
% 2.52/1.58 % (12459)------------------------------
% 2.52/1.59 % (12458)Instruction limit reached!
% 2.52/1.59 % (12458)------------------------------
% 2.52/1.59 % (12458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (12458)Termination reason: Unknown
% 2.52/1.59 % (12458)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (12458)Memory used [KB]: 1663
% 2.52/1.59 % (12458)Time elapsed: 0.010 s
% 2.52/1.59 % (12458)Instructions burned: 19 (million)
% 2.52/1.59 % (12458)------------------------------
% 2.52/1.59 % (12458)------------------------------
% 2.52/1.59 % (12460)Instruction limit reached!
% 2.52/1.59 % (12460)------------------------------
% 2.52/1.59 % (12460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (12460)Termination reason: Unknown
% 2.52/1.59 % (12460)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (12460)Memory used [KB]: 1663
% 2.52/1.59 % (12460)Time elapsed: 0.010 s
% 2.52/1.59 % (12460)Instructions burned: 17 (million)
% 2.52/1.59 % (12460)------------------------------
% 2.52/1.59 % (12460)------------------------------
% 2.52/1.59 % (12461)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12325 for (2998ds/3Mi)
% 2.52/1.59 % (12461)Instruction limit reached!
% 2.52/1.59 % (12461)------------------------------
% 2.52/1.59 % (12461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (12461)Termination reason: Unknown
% 2.52/1.59 % (12461)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (12461)Memory used [KB]: 1407
% 2.52/1.59 % (12461)Time elapsed: 0.002 s
% 2.52/1.59 % (12461)Instructions burned: 3 (million)
% 2.52/1.59 % (12461)------------------------------
% 2.52/1.59 % (12461)------------------------------
% 2.52/1.60 % (12452)Refutation not found, incomplete strategy
% 2.52/1.60 % (12452)------------------------------
% 2.52/1.60 % (12452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.60 % (12452)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.60
% 2.52/1.60
% 2.52/1.60 % (12452)Memory used [KB]: 6908
% 2.52/1.60 % (12452)Time elapsed: 0.048 s
% 2.52/1.60 % (12452)Instructions burned: 93 (million)
% 2.52/1.60 % (12452)------------------------------
% 2.52/1.60 % (12452)------------------------------
% 2.52/1.60 % (12462)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_12325 for (2998ds/30Mi)
% 2.52/1.60 % (12464)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_12325 for (2998ds/100Mi)
% 2.52/1.61 % (12463)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_12325 for (2998ds/127Mi)
% 2.52/1.62 % (12462)Instruction limit reached!
% 2.52/1.62 % (12462)------------------------------
% 2.52/1.62 % (12462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.62 % (12462)Termination reason: Unknown
% 2.52/1.62 % (12462)Termination phase: Preprocessing 1
% 2.52/1.62
% 2.52/1.62 % (12462)Memory used [KB]: 1663
% 2.52/1.62 % (12462)Time elapsed: 0.017 s
% 2.52/1.62 % (12462)Instructions burned: 31 (million)
% 2.52/1.62 % (12462)------------------------------
% 2.52/1.62 % (12462)------------------------------
% 2.52/1.62 % (12465)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_12325 for (2998ds/3Mi)
% 2.52/1.62 % (12465)Instruction limit reached!
% 2.52/1.62 % (12465)------------------------------
% 2.52/1.62 % (12465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.62 % (12465)Termination reason: Unknown
% 2.52/1.62 % (12465)Termination phase: shuffling
% 2.52/1.62
% 2.52/1.62 % (12465)Memory used [KB]: 1407
% 2.52/1.62 % (12465)Time elapsed: 0.003 s
% 2.52/1.62 % (12465)Instructions burned: 3 (million)
% 2.52/1.62 % (12465)------------------------------
% 2.52/1.62 % (12465)------------------------------
% 2.52/1.63 % (12466)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_12325 for (2998ds/20Mi)
% 2.52/1.64 % (12467)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_12325 for (2998ds/86Mi)
% 2.52/1.64 % (12466)Instruction limit reached!
% 2.52/1.64 % (12466)------------------------------
% 2.52/1.64 % (12466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.64 % (12466)Termination reason: Unknown
% 2.52/1.64 % (12466)Termination phase: Property scanning
% 2.52/1.64
% 2.52/1.64 % (12466)Memory used [KB]: 1663
% 2.52/1.64 % (12466)Time elapsed: 0.012 s
% 2.52/1.64 % (12466)Instructions burned: 21 (million)
% 2.52/1.64 % (12466)------------------------------
% 2.52/1.64 % (12466)------------------------------
% 2.52/1.65 % (12464)Instruction limit reached!
% 2.52/1.65 % (12464)------------------------------
% 2.52/1.65 % (12464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.65 % (12464)Termination reason: Unknown
% 2.52/1.65 % (12464)Termination phase: Saturation
% 2.52/1.65
% 2.52/1.65 % (12464)Memory used [KB]: 6652
% 2.52/1.65 % (12464)Time elapsed: 0.044 s
% 2.52/1.65 % (12464)Instructions burned: 100 (million)
% 2.52/1.65 % (12464)------------------------------
% 2.52/1.65 % (12464)------------------------------
% 2.52/1.66 % (12468)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_12325 for (2998ds/107Mi)
% 2.52/1.66 % (12469)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_12325 for (2998ds/39Mi)
% 2.52/1.67 % (12463)Instruction limit reached!
% 2.52/1.67 % (12463)------------------------------
% 2.52/1.67 % (12463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.67 % (12463)Termination reason: Unknown
% 2.52/1.67 % (12463)Termination phase: Saturation
% 2.52/1.67
% 2.52/1.67 % (12463)Memory used [KB]: 6908
% 2.52/1.67 % (12463)Time elapsed: 0.086 s
% 2.52/1.67 % (12463)Instructions burned: 127 (million)
% 2.52/1.67 % (12463)------------------------------
% 2.52/1.67 % (12463)------------------------------
% 2.52/1.68 % (12467)Instruction limit reached!
% 2.52/1.68 % (12467)------------------------------
% 2.52/1.68 % (12467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.68 % (12467)Termination reason: Unknown
% 2.52/1.68 % (12467)Termination phase: Saturation
% 2.52/1.68
% 2.52/1.68 % (12467)Memory used [KB]: 6652
% 2.52/1.68 % (12467)Time elapsed: 0.043 s
% 2.52/1.68 % (12467)Instructions burned: 86 (million)
% 2.52/1.68 % (12467)------------------------------
% 2.52/1.68 % (12467)------------------------------
% 2.52/1.68 % (12469)Instruction limit reached!
% 2.52/1.68 % (12469)------------------------------
% 2.52/1.68 % (12469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.68 % (12469)Termination reason: Unknown
% 2.52/1.68 % (12469)Termination phase: Property scanning
% 2.52/1.68
% 2.52/1.68 % (12469)Memory used [KB]: 2046
% 2.52/1.68 % (12469)Time elapsed: 0.020 s
% 2.52/1.68 % (12469)Instructions burned: 40 (million)
% 2.52/1.68 % (12469)------------------------------
% 2.52/1.68 % (12469)------------------------------
% 2.52/1.69 % (12470)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_12325 for (2997ds/448Mi)
% 2.52/1.69 % (12471)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_12325 for (2997ds/46Mi)
% 2.52/1.70 % (12472)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_12325 for (2997ds/98Mi)
% 2.52/1.72 % (12468)Instruction limit reached!
% 2.52/1.72 % (12468)------------------------------
% 2.52/1.72 % (12468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.72 % (12468)Termination reason: Unknown
% 2.52/1.72 % (12468)Termination phase: Saturation
% 2.52/1.72
% 2.52/1.72 % (12468)Memory used [KB]: 6652
% 2.52/1.72 % (12468)Time elapsed: 0.056 s
% 2.52/1.72 % (12468)Instructions burned: 108 (million)
% 2.52/1.72 % (12468)------------------------------
% 2.52/1.72 % (12468)------------------------------
% 2.52/1.72 % (12471)Instruction limit reached!
% 2.52/1.72 % (12471)------------------------------
% 2.52/1.72 % (12471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.72 % (12471)Termination reason: Unknown
% 2.52/1.72 % (12471)Termination phase: Property scanning
% 2.52/1.72
% 2.52/1.72 % (12471)Memory used [KB]: 1918
% 2.52/1.72 % (12471)Time elapsed: 0.025 s
% 2.52/1.72 % (12471)Instructions burned: 47 (million)
% 2.52/1.72 % (12471)------------------------------
% 2.52/1.72 % (12471)------------------------------
% 2.52/1.73 % (12473)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_12325 for (2997ds/507Mi)
% 2.52/1.73 % (12474)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_12325 for (2997ds/149Mi)
% 2.52/1.74 % (12472)Instruction limit reached!
% 2.52/1.74 % (12472)------------------------------
% 2.52/1.74 % (12472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.74 % (12472)Termination reason: Unknown
% 2.52/1.74 % (12472)Termination phase: Saturation
% 2.52/1.74
% 2.52/1.74 % (12472)Memory used [KB]: 2302
% 2.52/1.74 % (12472)Time elapsed: 0.046 s
% 2.52/1.74 % (12472)Instructions burned: 99 (million)
% 2.52/1.74 % (12472)------------------------------
% 2.52/1.74 % (12472)------------------------------
% 2.52/1.76 % (12475)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_12325 for (2997ds/39Mi)
% 2.52/1.76 % (12456)Instruction limit reached!
% 2.52/1.76 % (12456)------------------------------
% 2.52/1.76 % (12456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.76 % (12456)Termination reason: Unknown
% 2.52/1.76 % (12456)Termination phase: Saturation
% 2.52/1.76
% 2.52/1.76 % (12456)Memory used [KB]: 7675
% 2.52/1.76 % (12456)Time elapsed: 0.194 s
% 2.52/1.76 % (12456)Instructions burned: 377 (million)
% 2.52/1.76 % (12456)------------------------------
% 2.52/1.76 % (12456)------------------------------
% 2.52/1.78 % (12475)Instruction limit reached!
% 2.52/1.78 % (12475)------------------------------
% 2.52/1.78 % (12475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.78 % (12475)Termination reason: Unknown
% 2.52/1.78 % (12475)Termination phase: Preprocessing 3
% 2.52/1.78
% 2.52/1.78 % (12475)Memory used [KB]: 1918
% 2.52/1.78 % (12475)Time elapsed: 0.021 s
% 2.52/1.78 % (12475)Instructions burned: 39 (million)
% 2.52/1.78 % (12475)------------------------------
% 2.52/1.78 % (12475)------------------------------
% 2.52/1.78 % (12476)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/6Mi)
% 2.52/1.78 % (12473)Refutation not found, incomplete strategy
% 2.52/1.78 % (12473)------------------------------
% 2.52/1.78 % (12473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.78 % (12473)Termination reason: Refutation not found, incomplete strategy
% 2.52/1.78
% 2.52/1.78
% 2.52/1.78 % (12473)Memory used [KB]: 7164
% 2.52/1.78 % (12473)Time elapsed: 0.054 s
% 2.52/1.78 % (12473)Instructions burned: 103 (million)
% 2.52/1.78 % (12473)------------------------------
% 2.52/1.78 % (12473)------------------------------
% 2.52/1.78 % (12476)Instruction limit reached!
% 2.52/1.78 % (12476)------------------------------
% 2.52/1.78 % (12476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.78 % (12476)Termination reason: Unknown
% 2.52/1.78 % (12476)Termination phase: shuffling
% 2.52/1.78
% 2.52/1.78 % (12476)Memory used [KB]: 1407
% 2.52/1.78 % (12476)Time elapsed: 0.005 s
% 2.52/1.78 % (12476)Instructions burned: 7 (million)
% 2.52/1.78 % (12476)------------------------------
% 2.52/1.78 % (12476)------------------------------
% 2.52/1.80 % (12477)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/1087Mi)
% 2.52/1.80 % (12478)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/47Mi)
% 2.52/1.81 % (12479)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/44Mi)
% 2.52/1.81 % (12474)Instruction limit reached!
% 2.52/1.81 % (12474)------------------------------
% 2.52/1.81 % (12474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.81 % (12474)Termination reason: Unknown
% 2.52/1.81 % (12474)Termination phase: Saturation
% 2.52/1.81
% 2.52/1.81 % (12474)Memory used [KB]: 6908
% 2.52/1.81 % (12474)Time elapsed: 0.075 s
% 2.52/1.81 % (12474)Instructions burned: 150 (million)
% 2.52/1.81 % (12474)------------------------------
% 2.52/1.81 % (12474)------------------------------
% 2.52/1.82 % (12480)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/432Mi)
% 2.52/1.82 % (12478)Instruction limit reached!
% 2.52/1.82 % (12478)------------------------------
% 2.52/1.82 % (12478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.82 % (12478)Termination reason: Unknown
% 2.52/1.82 % (12478)Termination phase: Property scanning
% 2.52/1.82
% 2.52/1.82 % (12478)Memory used [KB]: 2046
% 2.52/1.82 % (12478)Time elapsed: 0.025 s
% 2.52/1.82 % (12478)Instructions burned: 48 (million)
% 2.52/1.82 % (12478)------------------------------
% 2.52/1.82 % (12478)------------------------------
% 2.52/1.83 % (12479)Instruction limit reached!
% 2.52/1.83 % (12479)------------------------------
% 2.52/1.83 % (12479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.83 % (12479)Termination reason: Unknown
% 2.52/1.83 % (12479)Termination phase: Property scanning
% 2.52/1.83
% 2.52/1.83 % (12479)Memory used [KB]: 2046
% 2.52/1.83 % (12479)Time elapsed: 0.014 s
% 2.52/1.83 % (12479)Instructions burned: 44 (million)
% 2.52/1.83 % (12479)------------------------------
% 2.52/1.83 % (12479)------------------------------
% 2.52/1.84 % (12481)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/62Mi)
% 2.52/1.85 % (12482)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/31Mi)
% 2.52/1.86 % (12482)Instruction limit reached!
% 2.52/1.86 % (12482)------------------------------
% 2.52/1.86 % (12482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.86 % (12482)Termination reason: Unknown
% 2.52/1.86 % (12482)Termination phase: Preprocessing 3
% 2.52/1.86
% 2.52/1.86 % (12482)Memory used [KB]: 1918
% 2.52/1.86 % (12482)Time elapsed: 0.017 s
% 2.52/1.86 % (12482)Instructions burned: 31 (million)
% 2.52/1.86 % (12482)------------------------------
% 2.52/1.86 % (12482)------------------------------
% 2.52/1.87 % (12481)Instruction limit reached!
% 2.52/1.87 % (12481)------------------------------
% 2.52/1.87 % (12481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.87 % (12481)Termination reason: Unknown
% 2.52/1.87 % (12481)Termination phase: Preprocessing 3
% 2.52/1.87
% 2.52/1.87 % (12481)Memory used [KB]: 2174
% 2.52/1.87 % (12481)Time elapsed: 0.030 s
% 2.52/1.87 % (12481)Instructions burned: 63 (million)
% 2.52/1.87 % (12481)------------------------------
% 2.52/1.87 % (12481)------------------------------
% 2.52/1.88 % (12483)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_12325 for (2996ds/96Mi)
% 2.52/1.89 % (12484)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/453Mi)
% 2.52/1.91 % (12470)Instruction limit reached!
% 2.52/1.91 % (12470)------------------------------
% 2.52/1.91 % (12470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.91 % (12470)Termination reason: Unknown
% 2.52/1.91 % (12470)Termination phase: Saturation
% 2.52/1.91
% 2.52/1.91 % (12470)Memory used [KB]: 8187
% 2.52/1.91 % (12470)Time elapsed: 0.222 s
% 2.52/1.91 % (12470)Instructions burned: 449 (million)
% 2.52/1.91 % (12470)------------------------------
% 2.52/1.91 % (12470)------------------------------
% 2.52/1.92 % (12485)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/71Mi)
% 2.52/1.93 % (12441)Instruction limit reached!
% 2.52/1.93 % (12441)------------------------------
% 2.52/1.93 % (12441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.93 % (12441)Termination reason: Unknown
% 2.52/1.93 % (12441)Termination phase: Saturation
% 2.52/1.93
% 2.52/1.93 % (12441)Memory used [KB]: 9722
% 2.52/1.93 % (12441)Time elapsed: 0.415 s
% 2.52/1.93 % (12441)Instructions burned: 1041 (million)
% 2.52/1.93 % (12441)------------------------------
% 2.52/1.93 % (12441)------------------------------
% 2.52/1.93 % (12483)Instruction limit reached!
% 2.52/1.93 % (12483)------------------------------
% 2.52/1.93 % (12483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.93 % (12483)Termination reason: Unknown
% 2.52/1.93 % (12483)Termination phase: Saturation
% 2.52/1.93
% 2.52/1.93 % (12483)Memory used [KB]: 6780
% 2.52/1.93 % (12483)Time elapsed: 0.048 s
% 2.52/1.93 % (12483)Instructions burned: 96 (million)
% 2.52/1.93 % (12483)------------------------------
% 2.52/1.93 % (12483)------------------------------
% 2.52/1.94 % (12486)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/83Mi)
% 2.52/1.94 % (12487)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/11Mi)
% 2.52/1.95 % (12487)Instruction limit reached!
% 2.52/1.95 % (12487)------------------------------
% 2.52/1.95 % (12487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.95 % (12487)Termination reason: Unknown
% 2.52/1.95 % (12487)Termination phase: shuffling
% 2.52/1.95
% 2.52/1.95 % (12487)Memory used [KB]: 1535
% 2.52/1.95 % (12487)Time elapsed: 0.007 s
% 2.52/1.95 % (12487)Instructions burned: 12 (million)
% 2.52/1.95 % (12487)------------------------------
% 2.52/1.95 % (12487)------------------------------
% 2.52/1.96 % (12485)Instruction limit reached!
% 2.52/1.96 % (12485)------------------------------
% 2.52/1.96 % (12485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.96 % (12485)Termination reason: Unknown
% 2.52/1.96 % (12485)Termination phase: Saturation
% 2.52/1.96
% 2.52/1.96 % (12485)Memory used [KB]: 6524
% 2.52/1.96 % (12485)Time elapsed: 0.036 s
% 2.52/1.96 % (12485)Instructions burned: 72 (million)
% 2.52/1.96 % (12485)------------------------------
% 2.52/1.96 % (12485)------------------------------
% 2.52/1.96 % (12457)Instruction limit reached!
% 2.52/1.96 % (12457)------------------------------
% 2.52/1.96 % (12457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.96 % (12457)Termination reason: Unknown
% 2.52/1.96 % (12457)Termination phase: Saturation
% 2.52/1.96
% 2.52/1.96 % (12457)Memory used [KB]: 7931
% 2.52/1.96 % (12457)Time elapsed: 0.386 s
% 2.52/1.96 % (12457)Instructions burned: 780 (million)
% 2.52/1.96 % (12457)------------------------------
% 2.52/1.96 % (12457)------------------------------
% 2.52/1.97 % (12488)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/41Mi)
% 2.52/1.97 % (12489)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/6Mi)
% 2.52/1.98 % (12489)Instruction limit reached!
% 2.52/1.98 % (12489)------------------------------
% 2.52/1.98 % (12489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.98 % (12489)Termination reason: Unknown
% 2.52/1.98 % (12489)Termination phase: shuffling
% 2.52/1.98
% 2.52/1.98 % (12489)Memory used [KB]: 1407
% 2.52/1.98 % (12489)Time elapsed: 0.005 s
% 2.52/1.98 % (12489)Instructions burned: 7 (million)
% 2.52/1.98 % (12489)------------------------------
% 2.52/1.98 % (12489)------------------------------
% 2.52/1.98 % (12490)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_12325 for (2995ds/150Mi)
% 2.52/1.98 % (12486)Instruction limit reached!
% 2.52/1.98 % (12486)------------------------------
% 2.52/1.98 % (12486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.98 % (12486)Termination reason: Unknown
% 2.52/1.98 % (12486)Termination phase: Saturation
% 2.52/1.98
% 2.52/1.98 % (12486)Memory used [KB]: 6524
% 2.52/1.98 % (12486)Time elapsed: 0.044 s
% 2.52/1.98 % (12486)Instructions burned: 83 (million)
% 2.52/1.98 % (12486)------------------------------
% 2.52/1.98 % (12486)------------------------------
% 2.52/1.99 % (12491)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_12325 for (2994ds/343Mi)
% 2.52/2.00 % (12488)Instruction limit reached!
% 2.52/2.00 % (12488)------------------------------
% 2.52/2.00 % (12488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/2.00 % (12488)Termination reason: Unknown
% 2.52/2.00 % (12488)Termination phase: Property scanning
% 2.52/2.00
% 2.52/2.00 % (12488)Memory used [KB]: 2046
% 2.52/2.00 % (12488)Time elapsed: 0.014 s
% 2.52/2.00 % (12488)Instructions burned: 44 (million)
% 2.52/2.00 % (12488)------------------------------
% 2.52/2.00 % (12488)------------------------------
% 2.52/2.00 % (12492)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_12325 for (2994ds/109Mi)
% 2.52/2.02 % (12493)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_12325 for (2994ds/253Mi)
% 6.12/2.04 % (12480)Instruction limit reached!
% 6.12/2.04 % (12480)------------------------------
% 6.12/2.04 % (12480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.12/2.04 % (12480)Termination reason: Unknown
% 6.12/2.04 % (12480)Termination phase: Saturation
% 6.12/2.04
% 6.12/2.04 % (12480)Memory used [KB]: 3709
% 6.12/2.04 % (12480)Time elapsed: 0.218 s
% 6.12/2.04 % (12480)Instructions burned: 432 (million)
% 6.12/2.04 % (12480)------------------------------
% 6.12/2.04 % (12480)------------------------------
% 6.12/2.05 % (12490)Instruction limit reached!
% 6.12/2.05 % (12490)------------------------------
% 6.12/2.05 % (12490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.12/2.05 % (12490)Termination reason: Unknown
% 6.12/2.05 % (12490)Termination phase: Saturation
% 6.12/2.05
% 6.12/2.05 % (12490)Memory used [KB]: 6908
% 6.12/2.05 % (12490)Time elapsed: 0.076 s
% 6.12/2.05 % (12490)Instructions burned: 150 (million)
% 6.12/2.05 % (12490)------------------------------
% 6.12/2.05 % (12490)------------------------------
% 6.12/2.05 % (12492)Instruction limit reached!
% 6.12/2.05 % (12492)------------------------------
% 6.12/2.05 % (12492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.12/2.05 % (12492)Termination reason: Unknown
% 6.12/2.05 % (12492)Termination phase: Saturation
% 6.12/2.05
% 6.12/2.05 % (12492)Memory used [KB]: 6780
% 6.12/2.05 % (12492)Time elapsed: 0.055 s
% 6.12/2.05 % (12492)Instructions burned: 109 (million)
% 6.12/2.05 % (12492)------------------------------
% 6.12/2.05 % (12492)------------------------------
% 6.12/2.06 % (12494)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_12325 for (2994ds/140Mi)
% 6.12/2.07 % (12495)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_12325 for (2994ds/614Mi)
% 6.12/2.07 % (12496)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_12325 for (2994ds/202Mi)
% 6.12/2.07 % (12494)Refutation not found, incomplete strategy
% 6.12/2.07 % (12494)------------------------------
% 6.12/2.07 % (12494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.12/2.08 % (12494)Termination reason: Refutation not found, incomplete strategy
% 6.12/2.08
% 6.12/2.08
% 6.12/2.08 % (12494)Memory used [KB]: 6268
% 6.12/2.08 % (12494)Time elapsed: 0.034 s
% 6.12/2.08 % (12494)Instructions burned: 32 (million)
% 6.12/2.08 % (12494)------------------------------
% 6.12/2.08 % (12494)------------------------------
% 6.12/2.08 % (12497)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_12325 for (2993ds/339Mi)
% 6.12/2.09 % (12496)Refutation not found, incomplete strategy
% 6.12/2.09 % (12496)------------------------------
% 6.12/2.09 % (12496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.12/2.09 % (12496)Termination reason: Refutation not found, incomplete strategy
% 6.12/2.09
% 6.12/2.09
% 6.12/2.09 % (12496)Memory used [KB]: 6140
% 6.12/2.09 % (12496)Time elapsed: 0.017 s
% 6.12/2.09 % (12496)Instructions burned: 31 (million)
% 6.12/2.09 % (12496)------------------------------
% 6.12/2.09 % (12496)------------------------------
% 6.12/2.09 % (12484)Instruction limit reached!
% 6.12/2.09 % (12484)------------------------------
% 6.12/2.09 % (12484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.12/2.09 % (12484)Termination reason: Unknown
% 6.12/2.09 % (12484)Termination phase: Saturation
% 6.12/2.09
% 6.12/2.09 % (12484)Memory used [KB]: 7675
% 6.12/2.09 % (12484)Time elapsed: 0.211 s
% 6.12/2.09 % (12484)Instructions burned: 453 (million)
% 6.12/2.09 % (12484)------------------------------
% 6.12/2.09 % (12484)------------------------------
% 6.12/2.10 % (12498)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=283:si=on:rtra=on_0 on DTF2THF_12325 for (2993ds/283Mi)
% 6.12/2.12 % (12499)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_12325 for (2993ds/134Mi)
% 6.57/2.15 % (12493)Instruction limit reached!
% 6.57/2.15 % (12493)------------------------------
% 6.57/2.15 % (12493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.57/2.15 % (12493)Termination reason: Unknown
% 6.57/2.15 % (12493)Termination phase: Saturation
% 6.57/2.15
% 6.57/2.15 % (12493)Memory used [KB]: 6908
% 6.57/2.15 % (12493)Time elapsed: 0.152 s
% 6.57/2.15 % (12493)Instructions burned: 253 (million)
% 6.57/2.15 % (12493)------------------------------
% 6.57/2.15 % (12493)------------------------------
% 6.57/2.16 % (12491)Instruction limit reached!
% 6.57/2.16 % (12491)------------------------------
% 6.57/2.16 % (12491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.57/2.16 % (12491)Termination reason: Unknown
% 6.57/2.16 % (12491)Termination phase: Saturation
% 6.57/2.16
% 6.57/2.16 % (12491)Memory used [KB]: 6908
% 6.57/2.16 % (12491)Time elapsed: 0.166 s
% 6.57/2.16 % (12491)Instructions burned: 344 (million)
% 6.57/2.16 % (12491)------------------------------
% 6.57/2.16 % (12491)------------------------------
% 6.57/2.17 % (12501)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_12325 for (2993ds/298Mi)
% 6.57/2.17 % (12502)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_12325 for (2993ds/192Mi)
% 6.57/2.17 % (12499)Instruction limit reached!
% 6.57/2.17 % (12499)------------------------------
% 6.57/2.17 % (12499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.57/2.17 % (12499)Termination reason: Unknown
% 6.57/2.17 % (12499)Termination phase: Saturation
% 6.57/2.17
% 6.57/2.17 % (12499)Memory used [KB]: 6908
% 6.57/2.17 % (12499)Time elapsed: 0.087 s
% 6.57/2.17 % (12499)Instructions burned: 135 (million)
% 6.57/2.17 % (12499)------------------------------
% 6.57/2.17 % (12499)------------------------------
% 6.57/2.19 % (12503)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_12325 for (2993ds/258Mi)
% 6.87/2.23 % (12502)Instruction limit reached!
% 6.87/2.23 % (12502)------------------------------
% 6.87/2.23 % (12502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.23 % (12502)Termination reason: Unknown
% 6.87/2.23 % (12502)Termination phase: Saturation
% 6.87/2.23
% 6.87/2.23 % (12502)Memory used [KB]: 7803
% 6.87/2.23 % (12502)Time elapsed: 0.062 s
% 6.87/2.23 % (12502)Instructions burned: 194 (million)
% 6.87/2.23 % (12502)------------------------------
% 6.87/2.23 % (12502)------------------------------
% 6.87/2.23 % (12497)Instruction limit reached!
% 6.87/2.23 % (12497)------------------------------
% 6.87/2.23 % (12497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.23 % (12497)Termination reason: Unknown
% 6.87/2.23 % (12497)Termination phase: Saturation
% 6.87/2.23
% 6.87/2.23 % (12497)Memory used [KB]: 7164
% 6.87/2.23 % (12497)Time elapsed: 0.147 s
% 6.87/2.23 % (12497)Instructions burned: 341 (million)
% 6.87/2.23 % (12497)------------------------------
% 6.87/2.23 % (12497)------------------------------
% 6.87/2.24 % (12504)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_12325 for (2992ds/90Mi)
% 6.87/2.25 % (12505)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_12325 for (2992ds/231Mi)
% 6.87/2.25 % (12498)Instruction limit reached!
% 6.87/2.25 % (12498)------------------------------
% 6.87/2.25 % (12498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.25 % (12498)Termination reason: Unknown
% 6.87/2.25 % (12498)Termination phase: Saturation
% 6.87/2.25
% 6.87/2.25 % (12498)Memory used [KB]: 7547
% 6.87/2.25 % (12498)Time elapsed: 0.145 s
% 6.87/2.25 % (12498)Instructions burned: 283 (million)
% 6.87/2.25 % (12498)------------------------------
% 6.87/2.25 % (12498)------------------------------
% 6.87/2.26 % (12506)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_12325 for (2992ds/458Mi)
% 6.87/2.28 % (12504)Instruction limit reached!
% 6.87/2.28 % (12504)------------------------------
% 6.87/2.28 % (12504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.87/2.28 % (12504)Termination reason: Unknown
% 6.87/2.28 % (12504)Termination phase: Saturation
% 6.87/2.28
% 6.87/2.28 % (12504)Memory used [KB]: 6652
% 6.87/2.28 % (12504)Time elapsed: 0.045 s
% 6.87/2.28 % (12504)Instructions burned: 91 (million)
% 6.87/2.28 % (12504)------------------------------
% 6.87/2.28 % (12504)------------------------------
% 7.42/2.30 % (12507)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_12325 for (2991ds/31Mi)
% 7.42/2.31 % (12505)Instruction limit reached!
% 7.42/2.31 % (12505)------------------------------
% 7.42/2.31 % (12505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.42/2.31 % (12505)Termination reason: Unknown
% 7.42/2.31 % (12505)Termination phase: Saturation
% 7.42/2.31
% 7.42/2.31 % (12505)Memory used [KB]: 7036
% 7.42/2.31 % (12505)Time elapsed: 0.091 s
% 7.42/2.31 % (12505)Instructions burned: 232 (million)
% 7.42/2.31 % (12505)------------------------------
% 7.42/2.31 % (12505)------------------------------
% 7.42/2.31 % (12501)Instruction limit reached!
% 7.42/2.31 % (12501)------------------------------
% 7.42/2.31 % (12501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.42/2.31 % (12501)Termination reason: Unknown
% 7.42/2.31 % (12501)Termination phase: Saturation
% 7.42/2.31
% 7.42/2.31 % (12501)Memory used [KB]: 7931
% 7.42/2.31 % (12501)Time elapsed: 0.167 s
% 7.42/2.31 % (12501)Instructions burned: 299 (million)
% 7.42/2.31 % (12501)------------------------------
% 7.42/2.31 % (12501)------------------------------
% 7.42/2.32 % (12507)Instruction limit reached!
% 7.42/2.32 % (12507)------------------------------
% 7.42/2.32 % (12507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.42/2.32 % (12507)Termination reason: Unknown
% 7.42/2.32 % (12507)Termination phase: Preprocessing 3
% 7.42/2.32
% 7.42/2.32 % (12507)Memory used [KB]: 1918
% 7.42/2.32 % (12507)Time elapsed: 0.018 s
% 7.42/2.32 % (12507)Instructions burned: 32 (million)
% 7.42/2.32 % (12507)------------------------------
% 7.42/2.32 % (12507)------------------------------
% 7.42/2.32 % (12508)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_12325 for (2991ds/297Mi)
% 7.42/2.32 % (12477)Instruction limit reached!
% 7.42/2.32 % (12477)------------------------------
% 7.42/2.32 % (12477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.42/2.32 % (12477)Termination reason: Unknown
% 7.42/2.32 % (12477)Termination phase: Saturation
% 7.42/2.32
% 7.42/2.32 % (12477)Memory used [KB]: 7419
% 7.42/2.32 % (12477)Time elapsed: 0.524 s
% 7.42/2.32 % (12477)Instructions burned: 1087 (million)
% 7.42/2.32 % (12477)------------------------------
% 7.42/2.32 % (12477)------------------------------
% 7.42/2.33 % (12509)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_12325 for (2991ds/987Mi)
% 7.42/2.33 % (12503)Instruction limit reached!
% 7.42/2.33 % (12503)------------------------------
% 7.42/2.33 % (12503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.42/2.33 % (12503)Termination reason: Unknown
% 7.42/2.33 % (12503)Termination phase: Saturation
% 7.42/2.33
% 7.42/2.33 % (12503)Memory used [KB]: 6652
% 7.42/2.33 % (12503)Time elapsed: 0.160 s
% 7.42/2.33 % (12503)Instructions burned: 259 (million)
% 7.42/2.33 % (12503)------------------------------
% 7.42/2.33 % (12503)------------------------------
% 7.42/2.33 % (12510)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_12325 for (2991ds/182Mi)
% 7.42/2.34 % (12511)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_12325 for (2991ds/182Mi)
% 7.42/2.34 % (12512)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_12325 for (2991ds/154Mi)
% 7.42/2.37 % (12495)Instruction limit reached!
% 7.42/2.37 % (12495)------------------------------
% 7.42/2.37 % (12495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.42/2.37 % (12495)Termination reason: Unknown
% 7.42/2.37 % (12495)Termination phase: Saturation
% 7.42/2.37
% 7.42/2.37 % (12495)Memory used [KB]: 8443
% 7.42/2.37 % (12495)Time elapsed: 0.301 s
% 7.42/2.37 % (12495)Instructions burned: 615 (million)
% 7.42/2.37 % (12495)------------------------------
% 7.42/2.37 % (12495)------------------------------
% 7.42/2.39 % (12513)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_12325 for (2990ds/79Mi)
% 8.18/2.40 % (12510)Instruction limit reached!
% 8.18/2.40 % (12510)------------------------------
% 8.18/2.40 % (12510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.40 % (12510)Termination reason: Unknown
% 8.18/2.40 % (12510)Termination phase: Saturation
% 8.18/2.40
% 8.18/2.40 % (12510)Memory used [KB]: 6908
% 8.18/2.40 % (12510)Time elapsed: 0.072 s
% 8.18/2.40 % (12510)Instructions burned: 182 (million)
% 8.18/2.40 % (12510)------------------------------
% 8.18/2.40 % (12510)------------------------------
% 8.18/2.41 % (12512)Instruction limit reached!
% 8.18/2.41 % (12512)------------------------------
% 8.18/2.41 % (12512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.41 % (12512)Termination reason: Unknown
% 8.18/2.41 % (12512)Termination phase: Saturation
% 8.18/2.41
% 8.18/2.41 % (12512)Memory used [KB]: 2814
% 8.18/2.41 % (12512)Time elapsed: 0.073 s
% 8.18/2.41 % (12512)Instructions burned: 154 (million)
% 8.18/2.41 % (12512)------------------------------
% 8.18/2.41 % (12512)------------------------------
% 8.18/2.41 % (12514)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_12325 for (2990ds/216Mi)
% 8.18/2.42 % (12513)Instruction limit reached!
% 8.18/2.42 % (12513)------------------------------
% 8.18/2.42 % (12513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.42 % (12513)Termination reason: Unknown
% 8.18/2.42 % (12513)Termination phase: Saturation
% 8.18/2.42
% 8.18/2.42 % (12513)Memory used [KB]: 6652
% 8.18/2.43 % (12513)Time elapsed: 0.040 s
% 8.18/2.43 % (12513)Instructions burned: 80 (million)
% 8.18/2.43 % (12513)------------------------------
% 8.18/2.43 % (12513)------------------------------
% 8.18/2.43 % (12511)Instruction limit reached!
% 8.18/2.43 % (12511)------------------------------
% 8.18/2.43 % (12511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.43 % (12511)Termination reason: Unknown
% 8.18/2.43 % (12511)Termination phase: Saturation
% 8.18/2.43
% 8.18/2.43 % (12511)Memory used [KB]: 7164
% 8.18/2.43 % (12511)Time elapsed: 0.093 s
% 8.18/2.43 % (12511)Instructions burned: 182 (million)
% 8.18/2.43 % (12511)------------------------------
% 8.18/2.43 % (12511)------------------------------
% 8.18/2.43 % (12508)Instruction limit reached!
% 8.18/2.43 % (12508)------------------------------
% 8.18/2.43 % (12508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.43 % (12508)Termination reason: Unknown
% 8.18/2.43 % (12508)Termination phase: Saturation
% 8.18/2.43
% 8.18/2.43 % (12508)Memory used [KB]: 7164
% 8.18/2.43 % (12508)Time elapsed: 0.112 s
% 8.18/2.43 % (12508)Instructions burned: 299 (million)
% 8.18/2.43 % (12508)------------------------------
% 8.18/2.43 % (12508)------------------------------
% 8.18/2.43 % (12515)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_12325 for (2990ds/114Mi)
% 8.18/2.44 % (12517)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_12325 for (2990ds/182Mi)
% 8.18/2.45 % (12516)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_12325 for (2990ds/922Mi)
% 8.18/2.45 % (12518)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_12325 for (2990ds/164Mi)
% 8.18/2.48 % (12514)Instruction limit reached!
% 8.18/2.48 % (12514)------------------------------
% 8.18/2.48 % (12514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.48 % (12514)Termination reason: Unknown
% 8.18/2.48 % (12514)Termination phase: Saturation
% 8.18/2.48
% 8.18/2.48 % (12514)Memory used [KB]: 7419
% 8.18/2.48 % (12514)Time elapsed: 0.067 s
% 8.18/2.48 % (12514)Instructions burned: 217 (million)
% 8.18/2.48 % (12514)------------------------------
% 8.18/2.48 % (12514)------------------------------
% 8.18/2.49 % (12515)Instruction limit reached!
% 8.18/2.49 % (12515)------------------------------
% 8.18/2.49 % (12515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.18/2.49 % (12515)Termination reason: Unknown
% 8.18/2.49 % (12515)Termination phase: Saturation
% 8.18/2.49
% 8.18/2.49 % (12515)Memory used [KB]: 6780
% 8.18/2.49 % (12515)Time elapsed: 0.057 s
% 8.18/2.49 % (12515)Instructions burned: 114 (million)
% 8.18/2.49 % (12515)------------------------------
% 8.18/2.49 % (12515)------------------------------
% 8.18/2.49 % (12519)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/50Mi)
% 8.79/2.50 % (12520)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/368Mi)
% 8.79/2.50 % (12506)Instruction limit reached!
% 8.79/2.50 % (12506)------------------------------
% 8.79/2.50 % (12506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.50 % (12506)Termination reason: Unknown
% 8.79/2.50 % (12506)Termination phase: Saturation
% 8.79/2.50
% 8.79/2.50 % (12506)Memory used [KB]: 7675
% 8.79/2.50 % (12506)Time elapsed: 0.236 s
% 8.79/2.50 % (12506)Instructions burned: 459 (million)
% 8.79/2.50 % (12506)------------------------------
% 8.79/2.50 % (12506)------------------------------
% 8.79/2.51 % (12519)Instruction limit reached!
% 8.79/2.51 % (12519)------------------------------
% 8.79/2.51 % (12519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.51 % (12519)Termination reason: Unknown
% 8.79/2.51 % (12519)Termination phase: Property scanning
% 8.79/2.51
% 8.79/2.51 % (12519)Memory used [KB]: 2046
% 8.79/2.51 % (12519)Time elapsed: 0.023 s
% 8.79/2.51 % (12519)Instructions burned: 51 (million)
% 8.79/2.51 % (12519)------------------------------
% 8.79/2.51 % (12519)------------------------------
% 8.79/2.51 % (12521)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/1021Mi)
% 8.79/2.52 % (12517)Instruction limit reached!
% 8.79/2.52 % (12517)------------------------------
% 8.79/2.52 % (12517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.52 % (12517)Termination reason: Unknown
% 8.79/2.52 % (12517)Termination phase: Saturation
% 8.79/2.52
% 8.79/2.52 % (12517)Memory used [KB]: 6908
% 8.79/2.52 % (12517)Time elapsed: 0.079 s
% 8.79/2.52 % (12517)Instructions burned: 183 (million)
% 8.79/2.52 % (12517)------------------------------
% 8.79/2.52 % (12517)------------------------------
% 8.79/2.52 % (12518)Instruction limit reached!
% 8.79/2.52 % (12518)------------------------------
% 8.79/2.52 % (12518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.52 % (12518)Termination reason: Unknown
% 8.79/2.52 % (12518)Termination phase: Saturation
% 8.79/2.52
% 8.79/2.52 % (12518)Memory used [KB]: 2302
% 8.79/2.52 % (12518)Time elapsed: 0.077 s
% 8.79/2.52 % (12518)Instructions burned: 164 (million)
% 8.79/2.52 % (12518)------------------------------
% 8.79/2.52 % (12518)------------------------------
% 8.79/2.53 % (12522)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/267Mi)
% 8.79/2.53 % (12523)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/204Mi)
% 8.79/2.54 % (12524)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/26Mi)
% 8.79/2.55 % (12524)Instruction limit reached!
% 8.79/2.55 % (12524)------------------------------
% 8.79/2.55 % (12524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.55 % (12524)Termination reason: Unknown
% 8.79/2.55 % (12524)Termination phase: shuffling
% 8.79/2.55
% 8.79/2.55 % (12524)Memory used [KB]: 1918
% 8.79/2.55 % (12524)Time elapsed: 0.014 s
% 8.79/2.55 % (12524)Instructions burned: 27 (million)
% 8.79/2.55 % (12524)------------------------------
% 8.79/2.55 % (12524)------------------------------
% 8.79/2.57 % (12525)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_12325 for (2989ds/53Mi)
% 8.79/2.59 % (12525)Instruction limit reached!
% 8.79/2.59 % (12525)------------------------------
% 8.79/2.59 % (12525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.59 % (12525)Termination reason: Unknown
% 8.79/2.59 % (12525)Termination phase: Property scanning
% 8.79/2.59
% 8.79/2.59 % (12525)Memory used [KB]: 2046
% 8.79/2.59 % (12525)Time elapsed: 0.026 s
% 8.79/2.59 % (12525)Instructions burned: 53 (million)
% 8.79/2.59 % (12525)------------------------------
% 8.79/2.59 % (12525)------------------------------
% 8.79/2.60 % (12523)Instruction limit reached!
% 8.79/2.60 % (12523)------------------------------
% 8.79/2.60 % (12523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.79/2.60 % (12523)Termination reason: Unknown
% 8.79/2.60 % (12523)Termination phase: Saturation
% 8.79/2.60
% 8.79/2.60 % (12523)Memory used [KB]: 7291
% 8.79/2.60 % (12523)Time elapsed: 0.067 s
% 8.79/2.60 % (12523)Instructions burned: 205 (million)
% 8.79/2.60 % (12523)------------------------------
% 8.79/2.60 % (12523)------------------------------
% 9.32/2.60 % (12527)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_12325 for (2988ds/4436Mi)
% 9.32/2.61 % (12528)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_12325 for (2988ds/1539Mi)
% 9.32/2.66 % (12522)Instruction limit reached!
% 9.32/2.66 % (12522)------------------------------
% 9.32/2.66 % (12522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.32/2.66 % (12522)Termination reason: Unknown
% 9.32/2.66 % (12522)Termination phase: Saturation
% 9.32/2.66
% 9.32/2.66 % (12522)Memory used [KB]: 7547
% 9.32/2.66 % (12522)Time elapsed: 0.137 s
% 9.32/2.66 % (12522)Instructions burned: 267 (million)
% 9.32/2.66 % (12522)------------------------------
% 9.32/2.66 % (12522)------------------------------
% 9.32/2.67 % (12520)Instruction limit reached!
% 9.32/2.67 % (12520)------------------------------
% 9.32/2.67 % (12520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.32/2.67 % (12520)Termination reason: Unknown
% 9.32/2.67 % (12520)Termination phase: Saturation
% 9.32/2.67
% 9.32/2.67 % (12520)Memory used [KB]: 8059
% 9.32/2.67 % (12520)Time elapsed: 0.179 s
% 9.32/2.67 % (12520)Instructions burned: 369 (million)
% 9.32/2.67 % (12520)------------------------------
% 9.32/2.67 % (12520)------------------------------
% 9.32/2.69 % (12530)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_12325 for (2987ds/5205Mi)
% 9.32/2.69 % (12529)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_12325 for (2988ds/408Mi)
% 9.74/2.74 % (12516)First to succeed.
% 9.74/2.75 % (12516)Refutation found. Thanks to Tanya!
% 9.74/2.75 % SZS status Theorem for DTF2THF_12325
% 9.74/2.75 % SZS output start Proof for DTF2THF_12325
% 9.74/2.75 thf(type_def_5, type, itself: ($tType) > $tType).
% 9.74/2.75 thf(type_def_7, type, int: $tType).
% 9.74/2.75 thf(type_def_8, type, coinductive_llist: ($tType) > $tType).
% 9.74/2.75 thf(type_def_9, type, nat: $tType).
% 9.74/2.75 thf(type_def_10, type, set: ($tType) > $tType).
% 9.74/2.75 thf(type_def_11, type, option: ($tType) > $tType).
% 9.74/2.75 thf(type_def_12, type, num: $tType).
% 9.74/2.75 thf(type_def_13, type, a: $tType).
% 9.74/2.75 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 9.74/2.75 thf(func_def_1, type, option: ($tType) > $tType).
% 9.74/2.75 thf(func_def_2, type, set: ($tType) > $tType).
% 9.74/2.75 thf(func_def_3, type, num: $tType).
% 9.74/2.75 thf(func_def_4, type, nat: $tType).
% 9.74/2.75 thf(func_def_5, type, int: $tType).
% 9.74/2.75 thf(func_def_6, type, itself: ($tType) > $tType).
% 9.74/2.75 thf(func_def_7, type, a: $tType).
% 9.74/2.75 thf(func_def_8, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_9, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_10, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_11, type, semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_12, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_13, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_14, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_15, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_16, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_17, type, archim1727834104eiling: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_18, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_19, type, archim1804426504_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_20, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_21, type, ordere236663937imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_22, type, ordere516151231imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 9.74/2.75 thf(func_def_23, type, archimedean_round: !>[X0: $tType]:(X0 > int)).
% 9.74/2.75 thf(func_def_24, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_25, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 9.74/2.75 thf(func_def_26, type, zero_zero: !>[X0: $tType]:(X0)).
% 9.74/2.75 thf(func_def_27, type, hilbert_GreatestM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 9.74/2.75 thf(func_def_28, type, hilbert_LeastM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 9.74/2.75 thf(func_def_29, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 9.74/2.75 thf(func_def_30, type, nat2: int > nat).
% 9.74/2.75 thf(func_def_31, type, lList2435255213lllsts: !>[X0: $tType]:(set @ X0 > set @ coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_32, type, lList22096119349pslsts: !>[X0: $tType]:(set @ X0 > set @ coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_33, type, lList21612149805nflsts: !>[X0: $tType]:(set @ X0 > set @ coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_34, type, lList2370560421utlast: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_35, type, lList2228235758e_ll2f: !>[X0: $tType]:(coinductive_llist @ X0 > nat > option @ X0)).
% 9.74/2.75 thf(func_def_36, type, lList21232602520length: !>[X0: $tType]:(coinductive_llist @ X0 > nat)).
% 9.74/2.75 thf(func_def_37, type, lList22119844313_ltake: !>[X0: $tType]:(coinductive_llist @ X0 > nat > coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_38, type, lList21148268032oslsts: !>[X0: $tType]:(set @ X0 > set @ coinductive_llist @ X0)).
% 9.74/2.75 thf(func_def_39, type, suc: nat > nat).
% 9.74/2.75 thf(func_def_40, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 9.74/2.75 thf(func_def_41, type, semiri532925092at_aux: !>[X0: $tType]:((X0 > X0) > nat > X0 > X0)).
% 9.74/2.75 thf(func_def_42, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 9.74/2.75 thf(func_def_43, type, nat_tr1645093318rphism: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > $o)).
% 9.74/2.75 thf(func_def_44, type, nat_tsub: int > int > int).
% 9.74/2.75 thf(func_def_45, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 9.74/2.75 thf(func_def_46, type, bind: !>[X0: $tType, X1: $tType]:(option @ X0 > (X0 > option @ X1) > option @ X1)).
% 9.74/2.75 thf(func_def_47, type, combine_options: !>[X0: $tType]:((X0 > X0 > X0) > option @ X0 > option @ X0 > option @ X0)).
% 9.74/2.75 thf(func_def_48, type, is_none: !>[X0: $tType]:(option @ X0 > $o)).
% 9.74/2.75 thf(func_def_49, type, none: !>[X0: $tType]:(option @ X0)).
% 9.74/2.75 thf(func_def_50, type, some: !>[X0: $tType]:(X0 > option @ X0)).
% 9.74/2.75 thf(func_def_51, type, case_option: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > option @ X1 > X0)).
% 9.74/2.75 thf(func_def_52, type, pred_option: !>[X0: $tType]:((X0 > $o) > option @ X0 > $o)).
% 9.74/2.75 thf(func_def_53, type, rec_option: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > option @ X1 > X0)).
% 9.74/2.75 thf(func_def_54, type, set_option: !>[X0: $tType]:(option @ X0 > set @ X0)).
% 9.74/2.75 thf(func_def_55, type, size_option: !>[X0: $tType]:((X0 > nat) > option @ X0 > nat)).
% 9.74/2.75 thf(func_def_56, type, the: !>[X0: $tType]:(option @ X0 > X0)).
% 9.74/2.75 thf(func_def_57, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 9.74/2.75 thf(func_def_58, type, type2: !>[X0: $tType]:(itself @ X0)).
% 9.74/2.75 thf(func_def_59, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 9.74/2.75 thf(func_def_60, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 9.74/2.75 thf(func_def_62, type, i: nat).
% 9.74/2.75 thf(func_def_63, type, j: nat).
% 9.74/2.75 thf(func_def_64, type, t: coinductive_llist @ a).
% 9.74/2.75 thf(func_def_65, type, x: a).
% 9.74/2.75 thf(func_def_79, type, sP0: !>[X0: $tType, X1: $tType]:((option @ X1 > option @ X0 > $o) > option @ X1 > option @ X0 > $o)).
% 9.74/2.75 thf(func_def_80, type, sK1: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > X0)).
% 9.74/2.75 thf(func_def_81, type, sK2: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat)).
% 9.74/2.75 thf(func_def_82, type, sK3: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat > X0)).
% 9.74/2.75 thf(func_def_83, type, sK4: (nat > $o) > nat).
% 9.74/2.75 thf(func_def_84, type, sK5: (nat > $o) > nat).
% 9.74/2.75 thf(func_def_85, type, sK6: (nat > $o) > nat).
% 9.74/2.75 thf(func_def_86, type, sK7: !>[X0: $tType, X1: $tType]:((option @ X1 > option @ X0 > $o) > option @ X1 > option @ X0 > X1)).
% 9.74/2.75 thf(func_def_87, type, sK8: !>[X0: $tType, X1: $tType]:((option @ X1 > option @ X0 > $o) > option @ X1 > option @ X0 > X0)).
% 9.74/2.75 thf(func_def_88, type, sK9: (nat > $o) > nat).
% 9.74/2.75 thf(func_def_89, type, sK10: nat > nat).
% 9.74/2.75 thf(func_def_90, type, sK11: !>[X0: $tType]:(option @ X0 > X0)).
% 9.74/2.75 thf(func_def_91, type, sK12: !>[X0: $tType]:(nat > coinductive_llist @ X0 > X0)).
% 9.74/2.75 thf(func_def_92, type, sK13: !>[X0: $tType]:(option @ X0 > X0)).
% 9.74/2.75 thf(func_def_93, type, sK14: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 9.74/2.75 thf(func_def_94, type, sK15: nat > nat).
% 9.74/2.75 thf(func_def_95, type, sK16: !>[X0: $tType]:(option @ X0 > X0)).
% 9.74/2.75 thf(func_def_96, type, sK17: !>[X0: $tType]:((option @ X0 > $o) > X0)).
% 9.74/2.75 thf(func_def_97, type, sK18: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 9.74/2.75 thf(func_def_98, type, sK19: nat > nat).
% 9.74/2.75 thf(func_def_99, type, sK20: (nat > nat > $o) > nat).
% 9.74/2.75 thf(func_def_100, type, sK21: (nat > nat > $o) > nat).
% 9.74/2.75 thf(func_def_101, type, sK22: (nat > nat > $o) > nat).
% 9.74/2.75 thf(func_def_102, type, sK23: (nat > nat > $o) > nat).
% 9.74/2.75 thf(func_def_103, type, sK24: (nat > $o) > nat).
% 9.74/2.75 thf(func_def_104, type, sK25: nat > nat).
% 9.74/2.75 thf(func_def_105, type, sK26: !>[X0: $tType]:(nat > coinductive_llist @ X0 > X0)).
% 9.74/2.75 thf(func_def_108, type, ph28: !>[X0: $tType]:(X0)).
% 9.74/2.75 thf(f2821,plain,(
% 9.74/2.75 $false),
% 9.74/2.75 inference(avatar_sat_refutation,[status(thm)],[f991,f1001,f1006,f1010,f2078,f2820])).
% 9.74/2.75 thf(f2820,plain,(
% 9.74/2.75 ~spl27_1 | ~spl27_3 | ~spl27_39),
% 9.74/2.75 inference(avatar_contradiction_clause,[status(thm)],[f2819])).
% 9.74/2.75 thf(f2819,plain,(
% 9.74/2.75 $false | (~spl27_1 | ~spl27_3 | ~spl27_39)),
% 9.74/2.75 inference(subsumption_resolution,[status(thm)],[f2818,f990])).
% 9.74/2.75 thf(f990,plain,(
% 9.74/2.75 ((ord_less_eq @ nat @ j @ i) = $true) | ~spl27_1),
% 9.74/2.75 inference(avatar_component_clause,[status(thm)],[f988])).
% 9.74/2.75 thf(f988,definition,(
% 9.74/2.75 spl27_1 <=> ((ord_less_eq @ nat @ j @ i) = $true)),
% 9.74/2.75 introduced(definition,[new_symbols(naming,[spl27_1])],[avatar_definition])).
% 9.74/2.75 thf(f2818,plain,(
% 9.74/2.75 ((ord_less_eq @ nat @ j @ i) != $true) | (~spl27_3 | ~spl27_39)),
% 9.74/2.75 inference(trivial_inequality_removal,[status(thm)],[f2813])).
% 9.74/2.75 thf(f2813,plain,(
% 9.74/2.75 ((ord_less_eq @ nat @ j @ i) != $true) | (none @ a != none @ a) | (~spl27_3 | ~spl27_39)),
% 9.74/2.75 inference(superposition,[status(thm)],[f1685,f2077])).
% 9.74/2.75 thf(f2077,plain,(
% 9.74/2.75 ((lList2228235758e_ll2f @ a @ t @ j) = none @ a) | ~spl27_39),
% 9.74/2.75 inference(avatar_component_clause,[status(thm)],[f2075])).
% 9.74/2.75 thf(f2075,definition,(
% 9.74/2.75 spl27_39 <=> ((lList2228235758e_ll2f @ a @ t @ j) = none @ a)),
% 9.74/2.75 introduced(definition,[new_symbols(naming,[spl27_39])],[avatar_definition])).
% 9.74/2.75 thf(f1685,plain,(
% 9.74/2.75 ( ! [X0 : nat] : ((none @ a != (lList2228235758e_ll2f @ a @ t @ X0)) | ((ord_less_eq @ nat @ X0 @ i) != $true)) ) | ~spl27_3),
% 9.74/2.75 inference(subsumption_resolution,[status(thm)],[f1683,f936])).
% 9.74/2.75 thf(f936,plain,(
% 9.74/2.75 ( ! [X0 : $tType,X1 : X0] : ((none @ X0 != (some @ X0 @ X1))) )),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f788])).
% 9.74/2.75 thf(f788,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : X0] : (none @ X0 != (some @ X0 @ X1))),
% 9.74/2.75 inference(rectify,[status(thm)],[f95])).
% 9.74/2.75 thf(f95,axiom,(
% 9.74/2.75 ! [X0 : $tType,X4 : X0] : ((some @ X0 @ X4) != none @ X0)),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',fact_94_option_Odistinct_I1_J)).
% 9.74/2.75 thf(f1683,plain,(
% 9.74/2.75 ( ! [X0 : nat] : (((ord_less_eq @ nat @ X0 @ i) != $true) | ((some @ a @ x) = none @ a) | (none @ a != (lList2228235758e_ll2f @ a @ t @ X0))) ) | ~spl27_3),
% 9.74/2.75 inference(superposition,[status(thm)],[f1000,f948])).
% 9.74/2.75 thf(f948,plain,(
% 9.74/2.75 ( ! [X0 : $tType,X2 : nat,X3 : nat,X1 : coinductive_llist @ X0] : ((none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X3)) | ((ord_less_eq @ nat @ X2 @ X3) != $true) | (none @ X0 != (lList2228235758e_ll2f @ X0 @ X1 @ X2))) )),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f800])).
% 9.74/2.75 thf(f800,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : nat,X3 : nat] : (((ord_less_eq @ nat @ X2 @ X3) != $true) | (none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X3)) | (none @ X0 != (lList2228235758e_ll2f @ X0 @ X1 @ X2)))),
% 9.74/2.75 inference(flattening,[status(thm)],[f799])).
% 9.74/2.75 thf(f799,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : nat,X3 : nat] : (((none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X3)) | ((ord_less_eq @ nat @ X2 @ X3) != $true)) | (none @ X0 != (lList2228235758e_ll2f @ X0 @ X1 @ X2)))),
% 9.74/2.75 inference(ennf_transformation,[status(thm)],[f386])).
% 9.74/2.75 thf(f386,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : nat,X3 : nat] : ((none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X2)) => (((ord_less_eq @ nat @ X2 @ X3) = $true) => (none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X3))))),
% 9.74/2.75 inference(fool_elimination,[status(thm)],[f385])).
% 9.74/2.75 thf(f385,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : nat,X3 : nat] : ((none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X2)) => ((ord_less_eq @ nat @ X2 @ X3) => (none @ X0 = (lList2228235758e_ll2f @ X0 @ X1 @ X3))))),
% 9.74/2.75 inference(rectify,[status(thm)],[f57])).
% 9.74/2.75 thf(f57,axiom,(
% 9.74/2.75 ! [X0 : $tType,X31 : coinductive_llist @ X0,X9 : nat,X8 : nat] : (((lList2228235758e_ll2f @ X0 @ X31 @ X9) = none @ X0) => ((ord_less_eq @ nat @ X9 @ X8) => ((lList2228235758e_ll2f @ X0 @ X31 @ X8) = none @ X0)))),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',fact_56_ll2f__None__le)).
% 9.74/2.75 thf(f1000,plain,(
% 9.74/2.75 ((lList2228235758e_ll2f @ a @ t @ i) = (some @ a @ x)) | ~spl27_3),
% 9.74/2.75 inference(avatar_component_clause,[status(thm)],[f998])).
% 9.74/2.75 thf(f998,definition,(
% 9.74/2.75 spl27_3 <=> ((lList2228235758e_ll2f @ a @ t @ i) = (some @ a @ x))),
% 9.74/2.75 introduced(definition,[new_symbols(naming,[spl27_3])],[avatar_definition])).
% 9.74/2.75 thf(f2078,plain,(
% 9.74/2.75 spl27_39 | ~spl27_5),
% 9.74/2.75 inference(avatar_split_clause,[status(thm)],[f2073,f1008,f2075])).
% 9.74/2.75 thf(f1008,definition,(
% 9.74/2.75 spl27_5 <=> ! [X0 : a] : ((lList2228235758e_ll2f @ a @ t @ j) != (some @ a @ X0))),
% 9.74/2.75 introduced(definition,[new_symbols(naming,[spl27_5])],[avatar_definition])).
% 9.74/2.75 thf(f2073,plain,(
% 9.74/2.75 ((lList2228235758e_ll2f @ a @ t @ j) = none @ a) | ~spl27_5),
% 9.74/2.75 inference(equality_resolution,[status(thm)],[f1075])).
% 9.74/2.75 thf(f1075,plain,(
% 9.74/2.75 ( ! [X0 : option @ a] : (((lList2228235758e_ll2f @ a @ t @ j) != X0) | (none @ a = X0)) ) | ~spl27_5),
% 9.74/2.75 inference(superposition,[status(thm)],[f1009,f946])).
% 9.74/2.75 thf(f946,plain,(
% 9.74/2.75 ( ! [X0 : $tType,X1 : option @ X0] : (((some @ X0 @ (sK16 @ X0 @ X1)) = X1) | (none @ X0 = X1)) )),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f872])).
% 9.74/2.75 thf(f872,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : option @ X0] : (((none @ X0 = X1) | ((some @ X0 @ (sK16 @ X0 @ X1)) = X1)) & (! [X3 : X0] : ((some @ X0 @ X3) != X1) | (none @ X0 != X1)))),
% 9.74/2.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK16])],[f870,f871])).
% 9.74/2.75 thf(f871,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : option @ X0] : (? [X2 : X0] : ((some @ X0 @ X2) = X1) => ((some @ X0 @ (sK16 @ X0 @ X1)) = X1))),
% 9.74/2.75 introduced(definition,[],[choice_axiom])).
% 9.74/2.75 thf(f870,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : option @ X0] : (((none @ X0 = X1) | ? [X2 : X0] : ((some @ X0 @ X2) = X1)) & (! [X3 : X0] : ((some @ X0 @ X3) != X1) | (none @ X0 != X1)))),
% 9.74/2.75 inference(rectify,[status(thm)],[f869])).
% 9.74/2.75 thf(f869,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : option @ X0] : (((none @ X0 = X1) | ? [X2 : X0] : ((some @ X0 @ X2) = X1)) & (! [X2 : X0] : ((some @ X0 @ X2) != X1) | (none @ X0 != X1)))),
% 9.74/2.75 inference(nnf_transformation,[status(thm)],[f696])).
% 9.74/2.75 thf(f696,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : option @ X0] : ((none @ X0 = X1) <=> ! [X2 : X0] : ((some @ X0 @ X2) != X1))),
% 9.74/2.75 inference(fool_elimination,[status(thm)],[f695])).
% 9.74/2.75 thf(f695,plain,(
% 9.74/2.75 ! [X0 : $tType,X1 : option @ X0] : ((none @ X0 = X1) = ! [X2 : X0] : ((some @ X0 @ X2) != X1))),
% 9.74/2.75 inference(rectify,[status(thm)],[f65])).
% 9.74/2.75 thf(f65,axiom,(
% 9.74/2.75 ! [X0 : $tType,X6 : option @ X0] : ((none @ X0 = X6) = ! [X28 : X0] : ((some @ X0 @ X28) != X6))),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',fact_64_not__Some__eq)).
% 9.74/2.75 thf(f1009,plain,(
% 9.74/2.75 ( ! [X0 : a] : (((lList2228235758e_ll2f @ a @ t @ j) != (some @ a @ X0))) ) | ~spl27_5),
% 9.74/2.75 inference(avatar_component_clause,[status(thm)],[f1008])).
% 9.74/2.75 thf(f1010,plain,(
% 9.74/2.75 spl27_5 | spl27_4),
% 9.74/2.75 inference(avatar_split_clause,[status(thm)],[f909,f1003,f1008])).
% 9.74/2.75 thf(f1003,definition,(
% 9.74/2.75 spl27_4 <=> (q = $true)),
% 9.74/2.75 introduced(definition,[new_symbols(naming,[spl27_4])],[avatar_definition])).
% 9.74/2.75 thf(f909,plain,(
% 9.74/2.75 ( ! [X0 : a] : (((lList2228235758e_ll2f @ a @ t @ j) != (some @ a @ X0)) | (q = $true)) )),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f791])).
% 9.74/2.75 thf(f791,plain,(
% 9.74/2.75 ! [X0 : a] : (((lList2228235758e_ll2f @ a @ t @ j) != (some @ a @ X0)) | (q = $true))),
% 9.74/2.75 inference(ennf_transformation,[status(thm)],[f474])).
% 9.74/2.75 thf(f474,plain,(
% 9.74/2.75 ! [X0 : a] : (((lList2228235758e_ll2f @ a @ t @ j) = (some @ a @ X0)) => (q = $true))),
% 9.74/2.75 inference(fool_elimination,[status(thm)],[f473])).
% 9.74/2.75 thf(f473,plain,(
% 9.74/2.75 ! [X0 : a] : (((lList2228235758e_ll2f @ a @ t @ j) = (some @ a @ X0)) => q)),
% 9.74/2.75 inference(rectify,[status(thm)],[f1])).
% 9.74/2.75 thf(f1,axiom,(
% 9.74/2.75 ! [X3 : a] : (((lList2228235758e_ll2f @ a @ t @ j) = (some @ a @ X3)) => q)),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',fact_0_H)).
% 9.74/2.75 thf(f1006,plain,(
% 9.74/2.75 ~spl27_4),
% 9.74/2.75 inference(avatar_split_clause,[status(thm)],[f941,f1003])).
% 9.74/2.75 thf(f941,plain,(
% 9.74/2.75 (q != $true)),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f763])).
% 9.74/2.75 thf(f763,plain,(
% 9.74/2.75 (q != $true)),
% 9.74/2.75 inference(flattening,[status(thm)],[f678])).
% 9.74/2.75 thf(f678,plain,(
% 9.74/2.75 ~(q = $true)),
% 9.74/2.75 inference(fool_elimination,[status(thm)],[f677])).
% 9.74/2.75 thf(f677,plain,(
% 9.74/2.75 ~q),
% 9.74/2.75 inference(rectify,[status(thm)],[f295])).
% 9.74/2.75 thf(f295,negated_conjecture,(
% 9.74/2.75 ~q),
% 9.74/2.75 inference(negated_conjecture,[status(cth)],[f294])).
% 9.74/2.75 thf(f294,conjecture,(
% 9.74/2.75 q),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',conj_0)).
% 9.74/2.75 thf(f1001,plain,(
% 9.74/2.75 spl27_3),
% 9.74/2.75 inference(avatar_split_clause,[status(thm)],[f933,f998])).
% 9.74/2.75 thf(f933,plain,(
% 9.74/2.75 ((lList2228235758e_ll2f @ a @ t @ i) = (some @ a @ x))),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f3])).
% 9.74/2.75 thf(f3,axiom,(
% 9.74/2.75 ((lList2228235758e_ll2f @ a @ t @ i) = (some @ a @ x))),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',fact_2_tisome)).
% 9.74/2.75 thf(f991,plain,(
% 9.74/2.75 spl27_1),
% 9.74/2.75 inference(avatar_split_clause,[status(thm)],[f971,f988])).
% 9.74/2.75 thf(f971,plain,(
% 9.74/2.75 ((ord_less_eq @ nat @ j @ i) = $true)),
% 9.74/2.75 inference(cnf_transformation,[status(thm)],[f662])).
% 9.74/2.75 thf(f662,plain,(
% 9.74/2.75 ((ord_less_eq @ nat @ j @ i) = $true)),
% 9.74/2.75 inference(fool_elimination,[status(thm)],[f661])).
% 9.74/2.75 thf(f661,plain,(
% 9.74/2.75 (ord_less_eq @ nat @ j @ i)),
% 9.74/2.75 inference(rectify,[status(thm)],[f2])).
% 9.74/2.75 thf(f2,axiom,(
% 9.74/2.75 (ord_less_eq @ nat @ j @ i)),
% 9.74/2.75 file('/export/starexec/sandbox/tmp/tmp.75MMyhqhBz/DTF2THF_12325.p',fact_1_jlei)).
% 9.74/2.75 % SZS output end Proof for DTF2THF_12325
% 9.74/2.75 % (12516)------------------------------
% 9.74/2.75 % (12516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.74/2.75 % (12516)Termination reason: Refutation
% 9.74/2.75
% 9.74/2.75 % (12516)Memory used [KB]: 7547
% 9.74/2.75 % (12516)Time elapsed: 0.301 s
% 9.74/2.75 % (12516)Instructions burned: 591 (million)
% 9.74/2.75 % (12516)------------------------------
% 9.74/2.75 % (12516)------------------------------
% 9.74/2.75 % (12429)Success in time 1.274 s
%------------------------------------------------------------------------------