%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT197^1 : 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 : n007.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 08:45:06 AM UTC 2026
% Result : Theorem 3.39s 1.94s
% Output : Refutation 3.39s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT197^1 : TPTP v9.2.1. Released v7.0.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.32 % Computer : n007.cluster.edu
% 0.16/0.32 % Model : x86_64 x86_64
% 0.16/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.32 % Memory : 8042.1875MB
% 0.16/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.32 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 06:09:53 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : DAT197^1 : TPTP v9.2.1. Released v7.0.0.
% 0.19/0.45 % Domain : Data Structures
% 0.19/0.45 % Problem : Lazy list mirror 137
% 0.19/0.45 % Version : [Bla16] axioms : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.19/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.19/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Bla16]
% 0.19/0.45 % Names : lmirror__137.p [Bla16]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.33 v9.1.0, 0.00 v8.1.0, 0.25 v7.5.0, 0.33 v7.3.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.19/0.45 % Syntax : Number of formulae : 353 ( 131 unt; 67 typ; 0 def)
% 0.19/0.45 % Number of atoms : 806 ( 309 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 15 ( 2 avg)
% 0.19/0.45 % Number of connectives : 4610 ( 27 ~; 6 |; 47 &;4261 @)
% 0.19/0.45 % ( 0 <=>; 269 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 25 ( 9 avg)
% 0.19/0.45 % Number of types : 6 ( 5 usr)
% 0.19/0.45 % Number of type conns : 918 ( 918 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 65 ( 62 usr; 6 con; 0-8 aty)
% 0.19/0.45 % Number of variables : 1359 ( 122 ^;1152 !; 20 ?;1359 :)
% 0.19/0.45 % ( 65 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.45 % SPC : TH1_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.19/0.45 % 2016-07-13 14:41:49.835
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 %----Could-be-implicit typings (8)
% 0.19/0.45 thf(ty_t_Coinductive__List_Ollist,type,
% 0.19/0.45 coinductive_llist: $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Code__Numeral_Onatural,type,
% 0.19/0.45 code_natural: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Extended__Nat_Oenat,type,
% 0.19/0.45 extended_enat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Set_Oset,type,
% 0.19/0.45 set: $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Nat_Onat,type,
% 0.19/0.45 nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_itself,type,
% 0.19/0.45 itself: $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_b,type,
% 0.19/0.45 b: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_a,type,
% 0.19/0.45 a: $tType ).
% 0.19/0.45
% 0.19/0.45 %----Explicit typings (59)
% 0.19/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.19/0.45 zero:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.45 group_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Omonoid__add,type,
% 0.19/0.45 monoid_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.19/0.45 semigroup_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.19/0.45 comm_semiring_1:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.19/0.45 comm_monoid_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.19/0.45 ab_semigroup_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.19/0.45 cancel_semigroup_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.19/0.45 cancel1352612707id_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.19/0.45 linord219039673up_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.19/0.45 ordere779506340up_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.45 canoni770627133id_add:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 0.19/0.45 semiri456707255roduct:
% 0.19/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.19/0.45 bNF_rel_fun:
% 0.19/0.45 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 0.19/0.45 code_case_natural:
% 0.19/0.45 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 0.19/0.45 code_rec_natural:
% 0.19/0.45 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
% 0.19/0.45 code_size_natural: code_natural > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 0.19/0.45 coindu328551480prefix:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ogen__llength,type,
% 0.19/0.45 coindu246717305length:
% 0.19/0.45 !>[A: $tType] : ( nat > ( coinductive_llist @ A ) > extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Oiterates,type,
% 0.19/0.45 coinductive_iterates:
% 0.19/0.45 !>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Olappend,type,
% 0.19/0.45 coinductive_lappend:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Oldrop,type,
% 0.19/0.45 coinductive_ldrop:
% 0.19/0.45 !>[A: $tType] : ( extended_enat > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_OldropWhile,type,
% 0.19/0.45 coindu218763757pWhile:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Olfilter,type,
% 0.19/0.45 coinductive_lfilter:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.19/0.45 coinductive_lfinite:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollast,type,
% 0.19/0.45 coinductive_llast:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollcp,type,
% 0.19/0.45 coinductive_llcp:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollength,type,
% 0.19/0.45 coinductive_llength:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.19/0.45 coinductive_LCons:
% 0.19/0.45 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 0.19/0.45 coindu1381640503_llist:
% 0.19/0.45 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 0.19/0.45 coindu1259883913_llist:
% 0.19/0.45 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_Ollist__all2,type,
% 0.19/0.45 coindu1486289336t_all2:
% 0.19/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_Olmap,type,
% 0.19/0.45 coinductive_lmap:
% 0.19/0.45 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.19/0.45 coinductive_lset:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
% 0.19/0.45 coindu543516966_llist:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Olmember,type,
% 0.19/0.45 coinductive_lmember:
% 0.19/0.45 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Olprefix,type,
% 0.19/0.45 coinductive_lprefix:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Oltake,type,
% 0.19/0.45 coinductive_ltake:
% 0.19/0.45 !>[A: $tType] : ( extended_enat > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 0.19/0.45 coindu501562517eWhile:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
% 0.19/0.45 coindu780009021istsum:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.19/0.45 coindu1441602521_llist:
% 0.19/0.45 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Omap__fun,type,
% 0.19/0.45 map_fun:
% 0.19/0.45 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.19/0.45 plus_plus:
% 0.19/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.45 zero_zero:
% 0.19/0.45 !>[A: $tType] : A ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_If,type,
% 0.19/0.45 if:
% 0.19/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
% 0.19/0.45 lMirro999291890or_aux:
% 0.19/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 0.19/0.45 case_nat:
% 0.19/0.45 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 0.19/0.45 rec_nat:
% 0.19/0.45 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.19/0.45 size_size:
% 0.19/0.45 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Pure_Otype,type,
% 0.19/0.45 type:
% 0.19/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Quotient_OBex1__rel,type,
% 0.19/0.45 bex1_rel:
% 0.19/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Quotient_OQuotient3,type,
% 0.19/0.45 quotient3:
% 0.19/0.45 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_OCollect,type,
% 0.19/0.45 collect:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member,type,
% 0.19/0.45 member:
% 0.19/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_P,type,
% 0.19/0.45 p: a > b > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_acc_Ha____,type,
% 0.19/0.45 acc_a: coinductive_llist @ b ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_acca____,type,
% 0.19/0.45 acca: coinductive_llist @ a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_xs_Ha____,type,
% 0.19/0.45 xs_a: coinductive_llist @ b ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_xsa____,type,
% 0.19/0.45 xsa: coinductive_llist @ a ).
% 0.19/0.45
% 0.19/0.45 %----Relevant facts (254)
% 0.19/0.45 thf(fact_0_local_OLNil_I2_J,axiom,
% 0.19/0.45 coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a ).
% 0.19/0.45
% 0.19/0.45 % local.LNil(2)
% 0.19/0.45 thf(fact_1_local_OLNil_I3_J,axiom,
% 0.19/0.45 coinductive_lfinite @ a @ acca ).
% 0.19/0.45
% 0.19/0.45 % local.LNil(3)
% 0.19/0.45 thf(fact_2_calculation,axiom,
% 0.19/0.45 ( ( coinductive_llength @ a @ ( lMirro999291890or_aux @ a @ acca @ xsa ) )
% 0.19/0.45 = ( coinductive_llength @ b @ ( lMirro999291890or_aux @ b @ acc_a @ xs_a ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % calculation
% 0.19/0.45 thf(fact_3_local_OLNil_I1_J,axiom,
% 0.19/0.45 coindu1486289336t_all2 @ a @ b @ p @ ( lMirro999291890or_aux @ a @ acca @ xsa ) @ ( lMirro999291890or_aux @ b @ acc_a @ xs_a ) ).
% 0.19/0.45
% 0.19/0.45 % local.LNil(1)
% 0.19/0.45 thf(fact_4_llist__all2__llengthD,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ( coinductive_llength @ A @ Xs )
% 0.19/0.45 = ( coinductive_llength @ B @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_llengthD
% 0.19/0.45 thf(fact_5_llcp__same__conv__length,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_llcp @ A @ Xs @ Xs )
% 0.19/0.45 = ( coinductive_llength @ A @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % llcp_same_conv_length
% 0.19/0.45 thf(fact_6_llength__lmap,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_llength @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 0.19/0.45 = ( coinductive_llength @ B @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % llength_lmap
% 0.19/0.45 thf(fact_7_lprefix__llength__eq__imp__eq,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( ( ( coinductive_llength @ A @ Xs )
% 0.19/0.45 = ( coinductive_llength @ A @ Ys ) )
% 0.19/0.45 => ( Xs = Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_llength_eq_imp_eq
% 0.19/0.45 thf(fact_8_llength__ltakeWhile__all,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.19/0.45 = ( coinductive_llength @ A @ Xs ) )
% 0.19/0.45 = ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.19/0.45 = Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % llength_ltakeWhile_all
% 0.19/0.45 thf(fact_9_llist__all2__llength__ltakeWhileD,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( P @ X @ Y )
% 0.19/0.45 => ( ( Q1 @ X )
% 0.19/0.45 = ( Q2 @ Y ) ) )
% 0.19/0.45 => ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ Q1 @ Xs ) )
% 0.19/0.45 = ( coinductive_llength @ B @ ( coindu501562517eWhile @ B @ Q2 @ Ys ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_llength_ltakeWhileD
% 0.19/0.45 thf(fact_10_llength__code,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coinductive_llength @ A )
% 0.19/0.45 = ( coindu246717305length @ A @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llength_code
% 0.19/0.45 thf(fact_11_llength__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ extended_enat @ extended_enat @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.19/0.45 @ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
% 0.19/0.45 @ ( coinductive_llength @ A )
% 0.19/0.45 @ ( coinductive_llength @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llength_transfer
% 0.19/0.45 thf(fact_12_lprefix__refl,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_refl
% 0.19/0.45 thf(fact_13_llist_Oleq__refl,axiom,
% 0.19/0.45 ! [A: $tType,X2: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X2 @ X2 ) ).
% 0.19/0.45
% 0.19/0.45 % llist.leq_refl
% 0.19/0.45 thf(fact_14_lfinite__lmap,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 0.19/0.45 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_lmap
% 0.19/0.45 thf(fact_15_lfinite__lmirror__aux,axiom,
% 0.19/0.45 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 0.19/0.45 = ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 & ( coinductive_lfinite @ A @ Acc ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_lmirror_aux
% 0.19/0.45 thf(fact_16_llcp__lprefix1,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( ( coinductive_llcp @ A @ Xs @ Ys )
% 0.19/0.45 = ( coinductive_llength @ A @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llcp_lprefix1
% 0.19/0.45 thf(fact_17_llcp__lprefix2,axiom,
% 0.19/0.45 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 0.19/0.45 => ( ( coinductive_llcp @ A @ Xs @ Ys )
% 0.19/0.45 = ( coinductive_llength @ A @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llcp_lprefix2
% 0.19/0.45 thf(fact_18_llcp__commute,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coinductive_llcp @ A )
% 0.19/0.45 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] : ( coinductive_llcp @ A @ Ys2 @ Xs2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llcp_commute
% 0.19/0.45 thf(fact_19_lmap__lprefix,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,F: A > B] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( coinductive_lprefix @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) @ ( coinductive_lmap @ A @ B @ F @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmap_lprefix
% 0.19/0.45 thf(fact_20_lmap__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ ( coindu1486289336t_all2 @ A @ C @ A2 ) @ ( coindu1486289336t_all2 @ B @ D @ B2 ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ C @ D ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmap_transfer
% 0.19/0.45 thf(fact_21_lprefix__trans,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.19/0.45 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_trans
% 0.19/0.45 thf(fact_22_llist__all2__rsp,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T2: B > B > $o,X2: coinductive_llist @ A,Y3: coinductive_llist @ B,A3: coinductive_llist @ A,B3: coinductive_llist @ B] :
% 0.19/0.45 ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( R @ X @ Y )
% 0.19/0.45 => ! [A4: A,B4: B] :
% 0.19/0.45 ( ( R @ A4 @ B4 )
% 0.19/0.45 => ( ( S @ X @ A4 )
% 0.19/0.45 = ( T2 @ Y @ B4 ) ) ) )
% 0.19/0.45 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
% 0.19/0.45 => ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
% 0.19/0.45 => ( ( coindu1486289336t_all2 @ A @ A @ S @ X2 @ A3 )
% 0.19/0.45 = ( coindu1486289336t_all2 @ B @ B @ T2 @ Y3 @ B3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_rsp
% 0.19/0.45 thf(fact_23_llist_Oleq__trans,axiom,
% 0.19/0.45 ! [A: $tType,X2: coinductive_llist @ A,Y3: coinductive_llist @ A,Z2: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ X2 @ Y3 )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Y3 @ Z2 )
% 0.19/0.45 => ( coinductive_lprefix @ A @ X2 @ Z2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.leq_trans
% 0.19/0.45 thf(fact_24_llist__all2__mono,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P2: A > B > $o] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( P @ X @ Y )
% 0.19/0.45 => ( P2 @ X @ Y ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_mono
% 0.19/0.45 thf(fact_25_lprefix__antisym,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 0.19/0.45 => ( Xs = Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_antisym
% 0.19/0.45 thf(fact_26_lprefix__lfiniteD,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( ( coinductive_lfinite @ A @ Ys )
% 0.19/0.45 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_lfiniteD
% 0.19/0.45 thf(fact_27_llist_Oleq__antisym,axiom,
% 0.19/0.45 ! [A: $tType,X2: coinductive_llist @ A,Y3: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ X2 @ Y3 )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Y3 @ X2 )
% 0.19/0.45 => ( X2 = Y3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.leq_antisym
% 0.19/0.45 thf(fact_28_lprefix__ltakeWhile,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ Xs ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_ltakeWhile
% 0.19/0.45 thf(fact_29_llist__all2__lfiniteD,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 = ( coinductive_lfinite @ B @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lfiniteD
% 0.19/0.45 thf(fact_30_lprefix__down__linear,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_down_linear
% 0.19/0.45 thf(fact_31_not__lfinite__lprefix__conv__eq,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 = ( Xs = Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_lfinite_lprefix_conv_eq
% 0.19/0.45 thf(fact_32_lmirror__aux__inf,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 0.19/0.45 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
% 0.19/0.45 = Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmirror_aux_inf
% 0.19/0.45 thf(fact_33_lmap__lmirror__aux,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,F: B > A,Acc: coinductive_llist @ B,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_lmap @ B @ A @ F @ ( lMirro999291890or_aux @ B @ Acc @ Xs ) )
% 0.19/0.45 = ( lMirro999291890or_aux @ A @ ( coinductive_lmap @ B @ A @ F @ Acc ) @ ( coinductive_lmap @ B @ A @ F @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmap_lmirror_aux
% 0.19/0.45 thf(fact_34_llist_Orel__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ A
% 0.19/0.45 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.19/0.45 = ( ^ [Y2: coinductive_llist @ A,Z: coinductive_llist @ A] : ( Y2 = Z ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_eq
% 0.19/0.45 thf(fact_35_llist_Orel__refl,axiom,
% 0.19/0.45 ! [B: $tType,Ra: B > B > $o,X2: coinductive_llist @ B] :
% 0.19/0.45 ( ! [X: B] : ( Ra @ X @ X )
% 0.19/0.45 => ( coindu1486289336t_all2 @ B @ B @ Ra @ X2 @ X2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_refl
% 0.19/0.45 thf(fact_36_llist__all2__lmirror__aux,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Acc: coinductive_llist @ A,Acc2: coinductive_llist @ B,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Acc @ Acc2 )
% 0.19/0.45 => ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Xs3 )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) @ ( lMirro999291890or_aux @ B @ Acc2 @ Xs3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lmirror_aux
% 0.19/0.45 thf(fact_37_llist_Omap__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb: A > E > $o,Sd: B > F2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F2 ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ E ) > ( coinductive_llist @ F2 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F2 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ F2 ) @ ( coindu1486289336t_all2 @ A @ E @ Rb ) @ ( coindu1486289336t_all2 @ B @ F2 @ Sd ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ E @ F2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.map_transfer
% 0.19/0.45 thf(fact_38_rel__funI,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 0.19/0.45 ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( A2 @ X @ Y )
% 0.19/0.45 => ( B2 @ ( F @ X ) @ ( G @ Y ) ) )
% 0.19/0.45 => ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_funI
% 0.19/0.45 thf(fact_39_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coinductive_lprefix @ A )
% 0.19/0.45 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_lfinite @ A @ Xs2 )
% 0.19/0.45 => ( coindu328551480prefix @ A @ Xs2 @ Ys2 ) )
% 0.19/0.45 & ( ~ ( coinductive_lfinite @ A @ Xs2 )
% 0.19/0.45 => ( Xs2 = Ys2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Coinductive_List.lprefix_nitpick_simps
% 0.19/0.45 thf(fact_40_llistsum__inf,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coindu780009021istsum @ A @ Xs )
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llistsum_inf
% 0.19/0.45 thf(fact_41_rel__funD,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
% 0.19/0.45 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 0.19/0.45 => ( ( A2 @ X2 @ Y3 )
% 0.19/0.45 => ( B2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_funD
% 0.19/0.45 thf(fact_42_let__rsp,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
% 0.19/0.45 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
% 0.19/0.45 @ ^ [S2: A,F3: A > C] : ( F3 @ S2 )
% 0.19/0.45 @ ^ [S2: B,F3: B > D] : ( F3 @ S2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % let_rsp
% 0.19/0.45 thf(fact_43_rel__funE,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
% 0.19/0.45 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 0.19/0.45 => ( ( A2 @ X2 @ Y3 )
% 0.19/0.45 => ( B2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_funE
% 0.19/0.45 thf(fact_44_mem__Collect__eq,axiom,
% 0.19/0.45 ! [A: $tType,A3: A,P: A > $o] :
% 0.19/0.45 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.19/0.45 = ( P @ A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % mem_Collect_eq
% 0.19/0.45 thf(fact_45_Collect__mem__eq,axiom,
% 0.19/0.45 ! [A: $tType,A2: set @ A] :
% 0.19/0.45 ( ( collect @ A
% 0.19/0.45 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 0.19/0.45 = A2 ) ).
% 0.19/0.45
% 0.19/0.45 % Collect_mem_eq
% 0.19/0.45 thf(fact_46_Collect__cong,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.45 ( ! [X: A] :
% 0.19/0.45 ( ( P @ X )
% 0.19/0.45 = ( Q @ X ) )
% 0.19/0.45 => ( ( collect @ A @ P )
% 0.19/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Collect_cong
% 0.19/0.45 thf(fact_47_ext,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.19/0.45 ( ! [X: A] :
% 0.19/0.45 ( ( F @ X )
% 0.19/0.45 = ( G @ X ) )
% 0.19/0.45 => ( F = G ) ) ).
% 0.19/0.45
% 0.19/0.45 % ext
% 0.19/0.45 thf(fact_48_apply__rsp_H,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
% 0.19/0.45 ( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
% 0.19/0.45 => ( ( R1 @ X2 @ Y3 )
% 0.19/0.45 => ( R2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % apply_rsp'
% 0.19/0.45 thf(fact_49_rel__funD2,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F: A > B,G: A > C,X2: A] :
% 0.19/0.45 ( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F @ G )
% 0.19/0.45 => ( ( A2 @ X2 @ X2 )
% 0.19/0.45 => ( B2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_funD2
% 0.19/0.45 thf(fact_50_rel__fun__mono,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F: A > C,G: B > D,Y4: A > B > $o,B2: C > D > $o] :
% 0.19/0.45 ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 0.19/0.45 => ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( Y4 @ X @ Y )
% 0.19/0.45 => ( X4 @ X @ Y ) )
% 0.19/0.45 => ( ! [X: C,Y: D] :
% 0.19/0.45 ( ( A2 @ X @ Y )
% 0.19/0.45 => ( B2 @ X @ Y ) )
% 0.19/0.45 => ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_fun_mono
% 0.19/0.45 thf(fact_51_rel__fun__mono_H,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,Y4: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 0.19/0.45 ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( Y4 @ X @ Y )
% 0.19/0.45 => ( X4 @ X @ Y ) )
% 0.19/0.45 => ( ! [X: C,Y: D] :
% 0.19/0.45 ( ( A2 @ X @ Y )
% 0.19/0.45 => ( B2 @ X @ Y ) )
% 0.19/0.45 => ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 0.19/0.45 => ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_fun_mono'
% 0.19/0.45 thf(fact_52_llist__all2__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) > $o )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o ) @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 0.19/0.45 @ ( coindu1486289336t_all2 @ A @ A )
% 0.19/0.45 @ ( coindu1486289336t_all2 @ B @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_transfer
% 0.19/0.45 thf(fact_53_llist_Orel__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) > $o )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.19/0.45 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ D ) > $o ) @ ( coindu1486289336t_all2 @ A @ C @ Sa )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ $o @ $o @ ( coindu1486289336t_all2 @ B @ D @ Sc )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 0.19/0.45 @ ( coindu1486289336t_all2 @ A @ B )
% 0.19/0.45 @ ( coindu1486289336t_all2 @ C @ D ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_transfer
% 0.19/0.45 thf(fact_54_ltakeWhile__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 0.19/0.45 @ ( coindu501562517eWhile @ A )
% 0.19/0.45 @ ( coindu501562517eWhile @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltakeWhile_transfer
% 0.19/0.45 thf(fact_55_lfinite__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 0.19/0.45 @ ( coinductive_lfinite @ A )
% 0.19/0.45 @ ( coinductive_lfinite @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_transfer
% 0.19/0.45 thf(fact_56_Coinductive__List_Ofinite__lprefix__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coindu328551480prefix @ A )
% 0.19/0.45 = ( coinductive_lprefix @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 % Coinductive_List.finite_lprefix_def
% 0.19/0.45 thf(fact_57_zero__reorient,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( zero @ A @ ( type @ A ) )
% 0.19/0.45 => ! [X2: A] :
% 0.19/0.45 ( ( ( zero_zero @ A )
% 0.19/0.45 = X2 )
% 0.19/0.45 = ( X2
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % zero_reorient
% 0.19/0.45 thf(fact_58_Let__transfer,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.19/0.45 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ B2 )
% 0.19/0.45 @ ^ [S2: A,F3: A > C] : ( F3 @ S2 )
% 0.19/0.45 @ ^ [S2: B,F3: B > D] : ( F3 @ S2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Let_transfer
% 0.19/0.45 thf(fact_59_rel__fun__def__butlast,axiom,
% 0.19/0.45 ! [B: $tType,D: $tType,C: $tType,E: $tType,F2: $tType,A: $tType,R: A > B > $o,S: C > E > $o,T2: D > F2 > $o,F: A > C > D,G: B > E > F2] :
% 0.19/0.45 ( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F2 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F2 @ S @ T2 ) @ F @ G )
% 0.19/0.45 = ( ! [X3: A,Y5: B] :
% 0.19/0.45 ( ( R @ X3 @ Y5 )
% 0.19/0.45 => ( bNF_rel_fun @ C @ E @ D @ F2 @ S @ T2 @ ( F @ X3 ) @ ( G @ Y5 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rel_fun_def_butlast
% 0.19/0.45 thf(fact_60_If__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 0.19/0.45 @ ( if @ A )
% 0.19/0.45 @ ( if @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % If_transfer
% 0.19/0.45 thf(fact_61_llist__corec__transfer,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > $o ) > ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > $o ) > ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > $o ) > ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > $o ) > ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 0.19/0.45 @ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) @ ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) ) ) ) )
% 0.19/0.45 @ ( coindu1259883913_llist @ A @ C )
% 0.19/0.45 @ ( coindu1259883913_llist @ B @ D ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_corec_transfer
% 0.19/0.45 thf(fact_62_llist_Ocorec__transfer,axiom,
% 0.19/0.45 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > B ) > ( D > $o ) > ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > $o ) > ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S @ R )
% 0.19/0.45 @ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( C > ( coinductive_llist @ A ) ) @ ( D > ( coinductive_llist @ B ) ) @ ( ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > D ) > D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ S @ ( coindu1486289336t_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( coinductive_llist @ A ) ) @ ( D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) @ ( bNF_rel_fun @ C @ D @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ S @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ) ) )
% 0.19/0.45 @ ( coindu1259883913_llist @ C @ A )
% 0.19/0.45 @ ( coindu1259883913_llist @ D @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.corec_transfer
% 0.19/0.45 thf(fact_63_unfold__llist__transfer,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) ) )
% 0.19/0.45 @ ( coindu1441602521_llist @ A @ C )
% 0.19/0.45 @ ( coindu1441602521_llist @ B @ D ) ) ).
% 0.19/0.45
% 0.19/0.45 % unfold_llist_transfer
% 0.19/0.45 thf(fact_64_iterates__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ A ) ) @ ( B > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ A2 @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_iterates @ A ) @ ( coinductive_iterates @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % iterates_transfer
% 0.19/0.45 thf(fact_65_case__llist__transfer,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,B2: A > B > $o,A2: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( ( C > ( coinductive_llist @ C ) > A ) > ( coinductive_llist @ C ) > A ) @ ( ( D > ( coinductive_llist @ D ) > B ) > ( coinductive_llist @ D ) > B ) @ B2 @ ( bNF_rel_fun @ ( C > ( coinductive_llist @ C ) > A ) @ ( D > ( coinductive_llist @ D ) > B ) @ ( ( coinductive_llist @ C ) > A ) @ ( ( coinductive_llist @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ ( ( coinductive_llist @ C ) > A ) @ ( ( coinductive_llist @ D ) > B ) @ A2 @ ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A @ B @ ( coindu1486289336t_all2 @ C @ D @ A2 ) @ B2 ) ) @ ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A @ B @ ( coindu1486289336t_all2 @ C @ D @ A2 ) @ B2 ) ) @ ( coindu1381640503_llist @ A @ C ) @ ( coindu1381640503_llist @ B @ D ) ) ).
% 0.19/0.45
% 0.19/0.45 % case_llist_transfer
% 0.19/0.45 thf(fact_66_llist_Ocase__transfer,axiom,
% 0.19/0.45 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( coinductive_llist @ A ) > C ) > ( coinductive_llist @ A ) > C ) @ ( ( B > ( coinductive_llist @ B ) > D ) > ( coinductive_llist @ B ) > D ) @ S @ ( bNF_rel_fun @ ( A > ( coinductive_llist @ A ) > C ) @ ( B > ( coinductive_llist @ B ) > D ) @ ( ( coinductive_llist @ A ) > C ) @ ( ( coinductive_llist @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > C ) @ ( ( coinductive_llist @ B ) > D ) @ R @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ D @ ( coindu1486289336t_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ D @ ( coindu1486289336t_all2 @ A @ B @ R ) @ S ) ) @ ( coindu1381640503_llist @ C @ A ) @ ( coindu1381640503_llist @ D @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.case_transfer
% 0.19/0.45 thf(fact_67_lfinite__iterates,axiom,
% 0.19/0.45 ! [A: $tType,F: A > A,X2: A] :
% 0.19/0.45 ~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F @ X2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_iterates
% 0.19/0.45 thf(fact_68_lmap__iterates,axiom,
% 0.19/0.45 ! [A: $tType,F: A > A,X2: A] :
% 0.19/0.45 ( ( coinductive_lmap @ A @ A @ F @ ( coinductive_iterates @ A @ F @ X2 ) )
% 0.19/0.45 = ( coinductive_iterates @ A @ F @ ( F @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmap_iterates
% 0.19/0.45 thf(fact_69_fun_Orel__transfer,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
% 0.19/0.45 @ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
% 0.19/0.45 @ ( bNF_rel_fun @ D @ D @ A @ C
% 0.19/0.45 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 0.19/0.45 @ Sa )
% 0.19/0.45 @ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
% 0.19/0.45 @ ( bNF_rel_fun @ D @ D @ B @ E
% 0.19/0.45 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 0.19/0.45 @ Sc )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 0.19/0.45 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.19/0.45 @ ^ [Y2: D,Z: D] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ D @ D @ C @ E
% 0.19/0.45 @ ^ [Y2: D,Z: D] : ( Y2 = Z ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fun.rel_transfer
% 0.19/0.45 thf(fact_70_ldropWhile__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 0.19/0.45 @ ( coindu218763757pWhile @ A )
% 0.19/0.45 @ ( coindu218763757pWhile @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldropWhile_transfer
% 0.19/0.45 thf(fact_71_map__fun__parametric,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,E: $tType,F2: $tType,H: $tType,G2: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G2 > $o,D2: F2 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F2 ) > ( B > E ) > A > F2 ) @ ( ( G2 > H ) > ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F2 ) @ ( G2 > H ) @ ( ( B > E ) > A > F2 ) @ ( ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ E @ G2 @ F2 @ H @ C2 @ D2 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G2 ) @ ( A > F2 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G2 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F2 @ H @ A2 @ D2 ) ) ) @ ( map_fun @ A @ B @ E @ F2 ) @ ( map_fun @ C @ D @ G2 @ H ) ) ).
% 0.19/0.45
% 0.19/0.45 % map_fun_parametric
% 0.19/0.45 thf(fact_72_lfilter__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 0.19/0.45 @ ( coinductive_lfilter @ A )
% 0.19/0.45 @ ( coinductive_lfilter @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_transfer
% 0.19/0.45 thf(fact_73_llist_Opred__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( coindu543516966_llist @ A )
% 0.19/0.45 @ ( coindu543516966_llist @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.pred_transfer
% 0.19/0.45 thf(fact_74_llist__all__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ( coindu543516966_llist @ A )
% 0.19/0.45 @ ( coindu543516966_llist @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all_transfer
% 0.19/0.45 thf(fact_75_if__rsp,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
% 0.19/0.45 @ ( if @ A )
% 0.19/0.45 @ ( if @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % if_rsp
% 0.19/0.45 thf(fact_76_lfilter__idem,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.19/0.45 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_idem
% 0.19/0.45 thf(fact_77_fun__quotient3,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs2: C > D,Rep2: D > C] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.19/0.45 => ( ( quotient3 @ C @ D @ R2 @ Abs2 @ Rep2 )
% 0.19/0.45 => ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs2 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fun_quotient3
% 0.19/0.45 thf(fact_78_let__prs,axiom,
% 0.19/0.45 ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.19/0.45 => ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
% 0.19/0.45 => ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep22 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs22 @ Rep12 ) @ Abs12 )
% 0.19/0.45 @ ^ [S2: C,F3: C > A] : ( F3 @ S2 ) )
% 0.19/0.45 = ( ^ [S2: D,F3: D > B] : ( F3 @ S2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % let_prs
% 0.19/0.45 thf(fact_79_cond__prs,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A3: $o,B3: B,C3: B] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.19/0.45 => ( ( A3
% 0.19/0.45 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.19/0.45 = B3 ) )
% 0.19/0.45 & ( ~ A3
% 0.19/0.45 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.19/0.45 = C3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % cond_prs
% 0.19/0.45 thf(fact_80_Quotient3I,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 0.19/0.45 ( ! [A4: A] :
% 0.19/0.45 ( ( Abs @ ( Rep @ A4 ) )
% 0.19/0.45 = A4 )
% 0.19/0.45 => ( ! [A4: A] : ( R @ ( Rep @ A4 ) @ ( Rep @ A4 ) )
% 0.19/0.45 => ( ! [R3: B,S3: B] :
% 0.19/0.45 ( ( R @ R3 @ S3 )
% 0.19/0.45 = ( ( R @ R3 @ R3 )
% 0.19/0.45 & ( R @ S3 @ S3 )
% 0.19/0.45 & ( ( Abs @ R3 )
% 0.19/0.45 = ( Abs @ S3 ) ) ) )
% 0.19/0.45 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3I
% 0.19/0.45 thf(fact_81_equals__rsp,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ Xa @ Xb )
% 0.19/0.45 => ( ( R @ Ya @ Yb )
% 0.19/0.45 => ( ( R @ Xa @ Ya )
% 0.19/0.45 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % equals_rsp
% 0.19/0.45 thf(fact_82_rep__abs__rsp,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ X1 @ X22 )
% 0.19/0.45 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rep_abs_rsp
% 0.19/0.45 thf(fact_83_Quotient3__def,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( quotient3 @ A @ B )
% 0.19/0.45 = ( ^ [R4: A > A > $o,Abs3: A > B,Rep3: B > A] :
% 0.19/0.45 ( ! [A5: B] :
% 0.19/0.45 ( ( Abs3 @ ( Rep3 @ A5 ) )
% 0.19/0.45 = A5 )
% 0.19/0.45 & ! [A5: B] : ( R4 @ ( Rep3 @ A5 ) @ ( Rep3 @ A5 ) )
% 0.19/0.45 & ! [R5: A,S2: A] :
% 0.19/0.45 ( ( R4 @ R5 @ S2 )
% 0.19/0.45 = ( ( R4 @ R5 @ R5 )
% 0.19/0.45 & ( R4 @ S2 @ S2 )
% 0.19/0.45 & ( ( Abs3 @ R5 )
% 0.19/0.45 = ( Abs3 @ S2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_def
% 0.19/0.45 thf(fact_84_Quotient3__rel,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( ( R @ R6 @ R6 )
% 0.19/0.45 & ( R @ S4 @ S4 )
% 0.19/0.45 & ( ( Abs @ R6 )
% 0.19/0.45 = ( Abs @ S4 ) ) )
% 0.19/0.45 = ( R @ R6 @ S4 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_rel
% 0.19/0.45 thf(fact_85_Quotient3__refl1,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ R6 @ S4 )
% 0.19/0.45 => ( R @ R6 @ R6 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_refl1
% 0.19/0.45 thf(fact_86_Quotient3__refl2,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ R6 @ S4 )
% 0.19/0.45 => ( R @ S4 @ S4 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_refl2
% 0.19/0.45 thf(fact_87_rep__abs__rsp__left,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ X1 @ X22 )
% 0.19/0.45 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % rep_abs_rsp_left
% 0.19/0.45 thf(fact_88_Quotient3__abs__rep,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( Abs @ ( Rep @ A3 ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_abs_rep
% 0.19/0.45 thf(fact_89_Quotient3__rel__abs,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ R6 @ S4 )
% 0.19/0.45 => ( ( Abs @ R6 )
% 0.19/0.45 = ( Abs @ S4 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_rel_abs
% 0.19/0.45 thf(fact_90_Quotient3__rel__rep,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B,B3: B] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 0.19/0.45 = ( A3 = B3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_rel_rep
% 0.19/0.45 thf(fact_91_Quotient3__rep__abs,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( R @ R6 @ R6 )
% 0.19/0.45 => ( R @ ( Rep @ ( Abs @ R6 ) ) @ R6 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_rep_abs
% 0.19/0.45 thf(fact_92_Quotient3__rep__reflp,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Quotient3_rep_reflp
% 0.19/0.45 thf(fact_93_lprefix__lfilterI,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( coinductive_lprefix @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_lfilterI
% 0.19/0.45 thf(fact_94_llist__all2__lfilterI,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( P @ X @ Y )
% 0.19/0.45 => ( ( Q1 @ X )
% 0.19/0.45 = ( Q2 @ Y ) ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lfilter @ A @ Q1 @ Xs ) @ ( coinductive_lfilter @ B @ Q2 @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lfilterI
% 0.19/0.45 thf(fact_95_apply__rspQ3_H_H,axiom,
% 0.19/0.45 ! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S: C > C > $o,F: A > C,X2: B] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S @ F @ F )
% 0.19/0.45 => ( S @ ( F @ ( Rep @ X2 ) ) @ ( F @ ( Rep @ X2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % apply_rspQ3''
% 0.19/0.45 thf(fact_96_apply__rspQ3,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F: A > C,G: A > C,X2: A,Y3: A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.19/0.45 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
% 0.19/0.45 => ( ( R1 @ X2 @ Y3 )
% 0.19/0.45 => ( R2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % apply_rspQ3
% 0.19/0.45 thf(fact_97_lfinite__lfilterI,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_lfilterI
% 0.19/0.45 thf(fact_98_llist__all2__ldropWhileI,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ! [X: A,Y: B] :
% 0.19/0.45 ( ( P @ X @ Y )
% 0.19/0.45 => ( ( Q1 @ X )
% 0.19/0.45 = ( Q2 @ Y ) ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coindu218763757pWhile @ A @ Q1 @ Xs ) @ ( coindu218763757pWhile @ B @ Q2 @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_ldropWhileI
% 0.19/0.45 thf(fact_99_quot__rel__rsp,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.19/0.45 => ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
% 0.19/0.45 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ R
% 0.19/0.45 @ R ) ) ).
% 0.19/0.45
% 0.19/0.45 % quot_rel_rsp
% 0.19/0.45 thf(fact_100_fun_Orel__refl,axiom,
% 0.19/0.45 ! [B: $tType,D: $tType,Ra: B > B > $o,X2: D > B] :
% 0.19/0.45 ( ! [X: B] : ( Ra @ X @ X )
% 0.19/0.45 => ( bNF_rel_fun @ D @ D @ B @ B
% 0.19/0.45 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 0.19/0.45 @ Ra
% 0.19/0.45 @ X2
% 0.19/0.45 @ X2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % fun.rel_refl
% 0.19/0.45 thf(fact_101_fun_Orel__eq,axiom,
% 0.19/0.45 ! [A: $tType,D: $tType] :
% 0.19/0.45 ( ( bNF_rel_fun @ D @ D @ A @ A
% 0.19/0.45 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 0.19/0.45 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.19/0.45 = ( ^ [Y2: D > A,Z: D > A] : ( Y2 = Z ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fun.rel_eq
% 0.19/0.45 thf(fact_102_map__fun__apply,axiom,
% 0.19/0.45 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 0.19/0.45 ( ( map_fun @ B @ C @ D @ A )
% 0.19/0.45 = ( ^ [F3: B > C,G3: D > A,H2: C > D,X3: B] : ( G3 @ ( H2 @ ( F3 @ X3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % map_fun_apply
% 0.19/0.45 thf(fact_103_bex1__rel__rsp,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
% 0.19/0.45 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.19/0.45 => ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
% 0.19/0.45 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 0.19/0.45 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 0.19/0.45 @ ( bex1_rel @ A @ R )
% 0.19/0.45 @ ( bex1_rel @ A @ R ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bex1_rel_rsp
% 0.19/0.45 thf(fact_104_llist_Octr__transfer_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ R @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) @ ( coinductive_LCons @ A ) @ ( coinductive_LCons @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.ctr_transfer(2)
% 0.19/0.45 thf(fact_105_LCons__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ A2 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_LCons @ A ) @ ( coinductive_LCons @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % LCons_transfer
% 0.19/0.45 thf(fact_106_ldropWhile__eq__ldrop,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coindu218763757pWhile @ A )
% 0.19/0.45 = ( ^ [P3: A > $o,Xs2: coinductive_llist @ A] : ( coinductive_ldrop @ A @ ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldropWhile_eq_ldrop
% 0.19/0.45 thf(fact_107_zero__natural_Orsp,axiom,
% 0.19/0.45 ( ( zero_zero @ nat )
% 0.19/0.45 = ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % zero_natural.rsp
% 0.19/0.45 thf(fact_108_llist_Oinject,axiom,
% 0.19/0.45 ! [A: $tType,X21: A,X222: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_LCons @ A @ X21 @ X222 )
% 0.19/0.45 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.19/0.45 = ( ( X21 = Y21 )
% 0.19/0.45 & ( X222 = Y22 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.inject
% 0.19/0.45 thf(fact_109_ldrop__0,axiom,
% 0.19/0.45 ! [B: $tType,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
% 0.19/0.45 = Xs ) ).
% 0.19/0.45
% 0.19/0.45 % ldrop_0
% 0.19/0.45 thf(fact_110_LCons__lprefix__LCons,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.19/0.45 = ( ( X2 = Y3 )
% 0.19/0.45 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % LCons_lprefix_LCons
% 0.19/0.45 thf(fact_111_llist__all2__LCons__LCons,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o,X21: A,X222: coinductive_llist @ A,Y21: B,Y22: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X222 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) )
% 0.19/0.45 = ( ( R @ X21 @ Y21 )
% 0.19/0.45 & ( coindu1486289336t_all2 @ A @ B @ R @ X222 @ Y22 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_LCons_LCons
% 0.19/0.45 thf(fact_112_lfinite__code_I2_J,axiom,
% 0.19/0.45 ! [B: $tType,X2: B,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) )
% 0.19/0.45 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_code(2)
% 0.19/0.45 thf(fact_113_lfinite__LCons,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_LCons
% 0.19/0.45 thf(fact_114_lfilter__LCons,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( P @ X2 )
% 0.19/0.45 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( coinductive_LCons @ A @ X2 @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 0.19/0.45 & ( ~ ( P @ X2 )
% 0.19/0.45 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_LCons
% 0.19/0.45 thf(fact_115_ldropWhile__LCons,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( P @ X2 )
% 0.19/0.45 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 0.19/0.45 & ( ~ ( P @ X2 )
% 0.19/0.45 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldropWhile_LCons
% 0.19/0.45 thf(fact_116_ldrop__lmap,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_ldrop @ A @ N @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 0.19/0.45 = ( coinductive_lmap @ B @ A @ F @ ( coinductive_ldrop @ B @ N @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldrop_lmap
% 0.19/0.45 thf(fact_117_lmirror__aux__simps_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X2: coinductive_llist @ A] :
% 0.19/0.45 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X2 ) )
% 0.19/0.45 = ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmirror_aux_simps(2)
% 0.19/0.45 thf(fact_118_unfold__llist__eq__LCons,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
% 0.19/0.45 = ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( ~ ( IS_LNIL @ B3 )
% 0.19/0.45 & ( X2
% 0.19/0.45 = ( LHD @ B3 ) )
% 0.19/0.45 & ( Xs
% 0.19/0.45 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unfold_llist_eq_LCons
% 0.19/0.45 thf(fact_119_llist_Opred__inject_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,A3: A,Aa2: coinductive_llist @ A] :
% 0.19/0.45 ( ( coindu543516966_llist @ A @ P @ ( coinductive_LCons @ A @ A3 @ Aa2 ) )
% 0.19/0.45 = ( ( P @ A3 )
% 0.19/0.45 & ( coindu543516966_llist @ A @ P @ Aa2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.pred_inject(2)
% 0.19/0.45 thf(fact_120_bex1__rel__aux2,axiom,
% 0.19/0.45 ! [A: $tType,R: A > A > $o,X2: A > $o,Y3: A > $o] :
% 0.19/0.45 ( ! [Xa2: A,Ya2: A] :
% 0.19/0.45 ( ( R @ Xa2 @ Ya2 )
% 0.19/0.45 => ( ( X2 @ Xa2 )
% 0.19/0.45 = ( Y3 @ Ya2 ) ) )
% 0.19/0.45 => ( ( bex1_rel @ A @ R @ Y3 )
% 0.19/0.45 => ( bex1_rel @ A @ R @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bex1_rel_aux2
% 0.19/0.45 thf(fact_121_bex1__rel__aux,axiom,
% 0.19/0.45 ! [A: $tType,R: A > A > $o,X2: A > $o,Y3: A > $o] :
% 0.19/0.45 ( ! [Xa2: A,Ya2: A] :
% 0.19/0.45 ( ( R @ Xa2 @ Ya2 )
% 0.19/0.45 => ( ( X2 @ Xa2 )
% 0.19/0.45 = ( Y3 @ Ya2 ) ) )
% 0.19/0.45 => ( ( bex1_rel @ A @ R @ X2 )
% 0.19/0.45 => ( bex1_rel @ A @ R @ Y3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bex1_rel_aux
% 0.19/0.45 thf(fact_122_Le__LCons,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X2: A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ X2 @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Le_LCons
% 0.19/0.45 thf(fact_123_LCons__lprefix__conv,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Ys )
% 0.19/0.45 = ( ? [Ys3: coinductive_llist @ A] :
% 0.19/0.45 ( ( Ys
% 0.19/0.45 = ( coinductive_LCons @ A @ X2 @ Ys3 ) )
% 0.19/0.45 & ( coinductive_lprefix @ A @ Xs @ Ys3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % LCons_lprefix_conv
% 0.19/0.45 thf(fact_124_llist_Orel__intros_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X222: coinductive_llist @ A,Y22: coinductive_llist @ B] :
% 0.19/0.45 ( ( R @ X21 @ Y21 )
% 0.19/0.45 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X222 @ Y22 )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X222 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_intros(2)
% 0.19/0.45 thf(fact_125_llist__all2__LCons1,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Ys )
% 0.19/0.45 = ( ? [Y5: B,Ys3: coinductive_llist @ B] :
% 0.19/0.45 ( ( Ys
% 0.19/0.45 = ( coinductive_LCons @ B @ Y5 @ Ys3 ) )
% 0.19/0.45 & ( P @ X2 @ Y5 )
% 0.19/0.45 & ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_LCons1
% 0.19/0.45 thf(fact_126_llist__all2__LCons2,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Y3: B,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
% 0.19/0.45 = ( ? [X3: A,Xs4: coinductive_llist @ A] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 = ( coinductive_LCons @ A @ X3 @ Xs4 ) )
% 0.19/0.45 & ( P @ X3 @ Y3 )
% 0.19/0.45 & ( coindu1486289336t_all2 @ A @ B @ P @ Xs4 @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_LCons2
% 0.19/0.45 thf(fact_127_lfinite__LConsI,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,X2: A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_LConsI
% 0.19/0.45 thf(fact_128_llist_Osimps_I13_J,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,F: A > B,X21: A,X222: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lmap @ A @ B @ F @ ( coinductive_LCons @ A @ X21 @ X222 ) )
% 0.19/0.45 = ( coinductive_LCons @ B @ ( F @ X21 ) @ ( coinductive_lmap @ A @ B @ F @ X222 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.simps(13)
% 0.19/0.45 thf(fact_129_lmap__eq__LCons__conv,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Y3: A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_lmap @ B @ A @ F @ Xs )
% 0.19/0.45 = ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.19/0.45 = ( ? [X3: B,Xs4: coinductive_llist @ B] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 = ( coinductive_LCons @ B @ X3 @ Xs4 ) )
% 0.19/0.45 & ( Y3
% 0.19/0.45 = ( F @ X3 ) )
% 0.19/0.45 & ( Ys
% 0.19/0.45 = ( coinductive_lmap @ B @ A @ F @ Xs4 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmap_eq_LCons_conv
% 0.19/0.45 thf(fact_130_llist__all2__ldropI,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,N: extended_enat] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_ldrop @ A @ N @ Xs ) @ ( coinductive_ldrop @ B @ N @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_ldropI
% 0.19/0.45 thf(fact_131_llist_Osimps_I5_J,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,F1: B,F22: A > ( coinductive_llist @ A ) > B,X21: A,X222: coinductive_llist @ A] :
% 0.19/0.45 ( ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ ( coinductive_LCons @ A @ X21 @ X222 ) )
% 0.19/0.45 = ( F22 @ X21 @ X222 ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.simps(5)
% 0.19/0.45 thf(fact_132_lfilter__LCons__found,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( P @ X2 )
% 0.19/0.45 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( coinductive_LCons @ A @ X2 @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_LCons_found
% 0.19/0.45 thf(fact_133_lfilter__LCons__seek,axiom,
% 0.19/0.45 ! [A: $tType,P4: A > $o,X2: A,L: coinductive_llist @ A] :
% 0.19/0.45 ( ~ ( P4 @ X2 )
% 0.19/0.45 => ( ( coinductive_lfilter @ A @ P4 @ ( coinductive_LCons @ A @ X2 @ L ) )
% 0.19/0.45 = ( coinductive_lfilter @ A @ P4 @ L ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_LCons_seek
% 0.19/0.45 thf(fact_134_iterates_Ocode,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coinductive_iterates @ A )
% 0.19/0.45 = ( ^ [F3: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_iterates @ A @ F3 @ ( F3 @ X3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % iterates.code
% 0.19/0.45 thf(fact_135_unfold__llist_Octr_I2_J,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,P4: A > $o,A3: A,G21: A > B,G22: A > A] :
% 0.19/0.45 ( ~ ( P4 @ A3 )
% 0.19/0.45 => ( ( coindu1441602521_llist @ A @ B @ P4 @ G21 @ G22 @ A3 )
% 0.19/0.45 = ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unfold_llist.ctr(2)
% 0.19/0.45 thf(fact_136_llist_Ocorec_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,P4: C > $o,A3: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
% 0.19/0.45 ( ~ ( P4 @ A3 )
% 0.19/0.45 => ( ( coindu1259883913_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ A3 )
% 0.19/0.45 = ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.corec(2)
% 0.19/0.45 thf(fact_137_iterates__equality,axiom,
% 0.19/0.45 ! [A: $tType,H3: A > ( coinductive_llist @ A ),F: A > A] :
% 0.19/0.45 ( ! [X: A] :
% 0.19/0.45 ( ( H3 @ X )
% 0.19/0.45 = ( coinductive_LCons @ A @ X @ ( coinductive_lmap @ A @ A @ F @ ( H3 @ X ) ) ) )
% 0.19/0.45 => ( H3
% 0.19/0.45 = ( coinductive_iterates @ A @ F ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % iterates_equality
% 0.19/0.45 thf(fact_138_iterates__lmap,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coinductive_iterates @ A )
% 0.19/0.45 = ( ^ [F3: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_lmap @ A @ A @ F3 @ ( coinductive_iterates @ A @ F3 @ X3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % iterates_lmap
% 0.19/0.45 thf(fact_139_ldrop__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ extended_enat @ extended_enat @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 0.19/0.45 @ ( coinductive_ldrop @ A )
% 0.19/0.45 @ ( coinductive_ldrop @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldrop_transfer
% 0.19/0.45 thf(fact_140_natural_Osize_I1_J,axiom,
% 0.19/0.45 ( ( code_size_natural @ ( zero_zero @ code_natural ) )
% 0.19/0.45 = ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % natural.size(1)
% 0.19/0.45 thf(fact_141_llistsum__LCons,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [Xs: coinductive_llist @ A,X2: A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coindu780009021istsum @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 = ( plus_plus @ A @ X2 @ ( coindu780009021istsum @ A @ Xs ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llistsum_LCons
% 0.19/0.45 thf(fact_142_llist__all2__lappend1D_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ Ys )
% 0.19/0.45 => ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ ( coinductive_ldrop @ B @ ( coinductive_llength @ A @ Xs ) @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lappend1D(2)
% 0.19/0.45 thf(fact_143_natural_Osize_I3_J,axiom,
% 0.19/0.45 ( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
% 0.19/0.45 = ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % natural.size(3)
% 0.19/0.45 thf(fact_144_lmember__code_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Y3: A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lmember @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 0.19/0.45 = ( ( X2 = Y3 )
% 0.19/0.45 | ( coinductive_lmember @ A @ X2 @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmember_code(2)
% 0.19/0.45 thf(fact_145_add__left__cancel,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A,C3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.19/0.45 = ( B3 = C3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_left_cancel
% 0.19/0.45 thf(fact_146_add__right__cancel,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [B3: A,A3: A,C3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.19/0.45 = ( plus_plus @ A @ C3 @ A3 ) )
% 0.19/0.45 = ( B3 = C3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_right_cancel
% 0.19/0.45 thf(fact_147_ldrop__ldrop,axiom,
% 0.19/0.45 ! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_ldrop @ A @ N @ ( coinductive_ldrop @ A @ M @ Xs ) )
% 0.19/0.45 = ( coinductive_ldrop @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldrop_ldrop
% 0.19/0.45 thf(fact_148_add__cancel__right__right,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A] :
% 0.19/0.45 ( ( A3
% 0.19/0.45 = ( plus_plus @ A @ A3 @ B3 ) )
% 0.19/0.45 = ( B3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_cancel_right_right
% 0.19/0.45 thf(fact_149_add__cancel__right__left,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A] :
% 0.19/0.45 ( ( A3
% 0.19/0.45 = ( plus_plus @ A @ B3 @ A3 ) )
% 0.19/0.45 = ( B3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_cancel_right_left
% 0.19/0.45 thf(fact_150_add__cancel__left__right,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.19/0.45 = A3 )
% 0.19/0.45 = ( B3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_cancel_left_right
% 0.19/0.45 thf(fact_151_add__cancel__left__left,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [B3: A,A3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.19/0.45 = A3 )
% 0.19/0.45 = ( B3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_cancel_left_left
% 0.19/0.45 thf(fact_152_double__zero__sym,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( ( zero_zero @ A )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ A3 ) )
% 0.19/0.45 = ( A3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % double_zero_sym
% 0.19/0.45 thf(fact_153_double__zero,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.19/0.45 = ( zero_zero @ A ) )
% 0.19/0.45 = ( A3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % double_zero
% 0.19/0.45 thf(fact_154_add_Oright__neutral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.right_neutral
% 0.19/0.45 thf(fact_155_add_Oleft__neutral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.left_neutral
% 0.19/0.45 thf(fact_156_lappend__code_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,Xa: A,X2: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X2 ) @ Ys )
% 0.19/0.45 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X2 @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_code(2)
% 0.19/0.45 thf(fact_157_llength__lappend,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_llength @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.19/0.45 = ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llength @ A @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llength_lappend
% 0.19/0.45 thf(fact_158_lfinite__lappend,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.19/0.45 = ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_lappend
% 0.19/0.45 thf(fact_159_lprefix__lappend__same,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 0.19/0.45 = ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_lappend_same
% 0.19/0.45 thf(fact_160_lfilter__lappend__lfinite,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.19/0.45 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_lappend_lfinite
% 0.19/0.45 thf(fact_161_lappend__ltakeWhile__ldropWhile,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.19/0.45 = Xs ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_ltakeWhile_ldropWhile
% 0.19/0.45 thf(fact_162_llcp__lappend__same,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_llcp @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 0.19/0.45 = ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llcp @ A @ Ys @ Zs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llcp_lappend_same
% 0.19/0.45 thf(fact_163_lprefix__lappend,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_lappend
% 0.19/0.45 thf(fact_164_lappend__lprefixE,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.19/0.45 => ~ ! [Zs2: coinductive_llist @ A] :
% 0.19/0.45 ( Zs
% 0.19/0.45 != ( coinductive_lappend @ A @ Xs @ Zs2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_lprefixE
% 0.19/0.45 thf(fact_165_lprefix__lappendD,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.19/0.45 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_lappendD
% 0.19/0.45 thf(fact_166_lprefix__conv__lappend,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( coinductive_lprefix @ A )
% 0.19/0.45 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 0.19/0.45 ? [Zs3: coinductive_llist @ A] :
% 0.19/0.45 ( Ys2
% 0.19/0.45 = ( coinductive_lappend @ A @ Xs2 @ Zs3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_conv_lappend
% 0.19/0.45 thf(fact_167_lprefix__lappend__sameI,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.19/0.45 => ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lprefix_lappend_sameI
% 0.19/0.45 thf(fact_168_comm__monoid__add__class_Oadd__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % comm_monoid_add_class.add_0
% 0.19/0.45 thf(fact_169_add_Ocomm__neutral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.comm_neutral
% 0.19/0.45 thf(fact_170_add_Ogroup__left__neutral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( group_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.group_left_neutral
% 0.19/0.45 thf(fact_171_add__eq__0__iff__both__eq__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [X2: A,Y3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ X2 @ Y3 )
% 0.19/0.45 = ( zero_zero @ A ) )
% 0.19/0.45 = ( ( X2
% 0.19/0.45 = ( zero_zero @ A ) )
% 0.19/0.45 & ( Y3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_eq_0_iff_both_eq_0
% 0.19/0.45 thf(fact_172_lappend__inf,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.19/0.45 = Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_inf
% 0.19/0.45 thf(fact_173_llistsum__lappend,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( monoid_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coinductive_lfinite @ A @ Ys )
% 0.19/0.45 => ( ( coindu780009021istsum @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.19/0.45 = ( plus_plus @ A @ ( coindu780009021istsum @ A @ Xs ) @ ( coindu780009021istsum @ A @ Ys ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llistsum_lappend
% 0.19/0.45 thf(fact_174_lmap__lappend__distrib,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_lmap @ B @ A @ F @ ( coinductive_lappend @ B @ Xs @ Ys ) )
% 0.19/0.45 = ( coinductive_lappend @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) @ ( coinductive_lmap @ B @ A @ F @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmap_lappend_distrib
% 0.19/0.45 thf(fact_175_lappend__assoc,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.19/0.45 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_assoc
% 0.19/0.45 thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A,C3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ab_semigroup_add_class.add_ac(1)
% 0.19/0.45 thf(fact_177_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.45 ( ( ( I = J )
% 0.19/0.45 & ( K = L ) )
% 0.19/0.45 => ( ( plus_plus @ A @ I @ K )
% 0.19/0.45 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_mono_thms_linordered_semiring(4)
% 0.19/0.45 thf(fact_178_add_Oassoc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A,C3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.assoc
% 0.19/0.45 thf(fact_179_add_Oleft__cancel,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( group_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A,C3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.19/0.45 = ( B3 = C3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.left_cancel
% 0.19/0.45 thf(fact_180_add_Oright__cancel,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( group_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [B3: A,A3: A,C3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.19/0.45 = ( plus_plus @ A @ C3 @ A3 ) )
% 0.19/0.45 = ( B3 = C3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.right_cancel
% 0.19/0.45 thf(fact_181_add_Ocommute,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ( ( plus_plus @ A )
% 0.19/0.45 = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.commute
% 0.19/0.45 thf(fact_182_add_Oleft__commute,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [B3: A,A3: A,C3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add.left_commute
% 0.19/0.45 thf(fact_183_add__left__imp__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A,B3: A,C3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ A3 @ B3 )
% 0.19/0.45 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.19/0.45 => ( B3 = C3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_left_imp_eq
% 0.19/0.45 thf(fact_184_add__right__imp__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.19/0.45 => ! [B3: A,A3: A,C3: A] :
% 0.19/0.45 ( ( ( plus_plus @ A @ B3 @ A3 )
% 0.19/0.45 = ( plus_plus @ A @ C3 @ A3 ) )
% 0.19/0.45 => ( B3 = C3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_right_imp_eq
% 0.19/0.45 thf(fact_185_lappend__iterates,axiom,
% 0.19/0.45 ! [A: $tType,F: A > A,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lappend @ A @ ( coinductive_iterates @ A @ F @ X2 ) @ Xs )
% 0.19/0.45 = ( coinductive_iterates @ A @ F @ X2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_iterates
% 0.19/0.45 thf(fact_186_lmirror__aux__acc,axiom,
% 0.19/0.45 ! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
% 0.19/0.45 = ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lmirror_aux_acc
% 0.19/0.45 thf(fact_187_llist__all2__lappendI,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coinductive_lfinite @ B @ Ys )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ Ys4 ) ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ ( coinductive_lappend @ B @ Ys @ Ys4 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lappendI
% 0.19/0.45 thf(fact_188_lappend__eq__lappend__conv,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Us: coinductive_llist @ A,Ys: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_llength @ A @ Xs )
% 0.19/0.45 = ( coinductive_llength @ A @ Us ) )
% 0.19/0.45 => ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.19/0.45 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 0.19/0.45 = ( ( Xs = Us )
% 0.19/0.45 & ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( Ys = Vs ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_eq_lappend_conv
% 0.19/0.45 thf(fact_189_lfilter__eq__lappend__lfiniteD,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.19/0.45 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.19/0.45 => ( ( coinductive_lfinite @ A @ Ys )
% 0.19/0.45 => ? [Us2: coinductive_llist @ A,Vs2: coinductive_llist @ A] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 = ( coinductive_lappend @ A @ Us2 @ Vs2 ) )
% 0.19/0.45 & ( coinductive_lfinite @ A @ Us2 )
% 0.19/0.45 & ( Ys
% 0.19/0.45 = ( coinductive_lfilter @ A @ P @ Us2 ) )
% 0.19/0.45 & ( Zs
% 0.19/0.45 = ( coinductive_lfilter @ A @ P @ Vs2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_eq_lappend_lfiniteD
% 0.19/0.45 thf(fact_190_lappend__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_lappend @ A ) @ ( coinductive_lappend @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_transfer
% 0.19/0.45 thf(fact_191_add__is__0,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ( plus_plus @ nat @ M @ N )
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 = ( ( M
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 & ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_is_0
% 0.19/0.45 thf(fact_192_Nat_Oadd__0__right,axiom,
% 0.19/0.45 ! [M: nat] :
% 0.19/0.45 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.19/0.45 = M ) ).
% 0.19/0.45
% 0.19/0.45 % Nat.add_0_right
% 0.19/0.45 thf(fact_193_natural_Osimps_I4_J,axiom,
% 0.19/0.45 ! [T: $tType,F1: T,F22: code_natural > T] :
% 0.19/0.45 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.19/0.45 = F1 ) ).
% 0.19/0.45
% 0.19/0.45 % natural.simps(4)
% 0.19/0.45 thf(fact_194_plus__natural_Orsp,axiom,
% 0.19/0.45 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 0.19/0.45 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 0.19/0.45 @ ( plus_plus @ nat )
% 0.19/0.45 @ ( plus_plus @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % plus_natural.rsp
% 0.19/0.45 thf(fact_195_add__eq__self__zero,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ( plus_plus @ nat @ M @ N )
% 0.19/0.45 = M )
% 0.19/0.45 => ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_eq_self_zero
% 0.19/0.45 thf(fact_196_plus__nat_Oadd__0,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 = N ) ).
% 0.19/0.45
% 0.19/0.45 % plus_nat.add_0
% 0.19/0.45 thf(fact_197_ind__euclid,axiom,
% 0.19/0.45 ! [P: nat > nat > $o,A3: nat,B3: nat] :
% 0.19/0.45 ( ! [A4: nat,B4: nat] :
% 0.19/0.45 ( ( P @ A4 @ B4 )
% 0.19/0.45 = ( P @ B4 @ A4 ) )
% 0.19/0.45 => ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
% 0.19/0.45 => ( ! [A4: nat,B4: nat] :
% 0.19/0.45 ( ( P @ A4 @ B4 )
% 0.19/0.45 => ( P @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
% 0.19/0.45 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ind_euclid
% 0.19/0.45 thf(fact_198_add__0__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiri456707255roduct @ A @ ( type @ A ) )
% 0.19/0.45 => ! [B3: A,A3: A] :
% 0.19/0.45 ( ( B3
% 0.19/0.45 = ( plus_plus @ A @ B3 @ A3 ) )
% 0.19/0.45 = ( A3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_0_iff
% 0.19/0.45 thf(fact_199_semiring__normalization__rules_I6_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % semiring_normalization_rules(6)
% 0.19/0.45 thf(fact_200_semiring__normalization__rules_I5_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % semiring_normalization_rules(5)
% 0.19/0.45 thf(fact_201_llast__lappend__LCons,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) )
% 0.19/0.45 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llast_lappend_LCons
% 0.19/0.45 thf(fact_202_rec__nat__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ A2
% 0.19/0.45 @ ( bNF_rel_fun @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B )
% 0.19/0.45 @ ( bNF_rel_fun @ nat @ nat @ ( A > A ) @ ( B > B )
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 0.19/0.45 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 0.19/0.45 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 0.19/0.45 @ A2 ) )
% 0.19/0.45 @ ( rec_nat @ A )
% 0.19/0.45 @ ( rec_nat @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % rec_nat_transfer
% 0.19/0.45 thf(fact_203_llast__LCons2,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Y3: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
% 0.19/0.45 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llast_LCons2
% 0.19/0.45 thf(fact_204_old_Onat_Osimps_I6_J,axiom,
% 0.19/0.45 ! [T: $tType,F1: T,F22: nat > T > T] :
% 0.19/0.45 ( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
% 0.19/0.45 = F1 ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.simps(6)
% 0.19/0.45 thf(fact_205_case__nat__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ A @ B @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) @ A2
% 0.19/0.45 @ ( bNF_rel_fun @ ( nat > A ) @ ( nat > B ) @ ( nat > A ) @ ( nat > B )
% 0.19/0.45 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 0.19/0.45 @ A2 )
% 0.19/0.45 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.19/0.45 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 0.19/0.45 @ A2 ) )
% 0.19/0.45 @ ( case_nat @ A )
% 0.19/0.45 @ ( case_nat @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % case_nat_transfer
% 0.19/0.45 thf(fact_206_natural_Osimps_I6_J,axiom,
% 0.19/0.45 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 0.19/0.45 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.19/0.45 = F1 ) ).
% 0.19/0.45
% 0.19/0.45 % natural.simps(6)
% 0.19/0.45 thf(fact_207_old_Onat_Osimps_I4_J,axiom,
% 0.19/0.45 ! [A: $tType,F1: A,F22: nat > A] :
% 0.19/0.45 ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
% 0.19/0.45 = F1 ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.simps(4)
% 0.19/0.45 thf(fact_208_ltake__transfer,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.19/0.45 ( bNF_rel_fun @ extended_enat @ extended_enat @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 0.19/0.45 @ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
% 0.19/0.45 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 0.19/0.45 @ ( coinductive_ltake @ A )
% 0.19/0.45 @ ( coinductive_ltake @ B ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltake_transfer
% 0.19/0.45 thf(fact_209_lfilter__eq__LConsD,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 0.19/0.45 = ( coinductive_LCons @ A @ X2 @ Xs ) )
% 0.19/0.45 => ? [Us2: coinductive_llist @ A,Vs2: coinductive_llist @ A] :
% 0.19/0.45 ( ( Ys
% 0.19/0.45 = ( coinductive_lappend @ A @ Us2 @ ( coinductive_LCons @ A @ X2 @ Vs2 ) ) )
% 0.19/0.45 & ( coinductive_lfinite @ A @ Us2 )
% 0.19/0.45 & ! [X5: A] :
% 0.19/0.45 ( ( member @ A @ X5 @ ( coinductive_lset @ A @ Us2 ) )
% 0.19/0.45 => ~ ( P @ X5 ) )
% 0.19/0.45 & ( P @ X2 )
% 0.19/0.45 & ( Xs
% 0.19/0.45 = ( coinductive_lfilter @ A @ P @ Vs2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_eq_LConsD
% 0.19/0.45 thf(fact_210_ltake__is__lprefix,axiom,
% 0.19/0.45 ! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ Xs ) ).
% 0.19/0.45
% 0.19/0.45 % ltake_is_lprefix
% 0.19/0.45 thf(fact_211_llist__all2__same,axiom,
% 0.19/0.45 ! [A: $tType,P: A > A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ A @ P @ Xs @ Xs )
% 0.19/0.45 = ( ! [X3: A] :
% 0.19/0.45 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( P @ X3 @ X3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_same
% 0.19/0.45 thf(fact_212_ltake__lmap,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coinductive_llist @ B] :
% 0.19/0.45 ( ( coinductive_ltake @ A @ N @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 0.19/0.45 = ( coinductive_lmap @ B @ A @ F @ ( coinductive_ltake @ B @ N @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltake_lmap
% 0.19/0.45 thf(fact_213_ldropWhile__cong,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.19/0.45 ( ( Xs = Ys )
% 0.19/0.45 => ( ! [X: A] :
% 0.19/0.45 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.19/0.45 => ( ( P @ X )
% 0.19/0.45 = ( Q @ X ) ) )
% 0.19/0.45 => ( ( coindu218763757pWhile @ A @ P @ Xs )
% 0.19/0.45 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ldropWhile_cong
% 0.19/0.45 thf(fact_214_in__lset__ldropWhileD,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 0.19/0.45 => ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % in_lset_ldropWhileD
% 0.19/0.45 thf(fact_215_llist_Opred__cong,axiom,
% 0.19/0.45 ! [A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,P: A > $o,Pa: A > $o] :
% 0.19/0.45 ( ( X2 = Ya )
% 0.19/0.45 => ( ! [Z3: A] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
% 0.19/0.45 => ( ( P @ Z3 )
% 0.19/0.45 = ( Pa @ Z3 ) ) )
% 0.19/0.45 => ( ( coindu543516966_llist @ A @ P @ X2 )
% 0.19/0.45 = ( coindu543516966_llist @ A @ Pa @ Ya ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.pred_cong
% 0.19/0.45 thf(fact_216_llist_Opred__mono__strong,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,X2: coinductive_llist @ A,Pa: A > $o] :
% 0.19/0.45 ( ( coindu543516966_llist @ A @ P @ X2 )
% 0.19/0.45 => ( ! [Z3: A] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 0.19/0.45 => ( ( P @ Z3 )
% 0.19/0.45 => ( Pa @ Z3 ) ) )
% 0.19/0.45 => ( coindu543516966_llist @ A @ Pa @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.pred_mono_strong
% 0.19/0.45 thf(fact_217_lset__ltakeWhileD,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 0.19/0.45 => ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 & ( P @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_ltakeWhileD
% 0.19/0.45 thf(fact_218_ltakeWhile__cong,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.19/0.45 ( ( Xs = Ys )
% 0.19/0.45 => ( ! [X: A] :
% 0.19/0.45 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.19/0.45 => ( ( P @ X )
% 0.19/0.45 = ( Q @ X ) ) )
% 0.19/0.45 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.19/0.45 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltakeWhile_cong
% 0.19/0.45 thf(fact_219_ltakeWhile__all,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.19/0.45 ( ! [X: A] :
% 0.19/0.45 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( P @ X ) )
% 0.19/0.45 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 0.19/0.45 = Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltakeWhile_all
% 0.19/0.45 thf(fact_220_lfilter__cong,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.19/0.45 ( ( Xs = Ys )
% 0.19/0.45 => ( ! [X: A] :
% 0.19/0.45 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.19/0.45 => ( ( P @ X )
% 0.19/0.45 = ( Q @ X ) ) )
% 0.19/0.45 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.19/0.45 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_cong
% 0.19/0.45 thf(fact_221_lfilter__id__conv,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.19/0.45 = Xs )
% 0.19/0.45 = ( ! [X3: A] :
% 0.19/0.45 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( P @ X3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfilter_id_conv
% 0.19/0.45 thf(fact_222_llist_Oinj__map__strong,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Xa: coinductive_llist @ A,F: A > B,Fa: A > B] :
% 0.19/0.45 ( ! [Z3: A,Za: A] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 0.19/0.45 => ( ( member @ A @ Za @ ( coinductive_lset @ A @ Xa ) )
% 0.19/0.45 => ( ( ( F @ Z3 )
% 0.19/0.45 = ( Fa @ Za ) )
% 0.19/0.45 => ( Z3 = Za ) ) ) )
% 0.19/0.45 => ( ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 0.19/0.45 = ( coinductive_lmap @ A @ B @ Fa @ Xa ) )
% 0.19/0.45 => ( X2 = Xa ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.inj_map_strong
% 0.19/0.45 thf(fact_223_llist_Omap__cong0,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,F: A > B,G: A > B] :
% 0.19/0.45 ( ! [Z3: A] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 0.19/0.45 => ( ( F @ Z3 )
% 0.19/0.45 = ( G @ Z3 ) ) )
% 0.19/0.45 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 0.19/0.45 = ( coinductive_lmap @ A @ B @ G @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.map_cong0
% 0.19/0.45 thf(fact_224_llist_Omap__cong,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
% 0.19/0.45 ( ( X2 = Ya )
% 0.19/0.45 => ( ! [Z3: A] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
% 0.19/0.45 => ( ( F @ Z3 )
% 0.19/0.45 = ( G @ Z3 ) ) )
% 0.19/0.45 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 0.19/0.45 = ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.map_cong
% 0.19/0.45 thf(fact_225_llist_Orel__refl__strong,axiom,
% 0.19/0.45 ! [A: $tType,X2: coinductive_llist @ A,Ra: A > A > $o] :
% 0.19/0.45 ( ! [Z3: A] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 0.19/0.45 => ( Ra @ Z3 @ Z3 ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ A @ Ra @ X2 @ X2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_refl_strong
% 0.19/0.45 thf(fact_226_llist_Orel__mono__strong,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,R: A > B > $o,X2: coinductive_llist @ A,Y3: coinductive_llist @ B,Ra: A > B > $o] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
% 0.19/0.45 => ( ! [Z3: A,Yb2: B] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 0.19/0.45 => ( ( member @ B @ Yb2 @ ( coinductive_lset @ B @ Y3 ) )
% 0.19/0.45 => ( ( R @ Z3 @ Yb2 )
% 0.19/0.45 => ( Ra @ Z3 @ Yb2 ) ) ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ Ra @ X2 @ Y3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_mono_strong
% 0.19/0.45 thf(fact_227_llist_Orel__cong,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,Y3: coinductive_llist @ B,Xa: coinductive_llist @ B,R: A > B > $o,Ra: A > B > $o] :
% 0.19/0.45 ( ( X2 = Ya )
% 0.19/0.45 => ( ( Y3 = Xa )
% 0.19/0.45 => ( ! [Z3: A,Yb2: B] :
% 0.19/0.45 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
% 0.19/0.45 => ( ( member @ B @ Yb2 @ ( coinductive_lset @ B @ Xa ) )
% 0.19/0.45 => ( ( R @ Z3 @ Yb2 )
% 0.19/0.45 = ( Ra @ Z3 @ Yb2 ) ) ) )
% 0.19/0.45 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
% 0.19/0.45 = ( coindu1486289336t_all2 @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.rel_cong
% 0.19/0.45 thf(fact_228_llist__all2__lsetD2,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Y3: B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ( member @ B @ Y3 @ ( coinductive_lset @ B @ Ys ) )
% 0.19/0.45 => ? [X: A] :
% 0.19/0.45 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 & ( P @ X @ Y3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lsetD2
% 0.19/0.45 thf(fact_229_llist__all2__lsetD1,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,X2: A] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ? [X: B] :
% 0.19/0.45 ( ( member @ B @ X @ ( coinductive_lset @ B @ Ys ) )
% 0.19/0.45 & ( P @ X2 @ X ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lsetD1
% 0.19/0.45 thf(fact_230_llist__all2__reflI,axiom,
% 0.19/0.45 ! [A: $tType,Xs: coinductive_llist @ A,P: A > A > $o] :
% 0.19/0.45 ( ! [X: A] :
% 0.19/0.45 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( P @ X @ X ) )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_reflI
% 0.19/0.45 thf(fact_231_llist__all2__ltakeI,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,N: extended_enat] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ltake @ B @ N @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_ltakeI
% 0.19/0.45 thf(fact_232_lset__intros_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,X6: A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_intros(2)
% 0.19/0.45 thf(fact_233_lset__intros_I1_J,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] : ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_intros(1)
% 0.19/0.45 thf(fact_234_llist_Oset__intros_I2_J,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,A22: coinductive_llist @ A,A1: A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ A22 ) )
% 0.19/0.45 => ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.set_intros(2)
% 0.19/0.45 thf(fact_235_llist_Oset__intros_I1_J,axiom,
% 0.19/0.45 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.set_intros(1)
% 0.19/0.45 thf(fact_236_lset__cases,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( ! [Xs5: coinductive_llist @ A] :
% 0.19/0.45 ( Xs
% 0.19/0.45 != ( coinductive_LCons @ A @ X2 @ Xs5 ) )
% 0.19/0.45 => ~ ! [X7: A,Xs5: coinductive_llist @ A] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 = ( coinductive_LCons @ A @ X7 @ Xs5 ) )
% 0.19/0.45 => ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs5 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_cases
% 0.19/0.45 thf(fact_237_lset__induct,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( ! [Xs6: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X2 @ Xs6 ) )
% 0.19/0.45 => ( ! [X7: A,Xs6: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs6 ) )
% 0.19/0.45 => ( ( X2 != X7 )
% 0.19/0.45 => ( ( P @ Xs6 )
% 0.19/0.45 => ( P @ ( coinductive_LCons @ A @ X7 @ Xs6 ) ) ) ) )
% 0.19/0.45 => ( P @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_induct
% 0.19/0.45 thf(fact_238_lset__induct_H,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( ! [Xs6: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X2 @ Xs6 ) )
% 0.19/0.45 => ( ! [X7: A,Xs6: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs6 ) )
% 0.19/0.45 => ( ( P @ Xs6 )
% 0.19/0.45 => ( P @ ( coinductive_LCons @ A @ X7 @ Xs6 ) ) ) )
% 0.19/0.45 => ( P @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_induct'
% 0.19/0.45 thf(fact_239_in__lset__ldropD,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,N: extended_enat,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_ldrop @ A @ N @ Xs ) ) )
% 0.19/0.45 => ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % in_lset_ldropD
% 0.19/0.45 thf(fact_240_llist_Oset__cases,axiom,
% 0.19/0.45 ! [A: $tType,E2: A,A3: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ E2 @ ( coinductive_lset @ A @ A3 ) )
% 0.19/0.45 => ( ! [Z22: coinductive_llist @ A] :
% 0.19/0.45 ( A3
% 0.19/0.45 != ( coinductive_LCons @ A @ E2 @ Z22 ) )
% 0.19/0.45 => ~ ! [Z1: A,Z22: coinductive_llist @ A] :
% 0.19/0.45 ( ( A3
% 0.19/0.45 = ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 0.19/0.45 => ~ ( member @ A @ E2 @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.set_cases
% 0.19/0.45 thf(fact_241_llist_Oset__induct,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,A3: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ A3 ) )
% 0.19/0.45 => ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 0.19/0.45 => ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
% 0.19/0.45 ( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
% 0.19/0.45 => ( ( P @ Xa2 @ Z22 )
% 0.19/0.45 => ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
% 0.19/0.45 => ( P @ X2 @ A3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist.set_induct
% 0.19/0.45 thf(fact_242_lset__lmirror__aux,axiom,
% 0.19/0.45 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lset @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 0.19/0.45 = ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Acc ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_lmirror_aux
% 0.19/0.45 thf(fact_243_lfinite__ldropWhile,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 0.19/0.45 = ( ? [X3: A] :
% 0.19/0.45 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 & ~ ( P @ X3 ) )
% 0.19/0.45 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_ldropWhile
% 0.19/0.45 thf(fact_244_lfinite__ltakeWhile,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 0.19/0.45 = ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 | ? [X3: A] :
% 0.19/0.45 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 & ~ ( P @ X3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lfinite_ltakeWhile
% 0.19/0.45 thf(fact_245_ltakeWhile__lappend1,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ( ~ ( P @ X2 )
% 0.19/0.45 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.19/0.45 = ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltakeWhile_lappend1
% 0.19/0.45 thf(fact_246_in__lset__lappend__iff,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.19/0.45 = ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 | ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.45 & ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % in_lset_lappend_iff
% 0.19/0.45 thf(fact_247_lappend__ltake__ldrop,axiom,
% 0.19/0.45 ! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_lappend @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ldrop @ A @ N @ Xs ) )
% 0.19/0.45 = Xs ) ).
% 0.19/0.45
% 0.19/0.45 % lappend_ltake_ldrop
% 0.19/0.45 thf(fact_248_lset__lmember,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 = ( coinductive_lmember @ A @ X2 @ Xs ) ) ).
% 0.19/0.45
% 0.19/0.45 % lset_lmember
% 0.19/0.45 thf(fact_249_ltake__ldrop,axiom,
% 0.19/0.45 ! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_ltake @ A @ N @ ( coinductive_ldrop @ A @ M @ Xs ) )
% 0.19/0.45 = ( coinductive_ldrop @ A @ M @ ( coinductive_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltake_ldrop
% 0.19/0.45 thf(fact_250_llist__all2__lappend1D_I1_J,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.19/0.45 ( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ Ys )
% 0.19/0.45 => ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_ltake @ B @ ( coinductive_llength @ A @ Xs ) @ Ys ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % llist_all2_lappend1D(1)
% 0.19/0.45 thf(fact_251_split__llist__first,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ? [Ys5: coinductive_llist @ A,Zs4: coinductive_llist @ A] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 = ( coinductive_lappend @ A @ Ys5 @ ( coinductive_LCons @ A @ X2 @ Zs4 ) ) )
% 0.19/0.45 & ( coinductive_lfinite @ A @ Ys5 )
% 0.19/0.45 & ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys5 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % split_llist_first
% 0.19/0.45 thf(fact_252_split__llist,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.19/0.45 => ? [Ys5: coinductive_llist @ A,Zs4: coinductive_llist @ A] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 = ( coinductive_lappend @ A @ Ys5 @ ( coinductive_LCons @ A @ X2 @ Zs4 ) ) )
% 0.19/0.45 & ( coinductive_lfinite @ A @ Ys5 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % split_llist
% 0.19/0.45 thf(fact_253_ltake__plus__conv__lappend,axiom,
% 0.19/0.45 ! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
% 0.19/0.45 ( ( coinductive_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs )
% 0.19/0.45 = ( coinductive_lappend @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ltake @ A @ M @ ( coinductive_ldrop @ A @ N @ Xs ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ltake_plus_conv_lappend
% 0.19/0.45
% 0.19/0.45 %----Type constructors (28)
% 0.19/0.45 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 0.19/0.45 semiri456707255roduct @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.19/0.45 canoni770627133id_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.19/0.45 ordere779506340up_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.19/0.45 cancel1352612707id_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.19/0.45 cancel_semigroup_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.19/0.45 ab_semigroup_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.19/0.45 comm_monoid_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.19/0.45 comm_semiring_1 @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.19/0.45 semigroup_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.19/0.45 monoid_add @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.19/0.45 zero @ nat @ ( type @ nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_1,axiom,
% 0.19/0.45 canoni770627133id_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_2,axiom,
% 0.19/0.45 ordere779506340up_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_3,axiom,
% 0.19/0.45 ab_semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_4,axiom,
% 0.19/0.45 comm_monoid_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_5,axiom,
% 0.19/0.45 comm_semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_6,axiom,
% 0.19/0.45 semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_7,axiom,
% 0.19/0.45 monoid_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_8,axiom,
% 0.19/0.45 zero @ extended_enat @ ( type @ extended_enat ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_9,axiom,
% 0.19/0.45 ordere779506340up_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_10,axiom,
% 0.19/0.45 cancel1352612707id_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_11,axiom,
% 0.19/0.45 cancel_semigroup_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_12,axiom,
% 0.19/0.45 ab_semigroup_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_13,axiom,
% 0.19/0.45 comm_monoid_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_14,axiom,
% 0.19/0.45 comm_semiring_1 @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_15,axiom,
% 0.19/0.45 semigroup_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_16,axiom,
% 0.19/0.45 monoid_add @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_17,axiom,
% 0.19/0.45 zero @ code_natural @ ( type @ code_natural ) ).
% 0.19/0.45
% 0.19/0.45 %----Helper facts (3)
% 0.19/0.45 thf(help_If_3_1_T,axiom,
% 0.19/0.45 ! [P: $o] :
% 0.19/0.45 ( ( P = $true )
% 0.19/0.45 | ( P = $false ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(help_If_2_1_T,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Y3: A] :
% 0.19/0.45 ( ( if @ A @ $false @ X2 @ Y3 )
% 0.19/0.45 = Y3 ) ).
% 0.19/0.45
% 0.19/0.45 thf(help_If_1_1_T,axiom,
% 0.19/0.45 ! [A: $tType,X2: A,Y3: A] :
% 0.19/0.45 ( ( if @ A @ $true @ X2 @ Y3 )
% 0.19/0.45 = X2 ) ).
% 0.19/0.45
% 0.19/0.45 %----Conjectures (1)
% 0.19/0.45 thf(conj_0,conjecture,
% 0.19/0.45 ( ( coinductive_llength @ a @ acca )
% 0.19/0.45 = ( coinductive_llength @ b @ acc_a ) ) ).
% 0.19/0.45
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 ------------------------
% 3.28/1.72 ---- Embedded in THF ---
% 3.28/1.72 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.28/1.72 %%% Generated on Tue Feb 24 06:09:54 EST 2026
% 3.28/1.72 %%% using '$$dhol' embedding, version 1.3.0.
% 3.28/1.72 %%% Logic specification used:
% 3.28/1.72 %%% thf(spec, logic, $$dhol).
% 3.28/1.72
% 3.28/1.72 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.DVE1XfhOnx/DTF2DTF15775.p
% 3.28/1.72 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 3.28/1.72 thf(ty_t_Code__Numeral_Onatural, type, code_natural: $tType).
% 3.28/1.72 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 3.28/1.72 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.28/1.72 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.28/1.72 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.28/1.72 thf(ty_tf_b, type, b: $tType).
% 3.28/1.72 thf(ty_tf_a, type, a: $tType).
% 3.28/1.72 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.72 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, type, semiri456707255roduct: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.28/1.73 thf(sy_c_BNF__Def_Orel__fun, type, bNF_rel_fun: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > (C > $o)) > ((B > (D > $o)) > ((A > B) > ((C > D) > $o))))))).
% 3.28/1.73 thf(sy_c_Code__Numeral_Onatural_Ocase__natural, type, code_case_natural: (!> [T:$tType]: ((T > ((code_natural > T) > (code_natural > T)))))).
% 3.28/1.73 thf(sy_c_Code__Numeral_Onatural_Orec__natural, type, code_rec_natural: (!> [T:$tType]: ((T > ((code_natural > (T > T)) > (code_natural > T)))))).
% 3.28/1.73 thf(sy_c_Code__Numeral_Onatural_Osize__natural, type, code_size_natural: (code_natural > nat)).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ofinite__lprefix, type, coindu328551480prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ogen__llength, type, coindu246717305length: (!> [A:$tType]: ((nat > ((coinductive_llist @ A) > extended_enat))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Oiterates, type, coinductive_iterates: (!> [A:$tType]: (((A > A) > (A > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Olappend, type, coinductive_lappend: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Oldrop, type, coinductive_ldrop: (!> [A:$tType]: ((extended_enat > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_OldropWhile, type, coindu218763757pWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Olfilter, type, coinductive_lfilter: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollast, type, coinductive_llast: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollcp, type, coinductive_llcp: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > extended_enat))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollength, type, coinductive_llength: (!> [A:$tType]: (((coinductive_llist @ A) > extended_enat)))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_Ocase__llist, type, coindu1381640503_llist: (!> [B:$tType,A:$tType]: ((B > ((A > ((coinductive_llist @ A) > B)) > ((coinductive_llist @ A) > B)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist, type, coindu1259883913_llist: (!> [C:$tType,A:$tType]: (((C > $o) > ((C > A) > ((C > $o) > ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A)))))))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_Ollist__all2, type, coindu1486289336t_all2: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((coinductive_llist @ A) > ((coinductive_llist @ B) > $o)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_Olmap, type, coinductive_lmap: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((coinductive_llist @ A) > (coinductive_llist @ Aa)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ollist_Opred__llist, type, coindu543516966_llist: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > $o))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Olprefix, type, coinductive_lprefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Oltake, type, coinductive_ltake: (!> [A:$tType]: ((extended_enat > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_OltakeWhile, type, coindu501562517eWhile: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum, type, coindu780009021istsum: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 3.28/1.73 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 3.28/1.73 thf(sy_c_Fun_Omap__fun, type, map_fun: (!> [C:$tType,A:$tType,B:$tType,D:$tType]: (((C > A) > ((B > D) > ((A > B) > (C > D))))))).
% 3.28/1.73 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 3.28/1.73 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 3.28/1.73 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.28/1.73 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux, type, lMirro999291890or_aux: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 3.28/1.73 thf(sy_c_Nat_Onat_Ocase__nat, type, case_nat: (!> [A:$tType]: ((A > ((nat > A) > (nat > A)))))).
% 3.28/1.73 thf(sy_c_Nat_Oold_Onat_Orec__nat, type, rec_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > T)))))).
% 3.28/1.73 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 3.28/1.73 thf(sy_c_Pure_Otype, type, type: (!> [A:$tType]: ((itself @ A)))).
% 3.28/1.73 thf(sy_c_Quotient_OBex1__rel, type, bex1_rel: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > $o))))).
% 3.28/1.73 thf(sy_c_Quotient_OQuotient3, type, quotient3: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > $o)))))).
% 3.28/1.73 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.28/1.73 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.28/1.73 thf(sy_v_P, type, p: (a > (b > $o))).
% 3.28/1.73 thf(sy_v_acc_Ha____, type, acc_a: (coinductive_llist @ b)).
% 3.28/1.73 thf(sy_v_acca____, type, acca: (coinductive_llist @ a)).
% 3.28/1.73 thf(sy_v_xs_Ha____, type, xs_a: (coinductive_llist @ b)).
% 3.28/1.73 thf(sy_v_xsa____, type, xsa: (coinductive_llist @ a)).
% 3.28/1.73 thf(fact_0_local_OLNil_I2_J, axiom, (((((coindu1486289336t_all2 @ a) @ b) @ p) @ acca) @ acc_a)).
% 3.28/1.73 thf(fact_1_local_OLNil_I3_J, axiom, ((coinductive_lfinite @ a) @ acca)).
% 3.28/1.73 thf(fact_2_calculation, axiom, (((coinductive_llength @ a) @ (((lMirro999291890or_aux @ a) @ acca) @ xsa)) = ((coinductive_llength @ b) @ (((lMirro999291890or_aux @ b) @ acc_a) @ xs_a)))).
% 3.28/1.73 thf(fact_3_local_OLNil_I1_J, axiom, (((((coindu1486289336t_all2 @ a) @ b) @ p) @ (((lMirro999291890or_aux @ a) @ acca) @ xsa)) @ (((lMirro999291890or_aux @ b) @ acc_a) @ xs_a))).
% 3.28/1.73 thf(fact_4_llist__all2__llengthD, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => (((coinductive_llength @ A) @ Xs) = ((coinductive_llength @ B) @ Ys)))))).
% 3.28/1.73 thf(fact_5_llcp__same__conv__length, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_llcp @ A) @ Xs) @ Xs) = ((coinductive_llength @ A) @ Xs))))).
% 3.28/1.73 thf(fact_6_llength__lmap, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs:(coinductive_llist @ B)]: ((((coinductive_llength @ A) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)) = ((coinductive_llength @ B) @ Xs))))).
% 3.28/1.73 thf(fact_7_lprefix__llength__eq__imp__eq, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_llength @ A) @ Xs) = ((coinductive_llength @ A) @ Ys)) => (Xs = Ys)))))).
% 3.28/1.73 thf(fact_8_llength__ltakeWhile__all, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_llength @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = ((coinductive_llength @ A) @ Xs)) = ((((coindu501562517eWhile @ A) @ P) @ Xs) = Xs))))).
% 3.28/1.73 thf(fact_9_llist__all2__llength__ltakeWhileD, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),Q1:(A > $o),Q2:(B > $o)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((! [X:A,Y:B]: ((((P @ X) @ Y) => ((Q1 @ X) = (Q2 @ Y))))) => (((coinductive_llength @ A) @ (((coindu501562517eWhile @ A) @ Q1) @ Xs)) = ((coinductive_llength @ B) @ (((coindu501562517eWhile @ B) @ Q2) @ Ys)))))))).
% 3.28/1.73 thf(fact_10_llength__code, axiom, (! [A:$tType]: (((coinductive_llength @ A) = ((coindu246717305length @ A) @ (zero_zero @ nat)))))).
% 3.28/1.73 thf(fact_11_llength__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ extended_enat) @ extended_enat) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (^ [Y2:extended_enat,Z:extended_enat]: ((Y2 = Z)))) @ (coinductive_llength @ A)) @ (coinductive_llength @ B))))).
% 3.28/1.73 thf(fact_12_lprefix__refl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ Xs)))).
% 3.28/1.73 thf(fact_13_llist_Oleq__refl, axiom, (! [A:$tType,X2:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ X2) @ X2)))).
% 3.28/1.73 thf(fact_14_lfinite__lmap, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ A) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 3.28/1.73 thf(fact_15_lfinite__lmirror__aux, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Acc)))))).
% 3.28/1.73 thf(fact_16_llcp__lprefix1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_llcp @ A) @ Xs) @ Ys) = ((coinductive_llength @ A) @ Xs)))))).
% 3.28/1.73 thf(fact_17_llcp__lprefix2, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Ys) @ Xs) => ((((coinductive_llcp @ A) @ Xs) @ Ys) = ((coinductive_llength @ A) @ Ys)))))).
% 3.28/1.73 thf(fact_18_llcp__commute, axiom, (! [A:$tType]: (((coinductive_llcp @ A) = (^ [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A)]: ((((coinductive_llcp @ A) @ Ys2) @ Xs2))))))).
% 3.28/1.73 thf(fact_19_lmap__lprefix, axiom, (! [B:$tType,A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),F:(A > B)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ B) @ ((((coinductive_lmap @ A) @ B) @ F) @ Xs)) @ ((((coinductive_lmap @ A) @ B) @ F) @ Ys)))))).
% 3.28/1.73 thf(fact_20_lmap__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (C > $o)),B2:(B > (D > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((coinductive_llist @ A) > (coinductive_llist @ B))) @ ((coinductive_llist @ C) > (coinductive_llist @ D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ C)) @ (coinductive_llist @ B)) @ (coinductive_llist @ D)) @ (((coindu1486289336t_all2 @ A) @ C) @ A2)) @ (((coindu1486289336t_all2 @ B) @ D) @ B2))) @ ((coinductive_lmap @ A) @ B)) @ ((coinductive_lmap @ C) @ D))))).
% 3.28/1.73 thf(fact_21_lprefix__trans, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Zs) => (((coinductive_lprefix @ A) @ Xs) @ Zs)))))).
% 3.28/1.73 thf(fact_22_llist__all2__rsp, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),S:(A > (A > $o)),T2:(B > (B > $o)),X2:(coinductive_llist @ A),Y3:(coinductive_llist @ B),A3:(coinductive_llist @ A),B3:(coinductive_llist @ B)]: (((! [X:A,Y:B]: ((((R @ X) @ Y) => (! [A4:A,B4:B]: ((((R @ A4) @ B4) => (((S @ X) @ A4) = ((T2 @ Y) @ B4)))))))) => ((((((coindu1486289336t_all2 @ A) @ B) @ R) @ X2) @ Y3) => ((((((coindu1486289336t_all2 @ A) @ B) @ R) @ A3) @ B3) => ((((((coindu1486289336t_all2 @ A) @ A) @ S) @ X2) @ A3) = (((((coindu1486289336t_all2 @ B) @ B) @ T2) @ Y3) @ B3)))))))).
% 3.28/1.73 thf(fact_23_llist_Oleq__trans, axiom, (! [A:$tType,X2:(coinductive_llist @ A),Y3:(coinductive_llist @ A),Z2:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X2) @ Y3) => ((((coinductive_lprefix @ A) @ Y3) @ Z2) => (((coinductive_lprefix @ A) @ X2) @ Z2)))))).
% 3.28/1.73 thf(fact_24_llist__all2__mono, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),P2:(A > (B > $o))]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((! [X:A,Y:B]: ((((P @ X) @ Y) => ((P2 @ X) @ Y)))) => (((((coindu1486289336t_all2 @ A) @ B) @ P2) @ Xs) @ Ys)))))).
% 3.28/1.73 thf(fact_25_lprefix__antisym, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Xs) => (Xs = Ys)))))).
% 3.28/1.73 thf(fact_26_lprefix__lfiniteD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ Xs)))))).
% 3.28/1.73 thf(fact_27_llist_Oleq__antisym, axiom, (! [A:$tType,X2:(coinductive_llist @ A),Y3:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X2) @ Y3) => ((((coinductive_lprefix @ A) @ Y3) @ X2) => (X2 = Y3)))))).
% 3.28/1.73 thf(fact_28_lprefix__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) @ Xs)))).
% 3.28/1.73 thf(fact_29_llist__all2__lfiniteD, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => (((coinductive_lfinite @ A) @ Xs) = ((coinductive_lfinite @ B) @ Ys)))))).
% 3.28/1.73 thf(fact_30_lprefix__down__linear, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Zs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Zs) => ((((coinductive_lprefix @ A) @ Ys) @ Zs) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) | (((coinductive_lprefix @ A) @ Ys) @ Xs))))))).
% 3.28/1.73 thf(fact_31_not__lfinite__lprefix__conv__eq, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = (Xs = Ys)))))).
% 3.28/1.73 thf(fact_32_lmirror__aux__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Acc:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((lMirro999291890or_aux @ A) @ Acc) @ Xs) = Xs))))).
% 3.28/1.73 thf(fact_33_lmap__lmirror__aux, axiom, (! [A:$tType,B:$tType,F:(B > A),Acc:(coinductive_llist @ B),Xs:(coinductive_llist @ B)]: ((((((coinductive_lmap @ B) @ A) @ F) @ (((lMirro999291890or_aux @ B) @ Acc) @ Xs)) = (((lMirro999291890or_aux @ A) @ ((((coinductive_lmap @ B) @ A) @ F) @ Acc)) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)))))).
% 3.28/1.73 thf(fact_34_llist_Orel__eq, axiom, (! [A:$tType]: (((((coindu1486289336t_all2 @ A) @ A) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) = (^ [Y2:(coinductive_llist @ A),Z:(coinductive_llist @ A)]: ((Y2 = Z))))))).
% 3.28/1.73 thf(fact_35_llist_Orel__refl, axiom, (! [B:$tType,Ra:(B > (B > $o)),X2:(coinductive_llist @ B)]: (((! [X:B]: (((Ra @ X) @ X))) => (((((coindu1486289336t_all2 @ B) @ B) @ Ra) @ X2) @ X2))))).
% 3.28/1.73 thf(fact_36_llist__all2__lmirror__aux, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Acc:(coinductive_llist @ A),Acc2:(coinductive_llist @ B),Xs:(coinductive_llist @ A),Xs3:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Acc) @ Acc2) => ((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Xs3) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)) @ (((lMirro999291890or_aux @ B) @ Acc2) @ Xs3))))))).
% 3.28/1.73 thf(fact_37_llist_Omap__transfer, axiom, (! [A:$tType,B:$tType,F2:$tType,E:$tType,Rb:(A > (E > $o)),Sd:(B > (F2 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (E > F2)) @ ((coinductive_llist @ A) > (coinductive_llist @ B))) @ ((coinductive_llist @ E) > (coinductive_llist @ F2))) @ ((((((bNF_rel_fun @ A) @ E) @ B) @ F2) @ Rb) @ Sd)) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ E)) @ (coinductive_llist @ B)) @ (coinductive_llist @ F2)) @ (((coindu1486289336t_all2 @ A) @ E) @ Rb)) @ (((coindu1486289336t_all2 @ B) @ F2) @ Sd))) @ ((coinductive_lmap @ A) @ B)) @ ((coinductive_lmap @ E) @ F2))))).
% 3.28/1.73 thf(fact_38_rel__funI, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D)]: (((! [X:A,Y:B]: ((((A2 @ X) @ Y) => ((B2 @ (F @ X)) @ (G @ Y))))) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F) @ G))))).
% 3.28/1.73 thf(fact_39_Coinductive__List_Olprefix__nitpick__simps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A)]: (((((coinductive_lfinite @ A) @ Xs2) => (((coindu328551480prefix @ A) @ Xs2) @ Ys2)) & ((~ (((coinductive_lfinite @ A) @ Xs2))) => (Xs2 = Ys2))))))))).
% 3.28/1.73 thf(fact_40_llistsum__inf, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu780009021istsum @ A) @ Xs) = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_41_rel__funD, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D),X2:A,Y3:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F) @ G) => (((A2 @ X2) @ Y3) => ((B2 @ (F @ X2)) @ (G @ Y3))))))).
% 3.28/1.73 thf(fact_42_let__rsp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (B > $o)),R2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D)) @ R1) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ C) @ D) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R1) @ R2)) @ R2)) @ (^ [S2:A,F3:(A > C)]: ((F3 @ S2)))) @ (^ [S2:B,F3:(B > D)]: ((F3 @ S2))))))).
% 3.28/1.73 thf(fact_43_rel__funE, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D),X2:A,Y3:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F) @ G) => (((A2 @ X2) @ Y3) => ((B2 @ (F @ X2)) @ (G @ Y3))))))).
% 3.28/1.73 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 3.28/1.73 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A2)))) = A2)))).
% 3.28/1.73 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.28/1.73 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 3.28/1.73 thf(fact_48_apply__rsp_H, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (B > $o)),R2:(C > (D > $o)),F:(A > C),G:(B > D),X2:A,Y3:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R1) @ R2) @ F) @ G) => (((R1 @ X2) @ Y3) => ((R2 @ (F @ X2)) @ (G @ Y3))))))).
% 3.28/1.73 thf(fact_49_rel__funD2, axiom, (! [B:$tType,C:$tType,A:$tType,A2:(A > (A > $o)),B2:(B > (C > $o)),F:(A > B),G:(A > C),X2:A]: ((((((((((bNF_rel_fun @ A) @ A) @ B) @ C) @ A2) @ B2) @ F) @ G) => (((A2 @ X2) @ X2) => ((B2 @ (F @ X2)) @ (G @ X2))))))).
% 3.28/1.73 thf(fact_50_rel__fun__mono, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,X4:(A > (B > $o)),A2:(C > (D > $o)),F:(A > C),G:(B > D),Y4:(A > (B > $o)),B2:(C > (D > $o))]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ X4) @ A2) @ F) @ G) => ((! [X:A,Y:B]: ((((Y4 @ X) @ Y) => ((X4 @ X) @ Y)))) => ((! [X:C,Y:D]: ((((A2 @ X) @ Y) => ((B2 @ X) @ Y)))) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ Y4) @ B2) @ F) @ G))))))).
% 3.28/1.73 thf(fact_51_rel__fun__mono_H, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,Y4:(A > (B > $o)),X4:(A > (B > $o)),A2:(C > (D > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D)]: (((! [X:A,Y:B]: ((((Y4 @ X) @ Y) => ((X4 @ X) @ Y)))) => ((! [X:C,Y:D]: ((((A2 @ X) @ Y) => ((B2 @ X) @ Y)))) => (((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ X4) @ A2) @ F) @ G) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ Y4) @ B2) @ F) @ G))))))).
% 3.28/1.73 thf(fact_52_llist__all2__transfer, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ ((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))) @ ((coinductive_llist @ B) > ((coinductive_llist @ B) > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ R) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ ((coinductive_llist @ A) > $o)) @ ((coinductive_llist @ B) > $o)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))))) @ ((coindu1486289336t_all2 @ A) @ A)) @ ((coindu1486289336t_all2 @ B) @ B))))).
% 3.28/1.73 thf(fact_53_llist_Orel__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,Sa:(A > (C > $o)),Sc:(B > (D > $o))]: (((((((((bNF_rel_fun @ (A > (B > $o))) @ (C > (D > $o))) @ ((coinductive_llist @ A) > ((coinductive_llist @ B) > $o))) @ ((coinductive_llist @ C) > ((coinductive_llist @ D) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ (B > $o)) @ (D > $o)) @ Sa) @ ((((((bNF_rel_fun @ B) @ D) @ $o) @ $o) @ Sc) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ C)) @ ((coinductive_llist @ B) > $o)) @ ((coinductive_llist @ D) > $o)) @ (((coindu1486289336t_all2 @ A) @ C) @ Sa)) @ ((((((bNF_rel_fun @ (coinductive_llist @ B)) @ (coinductive_llist @ D)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ B) @ D) @ Sc)) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))))) @ ((coindu1486289336t_all2 @ A) @ B)) @ ((coindu1486289336t_all2 @ C) @ D))))).
% 3.28/1.73 thf(fact_54_ltakeWhile__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coindu501562517eWhile @ A)) @ (coindu501562517eWhile @ B))))).
% 3.28/1.73 thf(fact_55_lfinite__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))) @ (coinductive_lfinite @ A)) @ (coinductive_lfinite @ B))))).
% 3.28/1.73 thf(fact_56_Coinductive__List_Ofinite__lprefix__def, axiom, (! [A:$tType]: (((coindu328551480prefix @ A) = (coinductive_lprefix @ A))))).
% 3.28/1.73 thf(fact_57_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type @ A)) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_58_Let__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D)) @ A2) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ C) @ D) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ B2)) @ (^ [S2:A,F3:(A > C)]: ((F3 @ S2)))) @ (^ [S2:B,F3:(B > D)]: ((F3 @ S2))))))).
% 3.28/1.73 thf(fact_59_rel__fun__def__butlast, axiom, (! [B:$tType,D:$tType,C:$tType,E:$tType,F2:$tType,A:$tType,R:(A > (B > $o)),S:(C > (E > $o)),T2:(D > (F2 > $o)),F:(A > (C > D)),G:(B > (E > F2))]: ((((((((((bNF_rel_fun @ A) @ B) @ (C > D)) @ (E > F2)) @ R) @ ((((((bNF_rel_fun @ C) @ E) @ D) @ F2) @ S) @ T2)) @ F) @ G) = (! [X3:A,Y5:B]: ((((R @ X3) @ Y5) => ((((((((bNF_rel_fun @ C) @ E) @ D) @ F2) @ S) @ T2) @ (F @ X3)) @ (G @ Y5))))))))).
% 3.28/1.73 thf(fact_60_If__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ $o) @ $o) @ (A > (A > A))) @ (B > (B > B))) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2))) @ (if @ A)) @ (if @ B))))).
% 3.28/1.73 thf(fact_61_llist__corec__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((A > C) > ((A > $o) > ((A > (coinductive_llist @ C)) > ((A > A) > (A > (coinductive_llist @ C))))))) @ ((B > D) > ((B > $o) > ((B > (coinductive_llist @ D)) > ((B > B) > (B > (coinductive_llist @ D))))))) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ ((A > $o) > ((A > (coinductive_llist @ C)) > ((A > A) > (A > (coinductive_llist @ C)))))) @ ((B > $o) > ((B > (coinductive_llist @ D)) > ((B > B) > (B > (coinductive_llist @ D)))))) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((A > (coinductive_llist @ C)) > ((A > A) > (A > (coinductive_llist @ C))))) @ ((B > (coinductive_llist @ D)) > ((B > B) > (B > (coinductive_llist @ D))))) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (A > (coinductive_llist @ C))) @ (B > (coinductive_llist @ D))) @ ((A > A) > (A > (coinductive_llist @ C)))) @ ((B > B) > (B > (coinductive_llist @ D)))) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A2) @ (((coindu1486289336t_all2 @ C) @ D) @ B2))) @ ((((((bNF_rel_fun @ (A > A)) @ (B > B)) @ (A > (coinductive_llist @ C))) @ (B > (coinductive_llist @ D))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2)) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A2) @ (((coindu1486289336t_all2 @ C) @ D) @ B2))))))) @ ((coindu1259883913_llist @ A) @ C)) @ ((coindu1259883913_llist @ B) @ D))))).
% 3.28/1.73 thf(fact_62_llist_Ocorec__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,S:(C > (D > $o)),R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (C > $o)) @ (D > $o)) @ ((C > A) > ((C > $o) > ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A))))))) @ ((D > B) > ((D > $o) > ((D > (coinductive_llist @ B)) > ((D > D) > (D > (coinductive_llist @ B))))))) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ S) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (C > A)) @ (D > B)) @ ((C > $o) > ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A)))))) @ ((D > $o) > ((D > (coinductive_llist @ B)) > ((D > D) > (D > (coinductive_llist @ B)))))) @ ((((((bNF_rel_fun @ C) @ D) @ A) @ B) @ S) @ R)) @ ((((((bNF_rel_fun @ (C > $o)) @ (D > $o)) @ ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A))))) @ ((D > (coinductive_llist @ B)) > ((D > D) > (D > (coinductive_llist @ B))))) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ S) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (C > (coinductive_llist @ A))) @ (D > (coinductive_llist @ B))) @ ((C > C) > (C > (coinductive_llist @ A)))) @ ((D > D) > (D > (coinductive_llist @ B)))) @ ((((((bNF_rel_fun @ C) @ D) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ S) @ (((coindu1486289336t_all2 @ A) @ B) @ R))) @ ((((((bNF_rel_fun @ (C > C)) @ (D > D)) @ (C > (coinductive_llist @ A))) @ (D > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ C) @ D) @ C) @ D) @ S) @ S)) @ ((((((bNF_rel_fun @ C) @ D) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ S) @ (((coindu1486289336t_all2 @ A) @ B) @ R))))))) @ ((coindu1259883913_llist @ C) @ A)) @ ((coindu1259883913_llist @ D) @ B))))).
% 3.28/1.73 thf(fact_63_unfold__llist__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((A > C) > ((A > A) > (A > (coinductive_llist @ C))))) @ ((B > D) > ((B > B) > (B > (coinductive_llist @ D))))) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ ((A > A) > (A > (coinductive_llist @ C)))) @ ((B > B) > (B > (coinductive_llist @ D)))) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (A > A)) @ (B > B)) @ (A > (coinductive_llist @ C))) @ (B > (coinductive_llist @ D))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2)) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A2) @ (((coindu1486289336t_all2 @ C) @ D) @ B2))))) @ ((coindu1441602521_llist @ A) @ C)) @ ((coindu1441602521_llist @ B) @ D))))).
% 3.28/1.73 thf(fact_64_iterates__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > A)) @ (B > B)) @ (A > (coinductive_llist @ A))) @ (B > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2)) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ A2) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_iterates @ A)) @ (coinductive_iterates @ B))))).
% 3.28/1.73 thf(fact_65_case__llist__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,B2:(A > (B > $o)),A2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((C > ((coinductive_llist @ C) > A)) > ((coinductive_llist @ C) > A))) @ ((D > ((coinductive_llist @ D) > B)) > ((coinductive_llist @ D) > B))) @ B2) @ ((((((bNF_rel_fun @ (C > ((coinductive_llist @ C) > A))) @ (D > ((coinductive_llist @ D) > B))) @ ((coinductive_llist @ C) > A)) @ ((coinductive_llist @ D) > B)) @ ((((((bNF_rel_fun @ C) @ D) @ ((coinductive_llist @ C) > A)) @ ((coinductive_llist @ D) > B)) @ A2) @ ((((((bNF_rel_fun @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A) @ B) @ (((coindu1486289336t_all2 @ C) @ D) @ A2)) @ B2))) @ ((((((bNF_rel_fun @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A) @ B) @ (((coindu1486289336t_all2 @ C) @ D) @ A2)) @ B2))) @ ((coindu1381640503_llist @ A) @ C)) @ ((coindu1381640503_llist @ B) @ D))))).
% 3.28/1.73 thf(fact_66_llist_Ocase__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,S:(C > (D > $o)),R:(A > (B > $o))]: (((((((((bNF_rel_fun @ C) @ D) @ ((A > ((coinductive_llist @ A) > C)) > ((coinductive_llist @ A) > C))) @ ((B > ((coinductive_llist @ B) > D)) > ((coinductive_llist @ B) > D))) @ S) @ ((((((bNF_rel_fun @ (A > ((coinductive_llist @ A) > C))) @ (B > ((coinductive_llist @ B) > D))) @ ((coinductive_llist @ A) > C)) @ ((coinductive_llist @ B) > D)) @ ((((((bNF_rel_fun @ A) @ B) @ ((coinductive_llist @ A) > C)) @ ((coinductive_llist @ B) > D)) @ R) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ C) @ D) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ S))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ C) @ D) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ S))) @ ((coindu1381640503_llist @ C) @ A)) @ ((coindu1381640503_llist @ D) @ B))))).
% 3.28/1.73 thf(fact_67_lfinite__iterates, axiom, (! [A:$tType,F:(A > A),X2:A]: ((~ (((coinductive_lfinite @ A) @ (((coinductive_iterates @ A) @ F) @ X2))))))).
% 3.28/1.73 thf(fact_68_lmap__iterates, axiom, (! [A:$tType,F:(A > A),X2:A]: ((((((coinductive_lmap @ A) @ A) @ F) @ (((coinductive_iterates @ A) @ F) @ X2)) = (((coinductive_iterates @ A) @ F) @ (F @ X2)))))).
% 3.28/1.73 thf(fact_69_fun_Orel__transfer, axiom, (! [B:$tType,A:$tType,C:$tType,E:$tType,D:$tType,Sa:(A > (C > $o)),Sc:(B > (E > $o))]: (((((((((bNF_rel_fun @ (A > (B > $o))) @ (C > (E > $o))) @ ((D > A) > ((D > B) > $o))) @ ((D > C) > ((D > E) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ (B > $o)) @ (E > $o)) @ Sa) @ ((((((bNF_rel_fun @ B) @ E) @ $o) @ $o) @ Sc) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))))) @ ((((((bNF_rel_fun @ (D > A)) @ (D > C)) @ ((D > B) > $o)) @ ((D > E) > $o)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y2:D,Z:D]: ((Y2 = Z)))) @ Sa)) @ ((((((bNF_rel_fun @ (D > B)) @ (D > E)) @ $o) @ $o) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ E) @ (^ [Y2:D,Z:D]: ((Y2 = Z)))) @ Sc)) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))))) @ (((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y2:D,Z:D]: ((Y2 = Z))))) @ (((((bNF_rel_fun @ D) @ D) @ C) @ E) @ (^ [Y2:D,Z:D]: ((Y2 = Z)))))))).
% 3.28/1.73 thf(fact_70_ldropWhile__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coindu218763757pWhile @ A)) @ (coindu218763757pWhile @ B))))).
% 3.28/1.73 thf(fact_71_map__fun__parametric, axiom, (! [A:$tType,B:$tType,E:$tType,F2:$tType,H:$tType,G2:$tType,D:$tType,C:$tType,A2:(A > (C > $o)),B2:(B > (D > $o)),C2:(E > (G2 > $o)),D2:(F2 > (H > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > F2) > ((B > E) > (A > F2)))) @ ((G2 > H) > ((D > G2) > (C > H)))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (E > F2)) @ (G2 > H)) @ ((B > E) > (A > F2))) @ ((D > G2) > (C > H))) @ ((((((bNF_rel_fun @ E) @ G2) @ F2) @ H) @ C2) @ D2)) @ ((((((bNF_rel_fun @ (B > E)) @ (D > G2)) @ (A > F2)) @ (C > H)) @ ((((((bNF_rel_fun @ B) @ D) @ E) @ G2) @ B2) @ C2)) @ ((((((bNF_rel_fun @ A) @ C) @ F2) @ H) @ A2) @ D2)))) @ ((((map_fun @ A) @ B) @ E) @ F2)) @ ((((map_fun @ C) @ D) @ G2) @ H))))).
% 3.28/1.73 thf(fact_72_lfilter__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_lfilter @ A)) @ (coinductive_lfilter @ B))))).
% 3.28/1.73 thf(fact_73_llist_Opred__transfer, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((coinductive_llist @ A) > $o)) @ ((coinductive_llist @ B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ (coindu543516966_llist @ A)) @ (coindu543516966_llist @ B))))).
% 3.28/1.73 thf(fact_74_llist__all__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((coinductive_llist @ A) > $o)) @ ((coinductive_llist @ B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ (coindu543516966_llist @ A)) @ (coindu543516966_llist @ B))))).
% 3.28/1.73 thf(fact_75_if__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((((((bNF_rel_fun @ $o) @ $o) @ (A > (A > A))) @ (A > (A > A))) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))) @ ((((((bNF_rel_fun @ A) @ A) @ (A > A)) @ (A > A)) @ R) @ ((((((bNF_rel_fun @ A) @ A) @ A) @ A) @ R) @ R))) @ (if @ A)) @ (if @ A)))))).
% 3.28/1.73 thf(fact_76_lfilter__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))).
% 3.28/1.73 thf(fact_77_fun__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R2:(C > (C > $o)),Abs2:(C > D),Rep2:(D > C)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => ((((((quotient3 @ C) @ D) @ R2) @ Abs2) @ Rep2) => (((((quotient3 @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R1) @ R2)) @ ((((((map_fun @ B) @ A) @ C) @ D) @ Rep1) @ Abs2)) @ ((((((map_fun @ A) @ B) @ D) @ C) @ Abs1) @ Rep2))))))).
% 3.28/1.73 thf(fact_78_let__prs, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),Abs22:(C > D),Rep22:(D > C)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ C) @ D) @ R2) @ Abs22) @ Rep22) => ((((((((map_fun @ D) @ C) @ ((C > A) > A)) @ ((D > B) > B)) @ Rep22) @ ((((((map_fun @ (D > B)) @ (C > A)) @ A) @ B) @ ((((((map_fun @ C) @ D) @ B) @ A) @ Abs22) @ Rep12)) @ Abs12)) @ (^ [S2:C,F3:(C > A)]: ((F3 @ S2)))) = (^ [S2:D,F3:(D > B)]: ((F3 @ S2))))))))).
% 3.28/1.73 thf(fact_79_cond__prs, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Absf:(A > B),Repf:(B > A),A3:$o,B3:B,C3:B]: (((((((quotient3 @ A) @ B) @ R) @ Absf) @ Repf) => ((A3 => ((Absf @ ((((if @ A) @ A3) @ (Repf @ B3)) @ (Repf @ C3))) = B3)) & ((~ (A3)) => ((Absf @ ((((if @ A) @ A3) @ (Repf @ B3)) @ (Repf @ C3))) = C3))))))).
% 3.28/1.73 thf(fact_80_Quotient3I, axiom, (! [B:$tType,A:$tType,Abs:(B > A),Rep:(A > B),R:(B > (B > $o))]: (((! [A4:A]: (((Abs @ (Rep @ A4)) = A4))) => ((! [A4:A]: (((R @ (Rep @ A4)) @ (Rep @ A4)))) => ((! [R3:B,S3:B]: ((((R @ R3) @ S3) = (((R @ R3) @ R3) & (((R @ S3) @ S3) & ((Abs @ R3) = (Abs @ S3))))))) => (((((quotient3 @ B) @ A) @ R) @ Abs) @ Rep))))))).
% 3.28/1.73 thf(fact_81_equals__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),Xa:A,Xb:A,Ya:A,Yb:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ Xa) @ Xb) => (((R @ Ya) @ Yb) => (((R @ Xa) @ Ya) = ((R @ Xb) @ Yb)))))))).
% 3.28/1.73 thf(fact_82_rep__abs__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ X1) @ (Rep @ (Abs @ X22)))))))).
% 3.28/1.73 thf(fact_83_Quotient3__def, axiom, (! [B:$tType,A:$tType]: ((((quotient3 @ A) @ B) = (^ [R4:(A > (A > $o)),Abs3:(A > B),Rep3:(B > A)]: (((! [A5:B]: (((Abs3 @ (Rep3 @ A5)) = A5))) & ((! [A5:B]: (((R4 @ (Rep3 @ A5)) @ (Rep3 @ A5)))) & (! [R5:A,S2:A]: ((((R4 @ R5) @ S2) = (((R4 @ R5) @ R5) & (((R4 @ S2) @ S2) & ((Abs3 @ R5) = (Abs3 @ S2))))))))))))))).
% 3.28/1.73 thf(fact_84_Quotient3__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R6:A,S4:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((R @ R6) @ R6) & (((R @ S4) @ S4) & ((Abs @ R6) = (Abs @ S4)))) = ((R @ R6) @ S4)))))).
% 3.28/1.73 thf(fact_85_Quotient3__refl1, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R6:A,S4:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R6) @ S4) => ((R @ R6) @ R6)))))).
% 3.28/1.73 thf(fact_86_Quotient3__refl2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R6:A,S4:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R6) @ S4) => ((R @ S4) @ S4)))))).
% 3.28/1.73 thf(fact_87_rep__abs__rsp__left, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ (Rep @ (Abs @ X1))) @ X22)))))).
% 3.28/1.73 thf(fact_88_Quotient3__abs__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((Abs @ (Rep @ A3)) = A3))))).
% 3.28/1.73 thf(fact_89_Quotient3__rel__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R6:A,S4:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R6) @ S4) => ((Abs @ R6) = (Abs @ S4))))))).
% 3.28/1.73 thf(fact_90_Quotient3__rel__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A3:B,B3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ (Rep @ A3)) @ (Rep @ B3)) = (A3 = B3)))))).
% 3.28/1.73 thf(fact_91_Quotient3__rep__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R6:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R6) @ R6) => ((R @ (Rep @ (Abs @ R6))) @ R6)))))).
% 3.28/1.73 thf(fact_92_Quotient3__rep__reflp, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((R @ (Rep @ A3)) @ (Rep @ A3)))))).
% 3.28/1.73 thf(fact_93_lprefix__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) @ (((coinductive_lfilter @ A) @ P) @ Ys)))))).
% 3.28/1.73 thf(fact_94_llist__all2__lfilterI, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),Q1:(A > $o),Q2:(B > $o)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((! [X:A,Y:B]: ((((P @ X) @ Y) => ((Q1 @ X) = (Q2 @ Y))))) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_lfilter @ A) @ Q1) @ Xs)) @ (((coinductive_lfilter @ B) @ Q2) @ Ys))))))).
% 3.28/1.73 thf(fact_95_apply__rspQ3_H_H, axiom, (! [C:$tType,A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),S:(C > (C > $o)),F:(A > C),X2:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R) @ S) @ F) @ F) => ((S @ (F @ (Rep @ X2))) @ (F @ (Rep @ X2)))))))).
% 3.28/1.73 thf(fact_96_apply__rspQ3, axiom, (! [B:$tType,C:$tType,A:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),F:(A > C),G:(A > C),X2:A,Y3:A]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => (((((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R1) @ R2) @ F) @ G) => (((R1 @ X2) @ Y3) => ((R2 @ (F @ X2)) @ (G @ Y3)))))))).
% 3.28/1.73 thf(fact_97_lfinite__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)))))).
% 3.28/1.73 thf(fact_98_llist__all2__ldropWhileI, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),Q1:(A > $o),Q2:(B > $o)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((! [X:A,Y:B]: ((((P @ X) @ Y) => ((Q1 @ X) = (Q2 @ Y))))) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coindu218763757pWhile @ A) @ Q1) @ Xs)) @ (((coindu218763757pWhile @ B) @ Q2) @ Ys))))))).
% 3.28/1.73 thf(fact_99_quot__rel__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((((((bNF_rel_fun @ A) @ A) @ (A > $o)) @ (A > $o)) @ R) @ ((((((bNF_rel_fun @ A) @ A) @ $o) @ $o) @ R) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ R) @ R))))).
% 3.28/1.73 thf(fact_100_fun_Orel__refl, axiom, (! [B:$tType,D:$tType,Ra:(B > (B > $o)),X2:(D > B)]: (((! [X:B]: (((Ra @ X) @ X))) => ((((((((bNF_rel_fun @ D) @ D) @ B) @ B) @ (^ [Y2:D,Z:D]: ((Y2 = Z)))) @ Ra) @ X2) @ X2))))).
% 3.28/1.73 thf(fact_101_fun_Orel__eq, axiom, (! [A:$tType,D:$tType]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ A) @ (^ [Y2:D,Z:D]: ((Y2 = Z)))) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) = (^ [Y2:(D > A),Z:(D > A)]: ((Y2 = Z))))))).
% 3.28/1.73 thf(fact_102_map__fun__apply, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType]: ((((((map_fun @ B) @ C) @ D) @ A) = (^ [F3:(B > C),G3:(D > A),H2:(C > D),X3:B]: ((G3 @ (H2 @ (F3 @ X3))))))))).
% 3.28/1.73 thf(fact_103_bex1__rel__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Absf:(A > B),Repf:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Absf) @ Repf) => ((((((((bNF_rel_fun @ (A > $o)) @ (A > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ A) @ $o) @ $o) @ R) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z))))) @ (^ [Y2:$o,Z:$o]: ((Y2 = Z)))) @ ((bex1_rel @ A) @ R)) @ ((bex1_rel @ A) @ R)))))).
% 3.28/1.73 thf(fact_104_llist_Octr__transfer_I2_J, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ R) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ (((coindu1486289336t_all2 @ A) @ B) @ R))) @ (coinductive_LCons @ A)) @ (coinductive_LCons @ B))))).
% 3.28/1.73 thf(fact_105_LCons__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ A2) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_LCons @ A)) @ (coinductive_LCons @ B))))).
% 3.28/1.73 thf(fact_106_ldropWhile__eq__ldrop, axiom, (! [A:$tType]: (((coindu218763757pWhile @ A) = (^ [P3:(A > $o),Xs2:(coinductive_llist @ A)]: ((((coinductive_ldrop @ A) @ ((coinductive_llength @ A) @ (((coindu501562517eWhile @ A) @ P3) @ Xs2))) @ Xs2))))))).
% 3.28/1.73 thf(fact_107_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 3.28/1.73 thf(fact_108_llist_Oinject, axiom, (! [A:$tType,X21:A,X222:(coinductive_llist @ A),Y21:A,Y22:(coinductive_llist @ A)]: ((((((coinductive_LCons @ A) @ X21) @ X222) = (((coinductive_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X222 = Y22)))))).
% 3.28/1.73 thf(fact_109_ldrop__0, axiom, (! [B:$tType,Xs:(coinductive_llist @ B)]: (((((coinductive_ldrop @ B) @ (zero_zero @ extended_enat)) @ Xs) = Xs)))).
% 3.28/1.73 thf(fact_110_LCons__lprefix__LCons, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = ((X2 = Y3) & (((coinductive_lprefix @ A) @ Xs) @ Ys)))))).
% 3.28/1.73 thf(fact_111_llist__all2__LCons__LCons, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X21:A,X222:(coinductive_llist @ A),Y21:B,Y22:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ R) @ (((coinductive_LCons @ A) @ X21) @ X222)) @ (((coinductive_LCons @ B) @ Y21) @ Y22)) = (((R @ X21) @ Y21) & (((((coindu1486289336t_all2 @ A) @ B) @ R) @ X222) @ Y22)))))).
% 3.28/1.73 thf(fact_112_lfinite__code_I2_J, axiom, (! [B:$tType,X2:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X2) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 3.28/1.73 thf(fact_113_lfinite__LCons, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 3.28/1.73 thf(fact_114_lfilter__LCons, axiom, (! [A:$tType,P:(A > $o),X2:A,Xs:(coinductive_llist @ A)]: ((((P @ X2) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = (((coinductive_LCons @ A) @ X2) @ (((coinductive_lfilter @ A) @ P) @ Xs)))) & ((~ ((P @ X2))) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 3.28/1.73 thf(fact_115_ldropWhile__LCons, axiom, (! [A:$tType,P:(A > $o),X2:A,Xs:(coinductive_llist @ A)]: ((((P @ X2) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = (((coindu218763757pWhile @ A) @ P) @ Xs))) & ((~ ((P @ X2))) => ((((coindu218763757pWhile @ A) @ P) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = (((coinductive_LCons @ A) @ X2) @ Xs))))))).
% 3.28/1.73 thf(fact_116_ldrop__lmap, axiom, (! [A:$tType,B:$tType,N:extended_enat,F:(B > A),Xs:(coinductive_llist @ B)]: (((((coinductive_ldrop @ A) @ N) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)) = ((((coinductive_lmap @ B) @ A) @ F) @ (((coinductive_ldrop @ B) @ N) @ Xs)))))).
% 3.28/1.73 thf(fact_117_lmirror__aux__simps_I2_J, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xa:A,X2:(coinductive_llist @ A)]: (((((lMirro999291890or_aux @ A) @ Acc) @ (((coinductive_LCons @ A) @ Xa) @ X2)) = (((coinductive_LCons @ A) @ Xa) @ (((lMirro999291890or_aux @ A) @ (((coinductive_LCons @ A) @ Xa) @ Acc)) @ X2)))))).
% 3.28/1.73 thf(fact_118_unfold__llist__eq__LCons, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B3:B,X2:A,Xs:(coinductive_llist @ A)]: (((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ B3) = (((coinductive_LCons @ A) @ X2) @ Xs)) = ((~ ((IS_LNIL @ B3))) & ((X2 = (LHD @ B3)) & (Xs = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B3))))))))).
% 3.28/1.73 thf(fact_119_llist_Opred__inject_I2_J, axiom, (! [A:$tType,P:(A > $o),A3:A,Aa2:(coinductive_llist @ A)]: (((((coindu543516966_llist @ A) @ P) @ (((coinductive_LCons @ A) @ A3) @ Aa2)) = ((P @ A3) & (((coindu543516966_llist @ A) @ P) @ Aa2)))))).
% 3.28/1.73 thf(fact_120_bex1__rel__aux2, axiom, (! [A:$tType,R:(A > (A > $o)),X2:(A > $o),Y3:(A > $o)]: (((! [Xa2:A,Ya2:A]: ((((R @ Xa2) @ Ya2) => ((X2 @ Xa2) = (Y3 @ Ya2))))) => ((((bex1_rel @ A) @ R) @ Y3) => (((bex1_rel @ A) @ R) @ X2)))))).
% 3.28/1.73 thf(fact_121_bex1__rel__aux, axiom, (! [A:$tType,R:(A > (A > $o)),X2:(A > $o),Y3:(A > $o)]: (((! [Xa2:A,Ya2:A]: ((((R @ Xa2) @ Ya2) => ((X2 @ Xa2) = (Y3 @ Ya2))))) => ((((bex1_rel @ A) @ R) @ X2) => (((bex1_rel @ A) @ R) @ Y3)))))).
% 3.28/1.73 thf(fact_122_Le__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),X2:A]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)) @ (((coinductive_LCons @ A) @ X2) @ Ys)))))).
% 3.28/1.73 thf(fact_123_LCons__lprefix__conv, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)) @ Ys) = (? [Ys3:(coinductive_llist @ A)]: (((Ys = (((coinductive_LCons @ A) @ X2) @ Ys3)) & (((coinductive_lprefix @ A) @ Xs) @ Ys3)))))))).
% 3.28/1.73 thf(fact_124_llist_Orel__intros_I2_J, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X21:A,Y21:B,X222:(coinductive_llist @ A),Y22:(coinductive_llist @ B)]: ((((R @ X21) @ Y21) => ((((((coindu1486289336t_all2 @ A) @ B) @ R) @ X222) @ Y22) => (((((coindu1486289336t_all2 @ A) @ B) @ R) @ (((coinductive_LCons @ A) @ X21) @ X222)) @ (((coinductive_LCons @ B) @ Y21) @ Y22))))))).
% 3.28/1.73 thf(fact_125_llist__all2__LCons1, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X2:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_LCons @ A) @ X2) @ Xs)) @ Ys) = (? [Y5:B,Ys3:(coinductive_llist @ B)]: (((Ys = (((coinductive_LCons @ B) @ Y5) @ Ys3)) & (((P @ X2) @ Y5) & (((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys3))))))))).
% 3.28/1.73 thf(fact_126_llist__all2__LCons2, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Y3:B,Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ (((coinductive_LCons @ B) @ Y3) @ Ys)) = (? [X3:A,Xs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X3) @ Xs4)) & (((P @ X3) @ Y3) & (((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs4) @ Ys))))))))).
% 3.28/1.73 thf(fact_127_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X2:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)))))).
% 3.28/1.73 thf(fact_128_llist_Osimps_I13_J, axiom, (! [B:$tType,A:$tType,F:(A > B),X21:A,X222:(coinductive_llist @ A)]: ((((((coinductive_lmap @ A) @ B) @ F) @ (((coinductive_LCons @ A) @ X21) @ X222)) = (((coinductive_LCons @ B) @ (F @ X21)) @ ((((coinductive_lmap @ A) @ B) @ F) @ X222)))))).
% 3.28/1.73 thf(fact_129_lmap__eq__LCons__conv, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs:(coinductive_llist @ B),Y3:A,Ys:(coinductive_llist @ A)]: (((((((coinductive_lmap @ B) @ A) @ F) @ Xs) = (((coinductive_LCons @ A) @ Y3) @ Ys)) = (? [X3:B,Xs4:(coinductive_llist @ B)]: (((Xs = (((coinductive_LCons @ B) @ X3) @ Xs4)) & ((Y3 = (F @ X3)) & (Ys = ((((coinductive_lmap @ B) @ A) @ F) @ Xs4)))))))))).
% 3.28/1.73 thf(fact_130_llist__all2__ldropI, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),N:extended_enat]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_ldrop @ A) @ N) @ Xs)) @ (((coinductive_ldrop @ B) @ N) @ Ys)))))).
% 3.28/1.73 thf(fact_131_llist_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F22:(A > ((coinductive_llist @ A) > B)),X21:A,X222:(coinductive_llist @ A)]: (((((((coindu1381640503_llist @ B) @ A) @ F1) @ F22) @ (((coinductive_LCons @ A) @ X21) @ X222)) = ((F22 @ X21) @ X222))))).
% 3.28/1.73 thf(fact_132_lfilter__LCons__found, axiom, (! [A:$tType,P:(A > $o),X2:A,Xs:(coinductive_llist @ A)]: (((P @ X2) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = (((coinductive_LCons @ A) @ X2) @ (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 3.28/1.73 thf(fact_133_lfilter__LCons__seek, axiom, (! [A:$tType,P4:(A > $o),X2:A,L:(coinductive_llist @ A)]: (((~ ((P4 @ X2))) => ((((coinductive_lfilter @ A) @ P4) @ (((coinductive_LCons @ A) @ X2) @ L)) = (((coinductive_lfilter @ A) @ P4) @ L)))))).
% 3.28/1.73 thf(fact_134_iterates_Ocode, axiom, (! [A:$tType]: (((coinductive_iterates @ A) = (^ [F3:(A > A),X3:A]: ((((coinductive_LCons @ A) @ X3) @ (((coinductive_iterates @ A) @ F3) @ (F3 @ X3))))))))).
% 3.28/1.73 thf(fact_135_unfold__llist_Octr_I2_J, axiom, (! [B:$tType,A:$tType,P4:(A > $o),A3:A,G21:(A > B),G22:(A > A)]: (((~ ((P4 @ A3))) => (((((((coindu1441602521_llist @ A) @ B) @ P4) @ G21) @ G22) @ A3) = (((coinductive_LCons @ B) @ (G21 @ A3)) @ ((((((coindu1441602521_llist @ A) @ B) @ P4) @ G21) @ G22) @ (G22 @ A3)))))))).
% 3.28/1.73 thf(fact_136_llist_Ocorec_I2_J, axiom, (! [A:$tType,C:$tType,P4:(C > $o),A3:C,G21:(C > A),Q22:(C > $o),G221:(C > (coinductive_llist @ A)),G222:(C > C)]: (((~ ((P4 @ A3))) => (((((((((coindu1259883913_llist @ C) @ A) @ P4) @ G21) @ Q22) @ G221) @ G222) @ A3) = (((coinductive_LCons @ A) @ (G21 @ A3)) @ ((((if @ (coinductive_llist @ A)) @ (Q22 @ A3)) @ (G221 @ A3)) @ ((((((((coindu1259883913_llist @ C) @ A) @ P4) @ G21) @ Q22) @ G221) @ G222) @ (G222 @ A3))))))))).
% 3.28/1.73 thf(fact_137_iterates__equality, axiom, (! [A:$tType,H3:(A > (coinductive_llist @ A)),F:(A > A)]: (((! [X:A]: (((H3 @ X) = (((coinductive_LCons @ A) @ X) @ ((((coinductive_lmap @ A) @ A) @ F) @ (H3 @ X)))))) => (H3 = ((coinductive_iterates @ A) @ F)))))).
% 3.28/1.73 thf(fact_138_iterates__lmap, axiom, (! [A:$tType]: (((coinductive_iterates @ A) = (^ [F3:(A > A),X3:A]: ((((coinductive_LCons @ A) @ X3) @ ((((coinductive_lmap @ A) @ A) @ F3) @ (((coinductive_iterates @ A) @ F3) @ X3))))))))).
% 3.28/1.73 thf(fact_139_ldrop__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ extended_enat) @ extended_enat) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ (^ [Y2:extended_enat,Z:extended_enat]: ((Y2 = Z)))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_ldrop @ A)) @ (coinductive_ldrop @ B))))).
% 3.28/1.73 thf(fact_140_natural_Osize_I1_J, axiom, ((code_size_natural @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 3.28/1.73 thf(fact_141_llistsum__LCons, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [Xs:(coinductive_llist @ A),X2:A]: ((((coinductive_lfinite @ A) @ Xs) => (((coindu780009021istsum @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)) = (((plus_plus @ A) @ X2) @ ((coindu780009021istsum @ A) @ Xs)))))))))).
% 3.28/1.73 thf(fact_142_llist__all2__lappend1D_I2_J, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Xs3:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Xs3)) @ Ys) => (((coinductive_lfinite @ A) @ Xs) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs3) @ (((coinductive_ldrop @ B) @ ((coinductive_llength @ A) @ Xs)) @ Ys))))))).
% 3.28/1.73 thf(fact_143_natural_Osize_I3_J, axiom, (((size_size @ code_natural) @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 3.28/1.73 thf(fact_144_lmember__code_I2_J, axiom, (! [A:$tType,X2:A,Y3:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lmember @ A) @ X2) @ (((coinductive_LCons @ A) @ Y3) @ Ys)) = ((X2 = Y3) | (((coinductive_lmember @ A) @ X2) @ Ys)))))).
% 3.28/1.73 thf(fact_145_add__left__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [A3:A,B3:A,C3:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C3)) = (B3 = C3)))))))).
% 3.28/1.73 thf(fact_146_add__right__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [B3:A,A3:A,C3:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C3) @ A3)) = (B3 = C3)))))))).
% 3.28/1.73 thf(fact_147_ldrop__ldrop, axiom, (! [A:$tType,N:extended_enat,M:extended_enat,Xs:(coinductive_llist @ A)]: (((((coinductive_ldrop @ A) @ N) @ (((coinductive_ldrop @ A) @ M) @ Xs)) = (((coinductive_ldrop @ A) @ (((plus_plus @ extended_enat) @ N) @ M)) @ Xs))))).
% 3.28/1.73 thf(fact_148_add__cancel__right__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [A3:A,B3:A]: (((A3 = (((plus_plus @ A) @ A3) @ B3)) = (B3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_149_add__cancel__right__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [A3:A,B3:A]: (((A3 = (((plus_plus @ A) @ B3) @ A3)) = (B3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_150_add__cancel__left__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [A3:A,B3:A]: ((((((plus_plus @ A) @ A3) @ B3) = A3) = (B3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_151_add__cancel__left__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [B3:A,A3:A]: ((((((plus_plus @ A) @ B3) @ A3) = A3) = (B3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_152_double__zero__sym, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A3:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A3) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_153_double__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A3:A]: ((((((plus_plus @ A) @ A3) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_154_add_Oright__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 3.28/1.73 thf(fact_155_add_Oleft__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 3.28/1.73 thf(fact_156_lappend__code_I2_J, axiom, (! [A:$tType,Xa:A,X2:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_LCons @ A) @ Xa) @ X2)) @ Ys) = (((coinductive_LCons @ A) @ Xa) @ (((coinductive_lappend @ A) @ X2) @ Ys)))))).
% 3.28/1.73 thf(fact_157_llength__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_llength @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((plus_plus @ extended_enat) @ ((coinductive_llength @ A) @ Xs)) @ ((coinductive_llength @ A) @ Ys)))))).
% 3.28/1.73 thf(fact_158_lfinite__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Ys)))))).
% 3.28/1.73 thf(fact_159_lprefix__lappend__same, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ (((coinductive_lappend @ A) @ Xs) @ Zs)) = (((coinductive_lfinite @ A) @ Xs) => (((coinductive_lprefix @ A) @ Ys) @ Zs)))))).
% 3.28/1.73 thf(fact_160_lfilter__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) @ (((coinductive_lfilter @ A) @ P) @ Ys))))))).
% 3.28/1.73 thf(fact_161_lappend__ltakeWhile__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = Xs)))).
% 3.28/1.73 thf(fact_162_llcp__lappend__same, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_llcp @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ (((coinductive_lappend @ A) @ Xs) @ Zs)) = (((plus_plus @ extended_enat) @ ((coinductive_llength @ A) @ Xs)) @ (((coinductive_llcp @ A) @ Ys) @ Zs)))))).
% 3.28/1.73 thf(fact_163_lprefix__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))).
% 3.28/1.73 thf(fact_164_lappend__lprefixE, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) => (~ ((! [Zs2:(coinductive_llist @ A)]: ((Zs != (((coinductive_lappend @ A) @ Xs) @ Zs2)))))))))).
% 3.28/1.73 thf(fact_165_lprefix__lappendD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Ys) @ Zs)) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) | (((coinductive_lprefix @ A) @ Ys) @ Xs)))))).
% 3.28/1.73 thf(fact_166_lprefix__conv__lappend, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A)]: ((? [Zs3:(coinductive_llist @ A)]: ((Ys2 = (((coinductive_lappend @ A) @ Xs2) @ Zs3)))))))))).
% 3.28/1.73 thf(fact_167_lprefix__lappend__sameI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Zs) @ Xs)) @ (((coinductive_lappend @ A) @ Zs) @ Ys)))))).
% 3.28/1.73 thf(fact_168_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 3.28/1.73 thf(fact_169_add_Ocomm__neutral, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 3.28/1.73 thf(fact_170_add_Ogroup__left__neutral, axiom, (! [A:$tType]: ((((group_add @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 3.28/1.73 thf(fact_171_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [X2:A,Y3:A]: ((((((plus_plus @ A) @ X2) @ Y3) = (zero_zero @ A)) = ((X2 = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 3.28/1.73 thf(fact_172_lappend__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 3.28/1.73 thf(fact_173_llistsum__lappend, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lfinite @ A) @ Ys) => (((coindu780009021istsum @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((plus_plus @ A) @ ((coindu780009021istsum @ A) @ Xs)) @ ((coindu780009021istsum @ A) @ Ys))))))))))).
% 3.28/1.73 thf(fact_174_lmap__lappend__distrib, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs:(coinductive_llist @ B),Ys:(coinductive_llist @ B)]: ((((((coinductive_lmap @ B) @ A) @ F) @ (((coinductive_lappend @ B) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)) @ ((((coinductive_lmap @ B) @ A) @ F) @ Ys)))))).
% 3.28/1.73 thf(fact_175_lappend__assoc, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) = (((coinductive_lappend @ A) @ Xs) @ (((coinductive_lappend @ A) @ Ys) @ Zs)))))).
% 3.28/1.73 thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type @ A)) => (! [A3:A,B3:A,C3:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ C3) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B3) @ C3))))))))).
% 3.28/1.73 thf(fact_177_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type @ A)) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 3.28/1.73 thf(fact_178_add_Oassoc, axiom, (! [A:$tType]: ((((semigroup_add @ A) @ (type @ A)) => (! [A3:A,B3:A,C3:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B3)) @ C3) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B3) @ C3))))))))).
% 3.28/1.73 thf(fact_179_add_Oleft__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type @ A)) => (! [A3:A,B3:A,C3:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C3)) = (B3 = C3)))))))).
% 3.28/1.73 thf(fact_180_add_Oright__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type @ A)) => (! [B3:A,A3:A,C3:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C3) @ A3)) = (B3 = C3)))))))).
% 3.28/1.73 thf(fact_181_add_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type @ A)) => ((plus_plus @ A) = (^ [A5:A,B5:A]: ((((plus_plus @ A) @ B5) @ A5)))))))).
% 3.28/1.73 thf(fact_182_add_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type @ A)) => (! [B3:A,A3:A,C3:A]: (((((plus_plus @ A) @ B3) @ (((plus_plus @ A) @ A3) @ C3)) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B3) @ C3))))))))).
% 3.28/1.73 thf(fact_183_add__left__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [A3:A,B3:A,C3:A]: ((((((plus_plus @ A) @ A3) @ B3) = (((plus_plus @ A) @ A3) @ C3)) => (B3 = C3)))))))).
% 3.28/1.73 thf(fact_184_add__right__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [B3:A,A3:A,C3:A]: ((((((plus_plus @ A) @ B3) @ A3) = (((plus_plus @ A) @ C3) @ A3)) => (B3 = C3)))))))).
% 3.28/1.73 thf(fact_185_lappend__iterates, axiom, (! [A:$tType,F:(A > A),X2:A,Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_iterates @ A) @ F) @ X2)) @ Xs) = (((coinductive_iterates @ A) @ F) @ X2))))).
% 3.28/1.73 thf(fact_186_lmirror__aux__acc, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((((lMirro999291890or_aux @ A) @ (((coinductive_lappend @ A) @ Ys) @ Zs)) @ Xs) = (((coinductive_lappend @ A) @ (((lMirro999291890or_aux @ A) @ Ys) @ Xs)) @ Zs))))).
% 3.28/1.73 thf(fact_187_llist__all2__lappendI, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),Xs3:(coinductive_llist @ A),Ys4:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lfinite @ B) @ Ys) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs3) @ Ys4))) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Xs3)) @ (((coinductive_lappend @ B) @ Ys) @ Ys4))))))).
% 3.28/1.73 thf(fact_188_lappend__eq__lappend__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Us:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((((coinductive_llength @ A) @ Xs) = ((coinductive_llength @ A) @ Us)) => (((((coinductive_lappend @ A) @ Xs) @ Ys) = (((coinductive_lappend @ A) @ Us) @ Vs)) = ((Xs = Us) & (((coinductive_lfinite @ A) @ Xs) => (Ys = Vs)))))))).
% 3.28/1.73 thf(fact_189_lfilter__eq__lappend__lfiniteD, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lappend @ A) @ Ys) @ Zs)) => (((coinductive_lfinite @ A) @ Ys) => (? [Us2:(coinductive_llist @ A),Vs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Us2) @ Vs2)) & (((coinductive_lfinite @ A) @ Us2) & ((Ys = (((coinductive_lfilter @ A) @ P) @ Us2)) & (Zs = (((coinductive_lfilter @ A) @ P) @ Vs2)))))))))))).
% 3.28/1.73 thf(fact_190_lappend__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_lappend @ A)) @ (coinductive_lappend @ B))))).
% 3.28/1.73 thf(fact_191_add__is__0, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 3.28/1.73 thf(fact_192_Nat_Oadd__0__right, axiom, (! [M:nat]: (((((plus_plus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 3.28/1.73 thf(fact_193_natural_Osimps_I4_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T)]: ((((((code_case_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 3.28/1.73 thf(fact_194_plus__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z)))) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z))))) @ (plus_plus @ nat)) @ (plus_plus @ nat))).
% 3.28/1.73 thf(fact_195_add__eq__self__zero, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = M) => (N = (zero_zero @ nat)))))).
% 3.28/1.73 thf(fact_196_plus__nat_Oadd__0, axiom, (! [N:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N) = N)))).
% 3.28/1.73 thf(fact_197_ind__euclid, axiom, (! [P:(nat > (nat > $o)),A3:nat,B3:nat]: (((! [A4:nat,B4:nat]: ((((P @ A4) @ B4) = ((P @ B4) @ A4)))) => ((! [A4:nat]: (((P @ A4) @ (zero_zero @ nat)))) => ((! [A4:nat,B4:nat]: ((((P @ A4) @ B4) => ((P @ A4) @ (((plus_plus @ nat) @ A4) @ B4))))) => ((P @ A3) @ B3))))))).
% 3.28/1.73 thf(fact_198_add__0__iff, axiom, (! [A:$tType]: ((((semiri456707255roduct @ A) @ (type @ A)) => (! [B3:A,A3:A]: (((B3 = (((plus_plus @ A) @ B3) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 3.28/1.73 thf(fact_199_semiring__normalization__rules_I6_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 3.28/1.73 thf(fact_200_semiring__normalization__rules_I5_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 3.28/1.73 thf(fact_201_llast__lappend__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y3:A,Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y3) @ Ys))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y3) @ Ys))))))).
% 3.28/1.73 thf(fact_202_rec__nat__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((nat > (A > A)) > (nat > A))) @ ((nat > (B > B)) > (nat > B))) @ A2) @ ((((((bNF_rel_fun @ (nat > (A > A))) @ (nat > (B > B))) @ (nat > A)) @ (nat > B)) @ ((((((bNF_rel_fun @ nat) @ nat) @ (A > A)) @ (B > B)) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z)))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2))) @ ((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z)))) @ A2))) @ (rec_nat @ A)) @ (rec_nat @ B))))).
% 3.28/1.73 thf(fact_203_llast__LCons2, axiom, (! [A:$tType,X2:A,Y3:A,Xs:(coinductive_llist @ A)]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X2) @ (((coinductive_LCons @ A) @ Y3) @ Xs))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y3) @ Xs)))))).
% 3.28/1.73 thf(fact_204_old_Onat_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(nat > (T > T))]: ((((((rec_nat @ T) @ F1) @ F22) @ (zero_zero @ nat)) = F1)))).
% 3.28/1.73 thf(fact_205_case__nat__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((nat > A) > (nat > A))) @ ((nat > B) > (nat > B))) @ A2) @ ((((((bNF_rel_fun @ (nat > A)) @ (nat > B)) @ (nat > A)) @ (nat > B)) @ ((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z)))) @ A2)) @ ((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y2:nat,Z:nat]: ((Y2 = Z)))) @ A2))) @ (case_nat @ A)) @ (case_nat @ B))))).
% 3.28/1.73 thf(fact_206_natural_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T))]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 3.28/1.73 thf(fact_207_old_Onat_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F22:(nat > A)]: ((((((case_nat @ A) @ F1) @ F22) @ (zero_zero @ nat)) = F1)))).
% 3.28/1.73 thf(fact_208_ltake__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ extended_enat) @ extended_enat) @ ((coinductive_llist @ A) > (coinductive_llist @ A))) @ ((coinductive_llist @ B) > (coinductive_llist @ B))) @ (^ [Y2:extended_enat,Z:extended_enat]: ((Y2 = Z)))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_ltake @ A)) @ (coinductive_ltake @ B))))).
% 3.28/1.73 thf(fact_209_lfilter__eq__LConsD, axiom, (! [A:$tType,P:(A > $o),Ys:(coinductive_llist @ A),X2:A,Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Ys) = (((coinductive_LCons @ A) @ X2) @ Xs)) => (? [Us2:(coinductive_llist @ A),Vs2:(coinductive_llist @ A)]: (((Ys = (((coinductive_lappend @ A) @ Us2) @ (((coinductive_LCons @ A) @ X2) @ Vs2))) & (((coinductive_lfinite @ A) @ Us2) & ((! [X5:A]: (((((member @ A) @ X5) @ ((coinductive_lset @ A) @ Us2)) => (~ ((P @ X5)))))) & ((P @ X2) & (Xs = (((coinductive_lfilter @ A) @ P) @ Vs2)))))))))))).
% 3.28/1.73 thf(fact_210_ltake__is__lprefix, axiom, (! [A:$tType,N:extended_enat,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (((coinductive_ltake @ A) @ N) @ Xs)) @ Xs)))).
% 3.28/1.73 thf(fact_211_llist__all2__same, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(coinductive_llist @ A)]: (((((((coindu1486289336t_all2 @ A) @ A) @ P) @ Xs) @ Xs) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => ((P @ X3) @ X3)))))))).
% 3.28/1.73 thf(fact_212_ltake__lmap, axiom, (! [A:$tType,B:$tType,N:extended_enat,F:(B > A),Xs:(coinductive_llist @ B)]: (((((coinductive_ltake @ A) @ N) @ ((((coinductive_lmap @ B) @ A) @ F) @ Xs)) = ((((coinductive_lmap @ B) @ A) @ F) @ (((coinductive_ltake @ B) @ N) @ Xs)))))).
% 3.28/1.73 thf(fact_213_ldropWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X) = (Q @ X))))) => ((((coindu218763757pWhile @ A) @ P) @ Xs) = (((coindu218763757pWhile @ A) @ Q) @ Ys))))))).
% 3.28/1.73 thf(fact_214_in__lset__ldropWhileD, axiom, (! [A:$tType,X2:A,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs))) => (((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)))))).
% 3.28/1.73 thf(fact_215_llist_Opred__cong, axiom, (! [A:$tType,X2:(coinductive_llist @ A),Ya:(coinductive_llist @ A),P:(A > $o),Pa:(A > $o)]: (((X2 = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ Ya)) => ((P @ Z3) = (Pa @ Z3))))) => ((((coindu543516966_llist @ A) @ P) @ X2) = (((coindu543516966_llist @ A) @ Pa) @ Ya))))))).
% 3.28/1.73 thf(fact_216_llist_Opred__mono__strong, axiom, (! [A:$tType,P:(A > $o),X2:(coinductive_llist @ A),Pa:(A > $o)]: (((((coindu543516966_llist @ A) @ P) @ X2) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ X2)) => ((P @ Z3) => (Pa @ Z3))))) => (((coindu543516966_llist @ A) @ Pa) @ X2)))))).
% 3.28/1.73 thf(fact_217_lset__ltakeWhileD, axiom, (! [A:$tType,X2:A,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs))) => ((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) & (P @ X2)))))).
% 3.28/1.73 thf(fact_218_ltakeWhile__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X) = (Q @ X))))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = (((coindu501562517eWhile @ A) @ Q) @ Ys))))))).
% 3.28/1.73 thf(fact_219_ltakeWhile__all, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (P @ X)))) => ((((coindu501562517eWhile @ A) @ P) @ Xs) = Xs))))).
% 3.28/1.73 thf(fact_220_lfilter__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X) = (Q @ X))))) => ((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lfilter @ A) @ Q) @ Ys))))))).
% 3.28/1.73 thf(fact_221_lfilter__id__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = Xs) = (! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (P @ X3)))))))).
% 3.28/1.73 thf(fact_222_llist_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X2:(coinductive_llist @ A),Xa:(coinductive_llist @ A),F:(A > B),Fa:(A > B)]: (((! [Z3:A,Za:A]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ X2)) => ((((member @ A) @ Za) @ ((coinductive_lset @ A) @ Xa)) => (((F @ Z3) = (Fa @ Za)) => (Z3 = Za)))))) => ((((((coinductive_lmap @ A) @ B) @ F) @ X2) = ((((coinductive_lmap @ A) @ B) @ Fa) @ Xa)) => (X2 = Xa)))))).
% 3.28/1.73 thf(fact_223_llist_Omap__cong0, axiom, (! [B:$tType,A:$tType,X2:(coinductive_llist @ A),F:(A > B),G:(A > B)]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ X2)) => ((F @ Z3) = (G @ Z3))))) => (((((coinductive_lmap @ A) @ B) @ F) @ X2) = ((((coinductive_lmap @ A) @ B) @ G) @ X2)))))).
% 3.28/1.73 thf(fact_224_llist_Omap__cong, axiom, (! [B:$tType,A:$tType,X2:(coinductive_llist @ A),Ya:(coinductive_llist @ A),F:(A > B),G:(A > B)]: (((X2 = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ Ya)) => ((F @ Z3) = (G @ Z3))))) => (((((coinductive_lmap @ A) @ B) @ F) @ X2) = ((((coinductive_lmap @ A) @ B) @ G) @ Ya))))))).
% 3.28/1.73 thf(fact_225_llist_Orel__refl__strong, axiom, (! [A:$tType,X2:(coinductive_llist @ A),Ra:(A > (A > $o))]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ X2)) => ((Ra @ Z3) @ Z3)))) => (((((coindu1486289336t_all2 @ A) @ A) @ Ra) @ X2) @ X2))))).
% 3.28/1.73 thf(fact_226_llist_Orel__mono__strong, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X2:(coinductive_llist @ A),Y3:(coinductive_llist @ B),Ra:(A > (B > $o))]: (((((((coindu1486289336t_all2 @ A) @ B) @ R) @ X2) @ Y3) => ((! [Z3:A,Yb2:B]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ X2)) => ((((member @ B) @ Yb2) @ ((coinductive_lset @ B) @ Y3)) => (((R @ Z3) @ Yb2) => ((Ra @ Z3) @ Yb2)))))) => (((((coindu1486289336t_all2 @ A) @ B) @ Ra) @ X2) @ Y3)))))).
% 3.28/1.73 thf(fact_227_llist_Orel__cong, axiom, (! [A:$tType,B:$tType,X2:(coinductive_llist @ A),Ya:(coinductive_llist @ A),Y3:(coinductive_llist @ B),Xa:(coinductive_llist @ B),R:(A > (B > $o)),Ra:(A > (B > $o))]: (((X2 = Ya) => ((Y3 = Xa) => ((! [Z3:A,Yb2:B]: (((((member @ A) @ Z3) @ ((coinductive_lset @ A) @ Ya)) => ((((member @ B) @ Yb2) @ ((coinductive_lset @ B) @ Xa)) => (((R @ Z3) @ Yb2) = ((Ra @ Z3) @ Yb2)))))) => ((((((coindu1486289336t_all2 @ A) @ B) @ R) @ X2) @ Y3) = (((((coindu1486289336t_all2 @ A) @ B) @ Ra) @ Ya) @ Xa)))))))).
% 3.28/1.73 thf(fact_228_llist__all2__lsetD2, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),Y3:B]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((((member @ B) @ Y3) @ ((coinductive_lset @ B) @ Ys)) => (? [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) & ((P @ X) @ Y3))))))))).
% 3.28/1.73 thf(fact_229_llist__all2__lsetD1, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),X2:A]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (? [X:B]: (((((member @ B) @ X) @ ((coinductive_lset @ B) @ Ys)) & ((P @ X2) @ X))))))))).
% 3.28/1.73 thf(fact_230_llist__all2__reflI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > (A > $o))]: (((! [X:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((P @ X) @ X)))) => (((((coindu1486289336t_all2 @ A) @ A) @ P) @ Xs) @ Xs))))).
% 3.28/1.73 thf(fact_231_llist__all2__ltakeI, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),N:extended_enat]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_ltake @ A) @ N) @ Xs)) @ (((coinductive_ltake @ B) @ N) @ Ys)))))).
% 3.28/1.73 thf(fact_232_lset__intros_I2_J, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),X6:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X6) @ Xs))))))).
% 3.28/1.73 thf(fact_233_lset__intros_I1_J, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: ((((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X2) @ Xs)))))).
% 3.28/1.73 thf(fact_234_llist_Oset__intros_I2_J, axiom, (! [A:$tType,X2:A,A22:(coinductive_llist @ A),A1:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ A22)) => (((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22))))))).
% 3.28/1.73 thf(fact_235_llist_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(coinductive_llist @ A)]: ((((member @ A) @ A1) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22)))))).
% 3.28/1.73 thf(fact_236_lset__cases, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs5:(coinductive_llist @ A)]: ((Xs != (((coinductive_LCons @ A) @ X2) @ Xs5)))) => (~ ((! [X7:A,Xs5:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X7) @ Xs5)) => (~ ((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs5)))))))))))))).
% 3.28/1.73 thf(fact_237_lset__induct, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs6:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X2) @ Xs6)))) => ((! [X7:A,Xs6:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs6)) => ((X2 != X7) => ((P @ Xs6) => (P @ (((coinductive_LCons @ A) @ X7) @ Xs6))))))) => (P @ Xs))))))).
% 3.28/1.73 thf(fact_238_lset__induct_H, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs6:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X2) @ Xs6)))) => ((! [X7:A,Xs6:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs6)) => ((P @ Xs6) => (P @ (((coinductive_LCons @ A) @ X7) @ Xs6)))))) => (P @ Xs))))))).
% 3.28/1.73 thf(fact_239_in__lset__ldropD, axiom, (! [A:$tType,X2:A,N:extended_enat,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coinductive_ldrop @ A) @ N) @ Xs))) => (((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)))))).
% 3.28/1.73 thf(fact_240_llist_Oset__cases, axiom, (! [A:$tType,E2:A,A3:(coinductive_llist @ A)]: (((((member @ A) @ E2) @ ((coinductive_lset @ A) @ A3)) => ((! [Z22:(coinductive_llist @ A)]: ((A3 != (((coinductive_LCons @ A) @ E2) @ Z22)))) => (~ ((! [Z1:A,Z22:(coinductive_llist @ A)]: (((A3 = (((coinductive_LCons @ A) @ Z1) @ Z22)) => (~ ((((member @ A) @ E2) @ ((coinductive_lset @ A) @ Z22)))))))))))))).
% 3.28/1.73 thf(fact_241_llist_Oset__induct, axiom, (! [A:$tType,X2:A,A3:(coinductive_llist @ A),P:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ A3)) => ((! [Z1:A,Z22:(coinductive_llist @ A)]: (((P @ Z1) @ (((coinductive_LCons @ A) @ Z1) @ Z22)))) => ((! [Z1:A,Z22:(coinductive_llist @ A),Xa2:A]: (((((member @ A) @ Xa2) @ ((coinductive_lset @ A) @ Z22)) => (((P @ Xa2) @ Z22) => ((P @ Xa2) @ (((coinductive_LCons @ A) @ Z1) @ Z22)))))) => ((P @ X2) @ A3))))))).
% 3.28/1.73 thf(fact_242_lset__lmirror__aux, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lset @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)) = ((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Acc)))))).
% 3.28/1.73 thf(fact_243_lfinite__ldropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu218763757pWhile @ A) @ P) @ Xs)) = ((? [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X3)))))) => ((coinductive_lfinite @ A) @ Xs)))))).
% 3.28/1.73 thf(fact_244_lfinite__ltakeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coindu501562517eWhile @ A) @ P) @ Xs)) = (((coinductive_lfinite @ A) @ Xs) | (? [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) & (~ ((P @ X3))))))))))).
% 3.28/1.73 thf(fact_245_ltakeWhile__lappend1, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => ((~ ((P @ X2))) => ((((coindu501562517eWhile @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coindu501562517eWhile @ A) @ P) @ Xs))))))).
% 3.28/1.73 thf(fact_246_in__lset__lappend__iff, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))) = ((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) | (((coinductive_lfinite @ A) @ Xs) & (((member @ A) @ X2) @ ((coinductive_lset @ A) @ Ys)))))))).
% 3.28/1.73 thf(fact_247_lappend__ltake__ldrop, axiom, (! [A:$tType,N:extended_enat,Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_ltake @ A) @ N) @ Xs)) @ (((coinductive_ldrop @ A) @ N) @ Xs)) = Xs)))).
% 3.28/1.73 thf(fact_248_lset__lmember, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X2) @ Xs))))).
% 3.28/1.73 thf(fact_249_ltake__ldrop, axiom, (! [A:$tType,N:extended_enat,M:extended_enat,Xs:(coinductive_llist @ A)]: (((((coinductive_ltake @ A) @ N) @ (((coinductive_ldrop @ A) @ M) @ Xs)) = (((coinductive_ldrop @ A) @ M) @ (((coinductive_ltake @ A) @ (((plus_plus @ extended_enat) @ N) @ M)) @ Xs)))))).
% 3.28/1.73 thf(fact_250_llist__all2__lappend1D_I1_J, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Xs3:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Xs3)) @ Ys) => (((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ (((coinductive_ltake @ B) @ ((coinductive_llength @ A) @ Xs)) @ Ys)))))).
% 3.28/1.73 thf(fact_251_split__llist__first, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys5:(coinductive_llist @ A),Zs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys5) @ (((coinductive_LCons @ A) @ X2) @ Zs4))) & (((coinductive_lfinite @ A) @ Ys5) & (~ ((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Ys5)))))))))))).
% 3.28/1.73 thf(fact_252_split__llist, axiom, (! [A:$tType,X2:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys5:(coinductive_llist @ A),Zs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys5) @ (((coinductive_LCons @ A) @ X2) @ Zs4))) & ((coinductive_lfinite @ A) @ Ys5)))))))).
% 3.28/1.73 thf(fact_253_ltake__plus__conv__lappend, axiom, (! [A:$tType,N:extended_enat,M:extended_enat,Xs:(coinductive_llist @ A)]: (((((coinductive_ltake @ A) @ (((plus_plus @ extended_enat) @ N) @ M)) @ Xs) = (((coinductive_lappend @ A) @ (((coinductive_ltake @ A) @ N) @ Xs)) @ (((coinductive_ltake @ A) @ M) @ (((coinductive_ldrop @ A) @ N) @ Xs))))))).
% 3.28/1.73 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, axiom, ((semiri456707255roduct @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, ((ordere779506340up_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, ((cancel1352612707id_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, ((cancel_semigroup_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, ((ab_semigroup_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, ((comm_monoid_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1, axiom, ((comm_semiring_1 @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, ((semigroup_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Omonoid__add, axiom, ((monoid_add @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Nat_Onat___Groups_Ozero, axiom, ((zero @ nat) @ (type @ nat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_1, axiom, ((canoni770627133id_add @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_2, axiom, ((ordere779506340up_add @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_3, axiom, ((ab_semigroup_add @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_4, axiom, ((comm_monoid_add @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_5, axiom, ((comm_semiring_1 @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_6, axiom, ((semigroup_add @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_7, axiom, ((monoid_add @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_8, axiom, ((zero @ extended_enat) @ (type @ extended_enat))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_9, axiom, ((ordere779506340up_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_10, axiom, ((cancel1352612707id_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_11, axiom, ((cancel_semigroup_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_12, axiom, ((ab_semigroup_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_13, axiom, ((comm_monoid_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_14, axiom, ((comm_semiring_1 @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_15, axiom, ((semigroup_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_16, axiom, ((monoid_add @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_17, axiom, ((zero @ code_natural) @ (type @ code_natural))).
% 3.28/1.73 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.28/1.73 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y3:A]: ((((((if @ A) @ $false) @ X2) @ Y3) = Y3)))).
% 3.28/1.73 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y3:A]: ((((((if @ A) @ $true) @ X2) @ Y3) = X2)))).
% 3.28/1.73 thf(conj_0, conjecture, (((coinductive_llength @ a) @ acca) = ((coinductive_llength @ b) @ acc_a))).
% 3.28/1.73 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.DVE1XfhOnx/DTF2DTF15775.p
% 3.28/1.73 ------------------------
% 3.28/1.78 ---- Cleaned THF ---
% 3.28/1.79 thf(ty_t_Coinductive__List_Ollist,type,
% 3.28/1.79 coinductive_llist: $tType > $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_t_Code__Numeral_Onatural,type,
% 3.28/1.79 code_natural: $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_t_Extended__Nat_Oenat,type,
% 3.28/1.79 extended_enat: $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_t_Set_Oset,type,
% 3.28/1.79 set: $tType > $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_t_Nat_Onat,type,
% 3.28/1.79 nat: $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_t_itself,type,
% 3.28/1.79 itself: $tType > $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_tf_b,type,
% 3.28/1.79 b: $tType ).
% 3.28/1.79
% 3.28/1.79 thf(ty_tf_a,type,
% 3.28/1.79 a: $tType ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Ozero,type,
% 3.28/1.79 zero:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Ogroup__add,type,
% 3.28/1.79 group_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Omonoid__add,type,
% 3.28/1.79 monoid_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Osemigroup__add,type,
% 3.28/1.79 semigroup_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 3.28/1.79 comm_semiring_1:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 3.28/1.79 comm_monoid_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 3.28/1.79 ab_semigroup_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 3.28/1.79 cancel_semigroup_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 3.28/1.79 cancel1352612707id_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 3.28/1.79 linord219039673up_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 3.28/1.79 ordere779506340up_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.28/1.79 canoni770627133id_add:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 3.28/1.79 semiri456707255roduct:
% 3.28/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_BNF__Def_Orel__fun,type,
% 3.28/1.79 bNF_rel_fun:
% 3.28/1.79 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 3.28/1.79 code_case_natural:
% 3.28/1.79 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 3.28/1.79 code_rec_natural:
% 3.28/1.79 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
% 3.28/1.79 code_size_natural: code_natural > nat ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 3.28/1.79 coindu328551480prefix:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ogen__llength,type,
% 3.28/1.79 coindu246717305length:
% 3.28/1.79 !>[A: $tType] : ( nat > ( coinductive_llist @ A ) > extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Oiterates,type,
% 3.28/1.79 coinductive_iterates:
% 3.28/1.79 !>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Olappend,type,
% 3.28/1.79 coinductive_lappend:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Oldrop,type,
% 3.28/1.79 coinductive_ldrop:
% 3.28/1.79 !>[A: $tType] : ( extended_enat > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_OldropWhile,type,
% 3.28/1.79 coindu218763757pWhile:
% 3.28/1.79 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Olfilter,type,
% 3.28/1.79 coinductive_lfilter:
% 3.28/1.79 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Olfinite,type,
% 3.28/1.79 coinductive_lfinite:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollast,type,
% 3.28/1.79 coinductive_llast:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollcp,type,
% 3.28/1.79 coinductive_llcp:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollength,type,
% 3.28/1.79 coinductive_llength:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 3.28/1.79 coinductive_LCons:
% 3.28/1.79 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 3.28/1.79 coindu1381640503_llist:
% 3.28/1.79 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 3.28/1.79 coindu1259883913_llist:
% 3.28/1.79 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_Ollist__all2,type,
% 3.28/1.79 coindu1486289336t_all2:
% 3.28/1.79 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_Olmap,type,
% 3.28/1.79 coinductive_lmap:
% 3.28/1.79 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 3.28/1.79 coinductive_lset:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
% 3.28/1.79 coindu543516966_llist:
% 3.28/1.79 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Olmember,type,
% 3.28/1.79 coinductive_lmember:
% 3.28/1.79 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Olprefix,type,
% 3.28/1.79 coinductive_lprefix:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Oltake,type,
% 3.28/1.79 coinductive_ltake:
% 3.28/1.79 !>[A: $tType] : ( extended_enat > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_OltakeWhile,type,
% 3.28/1.79 coindu501562517eWhile:
% 3.28/1.79 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
% 3.28/1.79 coindu780009021istsum:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 3.28/1.79 coindu1441602521_llist:
% 3.28/1.79 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Fun_Omap__fun,type,
% 3.28/1.79 map_fun:
% 3.28/1.79 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 3.28/1.79 plus_plus:
% 3.28/1.79 !>[A: $tType] : ( A > A > A ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.28/1.79 zero_zero:
% 3.28/1.79 !>[A: $tType] : A ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_If,type,
% 3.28/1.79 if:
% 3.28/1.79 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
% 3.28/1.79 lMirro999291890or_aux:
% 3.28/1.79 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 3.28/1.79 case_nat:
% 3.28/1.79 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 3.28/1.79 rec_nat:
% 3.28/1.79 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Nat_Osize__class_Osize,type,
% 3.28/1.79 size_size:
% 3.28/1.79 !>[A: $tType] : ( A > nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Pure_Otype,type,
% 3.28/1.79 type:
% 3.28/1.79 !>[A: $tType] : ( itself @ A ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Quotient_OBex1__rel,type,
% 3.28/1.79 bex1_rel:
% 3.28/1.79 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Quotient_OQuotient3,type,
% 3.28/1.79 quotient3:
% 3.28/1.79 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_Set_OCollect,type,
% 3.28/1.79 collect:
% 3.28/1.79 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_c_member,type,
% 3.28/1.79 member:
% 3.28/1.79 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.28/1.79
% 3.28/1.79 thf(sy_v_P,type,
% 3.28/1.79 p: a > b > $o ).
% 3.28/1.79
% 3.28/1.79 thf(sy_v_acc_Ha____,type,
% 3.28/1.79 acc_a: coinductive_llist @ b ).
% 3.28/1.79
% 3.28/1.79 thf(sy_v_acca____,type,
% 3.28/1.79 acca: coinductive_llist @ a ).
% 3.28/1.79
% 3.28/1.79 thf(sy_v_xs_Ha____,type,
% 3.28/1.79 xs_a: coinductive_llist @ b ).
% 3.28/1.79
% 3.28/1.79 thf(sy_v_xsa____,type,
% 3.28/1.79 xsa: coinductive_llist @ a ).
% 3.28/1.79
% 3.28/1.79 thf(fact_0_local_OLNil_I2_J,axiom,
% 3.28/1.79 coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a ).
% 3.28/1.79
% 3.28/1.79 thf(fact_1_local_OLNil_I3_J,axiom,
% 3.28/1.79 coinductive_lfinite @ a @ acca ).
% 3.28/1.79
% 3.28/1.79 thf(fact_2_calculation,axiom,
% 3.28/1.79 ( ( coinductive_llength @ a @ ( lMirro999291890or_aux @ a @ acca @ xsa ) )
% 3.28/1.79 = ( coinductive_llength @ b @ ( lMirro999291890or_aux @ b @ acc_a @ xs_a ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_3_local_OLNil_I1_J,axiom,
% 3.28/1.79 coindu1486289336t_all2 @ a @ b @ p @ ( lMirro999291890or_aux @ a @ acca @ xsa ) @ ( lMirro999291890or_aux @ b @ acc_a @ xs_a ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_4_llist__all2__llengthD,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ( coinductive_llength @ A @ Xs )
% 3.28/1.79 = ( coinductive_llength @ B @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_5_llcp__same__conv__length,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_llcp @ A @ Xs @ Xs )
% 3.28/1.79 = ( coinductive_llength @ A @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_6_llength__lmap,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_llength @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 3.28/1.79 = ( coinductive_llength @ B @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_7_lprefix__llength__eq__imp__eq,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( ( ( coinductive_llength @ A @ Xs )
% 3.28/1.79 = ( coinductive_llength @ A @ Ys ) )
% 3.28/1.79 => ( Xs = Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_8_llength__ltakeWhile__all,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 3.28/1.79 = ( coinductive_llength @ A @ Xs ) )
% 3.28/1.79 = ( ( coindu501562517eWhile @ A @ P @ Xs )
% 3.28/1.79 = Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_9_llist__all2__llength__ltakeWhileD,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( P @ X @ Y )
% 3.28/1.79 => ( ( Q1 @ X )
% 3.28/1.79 = ( Q2 @ Y ) ) )
% 3.28/1.79 => ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ Q1 @ Xs ) )
% 3.28/1.79 = ( coinductive_llength @ B @ ( coindu501562517eWhile @ B @ Q2 @ Ys ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_10_llength__code,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coinductive_llength @ A )
% 3.28/1.79 = ( coindu246717305length @ A @ ( zero_zero @ nat ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_11_llength__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ extended_enat @ extended_enat @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 3.28/1.79 @ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
% 3.28/1.79 @ ( coinductive_llength @ A )
% 3.28/1.79 @ ( coinductive_llength @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_12_lprefix__refl,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_13_llist_Oleq__refl,axiom,
% 3.28/1.79 ! [A: $tType,X2: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X2 @ X2 ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_14_lfinite__lmap,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 3.28/1.79 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_15_lfinite__lmirror__aux,axiom,
% 3.28/1.79 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 3.28/1.79 = ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 & ( coinductive_lfinite @ A @ Acc ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_16_llcp__lprefix1,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( ( coinductive_llcp @ A @ Xs @ Ys )
% 3.28/1.79 = ( coinductive_llength @ A @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_17_llcp__lprefix2,axiom,
% 3.28/1.79 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 3.28/1.79 => ( ( coinductive_llcp @ A @ Xs @ Ys )
% 3.28/1.79 = ( coinductive_llength @ A @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_18_llcp__commute,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coinductive_llcp @ A )
% 3.28/1.79 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] : ( coinductive_llcp @ A @ Ys2 @ Xs2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_19_lmap__lprefix,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,F: A > B] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( coinductive_lprefix @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) @ ( coinductive_lmap @ A @ B @ F @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_20_lmap__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ ( coindu1486289336t_all2 @ A @ C @ A2 ) @ ( coindu1486289336t_all2 @ B @ D @ B2 ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ C @ D ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_21_lprefix__trans,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 3.28/1.79 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_22_llist__all2__rsp,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T2: B > B > $o,X2: coinductive_llist @ A,Y3: coinductive_llist @ B,A3: coinductive_llist @ A,B3: coinductive_llist @ B] :
% 3.28/1.79 ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( R @ X @ Y )
% 3.28/1.79 => ! [A4: A,B4: B] :
% 3.28/1.79 ( ( R @ A4 @ B4 )
% 3.28/1.79 => ( ( S @ X @ A4 )
% 3.28/1.79 = ( T2 @ Y @ B4 ) ) ) )
% 3.28/1.79 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
% 3.28/1.79 => ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
% 3.28/1.79 => ( ( coindu1486289336t_all2 @ A @ A @ S @ X2 @ A3 )
% 3.28/1.79 = ( coindu1486289336t_all2 @ B @ B @ T2 @ Y3 @ B3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_23_llist_Oleq__trans,axiom,
% 3.28/1.79 ! [A: $tType,X2: coinductive_llist @ A,Y3: coinductive_llist @ A,Z2: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ X2 @ Y3 )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Y3 @ Z2 )
% 3.28/1.79 => ( coinductive_lprefix @ A @ X2 @ Z2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_24_llist__all2__mono,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P2: A > B > $o] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( P @ X @ Y )
% 3.28/1.79 => ( P2 @ X @ Y ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_25_lprefix__antisym,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 3.28/1.79 => ( Xs = Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_26_lprefix__lfiniteD,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( ( coinductive_lfinite @ A @ Ys )
% 3.28/1.79 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_27_llist_Oleq__antisym,axiom,
% 3.28/1.79 ! [A: $tType,X2: coinductive_llist @ A,Y3: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ X2 @ Y3 )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Y3 @ X2 )
% 3.28/1.79 => ( X2 = Y3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_28_lprefix__ltakeWhile,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ Xs ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_29_llist__all2__lfiniteD,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 = ( coinductive_lfinite @ B @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_30_lprefix__down__linear,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_31_not__lfinite__lprefix__conv__eq,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 = ( Xs = Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_32_lmirror__aux__inf,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 3.28/1.79 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
% 3.28/1.79 = Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_33_lmap__lmirror__aux,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,F: B > A,Acc: coinductive_llist @ B,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_lmap @ B @ A @ F @ ( lMirro999291890or_aux @ B @ Acc @ Xs ) )
% 3.28/1.79 = ( lMirro999291890or_aux @ A @ ( coinductive_lmap @ B @ A @ F @ Acc ) @ ( coinductive_lmap @ B @ A @ F @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_34_llist_Orel__eq,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ A
% 3.28/1.79 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 3.28/1.79 = ( ^ [Y2: coinductive_llist @ A,Z: coinductive_llist @ A] : ( Y2 = Z ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_35_llist_Orel__refl,axiom,
% 3.28/1.79 ! [B: $tType,Ra: B > B > $o,X2: coinductive_llist @ B] :
% 3.28/1.79 ( ! [X: B] : ( Ra @ X @ X )
% 3.28/1.79 => ( coindu1486289336t_all2 @ B @ B @ Ra @ X2 @ X2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_36_llist__all2__lmirror__aux,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Acc: coinductive_llist @ A,Acc2: coinductive_llist @ B,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Acc @ Acc2 )
% 3.28/1.79 => ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Xs3 )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) @ ( lMirro999291890or_aux @ B @ Acc2 @ Xs3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_37_llist_Omap__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb: A > E > $o,Sd: B > F2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F2 ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ E ) > ( coinductive_llist @ F2 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F2 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ F2 ) @ ( coindu1486289336t_all2 @ A @ E @ Rb ) @ ( coindu1486289336t_all2 @ B @ F2 @ Sd ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ E @ F2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_38_rel__funI,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 3.28/1.79 ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( A2 @ X @ Y )
% 3.28/1.79 => ( B2 @ ( F @ X ) @ ( G @ Y ) ) )
% 3.28/1.79 => ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_39_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coinductive_lprefix @ A )
% 3.28/1.79 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_lfinite @ A @ Xs2 )
% 3.28/1.79 => ( coindu328551480prefix @ A @ Xs2 @ Ys2 ) )
% 3.28/1.79 & ( ~ ( coinductive_lfinite @ A @ Xs2 )
% 3.28/1.79 => ( Xs2 = Ys2 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_40_llistsum__inf,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coindu780009021istsum @ A @ Xs )
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_41_rel__funD,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
% 3.28/1.79 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 3.28/1.79 => ( ( A2 @ X2 @ Y3 )
% 3.28/1.79 => ( B2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_42_let__rsp,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
% 3.28/1.79 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
% 3.28/1.79 @ ^ [S2: A,F3: A > C] : ( F3 @ S2 )
% 3.28/1.79 @ ^ [S2: B,F3: B > D] : ( F3 @ S2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_43_rel__funE,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
% 3.28/1.79 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 3.28/1.79 => ( ( A2 @ X2 @ Y3 )
% 3.28/1.79 => ( B2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_44_mem__Collect__eq,axiom,
% 3.28/1.79 ! [A: $tType,A3: A,P: A > $o] :
% 3.28/1.79 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.28/1.79 = ( P @ A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_45_Collect__mem__eq,axiom,
% 3.28/1.79 ! [A: $tType,A2: set @ A] :
% 3.28/1.79 ( ( collect @ A
% 3.28/1.79 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 3.28/1.79 = A2 ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_46_Collect__cong,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.28/1.79 ( ! [X: A] :
% 3.28/1.79 ( ( P @ X )
% 3.28/1.79 = ( Q @ X ) )
% 3.28/1.79 => ( ( collect @ A @ P )
% 3.28/1.79 = ( collect @ A @ Q ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_47_ext,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.28/1.79 ( ! [X: A] :
% 3.28/1.79 ( ( F @ X )
% 3.28/1.79 = ( G @ X ) )
% 3.28/1.79 => ( F = G ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_48_apply__rsp_H,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
% 3.28/1.79 ( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
% 3.28/1.79 => ( ( R1 @ X2 @ Y3 )
% 3.28/1.79 => ( R2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_49_rel__funD2,axiom,
% 3.28/1.79 ! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F: A > B,G: A > C,X2: A] :
% 3.28/1.79 ( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F @ G )
% 3.28/1.79 => ( ( A2 @ X2 @ X2 )
% 3.28/1.79 => ( B2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_50_rel__fun__mono,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F: A > C,G: B > D,Y4: A > B > $o,B2: C > D > $o] :
% 3.28/1.79 ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 3.28/1.79 => ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( Y4 @ X @ Y )
% 3.28/1.79 => ( X4 @ X @ Y ) )
% 3.28/1.79 => ( ! [X: C,Y: D] :
% 3.28/1.79 ( ( A2 @ X @ Y )
% 3.28/1.79 => ( B2 @ X @ Y ) )
% 3.28/1.79 => ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_51_rel__fun__mono_H,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,Y4: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 3.28/1.79 ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( Y4 @ X @ Y )
% 3.28/1.79 => ( X4 @ X @ Y ) )
% 3.28/1.79 => ( ! [X: C,Y: D] :
% 3.28/1.79 ( ( A2 @ X @ Y )
% 3.28/1.79 => ( B2 @ X @ Y ) )
% 3.28/1.79 => ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 3.28/1.79 => ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_52_llist__all2__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) > $o )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o ) @ ( coindu1486289336t_all2 @ A @ B @ R )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 3.28/1.79 @ ( coindu1486289336t_all2 @ A @ A )
% 3.28/1.79 @ ( coindu1486289336t_all2 @ B @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_53_llist_Orel__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) > $o )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 3.28/1.79 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ D ) > $o ) @ ( coindu1486289336t_all2 @ A @ C @ Sa )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ $o @ $o @ ( coindu1486289336t_all2 @ B @ D @ Sc )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 3.28/1.79 @ ( coindu1486289336t_all2 @ A @ B )
% 3.28/1.79 @ ( coindu1486289336t_all2 @ C @ D ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_54_ltakeWhile__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 3.28/1.79 @ ( coindu501562517eWhile @ A )
% 3.28/1.79 @ ( coindu501562517eWhile @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_55_lfinite__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 3.28/1.79 @ ( coinductive_lfinite @ A )
% 3.28/1.79 @ ( coinductive_lfinite @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_56_Coinductive__List_Ofinite__lprefix__def,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coindu328551480prefix @ A )
% 3.28/1.79 = ( coinductive_lprefix @ A ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_57_zero__reorient,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( zero @ A @ ( type @ A ) )
% 3.28/1.79 => ! [X2: A] :
% 3.28/1.79 ( ( ( zero_zero @ A )
% 3.28/1.79 = X2 )
% 3.28/1.79 = ( X2
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_58_Let__transfer,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 3.28/1.79 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ B2 )
% 3.28/1.79 @ ^ [S2: A,F3: A > C] : ( F3 @ S2 )
% 3.28/1.79 @ ^ [S2: B,F3: B > D] : ( F3 @ S2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_59_rel__fun__def__butlast,axiom,
% 3.28/1.79 ! [B: $tType,D: $tType,C: $tType,E: $tType,F2: $tType,A: $tType,R: A > B > $o,S: C > E > $o,T2: D > F2 > $o,F: A > C > D,G: B > E > F2] :
% 3.28/1.79 ( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F2 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F2 @ S @ T2 ) @ F @ G )
% 3.28/1.79 = ( ! [X3: A,Y5: B] :
% 3.28/1.79 ( ( R @ X3 @ Y5 )
% 3.28/1.79 => ( bNF_rel_fun @ C @ E @ D @ F2 @ S @ T2 @ ( F @ X3 ) @ ( G @ Y5 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_60_If__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 3.28/1.79 @ ( if @ A )
% 3.28/1.79 @ ( if @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_61_llist__corec__transfer,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > $o ) > ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > $o ) > ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > $o ) > ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > $o ) > ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 3.28/1.79 @ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) @ ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) ) ) ) )
% 3.28/1.79 @ ( coindu1259883913_llist @ A @ C )
% 3.28/1.79 @ ( coindu1259883913_llist @ B @ D ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_62_llist_Ocorec__transfer,axiom,
% 3.28/1.79 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > B ) > ( D > $o ) > ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > $o ) > ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S @ R )
% 3.28/1.79 @ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( C > ( coinductive_llist @ A ) ) @ ( D > ( coinductive_llist @ B ) ) @ ( ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > D ) > D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ S @ ( coindu1486289336t_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( coinductive_llist @ A ) ) @ ( D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) @ ( bNF_rel_fun @ C @ D @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ S @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ) ) )
% 3.28/1.79 @ ( coindu1259883913_llist @ C @ A )
% 3.28/1.79 @ ( coindu1259883913_llist @ D @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_63_unfold__llist__transfer,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) ) )
% 3.28/1.79 @ ( coindu1441602521_llist @ A @ C )
% 3.28/1.79 @ ( coindu1441602521_llist @ B @ D ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_64_iterates__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ A ) ) @ ( B > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ A2 @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_iterates @ A ) @ ( coinductive_iterates @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_65_case__llist__transfer,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,D: $tType,B: $tType,B2: A > B > $o,A2: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( ( C > ( coinductive_llist @ C ) > A ) > ( coinductive_llist @ C ) > A ) @ ( ( D > ( coinductive_llist @ D ) > B ) > ( coinductive_llist @ D ) > B ) @ B2 @ ( bNF_rel_fun @ ( C > ( coinductive_llist @ C ) > A ) @ ( D > ( coinductive_llist @ D ) > B ) @ ( ( coinductive_llist @ C ) > A ) @ ( ( coinductive_llist @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ ( ( coinductive_llist @ C ) > A ) @ ( ( coinductive_llist @ D ) > B ) @ A2 @ ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A @ B @ ( coindu1486289336t_all2 @ C @ D @ A2 ) @ B2 ) ) @ ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A @ B @ ( coindu1486289336t_all2 @ C @ D @ A2 ) @ B2 ) ) @ ( coindu1381640503_llist @ A @ C ) @ ( coindu1381640503_llist @ B @ D ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_66_llist_Ocase__transfer,axiom,
% 3.28/1.79 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( coinductive_llist @ A ) > C ) > ( coinductive_llist @ A ) > C ) @ ( ( B > ( coinductive_llist @ B ) > D ) > ( coinductive_llist @ B ) > D ) @ S @ ( bNF_rel_fun @ ( A > ( coinductive_llist @ A ) > C ) @ ( B > ( coinductive_llist @ B ) > D ) @ ( ( coinductive_llist @ A ) > C ) @ ( ( coinductive_llist @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > C ) @ ( ( coinductive_llist @ B ) > D ) @ R @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ D @ ( coindu1486289336t_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ D @ ( coindu1486289336t_all2 @ A @ B @ R ) @ S ) ) @ ( coindu1381640503_llist @ C @ A ) @ ( coindu1381640503_llist @ D @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_67_lfinite__iterates,axiom,
% 3.28/1.79 ! [A: $tType,F: A > A,X2: A] :
% 3.28/1.79 ~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F @ X2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_68_lmap__iterates,axiom,
% 3.28/1.79 ! [A: $tType,F: A > A,X2: A] :
% 3.28/1.79 ( ( coinductive_lmap @ A @ A @ F @ ( coinductive_iterates @ A @ F @ X2 ) )
% 3.28/1.79 = ( coinductive_iterates @ A @ F @ ( F @ X2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_69_fun_Orel__transfer,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
% 3.28/1.79 @ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
% 3.28/1.79 @ ( bNF_rel_fun @ D @ D @ A @ C
% 3.28/1.79 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 3.28/1.79 @ Sa )
% 3.28/1.79 @ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
% 3.28/1.79 @ ( bNF_rel_fun @ D @ D @ B @ E
% 3.28/1.79 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 3.28/1.79 @ Sc )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
% 3.28/1.79 @ ( bNF_rel_fun @ D @ D @ A @ B
% 3.28/1.79 @ ^ [Y2: D,Z: D] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ D @ D @ C @ E
% 3.28/1.79 @ ^ [Y2: D,Z: D] : ( Y2 = Z ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_70_ldropWhile__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 3.28/1.79 @ ( coindu218763757pWhile @ A )
% 3.28/1.79 @ ( coindu218763757pWhile @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_71_map__fun__parametric,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,E: $tType,F2: $tType,H: $tType,G2: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G2 > $o,D2: F2 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F2 ) > ( B > E ) > A > F2 ) @ ( ( G2 > H ) > ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F2 ) @ ( G2 > H ) @ ( ( B > E ) > A > F2 ) @ ( ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ E @ G2 @ F2 @ H @ C2 @ D2 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G2 ) @ ( A > F2 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G2 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F2 @ H @ A2 @ D2 ) ) ) @ ( map_fun @ A @ B @ E @ F2 ) @ ( map_fun @ C @ D @ G2 @ H ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_72_lfilter__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 3.28/1.79 @ ( coinductive_lfilter @ A )
% 3.28/1.79 @ ( coinductive_lfilter @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_73_llist_Opred__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( coindu543516966_llist @ A )
% 3.28/1.79 @ ( coindu543516966_llist @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_74_llist__all__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ( coindu543516966_llist @ A )
% 3.28/1.79 @ ( coindu543516966_llist @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_75_if__rsp,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
% 3.28/1.79 @ ( if @ A )
% 3.28/1.79 @ ( if @ A ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_76_lfilter__idem,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 3.28/1.79 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_77_fun__quotient3,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs2: C > D,Rep2: D > C] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 3.28/1.79 => ( ( quotient3 @ C @ D @ R2 @ Abs2 @ Rep2 )
% 3.28/1.79 => ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs2 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep2 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_78_let__prs,axiom,
% 3.28/1.79 ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 3.28/1.79 => ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
% 3.28/1.79 => ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep22 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs22 @ Rep12 ) @ Abs12 )
% 3.28/1.79 @ ^ [S2: C,F3: C > A] : ( F3 @ S2 ) )
% 3.28/1.79 = ( ^ [S2: D,F3: D > B] : ( F3 @ S2 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_79_cond__prs,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A3: $o,B3: B,C3: B] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 3.28/1.79 => ( ( A3
% 3.28/1.79 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 3.28/1.79 = B3 ) )
% 3.28/1.79 & ( ~ A3
% 3.28/1.79 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 3.28/1.79 = C3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_80_Quotient3I,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 3.28/1.79 ( ! [A4: A] :
% 3.28/1.79 ( ( Abs @ ( Rep @ A4 ) )
% 3.28/1.79 = A4 )
% 3.28/1.79 => ( ! [A4: A] : ( R @ ( Rep @ A4 ) @ ( Rep @ A4 ) )
% 3.28/1.79 => ( ! [R3: B,S3: B] :
% 3.28/1.79 ( ( R @ R3 @ S3 )
% 3.28/1.79 = ( ( R @ R3 @ R3 )
% 3.28/1.79 & ( R @ S3 @ S3 )
% 3.28/1.79 & ( ( Abs @ R3 )
% 3.28/1.79 = ( Abs @ S3 ) ) ) )
% 3.28/1.79 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_81_equals__rsp,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ Xa @ Xb )
% 3.28/1.79 => ( ( R @ Ya @ Yb )
% 3.28/1.79 => ( ( R @ Xa @ Ya )
% 3.28/1.79 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_82_rep__abs__rsp,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ X1 @ X22 )
% 3.28/1.79 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_83_Quotient3__def,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType] :
% 3.28/1.79 ( ( quotient3 @ A @ B )
% 3.28/1.79 = ( ^ [R4: A > A > $o,Abs3: A > B,Rep3: B > A] :
% 3.28/1.79 ( ! [A5: B] :
% 3.28/1.79 ( ( Abs3 @ ( Rep3 @ A5 ) )
% 3.28/1.79 = A5 )
% 3.28/1.79 & ! [A5: B] : ( R4 @ ( Rep3 @ A5 ) @ ( Rep3 @ A5 ) )
% 3.28/1.79 & ! [R5: A,S2: A] :
% 3.28/1.79 ( ( R4 @ R5 @ S2 )
% 3.28/1.79 = ( ( R4 @ R5 @ R5 )
% 3.28/1.79 & ( R4 @ S2 @ S2 )
% 3.28/1.79 & ( ( Abs3 @ R5 )
% 3.28/1.79 = ( Abs3 @ S2 ) ) ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_84_Quotient3__rel,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( ( R @ R6 @ R6 )
% 3.28/1.79 & ( R @ S4 @ S4 )
% 3.28/1.79 & ( ( Abs @ R6 )
% 3.28/1.79 = ( Abs @ S4 ) ) )
% 3.28/1.79 = ( R @ R6 @ S4 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_85_Quotient3__refl1,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ R6 @ S4 )
% 3.28/1.79 => ( R @ R6 @ R6 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_86_Quotient3__refl2,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ R6 @ S4 )
% 3.28/1.79 => ( R @ S4 @ S4 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_87_rep__abs__rsp__left,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ X1 @ X22 )
% 3.28/1.79 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_88_Quotient3__abs__rep,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( Abs @ ( Rep @ A3 ) )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_89_Quotient3__rel__abs,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ R6 @ S4 )
% 3.28/1.79 => ( ( Abs @ R6 )
% 3.28/1.79 = ( Abs @ S4 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_90_Quotient3__rel__rep,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B,B3: B] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 3.28/1.79 = ( A3 = B3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_91_Quotient3__rep__abs,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( R @ R6 @ R6 )
% 3.28/1.79 => ( R @ ( Rep @ ( Abs @ R6 ) ) @ R6 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_92_Quotient3__rep__reflp,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_93_lprefix__lfilterI,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( coinductive_lprefix @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_94_llist__all2__lfilterI,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( P @ X @ Y )
% 3.28/1.79 => ( ( Q1 @ X )
% 3.28/1.79 = ( Q2 @ Y ) ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lfilter @ A @ Q1 @ Xs ) @ ( coinductive_lfilter @ B @ Q2 @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_95_apply__rspQ3_H_H,axiom,
% 3.28/1.79 ! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S: C > C > $o,F: A > C,X2: B] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S @ F @ F )
% 3.28/1.79 => ( S @ ( F @ ( Rep @ X2 ) ) @ ( F @ ( Rep @ X2 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_96_apply__rspQ3,axiom,
% 3.28/1.79 ! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F: A > C,G: A > C,X2: A,Y3: A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 3.28/1.79 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
% 3.28/1.79 => ( ( R1 @ X2 @ Y3 )
% 3.28/1.79 => ( R2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_97_lfinite__lfilterI,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_98_llist__all2__ldropWhileI,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ! [X: A,Y: B] :
% 3.28/1.79 ( ( P @ X @ Y )
% 3.28/1.79 => ( ( Q1 @ X )
% 3.28/1.79 = ( Q2 @ Y ) ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coindu218763757pWhile @ A @ Q1 @ Xs ) @ ( coindu218763757pWhile @ B @ Q2 @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_99_quot__rel__rsp,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.28/1.79 => ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
% 3.28/1.79 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ R
% 3.28/1.79 @ R ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_100_fun_Orel__refl,axiom,
% 3.28/1.79 ! [B: $tType,D: $tType,Ra: B > B > $o,X2: D > B] :
% 3.28/1.79 ( ! [X: B] : ( Ra @ X @ X )
% 3.28/1.79 => ( bNF_rel_fun @ D @ D @ B @ B
% 3.28/1.79 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 3.28/1.79 @ Ra
% 3.28/1.79 @ X2
% 3.28/1.79 @ X2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_101_fun_Orel__eq,axiom,
% 3.28/1.79 ! [A: $tType,D: $tType] :
% 3.28/1.79 ( ( bNF_rel_fun @ D @ D @ A @ A
% 3.28/1.79 @ ^ [Y2: D,Z: D] : ( Y2 = Z )
% 3.28/1.79 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 3.28/1.79 = ( ^ [Y2: D > A,Z: D > A] : ( Y2 = Z ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_102_map__fun__apply,axiom,
% 3.28/1.79 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 3.28/1.79 ( ( map_fun @ B @ C @ D @ A )
% 3.28/1.79 = ( ^ [F3: B > C,G3: D > A,H2: C > D,X3: B] : ( G3 @ ( H2 @ ( F3 @ X3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_103_bex1__rel__rsp,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
% 3.28/1.79 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 3.28/1.79 => ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
% 3.28/1.79 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
% 3.28/1.79 @ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
% 3.28/1.79 @ ( bex1_rel @ A @ R )
% 3.28/1.79 @ ( bex1_rel @ A @ R ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_104_llist_Octr__transfer_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ R @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) @ ( coinductive_LCons @ A ) @ ( coinductive_LCons @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_105_LCons__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ A2 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_LCons @ A ) @ ( coinductive_LCons @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_106_ldropWhile__eq__ldrop,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coindu218763757pWhile @ A )
% 3.28/1.79 = ( ^ [P3: A > $o,Xs2: coinductive_llist @ A] : ( coinductive_ldrop @ A @ ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_107_zero__natural_Orsp,axiom,
% 3.28/1.79 ( ( zero_zero @ nat )
% 3.28/1.79 = ( zero_zero @ nat ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_108_llist_Oinject,axiom,
% 3.28/1.79 ! [A: $tType,X21: A,X222: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_LCons @ A @ X21 @ X222 )
% 3.28/1.79 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 3.28/1.79 = ( ( X21 = Y21 )
% 3.28/1.79 & ( X222 = Y22 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_109_ldrop__0,axiom,
% 3.28/1.79 ! [B: $tType,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
% 3.28/1.79 = Xs ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_110_LCons__lprefix__LCons,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 3.28/1.79 = ( ( X2 = Y3 )
% 3.28/1.79 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_111_llist__all2__LCons__LCons,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o,X21: A,X222: coinductive_llist @ A,Y21: B,Y22: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X222 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) )
% 3.28/1.79 = ( ( R @ X21 @ Y21 )
% 3.28/1.79 & ( coindu1486289336t_all2 @ A @ B @ R @ X222 @ Y22 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_112_lfinite__code_I2_J,axiom,
% 3.28/1.79 ! [B: $tType,X2: B,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) )
% 3.28/1.79 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_113_lfinite__LCons,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_114_lfilter__LCons,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( P @ X2 )
% 3.28/1.79 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( coinductive_LCons @ A @ X2 @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 3.28/1.79 & ( ~ ( P @ X2 )
% 3.28/1.79 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_115_ldropWhile__LCons,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( P @ X2 )
% 3.28/1.79 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 3.28/1.79 & ( ~ ( P @ X2 )
% 3.28/1.79 => ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_116_ldrop__lmap,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_ldrop @ A @ N @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 3.28/1.79 = ( coinductive_lmap @ B @ A @ F @ ( coinductive_ldrop @ B @ N @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_117_lmirror__aux__simps_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X2: coinductive_llist @ A] :
% 3.28/1.79 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X2 ) )
% 3.28/1.79 = ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_118_unfold__llist__eq__LCons,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
% 3.28/1.79 = ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( ~ ( IS_LNIL @ B3 )
% 3.28/1.79 & ( X2
% 3.28/1.79 = ( LHD @ B3 ) )
% 3.28/1.79 & ( Xs
% 3.28/1.79 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_119_llist_Opred__inject_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,A3: A,Aa2: coinductive_llist @ A] :
% 3.28/1.79 ( ( coindu543516966_llist @ A @ P @ ( coinductive_LCons @ A @ A3 @ Aa2 ) )
% 3.28/1.79 = ( ( P @ A3 )
% 3.28/1.79 & ( coindu543516966_llist @ A @ P @ Aa2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_120_bex1__rel__aux2,axiom,
% 3.28/1.79 ! [A: $tType,R: A > A > $o,X2: A > $o,Y3: A > $o] :
% 3.28/1.79 ( ! [Xa2: A,Ya2: A] :
% 3.28/1.79 ( ( R @ Xa2 @ Ya2 )
% 3.28/1.79 => ( ( X2 @ Xa2 )
% 3.28/1.79 = ( Y3 @ Ya2 ) ) )
% 3.28/1.79 => ( ( bex1_rel @ A @ R @ Y3 )
% 3.28/1.79 => ( bex1_rel @ A @ R @ X2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_121_bex1__rel__aux,axiom,
% 3.28/1.79 ! [A: $tType,R: A > A > $o,X2: A > $o,Y3: A > $o] :
% 3.28/1.79 ( ! [Xa2: A,Ya2: A] :
% 3.28/1.79 ( ( R @ Xa2 @ Ya2 )
% 3.28/1.79 => ( ( X2 @ Xa2 )
% 3.28/1.79 = ( Y3 @ Ya2 ) ) )
% 3.28/1.79 => ( ( bex1_rel @ A @ R @ X2 )
% 3.28/1.79 => ( bex1_rel @ A @ R @ Y3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_122_Le__LCons,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X2: A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ X2 @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_123_LCons__lprefix__conv,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Ys )
% 3.28/1.79 = ( ? [Ys3: coinductive_llist @ A] :
% 3.28/1.79 ( ( Ys
% 3.28/1.79 = ( coinductive_LCons @ A @ X2 @ Ys3 ) )
% 3.28/1.79 & ( coinductive_lprefix @ A @ Xs @ Ys3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_124_llist_Orel__intros_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X222: coinductive_llist @ A,Y22: coinductive_llist @ B] :
% 3.28/1.79 ( ( R @ X21 @ Y21 )
% 3.28/1.79 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X222 @ Y22 )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X222 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_125_llist__all2__LCons1,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Ys )
% 3.28/1.79 = ( ? [Y5: B,Ys3: coinductive_llist @ B] :
% 3.28/1.79 ( ( Ys
% 3.28/1.79 = ( coinductive_LCons @ B @ Y5 @ Ys3 ) )
% 3.28/1.79 & ( P @ X2 @ Y5 )
% 3.28/1.79 & ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_126_llist__all2__LCons2,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Y3: B,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
% 3.28/1.79 = ( ? [X3: A,Xs4: coinductive_llist @ A] :
% 3.28/1.79 ( ( Xs
% 3.28/1.79 = ( coinductive_LCons @ A @ X3 @ Xs4 ) )
% 3.28/1.79 & ( P @ X3 @ Y3 )
% 3.28/1.79 & ( coindu1486289336t_all2 @ A @ B @ P @ Xs4 @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_127_lfinite__LConsI,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,X2: A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_128_llist_Osimps_I13_J,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,F: A > B,X21: A,X222: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lmap @ A @ B @ F @ ( coinductive_LCons @ A @ X21 @ X222 ) )
% 3.28/1.79 = ( coinductive_LCons @ B @ ( F @ X21 ) @ ( coinductive_lmap @ A @ B @ F @ X222 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_129_lmap__eq__LCons__conv,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Y3: A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_lmap @ B @ A @ F @ Xs )
% 3.28/1.79 = ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 3.28/1.79 = ( ? [X3: B,Xs4: coinductive_llist @ B] :
% 3.28/1.79 ( ( Xs
% 3.28/1.79 = ( coinductive_LCons @ B @ X3 @ Xs4 ) )
% 3.28/1.79 & ( Y3
% 3.28/1.79 = ( F @ X3 ) )
% 3.28/1.79 & ( Ys
% 3.28/1.79 = ( coinductive_lmap @ B @ A @ F @ Xs4 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_130_llist__all2__ldropI,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,N: extended_enat] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_ldrop @ A @ N @ Xs ) @ ( coinductive_ldrop @ B @ N @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_131_llist_Osimps_I5_J,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,F1: B,F22: A > ( coinductive_llist @ A ) > B,X21: A,X222: coinductive_llist @ A] :
% 3.28/1.79 ( ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ ( coinductive_LCons @ A @ X21 @ X222 ) )
% 3.28/1.79 = ( F22 @ X21 @ X222 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_132_lfilter__LCons__found,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( P @ X2 )
% 3.28/1.79 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( coinductive_LCons @ A @ X2 @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_133_lfilter__LCons__seek,axiom,
% 3.28/1.79 ! [A: $tType,P4: A > $o,X2: A,L: coinductive_llist @ A] :
% 3.28/1.79 ( ~ ( P4 @ X2 )
% 3.28/1.79 => ( ( coinductive_lfilter @ A @ P4 @ ( coinductive_LCons @ A @ X2 @ L ) )
% 3.28/1.79 = ( coinductive_lfilter @ A @ P4 @ L ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_134_iterates_Ocode,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coinductive_iterates @ A )
% 3.28/1.79 = ( ^ [F3: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_iterates @ A @ F3 @ ( F3 @ X3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_135_unfold__llist_Octr_I2_J,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,P4: A > $o,A3: A,G21: A > B,G22: A > A] :
% 3.28/1.79 ( ~ ( P4 @ A3 )
% 3.28/1.79 => ( ( coindu1441602521_llist @ A @ B @ P4 @ G21 @ G22 @ A3 )
% 3.28/1.79 = ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_136_llist_Ocorec_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,C: $tType,P4: C > $o,A3: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
% 3.28/1.79 ( ~ ( P4 @ A3 )
% 3.28/1.79 => ( ( coindu1259883913_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ A3 )
% 3.28/1.79 = ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_137_iterates__equality,axiom,
% 3.28/1.79 ! [A: $tType,H3: A > ( coinductive_llist @ A ),F: A > A] :
% 3.28/1.79 ( ! [X: A] :
% 3.28/1.79 ( ( H3 @ X )
% 3.28/1.79 = ( coinductive_LCons @ A @ X @ ( coinductive_lmap @ A @ A @ F @ ( H3 @ X ) ) ) )
% 3.28/1.79 => ( H3
% 3.28/1.79 = ( coinductive_iterates @ A @ F ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_138_iterates__lmap,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coinductive_iterates @ A )
% 3.28/1.79 = ( ^ [F3: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_lmap @ A @ A @ F3 @ ( coinductive_iterates @ A @ F3 @ X3 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_139_ldrop__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ extended_enat @ extended_enat @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 3.28/1.79 @ ( coinductive_ldrop @ A )
% 3.28/1.79 @ ( coinductive_ldrop @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_140_natural_Osize_I1_J,axiom,
% 3.28/1.79 ( ( code_size_natural @ ( zero_zero @ code_natural ) )
% 3.28/1.79 = ( zero_zero @ nat ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_141_llistsum__LCons,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [Xs: coinductive_llist @ A,X2: A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coindu780009021istsum @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 = ( plus_plus @ A @ X2 @ ( coindu780009021istsum @ A @ Xs ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_142_llist__all2__lappend1D_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ Ys )
% 3.28/1.79 => ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ ( coinductive_ldrop @ B @ ( coinductive_llength @ A @ Xs ) @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_143_natural_Osize_I3_J,axiom,
% 3.28/1.79 ( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
% 3.28/1.79 = ( zero_zero @ nat ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_144_lmember__code_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Y3: A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lmember @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
% 3.28/1.79 = ( ( X2 = Y3 )
% 3.28/1.79 | ( coinductive_lmember @ A @ X2 @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_145_add__left__cancel,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A,C3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ C3 ) )
% 3.28/1.79 = ( B3 = C3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_146_add__right__cancel,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [B3: A,A3: A,C3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.28/1.79 = ( plus_plus @ A @ C3 @ A3 ) )
% 3.28/1.79 = ( B3 = C3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_147_ldrop__ldrop,axiom,
% 3.28/1.79 ! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_ldrop @ A @ N @ ( coinductive_ldrop @ A @ M @ Xs ) )
% 3.28/1.79 = ( coinductive_ldrop @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_148_add__cancel__right__right,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A] :
% 3.28/1.79 ( ( A3
% 3.28/1.79 = ( plus_plus @ A @ A3 @ B3 ) )
% 3.28/1.79 = ( B3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_149_add__cancel__right__left,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A] :
% 3.28/1.79 ( ( A3
% 3.28/1.79 = ( plus_plus @ A @ B3 @ A3 ) )
% 3.28/1.79 = ( B3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_150_add__cancel__left__right,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.28/1.79 = A3 )
% 3.28/1.79 = ( B3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_151_add__cancel__left__left,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [B3: A,A3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.28/1.79 = A3 )
% 3.28/1.79 = ( B3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_152_double__zero__sym,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( ( zero_zero @ A )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ A3 ) )
% 3.28/1.79 = ( A3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_153_double__zero,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ A3 @ A3 )
% 3.28/1.79 = ( zero_zero @ A ) )
% 3.28/1.79 = ( A3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_154_add_Oright__neutral,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_155_add_Oleft__neutral,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_156_lappend__code_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,Xa: A,X2: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X2 ) @ Ys )
% 3.28/1.79 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X2 @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_157_llength__lappend,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_llength @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.28/1.79 = ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llength @ A @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_158_lfinite__lappend,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.28/1.79 = ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_159_lprefix__lappend__same,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 3.28/1.79 = ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_160_lfilter__lappend__lfinite,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.28/1.79 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_161_lappend__ltakeWhile__ldropWhile,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 3.28/1.79 = Xs ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_162_llcp__lappend__same,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_llcp @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 3.28/1.79 = ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llcp @ A @ Ys @ Zs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_163_lprefix__lappend,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_164_lappend__lprefixE,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 3.28/1.79 => ~ ! [Zs2: coinductive_llist @ A] :
% 3.28/1.79 ( Zs
% 3.28/1.79 != ( coinductive_lappend @ A @ Xs @ Zs2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_165_lprefix__lappendD,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
% 3.28/1.79 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_166_lprefix__conv__lappend,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( coinductive_lprefix @ A )
% 3.28/1.79 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 3.28/1.79 ? [Zs3: coinductive_llist @ A] :
% 3.28/1.79 ( Ys2
% 3.28/1.79 = ( coinductive_lappend @ A @ Xs2 @ Zs3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_167_lprefix__lappend__sameI,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.28/1.79 => ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_168_comm__monoid__add__class_Oadd__0,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( comm_monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_169_add_Ocomm__neutral,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( comm_monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_170_add_Ogroup__left__neutral,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( group_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_171_add__eq__0__iff__both__eq__0,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [X2: A,Y3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ X2 @ Y3 )
% 3.28/1.79 = ( zero_zero @ A ) )
% 3.28/1.79 = ( ( X2
% 3.28/1.79 = ( zero_zero @ A ) )
% 3.28/1.79 & ( Y3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_172_lappend__inf,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.28/1.79 = Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_173_llistsum__lappend,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( monoid_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coinductive_lfinite @ A @ Ys )
% 3.28/1.79 => ( ( coindu780009021istsum @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.28/1.79 = ( plus_plus @ A @ ( coindu780009021istsum @ A @ Xs ) @ ( coindu780009021istsum @ A @ Ys ) ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_174_lmap__lappend__distrib,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_lmap @ B @ A @ F @ ( coinductive_lappend @ B @ Xs @ Ys ) )
% 3.28/1.79 = ( coinductive_lappend @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) @ ( coinductive_lmap @ B @ A @ F @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_175_lappend__assoc,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 3.28/1.79 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A,C3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_177_add__mono__thms__linordered__semiring_I4_J,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [I: A,J: A,K: A,L: A] :
% 3.28/1.79 ( ( ( I = J )
% 3.28/1.79 & ( K = L ) )
% 3.28/1.79 => ( ( plus_plus @ A @ I @ K )
% 3.28/1.79 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_178_add_Oassoc,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A,C3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_179_add_Oleft__cancel,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( group_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A,C3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ C3 ) )
% 3.28/1.79 = ( B3 = C3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_180_add_Oright__cancel,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( group_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [B3: A,A3: A,C3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.28/1.79 = ( plus_plus @ A @ C3 @ A3 ) )
% 3.28/1.79 = ( B3 = C3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_181_add_Ocommute,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ( ( plus_plus @ A )
% 3.28/1.79 = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_182_add_Oleft__commute,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [B3: A,A3: A,C3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_183_add__left__imp__eq,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A,B3: A,C3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ A3 @ B3 )
% 3.28/1.79 = ( plus_plus @ A @ A3 @ C3 ) )
% 3.28/1.79 => ( B3 = C3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_184_add__right__imp__eq,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 3.28/1.79 => ! [B3: A,A3: A,C3: A] :
% 3.28/1.79 ( ( ( plus_plus @ A @ B3 @ A3 )
% 3.28/1.79 = ( plus_plus @ A @ C3 @ A3 ) )
% 3.28/1.79 => ( B3 = C3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_185_lappend__iterates,axiom,
% 3.28/1.79 ! [A: $tType,F: A > A,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lappend @ A @ ( coinductive_iterates @ A @ F @ X2 ) @ Xs )
% 3.28/1.79 = ( coinductive_iterates @ A @ F @ X2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_186_lmirror__aux__acc,axiom,
% 3.28/1.79 ! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
% 3.28/1.79 = ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_187_llist__all2__lappendI,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coinductive_lfinite @ B @ Ys )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ Ys4 ) ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ ( coinductive_lappend @ B @ Ys @ Ys4 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_188_lappend__eq__lappend__conv,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Us: coinductive_llist @ A,Ys: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_llength @ A @ Xs )
% 3.28/1.79 = ( coinductive_llength @ A @ Us ) )
% 3.28/1.79 => ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.28/1.79 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 3.28/1.79 = ( ( Xs = Us )
% 3.28/1.79 & ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( Ys = Vs ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_189_lfilter__eq__lappend__lfiniteD,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 3.28/1.79 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 3.28/1.79 => ( ( coinductive_lfinite @ A @ Ys )
% 3.28/1.79 => ? [Us2: coinductive_llist @ A,Vs2: coinductive_llist @ A] :
% 3.28/1.79 ( ( Xs
% 3.28/1.79 = ( coinductive_lappend @ A @ Us2 @ Vs2 ) )
% 3.28/1.79 & ( coinductive_lfinite @ A @ Us2 )
% 3.28/1.79 & ( Ys
% 3.28/1.79 = ( coinductive_lfilter @ A @ P @ Us2 ) )
% 3.28/1.79 & ( Zs
% 3.28/1.79 = ( coinductive_lfilter @ A @ P @ Vs2 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_190_lappend__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_lappend @ A ) @ ( coinductive_lappend @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_191_add__is__0,axiom,
% 3.28/1.79 ! [M: nat,N: nat] :
% 3.28/1.79 ( ( ( plus_plus @ nat @ M @ N )
% 3.28/1.79 = ( zero_zero @ nat ) )
% 3.28/1.79 = ( ( M
% 3.28/1.79 = ( zero_zero @ nat ) )
% 3.28/1.79 & ( N
% 3.28/1.79 = ( zero_zero @ nat ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_192_Nat_Oadd__0__right,axiom,
% 3.28/1.79 ! [M: nat] :
% 3.28/1.79 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 3.28/1.79 = M ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_193_natural_Osimps_I4_J,axiom,
% 3.28/1.79 ! [T: $tType,F1: T,F22: code_natural > T] :
% 3.28/1.79 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 3.28/1.79 = F1 ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_194_plus__natural_Orsp,axiom,
% 3.28/1.79 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 3.28/1.79 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 3.28/1.79 @ ( plus_plus @ nat )
% 3.28/1.79 @ ( plus_plus @ nat ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_195_add__eq__self__zero,axiom,
% 3.28/1.79 ! [M: nat,N: nat] :
% 3.28/1.79 ( ( ( plus_plus @ nat @ M @ N )
% 3.28/1.79 = M )
% 3.28/1.79 => ( N
% 3.28/1.79 = ( zero_zero @ nat ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_196_plus__nat_Oadd__0,axiom,
% 3.28/1.79 ! [N: nat] :
% 3.28/1.79 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 3.28/1.79 = N ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_197_ind__euclid,axiom,
% 3.28/1.79 ! [P: nat > nat > $o,A3: nat,B3: nat] :
% 3.28/1.79 ( ! [A4: nat,B4: nat] :
% 3.28/1.79 ( ( P @ A4 @ B4 )
% 3.28/1.79 = ( P @ B4 @ A4 ) )
% 3.28/1.79 => ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
% 3.28/1.79 => ( ! [A4: nat,B4: nat] :
% 3.28/1.79 ( ( P @ A4 @ B4 )
% 3.28/1.79 => ( P @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
% 3.28/1.79 => ( P @ A3 @ B3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_198_add__0__iff,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( semiri456707255roduct @ A @ ( type @ A ) )
% 3.28/1.79 => ! [B3: A,A3: A] :
% 3.28/1.79 ( ( B3
% 3.28/1.79 = ( plus_plus @ A @ B3 @ A3 ) )
% 3.28/1.79 = ( A3
% 3.28/1.79 = ( zero_zero @ A ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_199_semiring__normalization__rules_I6_J,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_200_semiring__normalization__rules_I5_J,axiom,
% 3.28/1.79 ! [A: $tType] :
% 3.28/1.79 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 3.28/1.79 => ! [A3: A] :
% 3.28/1.79 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 3.28/1.79 = A3 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_201_llast__lappend__LCons,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) )
% 3.28/1.79 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_202_rec__nat__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ A2
% 3.28/1.79 @ ( bNF_rel_fun @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B )
% 3.28/1.79 @ ( bNF_rel_fun @ nat @ nat @ ( A > A ) @ ( B > B )
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 3.28/1.79 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 3.28/1.79 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 3.28/1.79 @ A2 ) )
% 3.28/1.79 @ ( rec_nat @ A )
% 3.28/1.79 @ ( rec_nat @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_203_llast__LCons2,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Y3: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
% 3.28/1.79 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_204_old_Onat_Osimps_I6_J,axiom,
% 3.28/1.79 ! [T: $tType,F1: T,F22: nat > T > T] :
% 3.28/1.79 ( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
% 3.28/1.79 = F1 ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_205_case__nat__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ A @ B @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) @ A2
% 3.28/1.79 @ ( bNF_rel_fun @ ( nat > A ) @ ( nat > B ) @ ( nat > A ) @ ( nat > B )
% 3.28/1.79 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 3.28/1.79 @ A2 )
% 3.28/1.79 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 3.28/1.79 @ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
% 3.28/1.79 @ A2 ) )
% 3.28/1.79 @ ( case_nat @ A )
% 3.28/1.79 @ ( case_nat @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_206_natural_Osimps_I6_J,axiom,
% 3.28/1.79 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 3.28/1.79 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 3.28/1.79 = F1 ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_207_old_Onat_Osimps_I4_J,axiom,
% 3.28/1.79 ! [A: $tType,F1: A,F22: nat > A] :
% 3.28/1.79 ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
% 3.28/1.79 = F1 ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_208_ltake__transfer,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 3.28/1.79 ( bNF_rel_fun @ extended_enat @ extended_enat @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
% 3.28/1.79 @ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
% 3.28/1.79 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
% 3.28/1.79 @ ( coinductive_ltake @ A )
% 3.28/1.79 @ ( coinductive_ltake @ B ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_209_lfilter__eq__LConsD,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 3.28/1.79 = ( coinductive_LCons @ A @ X2 @ Xs ) )
% 3.28/1.79 => ? [Us2: coinductive_llist @ A,Vs2: coinductive_llist @ A] :
% 3.28/1.79 ( ( Ys
% 3.28/1.79 = ( coinductive_lappend @ A @ Us2 @ ( coinductive_LCons @ A @ X2 @ Vs2 ) ) )
% 3.28/1.79 & ( coinductive_lfinite @ A @ Us2 )
% 3.28/1.79 & ! [X5: A] :
% 3.28/1.79 ( ( member @ A @ X5 @ ( coinductive_lset @ A @ Us2 ) )
% 3.28/1.79 => ~ ( P @ X5 ) )
% 3.28/1.79 & ( P @ X2 )
% 3.28/1.79 & ( Xs
% 3.28/1.79 = ( coinductive_lfilter @ A @ P @ Vs2 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_210_ltake__is__lprefix,axiom,
% 3.28/1.79 ! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ Xs ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_211_llist__all2__same,axiom,
% 3.28/1.79 ! [A: $tType,P: A > A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ A @ P @ Xs @ Xs )
% 3.28/1.79 = ( ! [X3: A] :
% 3.28/1.79 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( P @ X3 @ X3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_212_ltake__lmap,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coinductive_llist @ B] :
% 3.28/1.79 ( ( coinductive_ltake @ A @ N @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
% 3.28/1.79 = ( coinductive_lmap @ B @ A @ F @ ( coinductive_ltake @ B @ N @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_213_ldropWhile__cong,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 3.28/1.79 ( ( Xs = Ys )
% 3.28/1.79 => ( ! [X: A] :
% 3.28/1.79 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 3.28/1.79 => ( ( P @ X )
% 3.28/1.79 = ( Q @ X ) ) )
% 3.28/1.79 => ( ( coindu218763757pWhile @ A @ P @ Xs )
% 3.28/1.79 = ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_214_in__lset__ldropWhileD,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
% 3.28/1.79 => ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_215_llist_Opred__cong,axiom,
% 3.28/1.79 ! [A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,P: A > $o,Pa: A > $o] :
% 3.28/1.79 ( ( X2 = Ya )
% 3.28/1.79 => ( ! [Z3: A] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
% 3.28/1.79 => ( ( P @ Z3 )
% 3.28/1.79 = ( Pa @ Z3 ) ) )
% 3.28/1.79 => ( ( coindu543516966_llist @ A @ P @ X2 )
% 3.28/1.79 = ( coindu543516966_llist @ A @ Pa @ Ya ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_216_llist_Opred__mono__strong,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,X2: coinductive_llist @ A,Pa: A > $o] :
% 3.28/1.79 ( ( coindu543516966_llist @ A @ P @ X2 )
% 3.28/1.79 => ( ! [Z3: A] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 3.28/1.79 => ( ( P @ Z3 )
% 3.28/1.79 => ( Pa @ Z3 ) ) )
% 3.28/1.79 => ( coindu543516966_llist @ A @ Pa @ X2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_217_lset__ltakeWhileD,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
% 3.28/1.79 => ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 & ( P @ X2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_218_ltakeWhile__cong,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 3.28/1.79 ( ( Xs = Ys )
% 3.28/1.79 => ( ! [X: A] :
% 3.28/1.79 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 3.28/1.79 => ( ( P @ X )
% 3.28/1.79 = ( Q @ X ) ) )
% 3.28/1.79 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 3.28/1.79 = ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_219_ltakeWhile__all,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 3.28/1.79 ( ! [X: A] :
% 3.28/1.79 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( P @ X ) )
% 3.28/1.79 => ( ( coindu501562517eWhile @ A @ P @ Xs )
% 3.28/1.79 = Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_220_lfilter__cong,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 3.28/1.79 ( ( Xs = Ys )
% 3.28/1.79 => ( ! [X: A] :
% 3.28/1.79 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 3.28/1.79 => ( ( P @ X )
% 3.28/1.79 = ( Q @ X ) ) )
% 3.28/1.79 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 3.28/1.79 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_221_lfilter__id__conv,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 3.28/1.79 = Xs )
% 3.28/1.79 = ( ! [X3: A] :
% 3.28/1.79 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( P @ X3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_222_llist_Oinj__map__strong,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Xa: coinductive_llist @ A,F: A > B,Fa: A > B] :
% 3.28/1.79 ( ! [Z3: A,Za: A] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 3.28/1.79 => ( ( member @ A @ Za @ ( coinductive_lset @ A @ Xa ) )
% 3.28/1.79 => ( ( ( F @ Z3 )
% 3.28/1.79 = ( Fa @ Za ) )
% 3.28/1.79 => ( Z3 = Za ) ) ) )
% 3.28/1.79 => ( ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 3.28/1.79 = ( coinductive_lmap @ A @ B @ Fa @ Xa ) )
% 3.28/1.79 => ( X2 = Xa ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_223_llist_Omap__cong0,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,F: A > B,G: A > B] :
% 3.28/1.79 ( ! [Z3: A] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 3.28/1.79 => ( ( F @ Z3 )
% 3.28/1.79 = ( G @ Z3 ) ) )
% 3.28/1.79 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 3.28/1.79 = ( coinductive_lmap @ A @ B @ G @ X2 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_224_llist_Omap__cong,axiom,
% 3.28/1.79 ! [B: $tType,A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
% 3.28/1.79 ( ( X2 = Ya )
% 3.28/1.79 => ( ! [Z3: A] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
% 3.28/1.79 => ( ( F @ Z3 )
% 3.28/1.79 = ( G @ Z3 ) ) )
% 3.28/1.79 => ( ( coinductive_lmap @ A @ B @ F @ X2 )
% 3.28/1.79 = ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_225_llist_Orel__refl__strong,axiom,
% 3.28/1.79 ! [A: $tType,X2: coinductive_llist @ A,Ra: A > A > $o] :
% 3.28/1.79 ( ! [Z3: A] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 3.28/1.79 => ( Ra @ Z3 @ Z3 ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ A @ Ra @ X2 @ X2 ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_226_llist_Orel__mono__strong,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,R: A > B > $o,X2: coinductive_llist @ A,Y3: coinductive_llist @ B,Ra: A > B > $o] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
% 3.28/1.79 => ( ! [Z3: A,Yb2: B] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
% 3.28/1.79 => ( ( member @ B @ Yb2 @ ( coinductive_lset @ B @ Y3 ) )
% 3.28/1.79 => ( ( R @ Z3 @ Yb2 )
% 3.28/1.79 => ( Ra @ Z3 @ Yb2 ) ) ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ Ra @ X2 @ Y3 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_227_llist_Orel__cong,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,Y3: coinductive_llist @ B,Xa: coinductive_llist @ B,R: A > B > $o,Ra: A > B > $o] :
% 3.28/1.79 ( ( X2 = Ya )
% 3.28/1.79 => ( ( Y3 = Xa )
% 3.28/1.79 => ( ! [Z3: A,Yb2: B] :
% 3.28/1.79 ( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
% 3.28/1.79 => ( ( member @ B @ Yb2 @ ( coinductive_lset @ B @ Xa ) )
% 3.28/1.79 => ( ( R @ Z3 @ Yb2 )
% 3.28/1.79 = ( Ra @ Z3 @ Yb2 ) ) ) )
% 3.28/1.79 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
% 3.28/1.79 = ( coindu1486289336t_all2 @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_228_llist__all2__lsetD2,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Y3: B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ( member @ B @ Y3 @ ( coinductive_lset @ B @ Ys ) )
% 3.28/1.79 => ? [X: A] :
% 3.28/1.79 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 & ( P @ X @ Y3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_229_llist__all2__lsetD1,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,X2: A] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ? [X: B] :
% 3.28/1.79 ( ( member @ B @ X @ ( coinductive_lset @ B @ Ys ) )
% 3.28/1.79 & ( P @ X2 @ X ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_230_llist__all2__reflI,axiom,
% 3.28/1.79 ! [A: $tType,Xs: coinductive_llist @ A,P: A > A > $o] :
% 3.28/1.79 ( ! [X: A] :
% 3.28/1.79 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( P @ X @ X ) )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_231_llist__all2__ltakeI,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,N: extended_enat] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ltake @ B @ N @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_232_lset__intros_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,X6: A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_233_lset__intros_I1_J,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] : ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_234_llist_Oset__intros_I2_J,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,A22: coinductive_llist @ A,A1: A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ A22 ) )
% 3.28/1.79 => ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_235_llist_Oset__intros_I1_J,axiom,
% 3.28/1.79 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_236_lset__cases,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( ! [Xs5: coinductive_llist @ A] :
% 3.28/1.79 ( Xs
% 3.28/1.79 != ( coinductive_LCons @ A @ X2 @ Xs5 ) )
% 3.28/1.79 => ~ ! [X7: A,Xs5: coinductive_llist @ A] :
% 3.28/1.79 ( ( Xs
% 3.28/1.79 = ( coinductive_LCons @ A @ X7 @ Xs5 ) )
% 3.28/1.79 => ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs5 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_237_lset__induct,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( ! [Xs6: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X2 @ Xs6 ) )
% 3.28/1.79 => ( ! [X7: A,Xs6: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs6 ) )
% 3.28/1.79 => ( ( X2 != X7 )
% 3.28/1.79 => ( ( P @ Xs6 )
% 3.28/1.79 => ( P @ ( coinductive_LCons @ A @ X7 @ Xs6 ) ) ) ) )
% 3.28/1.79 => ( P @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_238_lset__induct_H,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( ! [Xs6: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X2 @ Xs6 ) )
% 3.28/1.79 => ( ! [X7: A,Xs6: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs6 ) )
% 3.28/1.79 => ( ( P @ Xs6 )
% 3.28/1.79 => ( P @ ( coinductive_LCons @ A @ X7 @ Xs6 ) ) ) )
% 3.28/1.79 => ( P @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_239_in__lset__ldropD,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,N: extended_enat,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_ldrop @ A @ N @ Xs ) ) )
% 3.28/1.79 => ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_240_llist_Oset__cases,axiom,
% 3.28/1.79 ! [A: $tType,E2: A,A3: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ E2 @ ( coinductive_lset @ A @ A3 ) )
% 3.28/1.79 => ( ! [Z22: coinductive_llist @ A] :
% 3.28/1.79 ( A3
% 3.28/1.79 != ( coinductive_LCons @ A @ E2 @ Z22 ) )
% 3.28/1.79 => ~ ! [Z1: A,Z22: coinductive_llist @ A] :
% 3.28/1.79 ( ( A3
% 3.28/1.79 = ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 3.28/1.79 => ~ ( member @ A @ E2 @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_241_llist_Oset__induct,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,A3: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ A3 ) )
% 3.28/1.79 => ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 3.28/1.79 => ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
% 3.28/1.79 ( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
% 3.28/1.79 => ( ( P @ Xa2 @ Z22 )
% 3.28/1.79 => ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
% 3.28/1.79 => ( P @ X2 @ A3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_242_lset__lmirror__aux,axiom,
% 3.28/1.79 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lset @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 3.28/1.79 = ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Acc ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_243_lfinite__ldropWhile,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
% 3.28/1.79 = ( ? [X3: A] :
% 3.28/1.79 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 & ~ ( P @ X3 ) )
% 3.28/1.79 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_244_lfinite__ltakeWhile,axiom,
% 3.28/1.79 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
% 3.28/1.79 = ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 | ? [X3: A] :
% 3.28/1.79 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 & ~ ( P @ X3 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_245_ltakeWhile__lappend1,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ( ~ ( P @ X2 )
% 3.28/1.79 => ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.28/1.79 = ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_246_in__lset__lappend__iff,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 3.28/1.79 = ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 | ( ( coinductive_lfinite @ A @ Xs )
% 3.28/1.79 & ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_247_lappend__ltake__ldrop,axiom,
% 3.28/1.79 ! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_lappend @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ldrop @ A @ N @ Xs ) )
% 3.28/1.79 = Xs ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_248_lset__lmember,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 = ( coinductive_lmember @ A @ X2 @ Xs ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_249_ltake__ldrop,axiom,
% 3.28/1.79 ! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_ltake @ A @ N @ ( coinductive_ldrop @ A @ M @ Xs ) )
% 3.28/1.79 = ( coinductive_ldrop @ A @ M @ ( coinductive_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_250_llist__all2__lappend1D_I1_J,axiom,
% 3.28/1.79 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 3.28/1.79 ( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ Ys )
% 3.28/1.79 => ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_ltake @ B @ ( coinductive_llength @ A @ Xs ) @ Ys ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_251_split__llist__first,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ? [Ys5: coinductive_llist @ A,Zs4: coinductive_llist @ A] :
% 3.28/1.79 ( ( Xs
% 3.28/1.79 = ( coinductive_lappend @ A @ Ys5 @ ( coinductive_LCons @ A @ X2 @ Zs4 ) ) )
% 3.28/1.79 & ( coinductive_lfinite @ A @ Ys5 )
% 3.28/1.79 & ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys5 ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_252_split__llist,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 3.28/1.79 => ? [Ys5: coinductive_llist @ A,Zs4: coinductive_llist @ A] :
% 3.28/1.79 ( ( Xs
% 3.28/1.79 = ( coinductive_lappend @ A @ Ys5 @ ( coinductive_LCons @ A @ X2 @ Zs4 ) ) )
% 3.28/1.79 & ( coinductive_lfinite @ A @ Ys5 ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(fact_253_ltake__plus__conv__lappend,axiom,
% 3.28/1.79 ! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
% 3.28/1.79 ( ( coinductive_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs )
% 3.28/1.79 = ( coinductive_lappend @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ltake @ A @ M @ ( coinductive_ldrop @ A @ N @ Xs ) ) ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 3.28/1.79 semiri456707255roduct @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.28/1.79 canoni770627133id_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 3.28/1.79 ordere779506340up_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 3.28/1.79 cancel1352612707id_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 3.28/1.79 cancel_semigroup_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 3.28/1.79 ab_semigroup_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 3.28/1.79 comm_monoid_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 3.28/1.79 comm_semiring_1 @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 3.28/1.79 semigroup_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 3.28/1.79 monoid_add @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 3.28/1.79 zero @ nat @ ( type @ nat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_1,axiom,
% 3.28/1.79 canoni770627133id_add @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_2,axiom,
% 3.28/1.79 ordere779506340up_add @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_3,axiom,
% 3.28/1.79 ab_semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_4,axiom,
% 3.28/1.79 comm_monoid_add @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_5,axiom,
% 3.28/1.79 comm_semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_6,axiom,
% 3.28/1.79 semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_7,axiom,
% 3.28/1.79 monoid_add @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_8,axiom,
% 3.28/1.79 zero @ extended_enat @ ( type @ extended_enat ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_9,axiom,
% 3.28/1.79 ordere779506340up_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_10,axiom,
% 3.28/1.79 cancel1352612707id_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_11,axiom,
% 3.28/1.79 cancel_semigroup_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_12,axiom,
% 3.28/1.79 ab_semigroup_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_13,axiom,
% 3.28/1.79 comm_monoid_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_14,axiom,
% 3.28/1.79 comm_semiring_1 @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_15,axiom,
% 3.28/1.79 semigroup_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_16,axiom,
% 3.28/1.79 monoid_add @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_17,axiom,
% 3.28/1.79 zero @ code_natural @ ( type @ code_natural ) ).
% 3.28/1.79
% 3.28/1.79 thf(help_If_3_1_T,axiom,
% 3.28/1.79 ! [P: $o] :
% 3.28/1.79 ( ( P = $true )
% 3.28/1.79 | ( P = $false ) ) ).
% 3.28/1.79
% 3.28/1.79 thf(help_If_2_1_T,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Y3: A] :
% 3.28/1.79 ( ( if @ A @ $false @ X2 @ Y3 )
% 3.28/1.79 = Y3 ) ).
% 3.28/1.79
% 3.28/1.79 thf(help_If_1_1_T,axiom,
% 3.28/1.79 ! [A: $tType,X2: A,Y3: A] :
% 3.28/1.79 ( ( if @ A @ $true @ X2 @ Y3 )
% 3.28/1.79 = X2 ) ).
% 3.28/1.79
% 3.28/1.79 thf(conj_0,conjecture,
% 3.28/1.79 ( ( coinductive_llength @ a @ acca )
% 3.28/1.79 = ( coinductive_llength @ b @ acc_a ) ) ).
% 3.28/1.79 ------------------------
% 3.39/1.83 % (15876)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/183Mi)
% 3.39/1.83 % (15877)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_15775 for (2999ds/4Mi)
% 3.39/1.83 % (15880)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_15775 for (2999ds/2Mi)
% 3.39/1.83 % (15879)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/2Mi)
% 3.39/1.83 % (15878)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_15775 for (2999ds/27Mi)
% 3.39/1.83 % (15882)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/18Mi)
% 3.39/1.83 % (15881)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_15775 for (2999ds/275Mi)
% 3.39/1.83 % (15879)Instruction limit reached!
% 3.39/1.83 % (15879)------------------------------
% 3.39/1.83 % (15879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.83 % (15879)Termination reason: Unknown
% 3.39/1.83 % (15879)Termination phase: shuffling
% 3.39/1.83
% 3.39/1.83 % (15880)Instruction limit reached!
% 3.39/1.83 % (15880)------------------------------
% 3.39/1.83 % (15880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.83 % (15880)Termination reason: Unknown
% 3.39/1.83 % (15880)Termination phase: shuffling
% 3.39/1.83
% 3.39/1.83 % (15880)Memory used [KB]: 1535
% 3.39/1.83 % (15880)Time elapsed: 0.003 s
% 3.39/1.83 % (15880)Instructions burned: 3 (million)
% 3.39/1.83 % (15880)------------------------------
% 3.39/1.83 % (15880)------------------------------
% 3.39/1.83 % (15879)Memory used [KB]: 1407
% 3.39/1.83 % (15879)Time elapsed: 0.003 s
% 3.39/1.83 % (15879)Instructions burned: 2 (million)
% 3.39/1.83 % (15879)------------------------------
% 3.39/1.83 % (15879)------------------------------
% 3.39/1.83 % (15877)Instruction limit reached!
% 3.39/1.83 % (15877)------------------------------
% 3.39/1.83 % (15877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.83 % (15877)Termination reason: Unknown
% 3.39/1.83 % (15877)Termination phase: shuffling
% 3.39/1.83
% 3.39/1.83 % (15877)Memory used [KB]: 1535
% 3.39/1.83 % (15877)Time elapsed: 0.004 s
% 3.39/1.83 % (15877)Instructions burned: 4 (million)
% 3.39/1.83 % (15877)------------------------------
% 3.39/1.83 % (15877)------------------------------
% 3.39/1.84 % (15882)Instruction limit reached!
% 3.39/1.84 % (15882)------------------------------
% 3.39/1.84 % (15882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.84 % (15882)Termination reason: Unknown
% 3.39/1.84 % (15882)Termination phase: shuffling
% 3.39/1.84
% 3.39/1.84 % (15882)Memory used [KB]: 1791
% 3.39/1.84 % (15882)Time elapsed: 0.011 s
% 3.39/1.84 % (15882)Instructions burned: 19 (million)
% 3.39/1.84 % (15882)------------------------------
% 3.39/1.84 % (15882)------------------------------
% 3.39/1.84 % (15878)Instruction limit reached!
% 3.39/1.84 % (15878)------------------------------
% 3.39/1.84 % (15878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.84 % (15878)Termination reason: Unknown
% 3.39/1.84 % (15878)Termination phase: shuffling
% 3.39/1.84
% 3.39/1.84 % (15878)Memory used [KB]: 1918
% 3.39/1.84 % (15878)Time elapsed: 0.015 s
% 3.39/1.84 % (15878)Instructions burned: 27 (million)
% 3.39/1.84 % (15878)------------------------------
% 3.39/1.84 % (15878)------------------------------
% 3.39/1.85 % (15883)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/3Mi)
% 3.39/1.85 % (15884)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_15775 for (2999ds/37Mi)
% 3.39/1.85 % (15885)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_15775 for (2999ds/15Mi)
% 3.39/1.85 % (15883)Instruction limit reached!
% 3.39/1.85 % (15883)------------------------------
% 3.39/1.85 % (15883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.85 % (15883)Termination reason: Unknown
% 3.39/1.85 % (15883)Termination phase: shuffling
% 3.39/1.85
% 3.39/1.85 % (15883)Memory used [KB]: 1535
% 3.39/1.85 % (15883)Time elapsed: 0.003 s
% 3.39/1.85 % (15883)Instructions burned: 3 (million)
% 3.39/1.85 % (15883)------------------------------
% 3.39/1.85 % (15883)------------------------------
% 3.39/1.86 % (15885)Instruction limit reached!
% 3.39/1.86 % (15885)------------------------------
% 3.39/1.86 % (15885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.86 % (15885)Termination reason: Unknown
% 3.39/1.86 % (15885)Termination phase: shuffling
% 3.39/1.86
% 3.39/1.86 % (15885)Memory used [KB]: 1791
% 3.39/1.86 % (15885)Time elapsed: 0.009 s
% 3.39/1.86 % (15885)Instructions burned: 15 (million)
% 3.39/1.86 % (15885)------------------------------
% 3.39/1.86 % (15885)------------------------------
% 3.39/1.86 % (15886)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_15775 for (2999ds/3Mi)
% 3.39/1.86 % (15886)Instruction limit reached!
% 3.39/1.86 % (15886)------------------------------
% 3.39/1.86 % (15886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.86 % (15886)Termination reason: Unknown
% 3.39/1.86 % (15886)Termination phase: shuffling
% 3.39/1.86
% 3.39/1.86 % (15886)Memory used [KB]: 1535
% 3.39/1.86 % (15886)Time elapsed: 0.003 s
% 3.39/1.86 % (15886)Instructions burned: 3 (million)
% 3.39/1.86 % (15886)------------------------------
% 3.39/1.86 % (15886)------------------------------
% 3.39/1.86 % (15887)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_15775 for (2999ds/1041Mi)
% 3.39/1.87 % (15888)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/7Mi)
% 3.39/1.87 % (15884)Instruction limit reached!
% 3.39/1.87 % (15884)------------------------------
% 3.39/1.87 % (15884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.87 % (15884)Termination reason: Unknown
% 3.39/1.87 % (15884)Termination phase: shuffling
% 3.39/1.87
% 3.39/1.87 % (15884)Memory used [KB]: 2174
% 3.39/1.87 % (15884)Time elapsed: 0.019 s
% 3.39/1.87 % (15884)Instructions burned: 38 (million)
% 3.39/1.87 % (15884)------------------------------
% 3.39/1.87 % (15884)------------------------------
% 3.39/1.87 % (15888)Instruction limit reached!
% 3.39/1.87 % (15888)------------------------------
% 3.39/1.87 % (15888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.87 % (15888)Termination reason: Unknown
% 3.39/1.87 % (15888)Termination phase: shuffling
% 3.39/1.87
% 3.39/1.87 % (15888)Memory used [KB]: 1535
% 3.39/1.87 % (15888)Time elapsed: 0.005 s
% 3.39/1.87 % (15888)Instructions burned: 7 (million)
% 3.39/1.87 % (15888)------------------------------
% 3.39/1.87 % (15888)------------------------------
% 3.39/1.87 % (15889)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/16Mi)
% 3.39/1.87 % (15890)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_15775 for (2999ds/3Mi)
% 3.39/1.87 % (15890)Instruction limit reached!
% 3.39/1.87 % (15890)------------------------------
% 3.39/1.87 % (15890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.87 % (15890)Termination reason: Unknown
% 3.39/1.87 % (15890)Termination phase: shuffling
% 3.39/1.87
% 3.39/1.87 % (15890)Memory used [KB]: 1535
% 3.39/1.87 % (15890)Time elapsed: 0.003 s
% 3.39/1.87 % (15890)Instructions burned: 3 (million)
% 3.39/1.87 % (15890)------------------------------
% 3.39/1.87 % (15890)------------------------------
% 3.39/1.88 % (15881)Refutation not found, incomplete strategy
% 3.39/1.88 % (15881)------------------------------
% 3.39/1.88 % (15881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.88 % (15881)Termination reason: Refutation not found, incomplete strategy
% 3.39/1.88
% 3.39/1.88
% 3.39/1.88 % (15881)Memory used [KB]: 7164
% 3.39/1.88 % (15881)Time elapsed: 0.050 s
% 3.39/1.88 % (15881)Instructions burned: 99 (million)
% 3.39/1.88 % (15881)------------------------------
% 3.39/1.88 % (15881)------------------------------
% 3.39/1.88 % (15889)Instruction limit reached!
% 3.39/1.88 % (15889)------------------------------
% 3.39/1.88 % (15889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.88 % (15889)Termination reason: Unknown
% 3.39/1.88 % (15889)Termination phase: shuffling
% 3.39/1.88
% 3.39/1.88 % (15889)Memory used [KB]: 1791
% 3.39/1.88 % (15889)Time elapsed: 0.010 s
% 3.39/1.88 % (15889)Instructions burned: 17 (million)
% 3.39/1.88 % (15889)------------------------------
% 3.39/1.88 % (15889)------------------------------
% 3.39/1.88 % (15891)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_15775 for (2999ds/3Mi)
% 3.39/1.88 % (15891)Instruction limit reached!
% 3.39/1.88 % (15891)------------------------------
% 3.39/1.88 % (15891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.88 % (15891)Termination reason: Unknown
% 3.39/1.88 % (15891)Termination phase: shuffling
% 3.39/1.88
% 3.39/1.88 % (15891)Memory used [KB]: 1535
% 3.39/1.88 % (15891)Time elapsed: 0.003 s
% 3.39/1.88 % (15891)Instructions burned: 3 (million)
% 3.39/1.88 % (15891)------------------------------
% 3.39/1.88 % (15891)------------------------------
% 3.39/1.88 % (15892)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/7Mi)
% 3.39/1.89 % (15892)Instruction limit reached!
% 3.39/1.89 % (15892)------------------------------
% 3.39/1.89 % (15892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.89 % (15892)Termination reason: Unknown
% 3.39/1.89 % (15892)Termination phase: shuffling
% 3.39/1.89
% 3.39/1.89 % (15892)Memory used [KB]: 1535
% 3.39/1.89 % (15892)Time elapsed: 0.005 s
% 3.39/1.89 % (15892)Instructions burned: 8 (million)
% 3.39/1.89 % (15892)------------------------------
% 3.39/1.89 % (15892)------------------------------
% 3.39/1.89 % (15893)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/3Mi)
% 3.39/1.89 % (15893)Instruction limit reached!
% 3.39/1.89 % (15893)------------------------------
% 3.39/1.89 % (15893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.89 % (15893)Termination reason: Unknown
% 3.39/1.89 % (15893)Termination phase: shuffling
% 3.39/1.89
% 3.39/1.89 % (15893)Memory used [KB]: 1535
% 3.39/1.89 % (15893)Time elapsed: 0.003 s
% 3.39/1.89 % (15893)Instructions burned: 3 (million)
% 3.39/1.89 % (15893)------------------------------
% 3.39/1.89 % (15893)------------------------------
% 3.39/1.90 % (15894)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/4Mi)
% 3.39/1.90 % (15895)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/18Mi)
% 3.39/1.90 % (15894)Instruction limit reached!
% 3.39/1.90 % (15894)------------------------------
% 3.39/1.90 % (15894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.90 % (15894)Termination reason: Unknown
% 3.39/1.90 % (15894)Termination phase: shuffling
% 3.39/1.90
% 3.39/1.90 % (15894)Memory used [KB]: 1535
% 3.39/1.90 % (15894)Time elapsed: 0.004 s
% 3.39/1.90 % (15894)Instructions burned: 5 (million)
% 3.39/1.90 % (15894)------------------------------
% 3.39/1.90 % (15894)------------------------------
% 3.39/1.90 % (15896)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/710Mi)
% 3.39/1.90 % (15897)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_15775 for (2999ds/6Mi)
% 3.39/1.91 % (15895)Instruction limit reached!
% 3.39/1.91 % (15895)------------------------------
% 3.39/1.91 % (15895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.91 % (15895)Termination reason: Unknown
% 3.39/1.91 % (15895)Termination phase: shuffling
% 3.39/1.91
% 3.39/1.91 % (15895)Memory used [KB]: 1791
% 3.39/1.91 % (15895)Time elapsed: 0.011 s
% 3.39/1.91 % (15895)Instructions burned: 19 (million)
% 3.39/1.91 % (15895)------------------------------
% 3.39/1.91 % (15895)------------------------------
% 3.39/1.91 % (15897)Instruction limit reached!
% 3.39/1.91 % (15897)------------------------------
% 3.39/1.91 % (15897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.91 % (15897)Termination reason: Unknown
% 3.39/1.91 % (15897)Termination phase: shuffling
% 3.39/1.91
% 3.39/1.91 % (15897)Memory used [KB]: 1535
% 3.39/1.91 % (15897)Time elapsed: 0.005 s
% 3.39/1.91 % (15897)Instructions burned: 7 (million)
% 3.39/1.91 % (15897)------------------------------
% 3.39/1.91 % (15897)------------------------------
% 3.39/1.91 % (15898)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_15775 for (2999ds/902Mi)
% 3.39/1.91 % (15899)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_15775 for (2999ds/21Mi)
% 3.39/1.92 % (15876)Instruction limit reached!
% 3.39/1.92 % (15876)------------------------------
% 3.39/1.92 % (15876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.92 % (15876)Termination reason: Unknown
% 3.39/1.92 % (15876)Termination phase: Saturation
% 3.39/1.92
% 3.39/1.92 % (15876)Memory used [KB]: 7419
% 3.39/1.92 % (15876)Time elapsed: 0.088 s
% 3.39/1.92 % (15876)Instructions burned: 183 (million)
% 3.39/1.92 % (15876)------------------------------
% 3.39/1.92 % (15876)------------------------------
% 3.39/1.92 % (15900)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_15775 for (2998ds/5Mi)
% 3.39/1.92 % (15901)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_15775 for (2998ds/6Mi)
% 3.39/1.92 % (15900)Instruction limit reached!
% 3.39/1.92 % (15900)------------------------------
% 3.39/1.92 % (15900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.92 % (15900)Termination reason: Unknown
% 3.39/1.92 % (15900)Termination phase: shuffling
% 3.39/1.92
% 3.39/1.92 % (15900)Memory used [KB]: 1535
% 3.39/1.92 % (15900)Time elapsed: 0.004 s
% 3.39/1.92 % (15900)Instructions burned: 5 (million)
% 3.39/1.92 % (15900)------------------------------
% 3.39/1.92 % (15900)------------------------------
% 3.39/1.92 % (15899)Instruction limit reached!
% 3.39/1.92 % (15899)------------------------------
% 3.39/1.92 % (15899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.92 % (15899)Termination reason: Unknown
% 3.39/1.92 % (15899)Termination phase: shuffling
% 3.39/1.92
% 3.39/1.92 % (15899)Memory used [KB]: 1791
% 3.39/1.92 % (15899)Time elapsed: 0.012 s
% 3.39/1.92 % (15899)Instructions burned: 22 (million)
% 3.39/1.92 % (15899)------------------------------
% 3.39/1.92 % (15899)------------------------------
% 3.39/1.93 % (15901)Instruction limit reached!
% 3.39/1.93 % (15901)------------------------------
% 3.39/1.93 % (15901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.93 % (15901)Termination reason: Unknown
% 3.39/1.93 % (15901)Termination phase: shuffling
% 3.39/1.93
% 3.39/1.93 % (15901)Memory used [KB]: 1663
% 3.39/1.93 % (15901)Time elapsed: 0.005 s
% 3.39/1.93 % (15901)Instructions burned: 7 (million)
% 3.39/1.93 % (15901)------------------------------
% 3.39/1.93 % (15901)------------------------------
% 3.39/1.93 % (15902)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_15775 for (2998ds/377Mi)
% 3.39/1.93 % (15896)First to succeed.
% 3.39/1.94 % (15896)Refutation found. Thanks to Tanya!
% 3.39/1.94 % SZS status Theorem for DTF2THF_15775
% 3.39/1.94 % SZS output start Proof for DTF2THF_15775
% 3.39/1.94 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.39/1.94 thf(type_def_7, type, code_natural: $tType).
% 3.39/1.94 thf(type_def_8, type, nat: $tType).
% 3.39/1.94 thf(type_def_9, type, coinductive_llist: ($tType) > $tType).
% 3.39/1.94 thf(type_def_10, type, extended_enat: $tType).
% 3.39/1.94 thf(type_def_11, type, set: ($tType) > $tType).
% 3.39/1.94 thf(type_def_12, type, a: $tType).
% 3.39/1.94 thf(type_def_13, type, b: $tType).
% 3.39/1.94 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 3.39/1.94 thf(func_def_1, type, code_natural: $tType).
% 3.39/1.94 thf(func_def_2, type, extended_enat: $tType).
% 3.39/1.94 thf(func_def_3, type, set: ($tType) > $tType).
% 3.39/1.94 thf(func_def_4, type, nat: $tType).
% 3.39/1.94 thf(func_def_5, type, itself: ($tType) > $tType).
% 3.39/1.94 thf(func_def_6, type, b: $tType).
% 3.39/1.94 thf(func_def_7, type, a: $tType).
% 3.39/1.94 thf(func_def_8, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_9, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_10, type, monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_11, type, semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_12, type, comm_semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_13, type, comm_monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_14, type, ab_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_15, type, cancel_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_16, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_17, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_18, type, ordere779506340up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_19, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_20, type, semiri456707255roduct: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.39/1.94 thf(func_def_21, type, bNF_rel_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > (X0 > X2) > (X1 > X3) > $o)).
% 3.39/1.94 thf(func_def_22, type, code_case_natural: !>[X0: $tType]:(X0 > (code_natural > X0) > code_natural > X0)).
% 3.39/1.94 thf(func_def_23, type, code_rec_natural: !>[X0: $tType]:(X0 > (code_natural > X0 > X0) > code_natural > X0)).
% 3.39/1.94 thf(func_def_24, type, code_size_natural: code_natural > nat).
% 3.39/1.94 thf(func_def_25, type, coindu328551480prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.39/1.94 thf(func_def_26, type, coindu246717305length: !>[X0: $tType]:(nat > coinductive_llist @ X0 > extended_enat)).
% 3.39/1.94 thf(func_def_27, type, coinductive_iterates: !>[X0: $tType]:((X0 > X0) > X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_28, type, coinductive_lappend: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_29, type, coinductive_ldrop: !>[X0: $tType]:(extended_enat > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_30, type, coindu218763757pWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_31, type, coinductive_lfilter: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_32, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.39/1.94 thf(func_def_33, type, coinductive_llast: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.39/1.94 thf(func_def_34, type, coinductive_llcp: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > extended_enat)).
% 3.39/1.94 thf(func_def_35, type, coinductive_llength: !>[X0: $tType]:(coinductive_llist @ X0 > extended_enat)).
% 3.39/1.94 thf(func_def_36, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_37, type, coindu1381640503_llist: !>[X0: $tType, X1: $tType]:(X0 > (X1 > coinductive_llist @ X1 > X0) > coinductive_llist @ X1 > X0)).
% 3.39/1.94 thf(func_def_38, type, coindu1259883913_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > $o) > (X0 > coinductive_llist @ X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 3.39/1.94 thf(func_def_39, type, coindu1486289336t_all2: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > coinductive_llist @ X0 > coinductive_llist @ X1 > $o)).
% 3.39/1.94 thf(func_def_40, type, coinductive_lmap: !>[X0: $tType, X1: $tType]:((X0 > X1) > coinductive_llist @ X0 > coinductive_llist @ X1)).
% 3.39/1.94 thf(func_def_41, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 3.39/1.94 thf(func_def_42, type, coindu543516966_llist: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > $o)).
% 3.39/1.94 thf(func_def_43, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 3.39/1.94 thf(func_def_44, type, coinductive_lprefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.39/1.94 thf(func_def_45, type, coinductive_ltake: !>[X0: $tType]:(extended_enat > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_46, type, coindu501562517eWhile: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_47, type, coindu780009021istsum: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.39/1.94 thf(func_def_48, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 3.39/1.94 thf(func_def_49, type, map_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (X1 > X2) > X0 > X3)).
% 3.39/1.94 thf(func_def_50, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.39/1.94 thf(func_def_51, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.39/1.94 thf(func_def_52, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.39/1.94 thf(func_def_53, type, lMirro999291890or_aux: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.39/1.94 thf(func_def_54, type, case_nat: !>[X0: $tType]:(X0 > (nat > X0) > nat > X0)).
% 3.39/1.94 thf(func_def_55, type, rec_nat: !>[X0: $tType]:(X0 > (nat > X0 > X0) > nat > X0)).
% 3.39/1.94 thf(func_def_56, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 3.39/1.94 thf(func_def_57, type, type: !>[X0: $tType]:(itself @ X0)).
% 3.39/1.94 thf(func_def_58, type, bex1_rel: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > $o)).
% 3.39/1.94 thf(func_def_59, type, quotient3: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > $o)).
% 3.39/1.94 thf(func_def_60, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.39/1.94 thf(func_def_61, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.39/1.94 thf(func_def_62, type, p: a > b > $o).
% 3.39/1.94 thf(func_def_63, type, acc_a: coinductive_llist @ b).
% 3.39/1.94 thf(func_def_64, type, acca: coinductive_llist @ a).
% 3.39/1.94 thf(func_def_65, type, xs_a: coinductive_llist @ b).
% 3.39/1.94 thf(func_def_66, type, xsa: coinductive_llist @ a).
% 3.39/1.94 thf(func_def_82, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.39/1.94 thf(func_def_84, type, ph2: !>[X0: $tType]:(X0)).
% 3.39/1.94 thf(f808,plain,(
% 3.39/1.94 $false),
% 3.39/1.94 inference(subsumption_resolution,[status(thm)],[f807,f779])).
% 3.39/1.94 thf(f779,plain,(
% 3.39/1.94 ((coinductive_llength @ a @ acca) != (coinductive_llength @ b @ acc_a))),
% 3.39/1.94 inference(cnf_transformation,[status(thm)],[f767])).
% 3.39/1.94 thf(f767,plain,(
% 3.39/1.94 ((coinductive_llength @ a @ acca) != (coinductive_llength @ b @ acc_a))),
% 3.39/1.94 inference(flattening,[status(thm)],[f287])).
% 3.39/1.94 thf(f287,negated_conjecture,(
% 3.39/1.94 ~((coinductive_llength @ a @ acca) = (coinductive_llength @ b @ acc_a))),
% 3.39/1.94 inference(negated_conjecture,[status(cth)],[f286])).
% 3.39/1.94 thf(f286,conjecture,(
% 3.39/1.94 ((coinductive_llength @ a @ acca) = (coinductive_llength @ b @ acc_a))),
% 3.39/1.94 file('/export/starexec/sandbox/tmp/tmp.DVE1XfhOnx/DTF2THF_15775.p',conj_0)).
% 3.39/1.94 thf(f807,plain,(
% 3.39/1.94 ((coinductive_llength @ a @ acca) = (coinductive_llength @ b @ acc_a))),
% 3.39/1.94 inference(trivial_inequality_removal,[status(thm)],[f806])).
% 3.39/1.94 thf(f806,plain,(
% 3.39/1.94 ($true != $true) | ((coinductive_llength @ a @ acca) = (coinductive_llength @ b @ acc_a))),
% 3.39/1.94 inference(superposition,[status(thm)],[f792,f782])).
% 3.39/1.94 thf(f782,plain,(
% 3.39/1.94 ((coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a) = $true)),
% 3.39/1.94 inference(cnf_transformation,[status(thm)],[f616])).
% 3.39/1.94 thf(f616,plain,(
% 3.39/1.94 ((coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a) = $true)),
% 3.39/1.94 inference(fool_elimination,[status(thm)],[f615])).
% 3.39/1.94 thf(f615,plain,(
% 3.39/1.94 (coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a)),
% 3.39/1.94 inference(rectify,[status(thm)],[f1])).
% 3.39/1.94 thf(f1,axiom,(
% 3.39/1.94 (coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a)),
% 3.39/1.94 file('/export/starexec/sandbox/tmp/tmp.DVE1XfhOnx/DTF2THF_15775.p',fact_0_local_OLNil_I2_J)).
% 3.39/1.94 thf(f792,plain,(
% 3.39/1.94 ( ! [X1 : $tType,X0 : $tType,X2 : X0 > X1 > $o,X3 : coinductive_llist @ X0,X4 : coinductive_llist @ X1] : (((coindu1486289336t_all2 @ X0 @ X1 @ X2 @ X3 @ X4) != $true) | ((coinductive_llength @ X1 @ X4) = (coinductive_llength @ X0 @ X3))) )),
% 3.39/1.94 inference(cnf_transformation,[status(thm)],[f770])).
% 3.39/1.94 thf(f770,plain,(
% 3.39/1.94 ! [X0 : $tType,X1 : $tType,X2 : X0 > X1 > $o,X3 : coinductive_llist @ X0,X4 : coinductive_llist @ X1] : (((coindu1486289336t_all2 @ X0 @ X1 @ X2 @ X3 @ X4) != $true) | ((coinductive_llength @ X1 @ X4) = (coinductive_llength @ X0 @ X3)))),
% 3.39/1.94 inference(ennf_transformation,[status(thm)],[f374])).
% 3.39/1.94 thf(f374,plain,(
% 3.39/1.94 ! [X0 : $tType,X1 : $tType,X2 : X0 > X1 > $o,X3 : coinductive_llist @ X0,X4 : coinductive_llist @ X1] : (((coindu1486289336t_all2 @ X0 @ X1 @ X2 @ X3 @ X4) = $true) => ((coinductive_llength @ X1 @ X4) = (coinductive_llength @ X0 @ X3)))),
% 3.39/1.94 inference(fool_elimination,[status(thm)],[f373])).
% 3.39/1.94 thf(f373,plain,(
% 3.39/1.94 ! [X0 : $tType,X1 : $tType,X2 : X0 > X1 > $o,X3 : coinductive_llist @ X0,X4 : coinductive_llist @ X1] : ((coindu1486289336t_all2 @ X0 @ X1 @ X2 @ X3 @ X4) => ((coinductive_llength @ X1 @ X4) = (coinductive_llength @ X0 @ X3)))),
% 3.39/1.94 inference(rectify,[status(thm)],[f5])).
% 3.39/1.94 thf(f5,axiom,(
% 3.39/1.94 ! [X0 : $tType,X2 : $tType,X6 : X0 > X2 > $o,X7 : coinductive_llist @ X0,X8 : coinductive_llist @ X2] : ((coindu1486289336t_all2 @ X0 @ X2 @ X6 @ X7 @ X8) => ((coinductive_llength @ X0 @ X7) = (coinductive_llength @ X2 @ X8)))),
% 3.39/1.94 file('/export/starexec/sandbox/tmp/tmp.DVE1XfhOnx/DTF2THF_15775.p',fact_4_llist__all2__llengthD)).
% 3.39/1.94 % SZS output end Proof for DTF2THF_15775
% 3.39/1.94 % (15896)------------------------------
% 3.39/1.94 % (15896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.94 % (15896)Termination reason: Refutation
% 3.39/1.94
% 3.39/1.94 % (15896)Memory used [KB]: 6780
% 3.39/1.94 % (15896)Time elapsed: 0.038 s
% 3.39/1.94 % (15896)Instructions burned: 71 (million)
% 3.39/1.94 % (15896)------------------------------
% 3.39/1.94 % (15896)------------------------------
% 3.39/1.94 % (15875)Success in time 0.134 s
%------------------------------------------------------------------------------