%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT148^1 : 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 : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:44:56 AM UTC 2026
% Result : Theorem 3.65s 2.00s
% Output : Refutation 3.65s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT148^1 : 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
% 0.19/0.34 % Computer : n016.cluster.edu
% 0.19/0.34 % Model : x86_64 x86_64
% 0.19/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.34 % Memory : 8042.1875MB
% 0.19/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.19/0.34 % CPULimit : 300
% 0.19/0.34 % WCLimit : 300
% 0.19/0.34 % DateTime : Tue Feb 24 06:09:36 EST 2026
% 0.19/0.34 % CPUTime :
% 0.19/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.21/0.45 ---- Original DTF file ---
% 0.21/0.45 thf(spec,logic,$$dhol).
% 0.21/0.45 %------------------------------------------------------------------------------
% 0.21/0.45 % File : DAT148^1 : TPTP v9.2.1. Released v7.0.0.
% 0.21/0.45 % Domain : Data Structures
% 0.21/0.45 % Problem : Coinductive stream 241
% 0.21/0.45 % Version : [Bla16] axioms : Especial.
% 0.21/0.45 % English :
% 0.21/0.45
% 0.21/0.45 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.21/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.21/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.21/0.45 % Source : [Bla16]
% 0.21/0.45 % Names : coinductive_stream__241.p [Bla16]
% 0.21/0.45
% 0.21/0.45 % Status : Theorem
% 0.21/0.45 % Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% 0.21/0.45 % Syntax : Number of formulae : 316 ( 108 unt; 55 typ; 0 def)
% 0.21/0.45 % Number of atoms : 1044 ( 303 equ; 8 cnn)
% 0.21/0.45 % Maximal formula atoms : 15 ( 4 avg)
% 0.21/0.45 % Number of connectives : 5131 ( 35 ~; 1 |; 33 &;4795 @)
% 0.21/0.45 % ( 0 <=>; 267 =>; 0 <=; 0 <~>)
% 0.21/0.45 % Maximal formula depth : 25 ( 11 avg)
% 0.21/0.45 % Number of types : 3 ( 2 usr)
% 0.21/0.45 % Number of type conns : 1735 (1735 >; 0 *; 0 +; 0 <<)
% 0.21/0.45 % Number of symbols : 57 ( 53 usr; 3 con; 0-8 aty)
% 0.21/0.45 % Number of variables : 1907 ( 418 ^;1384 !; 27 ?;1907 :)
% 0.21/0.45 % ( 78 !>; 0 ?*; 0 @-; 0 @+)
% 0.21/0.45 % SPC : TH1_THM_EQU_NAR
% 0.21/0.45
% 0.21/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.21/0.45 % 2016-07-13 15:13:48.760
% 0.21/0.45 %------------------------------------------------------------------------------
% 0.21/0.45 %----Could-be-implicit typings (6)
% 0.21/0.45 thf(ty_t_Coinductive__List_Ollist,type,
% 0.21/0.45 coinductive_llist: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_t_Stream_Ostream,type,
% 0.21/0.45 stream: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_t_Set_Oset,type,
% 0.21/0.45 set: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_t_itself,type,
% 0.21/0.45 itself: $tType > $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_tf_b,type,
% 0.21/0.45 b: $tType ).
% 0.21/0.45
% 0.21/0.45 thf(ty_tf_a,type,
% 0.21/0.45 a: $tType ).
% 0.21/0.45
% 0.21/0.45 %----Explicit typings (49)
% 0.21/0.45 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.21/0.45 bNF_rel_fun:
% 0.21/0.45 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Basic__BNFs_Opred__fun,type,
% 0.21/0.45 basic_pred_fun:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Oiterates,type,
% 0.21/0.45 coinductive_iterates:
% 0.21/0.45 !>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.21/0.45 coinductive_lfinite:
% 0.21/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 0.21/0.45 coindu1381640503_llist:
% 0.21/0.45 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 0.21/0.45 coindu1259883913_llist:
% 0.21/0.45 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Ollist_Ollist__all2,type,
% 0.21/0.45 coindu1486289336t_all2:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
% 0.21/0.45 coindu543516966_llist:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.21/0.45 coindu1441602521_llist:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
% 0.21/0.45 coindu1724414836stream:
% 0.21/0.45 !>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
% 0.21/0.45 coindu1183105481stream:
% 0.21/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Opcr__stream,type,
% 0.21/0.45 coindu773941317stream:
% 0.21/0.45 !>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( coinductive_llist @ C ) > ( stream @ B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
% 0.21/0.45 coindu2010755910_llist:
% 0.21/0.45 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
% 0.21/0.45 coindu139217191stream:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 0.21/0.45 comple1396247847notone:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Fun_Ocomp,type,
% 0.21/0.45 comp:
% 0.21/0.45 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Fun_Omap__fun,type,
% 0.21/0.45 map_fun:
% 0.21/0.45 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_If,type,
% 0.21/0.45 if:
% 0.21/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.21/0.45 inf_inf:
% 0.21/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Lifting_Orel__pred__comp,type,
% 0.21/0.45 rel_pred_comp:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.21/0.45 ord_less_eq:
% 0.21/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Partial__Function_Ofun__ord,type,
% 0.21/0.45 partial_fun_ord:
% 0.21/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Partial__Function_Oimg__ord,type,
% 0.21/0.45 partial_img_ord:
% 0.21/0.45 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Partial__Function_Omk__less,type,
% 0.21/0.45 partial_mk_less:
% 0.21/0.45 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Pure_Otype,type,
% 0.21/0.45 type:
% 0.21/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Quotient_OBex1__rel,type,
% 0.21/0.45 bex1_rel:
% 0.21/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Quotient_OQuotient3,type,
% 0.21/0.45 quotient3:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Relation_ODomainp,type,
% 0.21/0.45 domainp:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Relation_OPowp,type,
% 0.21/0.45 powp:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Relation_Oreflp,type,
% 0.21/0.45 reflp:
% 0.21/0.45 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Relation_Orelcompp,type,
% 0.21/0.45 relcompp:
% 0.21/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_OBall,type,
% 0.21/0.45 ball:
% 0.21/0.45 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_OBex,type,
% 0.21/0.45 bex:
% 0.21/0.45 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Set_OCollect,type,
% 0.21/0.45 collect:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Stream_Ositerate,type,
% 0.21/0.45 siterate:
% 0.21/0.45 !>[A: $tType] : ( ( A > A ) > A > ( stream @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Stream_Ostream_Ocase__stream,type,
% 0.21/0.45 case_stream:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Stream_Ostream_Ocorec__stream,type,
% 0.21/0.45 corec_stream:
% 0.21/0.45 !>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Stream_Ostream_Opred__stream,type,
% 0.21/0.45 pred_stream:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Stream_Ostream_Ostream__all2,type,
% 0.21/0.45 stream_all2:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream @ A ) > ( stream @ B ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Transfer_Obi__total,type,
% 0.21/0.45 bi_total:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Transfer_Oleft__total,type,
% 0.21/0.45 left_total:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Transfer_Orev__implies,type,
% 0.21/0.45 rev_implies: $o > $o > $o ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Transfer_Oright__total,type,
% 0.21/0.45 right_total:
% 0.21/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Transfer_Otransfer__bforall,type,
% 0.21/0.45 transfer_bforall:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Transfer_Otransfer__forall,type,
% 0.21/0.45 transfer_forall:
% 0.21/0.45 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_Typedef_Otype__definition,type,
% 0.21/0.45 type_definition:
% 0.21/0.45 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_c_member,type,
% 0.21/0.45 member:
% 0.21/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_xb,type,
% 0.21/0.45 xb: a > ( coinductive_llist @ b ) ).
% 0.21/0.45
% 0.21/0.45 thf(sy_v_yb,type,
% 0.21/0.45 yb: a > ( stream @ b ) ).
% 0.21/0.45
% 0.21/0.45 %----Relevant facts (256)
% 0.21/0.45 thf(fact_0_stream_ORep__inject,axiom,
% 0.21/0.45 ! [A: $tType,X: stream @ A,Y: stream @ A] :
% 0.21/0.45 ( ( ( coindu1724414836stream @ A @ X )
% 0.21/0.45 = ( coindu1724414836stream @ A @ Y ) )
% 0.21/0.45 = ( X = Y ) ) ).
% 0.21/0.45
% 0.21/0.45 % stream.Rep_inject
% 0.21/0.45 thf(fact_1_comp__apply,axiom,
% 0.21/0.45 ! [C: $tType,A: $tType,B: $tType] :
% 0.21/0.45 ( ( comp @ B @ A @ C )
% 0.21/0.45 = ( ^ [F: B > A,G: C > B,X2: C] : ( F @ ( G @ X2 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % comp_apply
% 0.21/0.45 thf(fact_2_rel__funI,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D] :
% 0.21/0.45 ( ! [X3: A,Y2: B] :
% 0.21/0.45 ( ( A2 @ X3 @ Y2 )
% 0.21/0.45 => ( B2 @ ( F2 @ X3 ) @ ( G2 @ Y2 ) ) )
% 0.21/0.45 => ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 ) ) ).
% 0.21/0.45
% 0.21/0.45 % rel_funI
% 0.21/0.45 thf(fact_3_fun_Orel__map_I2_J,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G2: B > C,Y: D > B] :
% 0.21/0.45 ( ( bNF_rel_fun @ D @ D @ A @ C
% 0.21/0.45 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.45 @ Sa
% 0.21/0.45 @ X
% 0.21/0.45 @ ( comp @ B @ C @ D @ G2 @ Y ) )
% 0.21/0.45 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.21/0.45 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.45 @ ^ [X2: A,Y4: B] : ( Sa @ X2 @ ( G2 @ Y4 ) )
% 0.21/0.45 @ X
% 0.21/0.45 @ Y ) ) ).
% 0.21/0.45
% 0.21/0.45 % fun.rel_map(2)
% 0.21/0.45 thf(fact_4_fun_Orel__map_I1_J,axiom,
% 0.21/0.45 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
% 0.21/0.45 ( ( bNF_rel_fun @ D @ D @ C @ B
% 0.21/0.45 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.45 @ Sb
% 0.21/0.45 @ ( comp @ A @ C @ D @ I @ X )
% 0.21/0.45 @ Y )
% 0.21/0.45 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.21/0.45 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.45 @ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
% 0.21/0.45 @ X
% 0.21/0.45 @ Y ) ) ).
% 0.21/0.45
% 0.21/0.45 % fun.rel_map(1)
% 0.21/0.45 thf(fact_5_o__rsp_I2_J,axiom,
% 0.21/0.45 ! [E: $tType,F3: $tType,H: $tType,G3: $tType,R1: E > F3 > $o] :
% 0.21/0.45 ( bNF_rel_fun @ ( G3 > H ) @ ( G3 > H ) @ ( ( E > G3 ) > E > H ) @ ( ( F3 > G3 ) > F3 > H )
% 0.21/0.45 @ ^ [Y3: G3 > H,Z: G3 > H] : ( Y3 = Z )
% 0.21/0.45 @ ( bNF_rel_fun @ ( E > G3 ) @ ( F3 > G3 ) @ ( E > H ) @ ( F3 > H )
% 0.21/0.45 @ ( bNF_rel_fun @ E @ F3 @ G3 @ G3 @ R1
% 0.21/0.45 @ ^ [Y3: G3,Z: G3] : ( Y3 = Z ) )
% 0.21/0.45 @ ( bNF_rel_fun @ E @ F3 @ H @ H @ R1
% 0.21/0.45 @ ^ [Y3: H,Z: H] : ( Y3 = Z ) ) )
% 0.21/0.45 @ ( comp @ G3 @ H @ E )
% 0.21/0.45 @ ( comp @ G3 @ H @ F3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % o_rsp(2)
% 0.21/0.45 thf(fact_6_o__rsp_I1_J,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R3: B > D > $o,R1: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R2 @ R3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R1 @ R3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % o_rsp(1)
% 0.21/0.45 thf(fact_7_fun_Omap__transfer,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType,D: $tType,G3: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G3 > $o] :
% 0.21/0.45 ( bNF_rel_fun @ ( A > B ) @ ( F3 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G3 @ Rb @ Sd )
% 0.21/0.45 @ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G3 )
% 0.21/0.45 @ ( bNF_rel_fun @ D @ D @ A @ F3
% 0.21/0.45 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.45 @ Rb )
% 0.21/0.45 @ ( bNF_rel_fun @ D @ D @ B @ G3
% 0.21/0.45 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.45 @ Sd ) )
% 0.21/0.45 @ ( comp @ A @ B @ D )
% 0.21/0.45 @ ( comp @ F3 @ G3 @ D ) ) ).
% 0.21/0.45
% 0.21/0.45 % fun.map_transfer
% 0.21/0.45 thf(fact_8_comp__transfer,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B2: A > C > $o,C2: B > D > $o,A2: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B2 @ C2 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A2 @ B2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A2 @ C2 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % comp_transfer
% 0.21/0.45 thf(fact_9_K__record__comp,axiom,
% 0.21/0.45 ! [C: $tType,B: $tType,A: $tType,C3: B,F2: A > C] :
% 0.21/0.45 ( ( comp @ C @ B @ A
% 0.21/0.45 @ ^ [X2: C] : C3
% 0.21/0.45 @ F2 )
% 0.21/0.45 = ( ^ [X2: A] : C3 ) ) ).
% 0.21/0.45
% 0.21/0.45 % K_record_comp
% 0.21/0.45 thf(fact_10_fun_Omap__ident,axiom,
% 0.21/0.45 ! [A: $tType,D: $tType,T: D > A] :
% 0.21/0.45 ( ( comp @ A @ A @ D
% 0.21/0.45 @ ^ [X2: A] : X2
% 0.21/0.45 @ T )
% 0.21/0.45 = T ) ).
% 0.21/0.45
% 0.21/0.45 % fun.map_ident
% 0.21/0.45 thf(fact_11_stream__from__llist__setup_Ocr__stream__def,axiom,
% 0.21/0.45 ! [A: $tType] :
% 0.21/0.45 ( ( coindu1183105481stream @ A )
% 0.21/0.45 = ( ^ [X2: coinductive_llist @ A,Y4: stream @ A] :
% 0.21/0.45 ( X2
% 0.21/0.45 = ( coindu1724414836stream @ A @ Y4 ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % stream_from_llist_setup.cr_stream_def
% 0.21/0.45 thf(fact_12_llist__of__stream__corec__stream,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType,SHD: B > A,EndORmore: B > $o,STL_more: B > ( stream @ A ),STL_end: B > B,X: B] :
% 0.21/0.45 ( ( coindu1724414836stream @ A @ ( corec_stream @ B @ A @ SHD @ EndORmore @ STL_more @ STL_end @ X ) )
% 0.21/0.45 = ( coindu1259883913_llist @ B @ A
% 0.21/0.45 @ ^ [Uu: B] : $false
% 0.21/0.45 @ SHD
% 0.21/0.45 @ EndORmore
% 0.21/0.45 @ ( comp @ ( stream @ A ) @ ( coinductive_llist @ A ) @ B @ ( coindu1724414836stream @ A ) @ STL_more )
% 0.21/0.45 @ STL_end
% 0.21/0.45 @ X ) ) ).
% 0.21/0.45
% 0.21/0.45 % llist_of_stream_corec_stream
% 0.21/0.45 thf(fact_13_rel__fun__def__butlast,axiom,
% 0.21/0.45 ! [B: $tType,D: $tType,C: $tType,E: $tType,F3: $tType,A: $tType,R: A > B > $o,S: C > E > $o,T2: D > F3 > $o,F2: A > C > D,G2: B > E > F3] :
% 0.21/0.45 ( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T2 ) @ F2 @ G2 )
% 0.21/0.45 = ( ! [X2: A,Y4: B] :
% 0.21/0.45 ( ( R @ X2 @ Y4 )
% 0.21/0.45 => ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T2 @ ( F2 @ X2 ) @ ( G2 @ Y4 ) ) ) ) ) ).
% 0.21/0.45
% 0.21/0.45 % rel_fun_def_butlast
% 0.21/0.45 thf(fact_14_If__transfer,axiom,
% 0.21/0.45 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.21/0.45 ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
% 0.21/0.45 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.21/0.45 @ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 0.21/0.46 @ ( if @ A )
% 0.21/0.46 @ ( if @ B ) ) ).
% 0.21/0.46
% 0.21/0.46 % If_transfer
% 0.21/0.46 thf(fact_15_fun_Orel__transfer,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
% 0.21/0.46 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
% 0.21/0.46 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
% 0.21/0.46 @ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.21/0.46 @ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
% 0.21/0.46 @ ( bNF_rel_fun @ D @ D @ A @ C
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.46 @ Sa )
% 0.21/0.46 @ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
% 0.21/0.46 @ ( bNF_rel_fun @ D @ D @ B @ E
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.46 @ Sc )
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.21/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z ) )
% 0.21/0.46 @ ( bNF_rel_fun @ D @ D @ C @ E
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % fun.rel_transfer
% 0.21/0.46 thf(fact_16_fun_Orel__refl,axiom,
% 0.21/0.46 ! [B: $tType,D: $tType,Ra: B > B > $o,X: D > B] :
% 0.21/0.46 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.21/0.46 => ( bNF_rel_fun @ D @ D @ B @ B
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.46 @ Ra
% 0.21/0.46 @ X
% 0.21/0.46 @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % fun.rel_refl
% 0.21/0.46 thf(fact_17_fun_Orel__eq,axiom,
% 0.21/0.46 ! [A: $tType,D: $tType] :
% 0.21/0.46 ( ( bNF_rel_fun @ D @ D @ A @ A
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.21/0.46 = ( ^ [Y3: D > A,Z: D > A] : ( Y3 = Z ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % fun.rel_eq
% 0.21/0.46 thf(fact_18_rel__fun__mono_H,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F2: A > C,G2: B > D] :
% 0.21/0.46 ( ! [X3: A,Y2: B] :
% 0.21/0.46 ( ( Y5 @ X3 @ Y2 )
% 0.21/0.46 => ( X4 @ X3 @ Y2 ) )
% 0.21/0.46 => ( ! [X3: C,Y2: D] :
% 0.21/0.46 ( ( A2 @ X3 @ Y2 )
% 0.21/0.46 => ( B2 @ X3 @ Y2 ) )
% 0.21/0.46 => ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F2 @ G2 )
% 0.21/0.46 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F2 @ G2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rel_fun_mono'
% 0.21/0.46 thf(fact_19_rel__fun__mono,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F2: A > C,G2: B > D,Y5: A > B > $o,B2: C > D > $o] :
% 0.21/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F2 @ G2 )
% 0.21/0.46 => ( ! [X3: A,Y2: B] :
% 0.21/0.46 ( ( Y5 @ X3 @ Y2 )
% 0.21/0.46 => ( X4 @ X3 @ Y2 ) )
% 0.21/0.46 => ( ! [X3: C,Y2: D] :
% 0.21/0.46 ( ( A2 @ X3 @ Y2 )
% 0.21/0.46 => ( B2 @ X3 @ Y2 ) )
% 0.21/0.46 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F2 @ G2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rel_fun_mono
% 0.21/0.46 thf(fact_20_let__rsp,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
% 0.21/0.46 ( 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 )
% 0.21/0.46 @ ^ [S2: A,F: A > C] : ( F @ S2 )
% 0.21/0.46 @ ^ [S2: B,F: B > D] : ( F @ S2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % let_rsp
% 0.21/0.46 thf(fact_21_rel__funD,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
% 0.21/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 )
% 0.21/0.46 => ( ( A2 @ X @ Y )
% 0.21/0.46 => ( B2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rel_funD
% 0.21/0.46 thf(fact_22_rewriteR__comp__comp2,axiom,
% 0.21/0.46 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H2: A > C,R12: D > B,R22: A > D,F2: B > E,L: D > E] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ G2 @ H2 )
% 0.21/0.46 = ( comp @ D @ B @ A @ R12 @ R22 ) )
% 0.21/0.46 => ( ( ( comp @ B @ E @ D @ F2 @ R12 )
% 0.21/0.46 = L )
% 0.21/0.46 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H2 )
% 0.21/0.46 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rewriteR_comp_comp2
% 0.21/0.46 thf(fact_23_rewriteL__comp__comp2,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H2: E > A,R4: E > D] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 0.21/0.46 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.21/0.46 => ( ( ( comp @ A @ D @ E @ L2 @ H2 )
% 0.21/0.46 = R4 )
% 0.21/0.46 => ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H2 ) )
% 0.21/0.46 = ( comp @ D @ B @ E @ L1 @ R4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rewriteL_comp_comp2
% 0.21/0.46 thf(fact_24_rewriteR__comp__comp,axiom,
% 0.21/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H2: A > C,R4: A > B,F2: B > D] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ G2 @ H2 )
% 0.21/0.46 = R4 )
% 0.21/0.46 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H2 )
% 0.21/0.46 = ( comp @ B @ D @ A @ F2 @ R4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rewriteR_comp_comp
% 0.21/0.46 thf(fact_25_rewriteL__comp__comp,axiom,
% 0.21/0.46 ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H2: D > A] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 0.21/0.46 = L )
% 0.21/0.46 => ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H2 ) )
% 0.21/0.46 = ( comp @ A @ B @ D @ L @ H2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rewriteL_comp_comp
% 0.21/0.46 thf(fact_26_fun_Omap__comp,axiom,
% 0.21/0.46 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
% 0.21/0.46 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
% 0.21/0.46 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% 0.21/0.46
% 0.21/0.46 % fun.map_comp
% 0.21/0.46 thf(fact_27_comp__apply__eq,axiom,
% 0.21/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X: C,H2: D > A,K: C > D] :
% 0.21/0.46 ( ( ( F2 @ ( G2 @ X ) )
% 0.21/0.46 = ( H2 @ ( K @ X ) ) )
% 0.21/0.46 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X )
% 0.21/0.46 = ( comp @ D @ A @ C @ H2 @ K @ X ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_apply_eq
% 0.21/0.46 thf(fact_28_comp__eq__dest__lhs,axiom,
% 0.21/0.46 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.21/0.46 = C3 )
% 0.21/0.46 => ( ( A3 @ ( B3 @ V ) )
% 0.21/0.46 = ( C3 @ V ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_eq_dest_lhs
% 0.21/0.46 thf(fact_29_comp__eq__elim,axiom,
% 0.21/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D2: A > D] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.21/0.46 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.21/0.46 => ! [V2: A] :
% 0.21/0.46 ( ( A3 @ ( B3 @ V2 ) )
% 0.21/0.46 = ( C3 @ ( D2 @ V2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_eq_elim
% 0.21/0.46 thf(fact_30_comp__eq__dest,axiom,
% 0.21/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D2: A > D,V: A] :
% 0.21/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.21/0.46 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.21/0.46 => ( ( A3 @ ( B3 @ V ) )
% 0.21/0.46 = ( C3 @ ( D2 @ V ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_eq_dest
% 0.21/0.46 thf(fact_31_comp__assoc,axiom,
% 0.21/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H2: A > C] :
% 0.21/0.46 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H2 )
% 0.21/0.46 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H2 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_assoc
% 0.21/0.46 thf(fact_32_comp__def,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.21/0.46 ( ( comp @ B @ C @ A )
% 0.21/0.46 = ( ^ [F: B > C,G: A > B,X2: A] : ( F @ ( G @ X2 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_def
% 0.21/0.46 thf(fact_33_Let__transfer,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.21/0.46 ( 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 )
% 0.21/0.46 @ ^ [S2: A,F: A > C] : ( F @ S2 )
% 0.21/0.46 @ ^ [S2: B,F: B > D] : ( F @ S2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % Let_transfer
% 0.21/0.46 thf(fact_34_stream_Ocorec__disc,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType] :
% 0.21/0.46 ( ( corec_stream @ C @ A )
% 0.21/0.46 = ( corec_stream @ C @ A ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.corec_disc
% 0.21/0.46 thf(fact_35_type__copy__map__cong0,axiom,
% 0.21/0.46 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X: C,N: D > A,H2: C > D,F2: A > E] :
% 0.21/0.46 ( ( ( M @ ( G2 @ X ) )
% 0.21/0.46 = ( N @ ( H2 @ X ) ) )
% 0.21/0.46 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X )
% 0.21/0.46 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H2 @ X ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % type_copy_map_cong0
% 0.21/0.46 thf(fact_36_comp__cong,axiom,
% 0.21/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X: C,F4: D > A,G4: E > D,X5: E] :
% 0.21/0.46 ( ( ( F2 @ ( G2 @ X ) )
% 0.21/0.46 = ( F4 @ ( G4 @ X5 ) ) )
% 0.21/0.46 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X )
% 0.21/0.46 = ( comp @ D @ A @ E @ F4 @ G4 @ X5 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % comp_cong
% 0.21/0.46 thf(fact_37_rel__funD2,axiom,
% 0.21/0.46 ! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F2: A > B,G2: A > C,X: A] :
% 0.21/0.46 ( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F2 @ G2 )
% 0.21/0.46 => ( ( A2 @ X @ X )
% 0.21/0.46 => ( B2 @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rel_funD2
% 0.21/0.46 thf(fact_38_apply__rsp_H,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
% 0.21/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F2 @ G2 )
% 0.21/0.46 => ( ( R1 @ X @ Y )
% 0.21/0.46 => ( R2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % apply_rsp'
% 0.21/0.46 thf(fact_39_rel__funE,axiom,
% 0.21/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
% 0.21/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 )
% 0.21/0.46 => ( ( A2 @ X @ Y )
% 0.21/0.46 => ( B2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rel_funE
% 0.21/0.46 thf(fact_40_stream__from__llist__setup_Ostream_Opcr__cr__eq,axiom,
% 0.21/0.46 ! [D: $tType] :
% 0.21/0.46 ( ( coindu773941317stream @ D @ D
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z ) )
% 0.21/0.46 = ( coindu1183105481stream @ D ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream_from_llist_setup.stream.pcr_cr_eq
% 0.21/0.46 thf(fact_41_stream__of__llist__llist__of__stream,axiom,
% 0.21/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.21/0.46 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.21/0.46 = Xs ) ).
% 0.21/0.46
% 0.21/0.46 % stream_of_llist_llist_of_stream
% 0.21/0.46 thf(fact_42_stream_ORep__inverse,axiom,
% 0.21/0.46 ! [A: $tType,X: stream @ A] :
% 0.21/0.46 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
% 0.21/0.46 = X ) ).
% 0.21/0.46
% 0.21/0.46 % stream.Rep_inverse
% 0.21/0.46 thf(fact_43_map__fun__parametric,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,E: $tType,F3: $tType,H: $tType,G3: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G3 > $o,D3: F3 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F3 ) > ( B > E ) > A > F3 ) @ ( ( G3 > H ) > ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F3 ) @ ( G3 > H ) @ ( ( B > E ) > A > F3 ) @ ( ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ E @ G3 @ F3 @ H @ C2 @ D3 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G3 ) @ ( A > F3 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G3 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F3 @ H @ A2 @ D3 ) ) ) @ ( map_fun @ A @ B @ E @ F3 ) @ ( map_fun @ C @ D @ G3 @ H ) ) ).
% 0.21/0.46
% 0.21/0.46 % map_fun_parametric
% 0.21/0.46 thf(fact_44_map__fun__apply,axiom,
% 0.21/0.46 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 0.21/0.46 ( ( map_fun @ B @ C @ D @ A )
% 0.21/0.46 = ( ^ [F: B > C,G: D > A,H3: C > D,X2: B] : ( G @ ( H3 @ ( F @ X2 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % map_fun_apply
% 0.21/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.21/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.21/0.46 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.21/0.46 = ( P @ A3 ) ) ).
% 0.21/0.46
% 0.21/0.46 % mem_Collect_eq
% 0.21/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.21/0.46 ! [A: $tType,A2: set @ A] :
% 0.21/0.46 ( ( collect @ A
% 0.21/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.21/0.46 = A2 ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_mem_eq
% 0.21/0.46 thf(fact_47_Collect__cong,axiom,
% 0.21/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.21/0.46 ( ! [X3: A] :
% 0.21/0.46 ( ( P @ X3 )
% 0.21/0.46 = ( Q @ X3 ) )
% 0.21/0.46 => ( ( collect @ A @ P )
% 0.21/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Collect_cong
% 0.21/0.46 thf(fact_48_ext,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.21/0.46 ( ! [X3: A] :
% 0.21/0.46 ( ( F2 @ X3 )
% 0.21/0.46 = ( G2 @ X3 ) )
% 0.21/0.46 => ( F2 = G2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % ext
% 0.21/0.46 thf(fact_49_map__fun__def,axiom,
% 0.21/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType] :
% 0.21/0.46 ( ( map_fun @ C @ A @ B @ D )
% 0.21/0.46 = ( ^ [F: C > A,G: B > D,H3: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G @ H3 ) @ F ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % map_fun_def
% 0.21/0.46 thf(fact_50_map__fun_Ocomp,axiom,
% 0.21/0.46 ! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F2: E > C,G2: D > F3,H2: C > A,I: B > D] :
% 0.21/0.46 ( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F2 @ G2 ) @ ( map_fun @ C @ A @ B @ D @ H2 @ I ) )
% 0.21/0.46 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F2 ) @ ( comp @ D @ F3 @ B @ G2 @ I ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % map_fun.comp
% 0.21/0.46 thf(fact_51_map__fun_Ocompositionality,axiom,
% 0.21/0.46 ! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F2: E > C,G2: D > F3,H2: C > A,I: B > D,Fun: A > B] :
% 0.21/0.46 ( ( map_fun @ E @ C @ D @ F3 @ F2 @ G2 @ ( map_fun @ C @ A @ B @ D @ H2 @ I @ Fun ) )
% 0.21/0.46 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F2 ) @ ( comp @ D @ F3 @ B @ G2 @ I ) @ Fun ) ) ).
% 0.21/0.46
% 0.21/0.46 % map_fun.compositionality
% 0.21/0.46 thf(fact_52_fun_Opred__transfer,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,D: $tType,R: A > B > $o] :
% 0.21/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
% 0.21/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.21/0.46 @ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
% 0.21/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.21/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.21/0.46 @ R )
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.21/0.46 @ ( basic_pred_fun @ D @ A
% 0.21/0.46 @ ^ [Uu: D] : $true )
% 0.21/0.46 @ ( basic_pred_fun @ D @ B
% 0.21/0.46 @ ^ [Uu: D] : $true ) ) ).
% 0.21/0.46
% 0.21/0.46 % fun.pred_transfer
% 0.21/0.46 thf(fact_53_if__rsp,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.21/0.46 @ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
% 0.21/0.46 @ ( if @ A )
% 0.21/0.46 @ ( if @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % if_rsp
% 0.21/0.46 thf(fact_54_stream_Ocorec__transfer,axiom,
% 0.21/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
% 0.21/0.46 ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > $o ) > ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S @ R )
% 0.21/0.46 @ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) )
% 0.21/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.21/0.46 @ ( bNF_rel_fun @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( ( C > C ) > C > ( stream @ A ) ) @ ( ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S @ ( stream_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S @ ( stream_all2 @ A @ B @ R ) ) ) ) )
% 0.21/0.46 @ ( corec_stream @ C @ A )
% 0.21/0.46 @ ( corec_stream @ D @ B ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.corec_transfer
% 0.21/0.46 thf(fact_55_unfold__stream__transfer,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType] :
% 0.21/0.46 ( bNF_rel_fun @ ( A > B ) @ ( A > B ) @ ( ( A > A ) > A > ( coinductive_llist @ B ) ) @ ( ( A > A ) > A > ( stream @ B ) )
% 0.21/0.46 @ ^ [Y3: A > B,Z: A > B] : ( Y3 = Z )
% 0.21/0.46 @ ( bNF_rel_fun @ ( A > A ) @ ( A > A ) @ ( A > ( coinductive_llist @ B ) ) @ ( A > ( stream @ B ) )
% 0.21/0.46 @ ^ [Y3: A > A,Z: A > A] : ( Y3 = Z )
% 0.21/0.46 @ ( bNF_rel_fun @ A @ A @ ( coinductive_llist @ B ) @ ( stream @ B )
% 0.21/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.21/0.46 @ ( coindu773941317stream @ B @ B
% 0.21/0.46 @ ^ [Y3: B,Z: B] : ( Y3 = Z ) ) ) )
% 0.21/0.46 @ ( coindu1441602521_llist @ A @ B
% 0.21/0.46 @ ^ [Uu: A] : $false )
% 0.21/0.46 @ ( coindu139217191stream @ A @ B ) ) ).
% 0.21/0.46
% 0.21/0.46 % unfold_stream_transfer
% 0.21/0.46 thf(fact_56_fun__ord__parametric,axiom,
% 0.21/0.46 ! [C: $tType,D: $tType,A: $tType,B: $tType,F3: $tType,E: $tType,C2: A > B > $o,A2: C > E > $o,B2: D > F3 > $o] :
% 0.21/0.46 ( ( bi_total @ A @ B @ C2 )
% 0.21/0.46 => ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F3 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F3 ) > $o )
% 0.21/0.46 @ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F3 > $o ) @ A2
% 0.21/0.46 @ ( bNF_rel_fun @ D @ F3 @ $o @ $o @ B2
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.21/0.46 @ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F3 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C2 @ A2 )
% 0.21/0.46 @ ( bNF_rel_fun @ ( A > D ) @ ( B > F3 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F3 @ C2 @ B2 )
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.21/0.46 @ ( partial_fun_ord @ C @ D @ A )
% 0.21/0.46 @ ( partial_fun_ord @ E @ F3 @ B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % fun_ord_parametric
% 0.21/0.46 thf(fact_57_llist__of__stream__stream__of__llist,axiom,
% 0.21/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.21/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.21/0.46 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.21/0.46 = Xs ) ) ).
% 0.21/0.46
% 0.21/0.46 % llist_of_stream_stream_of_llist
% 0.21/0.46 thf(fact_58_cr__streamI,axiom,
% 0.21/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.21/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.21/0.46 => ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % cr_streamI
% 0.21/0.46 thf(fact_59_stream_OAbs__inverse,axiom,
% 0.21/0.46 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.21/0.46 ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.21/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.21/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.21/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.21/0.46 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Y ) )
% 0.21/0.46 = Y ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.Abs_inverse
% 0.21/0.46 thf(fact_60_llist__of__stream__unfold__stream,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,SHD: B > A,STL: B > B,X: B] :
% 0.21/0.46 ( ( coindu1724414836stream @ A @ ( coindu139217191stream @ B @ A @ SHD @ STL @ X ) )
% 0.21/0.46 = ( coindu1441602521_llist @ B @ A
% 0.21/0.46 @ ^ [Uu: B] : $false
% 0.21/0.46 @ SHD
% 0.21/0.46 @ STL
% 0.21/0.46 @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % llist_of_stream_unfold_stream
% 0.21/0.46 thf(fact_61_stream_Odisc__transfer,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.21/0.46 ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
% 0.21/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.21/0.46 @ ^ [Stream: stream @ A] : ( Stream = Stream )
% 0.21/0.46 @ ^ [Stream: stream @ B] : ( Stream = Stream ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.disc_transfer
% 0.21/0.46 thf(fact_62_stream_Orel__refl,axiom,
% 0.21/0.46 ! [B: $tType,Ra: B > B > $o,X: stream @ B] :
% 0.21/0.46 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.21/0.46 => ( stream_all2 @ B @ B @ Ra @ X @ X ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.rel_refl
% 0.21/0.46 thf(fact_63_stream_Orel__eq,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( ( stream_all2 @ A @ A
% 0.21/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.21/0.46 = ( ^ [Y3: stream @ A,Z: stream @ A] : ( Y3 = Z ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.rel_eq
% 0.21/0.46 thf(fact_64_stream_Obi__total__rel,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.21/0.46 ( ( bi_total @ A @ B @ R )
% 0.21/0.46 => ( bi_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % stream.bi_total_rel
% 0.21/0.46 thf(fact_65_bi__total__def,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType] :
% 0.21/0.46 ( ( bi_total @ A @ B )
% 0.21/0.46 = ( ^ [R5: A > B > $o] :
% 0.21/0.46 ( ! [X2: A] :
% 0.21/0.46 ( ^ [P2: B > $o] :
% 0.21/0.46 ? [X6: B] : ( P2 @ X6 )
% 0.21/0.46 @ ( R5 @ X2 ) )
% 0.21/0.46 & ! [Y4: B] :
% 0.21/0.46 ? [X2: A] : ( R5 @ X2 @ Y4 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % bi_total_def
% 0.21/0.46 thf(fact_66_bi__total__eq,axiom,
% 0.21/0.46 ! [A: $tType] :
% 0.21/0.46 ( bi_total @ A @ A
% 0.21/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.21/0.46
% 0.21/0.46 % bi_total_eq
% 0.21/0.46 thf(fact_67_Quotient3__rep__reflp,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_rep_reflp
% 0.21/0.46 thf(fact_68_Quotient3__rep__abs,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ R4 @ R4 )
% 0.21/0.46 => ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_rep_abs
% 0.21/0.46 thf(fact_69_Quotient3__rel__rep,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B,B3: B] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 0.21/0.46 = ( A3 = B3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_rel_rep
% 0.21/0.46 thf(fact_70_Quotient3__rel__abs,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ R4 @ S3 )
% 0.21/0.46 => ( ( Abs @ R4 )
% 0.21/0.46 = ( Abs @ S3 ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_rel_abs
% 0.21/0.46 thf(fact_71_Quotient3__abs__rep,axiom,
% 0.21/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( Abs @ ( Rep @ A3 ) )
% 0.21/0.46 = A3 ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_abs_rep
% 0.21/0.46 thf(fact_72_rep__abs__rsp__left,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ X1 @ X22 )
% 0.21/0.46 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rep_abs_rsp_left
% 0.21/0.46 thf(fact_73_Quotient3__refl2,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ R4 @ S3 )
% 0.21/0.46 => ( R @ S3 @ S3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_refl2
% 0.21/0.46 thf(fact_74_Quotient3__refl1,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ R4 @ S3 )
% 0.21/0.46 => ( R @ R4 @ R4 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_refl1
% 0.21/0.46 thf(fact_75_Quotient3__rel,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( ( R @ R4 @ R4 )
% 0.21/0.46 & ( R @ S3 @ S3 )
% 0.21/0.46 & ( ( Abs @ R4 )
% 0.21/0.46 = ( Abs @ S3 ) ) )
% 0.21/0.46 = ( R @ R4 @ S3 ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_rel
% 0.21/0.46 thf(fact_76_Quotient3__def,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType] :
% 0.21/0.46 ( ( quotient3 @ A @ B )
% 0.21/0.46 = ( ^ [R5: A > A > $o,Abs2: A > B,Rep2: B > A] :
% 0.21/0.46 ( ! [A4: B] :
% 0.21/0.46 ( ( Abs2 @ ( Rep2 @ A4 ) )
% 0.21/0.46 = A4 )
% 0.21/0.46 & ! [A4: B] : ( R5 @ ( Rep2 @ A4 ) @ ( Rep2 @ A4 ) )
% 0.21/0.46 & ! [R6: A,S2: A] :
% 0.21/0.46 ( ( R5 @ R6 @ S2 )
% 0.21/0.46 = ( ( R5 @ R6 @ R6 )
% 0.21/0.46 & ( R5 @ S2 @ S2 )
% 0.21/0.46 & ( ( Abs2 @ R6 )
% 0.21/0.46 = ( Abs2 @ S2 ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % Quotient3_def
% 0.21/0.46 thf(fact_77_rep__abs__rsp,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ X1 @ X22 )
% 0.21/0.46 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % rep_abs_rsp
% 0.21/0.46 thf(fact_78_pred__funI,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,A2: A > $o,B2: B > $o,F2: A > B] :
% 0.21/0.46 ( ! [X3: A] :
% 0.21/0.46 ( ( A2 @ X3 )
% 0.21/0.46 => ( B2 @ ( F2 @ X3 ) ) )
% 0.21/0.46 => ( basic_pred_fun @ A @ B @ A2 @ B2 @ F2 ) ) ).
% 0.21/0.46
% 0.21/0.46 % pred_funI
% 0.21/0.46 thf(fact_79_equals__rsp,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 0.21/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.21/0.46 => ( ( R @ Xa @ Xb )
% 0.21/0.46 => ( ( R @ Ya @ Yb )
% 0.21/0.46 => ( ( R @ Xa @ Ya )
% 0.21/0.46 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 % equals_rsp
% 0.21/0.46 thf(fact_80_Quotient3I,axiom,
% 0.21/0.46 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 0.21/0.46 ( ! [A5: A] :
% 0.21/0.46 ( ( Abs @ ( Rep @ A5 ) )
% 0.30/0.46 = A5 )
% 0.30/0.46 => ( ! [A5: A] : ( R @ ( Rep @ A5 ) @ ( Rep @ A5 ) )
% 0.30/0.46 => ( ! [R7: B,S4: B] :
% 0.30/0.46 ( ( R @ R7 @ S4 )
% 0.30/0.46 = ( ( R @ R7 @ R7 )
% 0.30/0.46 & ( R @ S4 @ S4 )
% 0.30/0.46 & ( ( Abs @ R7 )
% 0.30/0.46 = ( Abs @ S4 ) ) ) )
% 0.30/0.46 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3I
% 0.30/0.46 thf(fact_81_cond__prs,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A3: $o,B3: B,C3: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.30/0.46 => ( ( A3
% 0.30/0.46 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.30/0.46 = B3 ) )
% 0.30/0.46 & ( ~ A3
% 0.30/0.46 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.30/0.46 = C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cond_prs
% 0.30/0.46 thf(fact_82_fun_Opred__True,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType] :
% 0.30/0.46 ( ( basic_pred_fun @ D @ A
% 0.30/0.46 @ ^ [Uu: D] : $true
% 0.30/0.46 @ ^ [Uu: A] : $true )
% 0.30/0.46 = ( ^ [Uu: D > A] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.pred_True
% 0.30/0.46 thf(fact_83_fun__quotient3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
% 0.30/0.46 => ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs22 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_quotient3
% 0.30/0.46 thf(fact_84_apply__rspQ3_H_H,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S: C > C > $o,F2: A > C,X: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S @ F2 @ F2 )
% 0.30/0.46 => ( S @ ( F2 @ ( Rep @ X ) ) @ ( F2 @ ( Rep @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % apply_rspQ3''
% 0.30/0.46 thf(fact_85_apply__rspQ3,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F2: A > C,G2: A > C,X: A,Y: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F2 @ G2 )
% 0.30/0.46 => ( ( R1 @ X @ Y )
% 0.30/0.46 => ( R2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % apply_rspQ3
% 0.30/0.46 thf(fact_86_let__prs,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep23 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs23 @ Rep12 ) @ Abs12 )
% 0.30/0.46 @ ^ [S2: C,F: C > A] : ( F @ S2 ) )
% 0.30/0.46 = ( ^ [S2: D,F: D > B] : ( F @ S2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % let_prs
% 0.30/0.46 thf(fact_87_lfinite__llist__of__stream,axiom,
% 0.30/0.46 ! [A: $tType,Xs: stream @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 0.30/0.46
% 0.30/0.46 % lfinite_llist_of_stream
% 0.30/0.46 thf(fact_88_lambda__prs1,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,F2: B > D] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs23 @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep23 @ F2 ) )
% 0.30/0.46 = F2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lambda_prs1
% 0.30/0.46 thf(fact_89_lambda__prs,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,F2: B > D] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs23
% 0.30/0.46 @ ^ [X2: A] : ( Rep23 @ ( F2 @ ( Abs12 @ X2 ) ) ) )
% 0.30/0.46 = F2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lambda_prs
% 0.30/0.46 thf(fact_90_stream_ORep__induct,axiom,
% 0.30/0.46 ! [A: $tType,Y: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.30/0.46 ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.30/0.46 => ( ! [X3: stream @ A] : ( P @ ( coindu1724414836stream @ A @ X3 ) )
% 0.30/0.46 => ( P @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Rep_induct
% 0.30/0.46 thf(fact_91_stream_ORep__cases,axiom,
% 0.30/0.46 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.30/0.46 ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.30/0.46 => ~ ! [X3: stream @ A] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( coindu1724414836stream @ A @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Rep_cases
% 0.30/0.46 thf(fact_92_stream_ORep,axiom,
% 0.30/0.46 ! [A: $tType,X: stream @ A] :
% 0.30/0.46 ( member @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A @ X )
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Rep
% 0.30/0.46 thf(fact_93_stream_OAbs__cases,axiom,
% 0.30/0.46 ! [A: $tType,X: stream @ A] :
% 0.30/0.46 ~ ! [Y2: coinductive_llist @ A] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( coindu2010755910_llist @ A @ Y2 ) )
% 0.30/0.46 => ~ ( member @ ( coinductive_llist @ A ) @ Y2
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Abs_cases
% 0.30/0.46 thf(fact_94_stream_OAbs__induct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
% 0.30/0.46 ( ! [Y2: coinductive_llist @ A] :
% 0.30/0.46 ( ( member @ ( coinductive_llist @ A ) @ Y2
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.30/0.46 => ( P @ ( coindu2010755910_llist @ A @ Y2 ) ) )
% 0.30/0.46 => ( P @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Abs_induct
% 0.30/0.46 thf(fact_95_stream_OAbs__inject,axiom,
% 0.30/0.46 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
% 0.30/0.46 ( ( member @ ( coinductive_llist @ A ) @ X
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.30/0.46 => ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.30/0.46 => ( ( ( coindu2010755910_llist @ A @ X )
% 0.30/0.46 = ( coindu2010755910_llist @ A @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Abs_inject
% 0.30/0.46 thf(fact_96_unfold__stream__ltl__unroll,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B3: B] :
% 0.30/0.46 ( ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B3 ) )
% 0.30/0.46 = ( coindu139217191stream @ B @ A @ ( comp @ B @ A @ B @ SHD @ STL ) @ STL @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % unfold_stream_ltl_unroll
% 0.30/0.46 thf(fact_97_stream_Orel__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( stream @ A ) > ( stream @ B ) > $o ) @ ( ( stream @ C ) > ( stream @ D ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.30/0.46 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ C ) @ ( ( stream @ B ) > $o ) @ ( ( stream @ D ) > $o ) @ ( stream_all2 @ A @ C @ Sa )
% 0.30/0.46 @ ( bNF_rel_fun @ ( stream @ B ) @ ( stream @ D ) @ $o @ $o @ ( stream_all2 @ B @ D @ Sc )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( stream_all2 @ A @ B )
% 0.30/0.46 @ ( stream_all2 @ C @ D ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_transfer
% 0.30/0.46 thf(fact_98_o__prs_I1_J,axiom,
% 0.30/0.46 ! [C: $tType,E: $tType,A: $tType,B: $tType,F3: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,R3: E > E > $o,Abs3: E > F3,Rep3: F3 > E] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ( quotient3 @ E @ F3 @ R3 @ Abs3 @ Rep3 )
% 0.30/0.46 => ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs23 @ Rep3 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep23 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep12 @ Abs3 ) ) @ ( comp @ C @ E @ A ) )
% 0.30/0.46 = ( comp @ D @ F3 @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % o_prs(1)
% 0.30/0.46 thf(fact_99_fun_Omap__cong__pred,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F2: A > B,G2: A > B] :
% 0.30/0.46 ( ( X = Ya )
% 0.30/0.46 => ( ( basic_pred_fun @ D @ A
% 0.30/0.46 @ ^ [Uu: D] : $true
% 0.30/0.46 @ ^ [Z2: A] :
% 0.30/0.46 ( ( F2 @ Z2 )
% 0.30/0.46 = ( G2 @ Z2 ) )
% 0.30/0.46 @ Ya )
% 0.30/0.46 => ( ( comp @ A @ B @ D @ F2 @ X )
% 0.30/0.46 = ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_cong_pred
% 0.30/0.46 thf(fact_100_fun_Opred__map,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D: $tType,Q: B > $o,F2: A > B,X: D > A] :
% 0.30/0.46 ( ( basic_pred_fun @ D @ B
% 0.30/0.46 @ ^ [Uu: D] : $true
% 0.30/0.46 @ Q
% 0.30/0.46 @ ( comp @ A @ B @ D @ F2 @ X ) )
% 0.30/0.46 = ( basic_pred_fun @ D @ A
% 0.30/0.46 @ ^ [Uu: D] : $true
% 0.30/0.46 @ ( comp @ B @ $o @ A @ Q @ F2 )
% 0.30/0.46 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.pred_map
% 0.30/0.46 thf(fact_101_quot__rel__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
% 0.30/0.46 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ R
% 0.30/0.46 @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % quot_rel_rsp
% 0.30/0.46 thf(fact_102_corec__llist__never__stop,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coinductive_llist @ A ),LTL: B > B,X: B] :
% 0.30/0.46 ( ( coindu1259883913_llist @ B @ A @ IS_LNIL @ LHD
% 0.30/0.46 @ ^ [Uu: B] : $false
% 0.30/0.46 @ MORE
% 0.30/0.46 @ LTL
% 0.30/0.46 @ X )
% 0.30/0.46 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % corec_llist_never_stop
% 0.30/0.46 thf(fact_103_unfold__llist__ltl__unroll,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B] :
% 0.30/0.46 ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) )
% 0.30/0.46 = ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % unfold_llist_ltl_unroll
% 0.30/0.46 thf(fact_104_monotone__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( comple1396247847notone @ A @ C )
% 0.30/0.46 @ ( comple1396247847notone @ B @ D ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_parametric
% 0.30/0.46 thf(fact_105_stream_Otype__definition__axioms,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( type_definition @ ( stream @ A ) @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A ) @ ( coindu2010755910_llist @ A )
% 0.30/0.46 @ ( collect @ ( coinductive_llist @ A )
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.type_definition_axioms
% 0.30/0.46 thf(fact_106_relcompp__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) ) )
% 0.30/0.46 @ ( relcompp @ C @ A @ E )
% 0.30/0.46 @ ( relcompp @ D @ B @ F3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_transfer
% 0.30/0.46 thf(fact_107_bex1__rel__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( bex1_rel @ A @ R )
% 0.30/0.46 @ ( bex1_rel @ A @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex1_rel_rsp
% 0.30/0.46 thf(fact_108_left__total__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( left_total @ A @ C )
% 0.30/0.46 @ ( left_total @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_parametric
% 0.30/0.46 thf(fact_109_fun_Orel__compp,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.30/0.46 ( ( bNF_rel_fun @ D @ D @ A @ C
% 0.30/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.30/0.46 @ ( relcompp @ A @ B @ C @ R @ S ) )
% 0.30/0.46 = ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.30/0.46 @ R )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ B @ C
% 0.30/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.30/0.46 @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_compp
% 0.30/0.46 thf(fact_110_conj__comp__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G2: A > B] :
% 0.30/0.46 ( ( comp @ B @ $o @ A
% 0.30/0.46 @ ^ [X2: B] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 & ( Q @ X2 ) )
% 0.30/0.46 @ G2 )
% 0.30/0.46 = ( ^ [X2: A] :
% 0.30/0.46 ( ( comp @ B @ $o @ A @ P @ G2 @ X2 )
% 0.30/0.46 & ( comp @ B @ $o @ A @ Q @ G2 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % conj_comp_iff
% 0.30/0.46 thf(fact_111_stream_Orel__compp,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.30/0.46 ( ( stream_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
% 0.30/0.46 = ( relcompp @ ( stream @ A ) @ ( stream @ B ) @ ( stream @ C ) @ ( stream_all2 @ A @ B @ R ) @ ( stream_all2 @ B @ C @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_compp
% 0.30/0.46 thf(fact_112_stream_Oleft__total__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ( left_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.left_total_rel
% 0.30/0.46 thf(fact_113_eq__comp__r,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( relcompp @ A @ A @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/0.46 @ ( relcompp @ A @ A @ A @ R
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % eq_comp_r
% 0.30/0.46 thf(fact_114_bex1__rel__aux,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 0.30/0.46 ( ! [Xa2: A,Ya2: A] :
% 0.30/0.46 ( ( R @ Xa2 @ Ya2 )
% 0.30/0.46 => ( ( X @ Xa2 )
% 0.30/0.46 = ( Y @ Ya2 ) ) )
% 0.30/0.46 => ( ( bex1_rel @ A @ R @ X )
% 0.30/0.46 => ( bex1_rel @ A @ R @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex1_rel_aux
% 0.30/0.46 thf(fact_115_bex1__rel__aux2,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 0.30/0.46 ( ! [Xa2: A,Ya2: A] :
% 0.30/0.46 ( ( R @ Xa2 @ Ya2 )
% 0.30/0.46 => ( ( X @ Xa2 )
% 0.30/0.46 = ( Y @ Ya2 ) ) )
% 0.30/0.46 => ( ( bex1_rel @ A @ R @ Y )
% 0.30/0.46 => ( bex1_rel @ A @ R @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex1_rel_aux2
% 0.30/0.46 thf(fact_116_left__totalE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ! [X7: A] :
% 0.30/0.46 ? [X12: B] : ( R @ X7 @ X12 ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_totalE
% 0.30/0.46 thf(fact_117_left__totalI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ? [X13: B] : ( R @ X3 @ X13 )
% 0.30/0.46 => ( left_total @ A @ B @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_totalI
% 0.30/0.46 thf(fact_118_left__total__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ( ( left_total @ B @ C @ S )
% 0.30/0.46 => ( left_total @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_OO
% 0.30/0.46 thf(fact_119_left__total__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( left_total @ A @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_eq
% 0.30/0.46 thf(fact_120_left__total__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( left_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ! [X2: A] :
% 0.30/0.46 ( ^ [P2: B > $o] :
% 0.30/0.46 ? [X6: B] : ( P2 @ X6 )
% 0.30/0.46 @ ( R5 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_def
% 0.30/0.46 thf(fact_121_bi__total__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ B @ C @ B2 )
% 0.30/0.46 => ( bi_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_OO
% 0.30/0.46 thf(fact_122_typedef__rep__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T2: A > B > $o] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( T2
% 0.30/0.46 = ( ^ [X2: A,Y4: B] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( Rep @ Y4 ) ) ) )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ A @ A @ T2
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/0.46 @ ^ [X2: A] : X2
% 0.30/0.46 @ Rep ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % typedef_rep_transfer
% 0.30/0.46 thf(fact_123_OOO__quotient3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: B > B > $o,Abs23: B > C,Rep23: C > B,R23: A > A > $o] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ B @ C @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ! [X3: A,Y2: A] :
% 0.30/0.46 ( ( R23 @ X3 @ Y2 )
% 0.30/0.46 => ( ( R1 @ X3 @ X3 )
% 0.30/0.46 => ( ( R1 @ Y2 @ Y2 )
% 0.30/0.46 => ( R2 @ ( Abs12 @ X3 ) @ ( Abs12 @ Y2 ) ) ) ) )
% 0.30/0.46 => ( ! [X3: B,Y2: B] :
% 0.30/0.46 ( ( R2 @ X3 @ Y2 )
% 0.30/0.46 => ( R23 @ ( Rep12 @ X3 ) @ ( Rep12 @ Y2 ) ) )
% 0.30/0.46 => ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R23 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs23 @ Abs12 ) @ ( comp @ B @ A @ C @ Rep12 @ Rep23 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % OOO_quotient3
% 0.30/0.46 thf(fact_124_OOO__eq__quotient3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,Abs23: B > C,Rep23: C > B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ B @ C
% 0.30/0.46 @ ^ [Y3: B,Z: B] : ( Y3 = Z )
% 0.30/0.46 @ Abs23
% 0.30/0.46 @ Rep23 )
% 0.30/0.46 => ( quotient3 @ A @ C
% 0.30/0.46 @ ( relcompp @ A @ A @ A @ R1
% 0.30/0.46 @ ( relcompp @ A @ A @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/0.46 @ R1 ) )
% 0.30/0.46 @ ( comp @ B @ C @ A @ Abs23 @ Abs12 )
% 0.30/0.46 @ ( comp @ B @ A @ C @ Rep12 @ Rep23 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % OOO_eq_quotient3
% 0.30/0.46 thf(fact_125_monotone__applyI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,X: C] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F5 )
% 0.30/0.46 => ( comple1396247847notone @ ( C > A ) @ B @ ( partial_fun_ord @ A @ A @ C @ Orda ) @ Ordb
% 0.30/0.46 @ ^ [F: C > A] : ( F5 @ ( F @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_applyI
% 0.30/0.46 thf(fact_126_monotone__fun__ord__apply,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,Orda: A > A > $o,Ordb: C > C > $o,F2: A > B > C] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ ( B > C ) @ Orda @ ( partial_fun_ord @ C @ C @ B @ Ordb ) @ F2 )
% 0.30/0.46 = ( ! [X2: B] :
% 0.30/0.46 ( comple1396247847notone @ A @ C @ Orda @ Ordb
% 0.30/0.46 @ ^ [Y4: A] : ( F2 @ Y4 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_fun_ord_apply
% 0.30/0.46 thf(fact_127_stream_Ocase__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,S: C > D > $o] : ( bNF_rel_fun @ ( A > ( stream @ A ) > C ) @ ( B > ( stream @ B ) > D ) @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ R @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S ) @ ( case_stream @ A @ C ) @ ( case_stream @ B @ D ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.case_transfer
% 0.30/0.46 thf(fact_128_monotone__if__fun,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,Orda: B > B > $o,Ordb: D > D > $o,F5: ( A > B ) > C > D,G5: ( A > B ) > C > D,C3: C > $o] :
% 0.30/0.46 ( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ F5 )
% 0.30/0.46 => ( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ G5 )
% 0.30/0.46 => ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb )
% 0.30/0.46 @ ^ [F: A > B,N2: C] : ( if @ D @ ( C3 @ N2 ) @ ( F5 @ F @ N2 ) @ ( G5 @ F @ N2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_if_fun
% 0.30/0.46 thf(fact_129_monotone__fun__apply__fun,axiom,
% 0.30/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,Ord: C > C > $o,T: A,G2: D > B] :
% 0.30/0.46 ( comple1396247847notone @ ( A > B > C ) @ ( D > C ) @ ( partial_fun_ord @ ( B > C ) @ ( B > C ) @ A @ ( partial_fun_ord @ C @ C @ B @ Ord ) ) @ ( partial_fun_ord @ C @ C @ D @ Ord )
% 0.30/0.46 @ ^ [F: A > B > C,N2: D] : ( F @ T @ ( G2 @ N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_fun_apply_fun
% 0.30/0.46 thf(fact_130_stream_Ocase__distrib,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,H2: B > C,F2: A > ( stream @ A ) > B,Stream2: stream @ A] :
% 0.30/0.46 ( ( H2 @ ( case_stream @ A @ B @ F2 @ Stream2 ) )
% 0.30/0.46 = ( case_stream @ A @ C
% 0.30/0.46 @ ^ [X14: A,X23: stream @ A] : ( H2 @ ( F2 @ X14 @ X23 ) )
% 0.30/0.46 @ Stream2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.case_distrib
% 0.30/0.46 thf(fact_131_monotone__id_H,axiom,
% 0.30/0.46 ! [A: $tType,Ord: A > A > $o] :
% 0.30/0.46 ( comple1396247847notone @ A @ A @ Ord @ Ord
% 0.30/0.46 @ ^ [X2: A] : X2 ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_id'
% 0.30/0.46 thf(fact_132_call__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Ord: B > B > $o,T: A] :
% 0.30/0.46 ( comple1396247847notone @ ( A > B ) @ B @ ( partial_fun_ord @ B @ B @ A @ Ord ) @ Ord
% 0.30/0.46 @ ^ [F: A > B] : ( F @ T ) ) ).
% 0.30/0.46
% 0.30/0.46 % call_mono
% 0.30/0.46 thf(fact_133_stream_Opred__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( stream @ A ) > $o ) @ ( ( stream @ B ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( pred_stream @ A )
% 0.30/0.46 @ ( pred_stream @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_transfer
% 0.30/0.46 thf(fact_134_transfer__forall__transfer_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( transfer_forall @ A )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(1)
% 0.30/0.46 thf(fact_135_Domainp__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( domainp @ C @ A )
% 0.30/0.46 @ ( domainp @ D @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_transfer
% 0.30/0.46 thf(fact_136_stream_Opred__True,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( pred_stream @ A
% 0.30/0.46 @ ^ [Uu: A] : $true )
% 0.30/0.46 = ( ^ [Uu: stream @ A] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_True
% 0.30/0.46 thf(fact_137_fun_ODomainp__rel,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( domainp @ ( C > A ) @ ( C > B )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ C @ A @ B
% 0.30/0.46 @ ^ [Y3: C,Z: C] : ( Y3 = Z )
% 0.30/0.46 @ R ) )
% 0.30/0.46 = ( basic_pred_fun @ C @ A
% 0.30/0.46 @ ^ [Uu: C] : $true
% 0.30/0.46 @ ( domainp @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.Domainp_rel
% 0.30/0.46 thf(fact_138_stream_ODomainp__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( domainp @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) )
% 0.30/0.46 = ( pred_stream @ A @ ( domainp @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.Domainp_rel
% 0.30/0.46 thf(fact_139_Domainp__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( domainp @ A @ B )
% 0.30/0.46 = ( ^ [T3: A > B > $o,X2: A] :
% 0.30/0.46 ( ^ [P2: B > $o] :
% 0.30/0.46 ? [X6: B] : ( P2 @ X6 )
% 0.30/0.46 @ ( T3 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_iff
% 0.30/0.46 thf(fact_140_Domainp__refl,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( domainp @ A @ B )
% 0.30/0.46 = ( domainp @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_refl
% 0.30/0.46 thf(fact_141_pcr__Domainp__total,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,A2: C > A > $o,P: C > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ B2 )
% 0.30/0.46 => ( ( ( domainp @ C @ A @ A2 )
% 0.30/0.46 = P )
% 0.30/0.46 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pcr_Domainp_total
% 0.30/0.46 thf(fact_142_stream__from__llist__setup_Ostream_Odomain__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( domainp @ ( coinductive_llist @ A ) @ ( stream @ A )
% 0.30/0.46 @ ( coindu773941317stream @ A @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) )
% 0.30/0.46 = ( ^ [Xs2: coinductive_llist @ A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.stream.domain_eq
% 0.30/0.46 thf(fact_143_fun__ord__def,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( partial_fun_ord @ A @ B @ C )
% 0.30/0.46 = ( ^ [Ord2: A > B > $o,F: C > A,G: C > B] :
% 0.30/0.46 ! [X2: C] : ( Ord2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_ord_def
% 0.30/0.46 thf(fact_144_pcr__Domainp__par__left__total,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o,P3: C > $o] :
% 0.30/0.46 ( ( ( domainp @ A @ B @ B2 )
% 0.30/0.46 = P )
% 0.30/0.46 => ( ( left_total @ C @ A @ A2 )
% 0.30/0.46 => ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ P3
% 0.30/0.46 @ P )
% 0.30/0.46 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = P3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pcr_Domainp_par_left_total
% 0.30/0.46 thf(fact_145_let__mono,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,Orda: B > B > $o,Ordb: C > C > $o,B3: B > A > C,T: A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( comple1396247847notone @ B @ C @ Orda @ Ordb
% 0.30/0.46 @ ^ [F: B] : ( B3 @ F @ X3 ) )
% 0.30/0.46 => ( comple1396247847notone @ B @ C @ Orda @ Ordb
% 0.30/0.46 @ ^ [F: B] : ( B3 @ F @ T ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % let_mono
% 0.30/0.46 thf(fact_146_if__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,G5: A > B,C3: $o] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F5 )
% 0.30/0.46 => ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ G5 )
% 0.30/0.46 => ( comple1396247847notone @ A @ B @ Orda @ Ordb
% 0.30/0.46 @ ^ [F: A] : ( if @ B @ C3 @ ( F5 @ F ) @ ( G5 @ F ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % if_mono
% 0.30/0.46 thf(fact_147_transfer__forall__transfer_I4_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ rev_implies
% 0.30/0.46 @ ( transfer_forall @ A )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(4)
% 0.30/0.46 thf(fact_148_transfer__forall__transfer_I5_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) @ rev_implies @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(5)
% 0.30/0.46 thf(fact_149_reflp__transfer_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( reflp @ A )
% 0.30/0.46 @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(1)
% 0.30/0.46 thf(fact_150_right__total__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( right_total @ A @ C )
% 0.30/0.46 @ ( right_total @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_parametric
% 0.30/0.46 thf(fact_151_fun_Orel__reflp,axiom,
% 0.30/0.46 ! [D: $tType,A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( reflp @ A @ R )
% 0.30/0.46 => ( reflp @ ( D > A )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ A
% 0.30/0.46 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.30/0.46 @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_reflp
% 0.30/0.46 thf(fact_152_DEADID_Orel__reflp,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( reflp @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % DEADID.rel_reflp
% 0.30/0.46 thf(fact_153_right__total__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( right_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ! [Y4: B] :
% 0.30/0.46 ? [X2: A] : ( R5 @ X2 @ Y4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_def
% 0.30/0.46 thf(fact_154_rev__implies__def,axiom,
% 0.30/0.46 ( rev_implies
% 0.30/0.46 = ( ^ [X2: $o,Y4: $o] :
% 0.30/0.46 ( Y4
% 0.30/0.46 => X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_implies_def
% 0.30/0.46 thf(fact_155_right__total__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( right_total @ A @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_eq
% 0.30/0.46 thf(fact_156_right__totalI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: B > A > $o] :
% 0.30/0.46 ( ! [Y2: A] :
% 0.30/0.46 ? [X7: B] : ( A2 @ X7 @ Y2 )
% 0.30/0.46 => ( right_total @ B @ A @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_totalI
% 0.30/0.46 thf(fact_157_right__totalE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o,Y: B] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ~ ! [X3: A] :
% 0.30/0.46 ~ ( A2 @ X3 @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_totalE
% 0.30/0.46 thf(fact_158_stream__from__llist__setup_Ostream_Oright__total,axiom,
% 0.30/0.46 ! [F3: $tType,E: $tType,T2: E > F3 > $o] :
% 0.30/0.46 ( ( right_total @ E @ F3 @ T2 )
% 0.30/0.46 => ( right_total @ ( coinductive_llist @ E ) @ ( stream @ F3 ) @ ( coindu773941317stream @ E @ F3 @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.stream.right_total
% 0.30/0.46 thf(fact_159_stream_Oright__total__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ R )
% 0.30/0.46 => ( right_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.right_total_rel
% 0.30/0.46 thf(fact_160_stream_Orel__reflp,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( reflp @ A @ R )
% 0.30/0.46 => ( reflp @ ( stream @ A ) @ ( stream_all2 @ A @ A @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_reflp
% 0.30/0.46 thf(fact_161_right__total__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_total @ B @ C @ B2 )
% 0.30/0.46 => ( right_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_OO
% 0.30/0.46 thf(fact_162_typedef__right__total,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T2: A > B > $o] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( T2
% 0.30/0.46 = ( ^ [X2: A,Y4: B] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( Rep @ Y4 ) ) ) )
% 0.30/0.46 => ( right_total @ A @ B @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % typedef_right_total
% 0.30/0.46 thf(fact_163_reflp__transfer_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) ) @ (=>) @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(2)
% 0.30/0.46 thf(fact_164_reflp__transfer_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ (=>)
% 0.30/0.46 @ ( reflp @ A )
% 0.30/0.46 @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(3)
% 0.30/0.46 thf(fact_165_bi__totalI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ( ( right_total @ A @ B @ R )
% 0.30/0.46 => ( bi_total @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_totalI
% 0.30/0.46 thf(fact_166_bi__total__alt__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_total @ A @ B )
% 0.30/0.46 = ( ^ [A6: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ A6 )
% 0.30/0.46 & ( right_total @ A @ B @ A6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_alt_def
% 0.30/0.46 thf(fact_167_reflp__transfer_I4_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) ) @ rev_implies @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(4)
% 0.30/0.46 thf(fact_168_reflp__transfer_I5_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ rev_implies
% 0.30/0.46 @ ( reflp @ A )
% 0.30/0.46 @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(5)
% 0.30/0.46 thf(fact_169_transfer__forall__transfer_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) @ (=>) @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(3)
% 0.30/0.46 thf(fact_170_transfer__forall__transfer_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ (=>)
% 0.30/0.46 @ ( transfer_forall @ A )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(2)
% 0.30/0.46 thf(fact_171_Domainp__forall__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( transfer_bforall @ A @ ( domainp @ A @ B @ A2 ) )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_forall_transfer
% 0.30/0.46 thf(fact_172_right__total__relcompp__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) ) )
% 0.30/0.46 @ ^ [R5: C > A > $o,S5: A > E > $o,X2: C,Z2: E] :
% 0.30/0.46 ? [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B2 ) ) )
% 0.30/0.46 & ( R5 @ X2 @ Y4 )
% 0.30/0.46 & ( S5 @ Y4 @ Z2 ) )
% 0.30/0.46 @ ( relcompp @ D @ B @ F3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_relcompp_transfer
% 0.30/0.46 thf(fact_173_right__total__Domainp__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [T3: C > A > $o,X2: C] :
% 0.30/0.46 ? [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B2 ) ) )
% 0.30/0.46 & ( T3 @ X2 @ Y4 ) )
% 0.30/0.46 @ ( domainp @ D @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_Domainp_transfer
% 0.30/0.46 thf(fact_174_bex__reg,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ R )
% 0.30/0.46 => ( ( P @ X3 )
% 0.30/0.46 => ( Q @ X3 ) ) )
% 0.30/0.46 => ( ? [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ R )
% 0.30/0.46 & ( P @ X7 ) )
% 0.30/0.46 => ? [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ R )
% 0.30/0.46 & ( Q @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex_reg
% 0.30/0.46 thf(fact_175_right__total__Ex__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( bex @ A @ ( collect @ A @ ( domainp @ A @ B @ A2 ) ) )
% 0.30/0.46 @ ^ [P2: B > $o] :
% 0.30/0.46 ? [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_Ex_transfer
% 0.30/0.46 thf(fact_176_Domainp__prod__fun__eq,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,T2: B > C > $o] :
% 0.30/0.46 ( ( domainp @ ( A > B ) @ ( A > C )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ A @ B @ C
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/0.46 @ T2 ) )
% 0.30/0.46 = ( ^ [F: A > B] :
% 0.30/0.46 ! [X2: A] : ( domainp @ B @ C @ T2 @ ( F @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_prod_fun_eq
% 0.30/0.46 thf(fact_177_stream__from__llist__setup_Ostream_Odomain__par__left__total,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,T2: C > B > $o,P3: ( coinductive_llist @ C ) > $o] :
% 0.30/0.46 ( ( left_total @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ C @ B @ T2 ) )
% 0.30/0.46 => ( ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ C @ B @ T2 )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ P3
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ B] :
% 0.30/0.46 ~ ( coinductive_lfinite @ B @ Xs2 ) )
% 0.30/0.46 => ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
% 0.30/0.46 = P3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.stream.domain_par_left_total
% 0.30/0.46 thf(fact_178_llist_Orel__reflp,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( reflp @ A @ R )
% 0.30/0.46 => ( reflp @ ( coinductive_llist @ A ) @ ( coindu1486289336t_all2 @ A @ A @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.rel_reflp
% 0.30/0.46 thf(fact_179_llist_Oright__total__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ R )
% 0.30/0.46 => ( right_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.right_total_rel
% 0.30/0.46 thf(fact_180_bex__reg__left,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A,Q: A > $o,P: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ R )
% 0.30/0.46 => ( ( Q @ X3 )
% 0.30/0.46 => ( P @ X3 ) ) )
% 0.30/0.46 => ( ? [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ R )
% 0.30/0.46 & ( Q @ X7 ) )
% 0.30/0.46 => ? [X12: A] : ( P @ X12 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex_reg_left
% 0.30/0.46 thf(fact_181_rel__fun__eq__rel,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,R: B > C > $o] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ A @ B @ C
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/0.46 @ R )
% 0.30/0.46 = ( ^ [F: A > B,G: A > C] :
% 0.30/0.46 ! [X2: A] : ( R @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_fun_eq_rel
% 0.30/0.46 thf(fact_182_rel__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ C @ B @ D )
% 0.30/0.46 = ( ^ [A6: A > C > $o,B4: B > D > $o,F: A > B,G: C > D] :
% 0.30/0.46 ! [X2: A,Y4: C] :
% 0.30/0.46 ( ( A6 @ X2 @ Y4 )
% 0.30/0.46 => ( B4 @ ( F @ X2 ) @ ( G @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_fun_def
% 0.30/0.46 thf(fact_183_stream__from__llist__setup_Ostream_Odomain,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,T2: C > B > $o] :
% 0.30/0.46 ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
% 0.30/0.46 = ( ^ [X2: coinductive_llist @ C] :
% 0.30/0.46 ? [Y4: coinductive_llist @ B] :
% 0.30/0.46 ( ( coindu1486289336t_all2 @ C @ B @ T2 @ X2 @ Y4 )
% 0.30/0.46 & ~ ( coinductive_lfinite @ B @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.stream.domain
% 0.30/0.46 thf(fact_184_llist__all2__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o ) @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.30/0.46 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( coindu1486289336t_all2 @ A @ A )
% 0.30/0.46 @ ( coindu1486289336t_all2 @ B @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_all2_transfer
% 0.30/0.46 thf(fact_185_llist_Orel__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.30/0.46 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ D ) > $o ) @ ( coindu1486289336t_all2 @ A @ C @ Sa )
% 0.30/0.46 @ ( bNF_rel_fun @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ $o @ $o @ ( coindu1486289336t_all2 @ B @ D @ Sc )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.30/0.46 @ ( coindu1486289336t_all2 @ A @ B )
% 0.30/0.46 @ ( coindu1486289336t_all2 @ C @ D ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.rel_transfer
% 0.30/0.46 thf(fact_186_lfinite__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( coinductive_lfinite @ A )
% 0.30/0.46 @ ( coinductive_lfinite @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % lfinite_transfer
% 0.30/0.46 thf(fact_187_transfer__forall__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( transfer_forall @ A )
% 0.30/0.46 = ( ^ [P2: A > $o] :
% 0.30/0.46 ! [X6: A] : ( P2 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_def
% 0.30/0.46 thf(fact_188_OO__def,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( relcompp @ A @ C @ B )
% 0.30/0.46 = ( ^ [R5: A > C > $o,S5: C > B > $o,X2: A,Z2: B] :
% 0.30/0.46 ? [Y4: C] :
% 0.30/0.46 ( ( R5 @ X2 @ Y4 )
% 0.30/0.46 & ( S5 @ Y4 @ Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % OO_def
% 0.30/0.46 thf(fact_189_llist_Obi__total__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ R )
% 0.30/0.46 => ( bi_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.bi_total_rel
% 0.30/0.46 thf(fact_190_llist__all2__lfiniteD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.30/0.46 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.30/0.46 => ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.46 = ( coinductive_lfinite @ B @ Ys ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_all2_lfiniteD
% 0.30/0.46 thf(fact_191_llist_Orel__refl,axiom,
% 0.30/0.46 ! [B: $tType,Ra: B > B > $o,X: coinductive_llist @ B] :
% 0.30/0.46 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.30/0.46 => ( coindu1486289336t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.rel_refl
% 0.30/0.46 thf(fact_192_llist_Orel__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( coindu1486289336t_all2 @ A @ A
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.30/0.46 = ( ^ [Y3: coinductive_llist @ A,Z: coinductive_llist @ A] : ( Y3 = Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.rel_eq
% 0.30/0.46 thf(fact_193_llist__all2__mono,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P3: A > B > $o] :
% 0.30/0.46 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.30/0.46 => ( ! [X3: A,Y2: B] :
% 0.30/0.46 ( ( P @ X3 @ Y2 )
% 0.30/0.46 => ( P3 @ X3 @ Y2 ) )
% 0.30/0.46 => ( coindu1486289336t_all2 @ A @ B @ P3 @ Xs @ Ys ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_all2_mono
% 0.30/0.46 thf(fact_194_llist__all2__rsp,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T2: B > B > $o,X: coinductive_llist @ A,Y: coinductive_llist @ B,A3: coinductive_llist @ A,B3: coinductive_llist @ B] :
% 0.30/0.46 ( ! [X3: A,Y2: B] :
% 0.30/0.46 ( ( R @ X3 @ Y2 )
% 0.30/0.46 => ! [A5: A,B5: B] :
% 0.30/0.46 ( ( R @ A5 @ B5 )
% 0.30/0.46 => ( ( S @ X3 @ A5 )
% 0.30/0.46 = ( T2 @ Y2 @ B5 ) ) ) )
% 0.30/0.46 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X @ Y )
% 0.30/0.46 => ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
% 0.30/0.46 => ( ( coindu1486289336t_all2 @ A @ A @ S @ X @ A3 )
% 0.30/0.46 = ( coindu1486289336t_all2 @ B @ B @ T2 @ Y @ B3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_all2_rsp
% 0.30/0.46 thf(fact_195_llist__all2__conj,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.30/0.46 ( ( coindu1486289336t_all2 @ A @ B
% 0.30/0.46 @ ^ [X2: A,Y4: B] :
% 0.30/0.46 ( ( P @ X2 @ Y4 )
% 0.30/0.46 & ( Q @ X2 @ Y4 ) )
% 0.30/0.46 @ Xs
% 0.30/0.46 @ Ys )
% 0.30/0.46 = ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.30/0.46 & ( coindu1486289336t_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_all2_conj
% 0.30/0.46 thf(fact_196_llist_Oleft__total__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ( left_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.left_total_rel
% 0.30/0.46 thf(fact_197_llist_Orel__compp,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.30/0.46 ( ( coindu1486289336t_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
% 0.30/0.46 = ( relcompp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ C ) @ ( coindu1486289336t_all2 @ A @ B @ R ) @ ( coindu1486289336t_all2 @ B @ C @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.rel_compp
% 0.30/0.46 thf(fact_198_pred__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( basic_pred_fun @ A @ B )
% 0.30/0.46 = ( ^ [A6: A > $o,B4: B > $o,F: A > B] :
% 0.30/0.46 ! [X2: A] :
% 0.30/0.46 ( ( A6 @ X2 )
% 0.30/0.46 => ( B4 @ ( F @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pred_fun_def
% 0.30/0.46 thf(fact_199_pcr__Domainp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o] :
% 0.30/0.46 ( ( ( domainp @ A @ B @ B2 )
% 0.30/0.46 = P )
% 0.30/0.46 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = ( ^ [X2: C] :
% 0.30/0.46 ? [Y4: A] :
% 0.30/0.46 ( ( A2 @ X2 @ Y4 )
% 0.30/0.46 & ( P @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pcr_Domainp
% 0.30/0.46 thf(fact_200_stream__from__llist__setup_Ostream_Orep__transfer,axiom,
% 0.30/0.46 ! [E: $tType,F3: $tType,T2: E > F3 > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( coinductive_llist @ E ) @ ( stream @ F3 ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ F3 ) @ ( coindu773941317stream @ E @ F3 @ T2 ) @ ( coindu1486289336t_all2 @ E @ F3 @ T2 )
% 0.30/0.46 @ ^ [X2: coinductive_llist @ E] : X2
% 0.30/0.46 @ ( coindu1724414836stream @ F3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.stream.rep_transfer
% 0.30/0.46 thf(fact_201_stream__from__llist__setup_Opcr__stream__def,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType] :
% 0.30/0.46 ( ( coindu773941317stream @ C @ B )
% 0.30/0.46 = ( ^ [T3: C > B > $o] : ( relcompp @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( stream @ B ) @ ( coindu1486289336t_all2 @ C @ B @ T3 ) @ ( coindu1183105481stream @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.pcr_stream_def
% 0.30/0.46 thf(fact_202_transfer__bforall__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( transfer_bforall @ A )
% 0.30/0.46 = ( ^ [P4: A > $o,Q2: A > $o] :
% 0.30/0.46 ! [X2: A] :
% 0.30/0.46 ( ( P4 @ X2 )
% 0.30/0.46 => ( Q2 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_bforall_def
% 0.30/0.46 thf(fact_203_unfold__llist__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( 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 ) ) ) )
% 0.30/0.46 @ ( coindu1441602521_llist @ A @ C )
% 0.30/0.46 @ ( coindu1441602521_llist @ B @ D ) ) ).
% 0.30/0.46
% 0.30/0.46 % unfold_llist_transfer
% 0.30/0.46 thf(fact_204_llist_Ocorec__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
% 0.30/0.46 ( 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 ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( 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 )
% 0.30/0.46 @ ( 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 ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( 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 ) ) ) ) ) )
% 0.30/0.46 @ ( coindu1259883913_llist @ C @ A )
% 0.30/0.46 @ ( coindu1259883913_llist @ D @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.corec_transfer
% 0.30/0.46 thf(fact_205_llist__corec__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( 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 ) )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( 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 )
% 0.30/0.46 @ ( 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 ) )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( 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 ) ) ) ) ) )
% 0.30/0.46 @ ( coindu1259883913_llist @ A @ C )
% 0.30/0.46 @ ( coindu1259883913_llist @ B @ D ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_corec_transfer
% 0.30/0.46 thf(fact_206_Ex__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ^ [P2: A > $o] :
% 0.30/0.46 ? [X6: A] : ( P2 @ X6 )
% 0.30/0.46 @ ^ [P2: B > $o] :
% 0.30/0.46 ? [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ex_transfer
% 0.30/0.46 thf(fact_207_All__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ^ [P2: A > $o] :
% 0.30/0.46 ! [X6: A] : ( P2 @ X6 )
% 0.30/0.46 @ ^ [P2: B > $o] :
% 0.30/0.46 ! [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % All_transfer
% 0.30/0.46 thf(fact_208_bi__total__alt__def2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ^ [P2: A > $o] :
% 0.30/0.46 ! [X6: A] : ( P2 @ X6 )
% 0.30/0.46 @ ^ [P2: B > $o] :
% 0.30/0.46 ! [X6: B] : ( P2 @ X6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_alt_def2
% 0.30/0.46 thf(fact_209_right__total__alt__def2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( right_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5 @ (=>) ) @ (=>)
% 0.30/0.46 @ ^ [P2: A > $o] :
% 0.30/0.46 ! [X6: A] : ( P2 @ X6 )
% 0.30/0.46 @ ^ [P2: B > $o] :
% 0.30/0.46 ! [X6: B] : ( P2 @ X6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_alt_def2
% 0.30/0.46 thf(fact_210_Bex__def__raw,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bex @ A )
% 0.30/0.46 = ( ^ [A6: set @ A,P4: A > $o] :
% 0.30/0.46 ? [X2: A] :
% 0.30/0.46 ( ( member @ A @ X2 @ A6 )
% 0.30/0.46 & ( P4 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Bex_def_raw
% 0.30/0.46 thf(fact_211_rel__pred__comp__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( rel_pred_comp @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o,P4: B > $o,X2: A] :
% 0.30/0.46 ? [Y4: B] :
% 0.30/0.46 ( ( R5 @ X2 @ Y4 )
% 0.30/0.46 & ( P4 @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_pred_comp_def
% 0.30/0.46 thf(fact_212_right__total__All__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ ( ball @ A @ ( collect @ A @ ( domainp @ A @ B @ A2 ) ) )
% 0.30/0.46 @ ^ [P2: B > $o] :
% 0.30/0.46 ! [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_All_transfer
% 0.30/0.46 thf(fact_213_ball__reg__right,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ R )
% 0.30/0.46 => ( ( P @ X3 )
% 0.30/0.46 => ( Q @ X3 ) ) )
% 0.30/0.46 => ( ! [X12: A] : ( P @ X12 )
% 0.30/0.46 => ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ R )
% 0.30/0.46 => ( Q @ X7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_reg_right
% 0.30/0.46 thf(fact_214_Ball__def__raw,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ball @ A )
% 0.30/0.46 = ( ^ [A6: set @ A,P4: A > $o] :
% 0.30/0.46 ! [X2: A] :
% 0.30/0.46 ( ( member @ A @ X2 @ A6 )
% 0.30/0.46 => ( P4 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ball_def_raw
% 0.30/0.46 thf(fact_215_Ball__comp__iff,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,A2: B > ( set @ C ),F2: C > $o,G2: A > B] :
% 0.30/0.46 ( ( comp @ B @ $o @ A
% 0.30/0.46 @ ^ [X2: B] :
% 0.30/0.46 ! [Y4: C] :
% 0.30/0.46 ( ( member @ C @ Y4 @ ( A2 @ X2 ) )
% 0.30/0.46 => ( F2 @ Y4 ) )
% 0.30/0.46 @ G2 )
% 0.30/0.46 = ( ^ [X2: A] :
% 0.30/0.46 ! [Y4: C] :
% 0.30/0.46 ( ( member @ C @ Y4 @ ( comp @ B @ ( set @ C ) @ A @ A2 @ G2 @ X2 ) )
% 0.30/0.46 => ( F2 @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ball_comp_iff
% 0.30/0.46 thf(fact_216_ball__reg,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ R )
% 0.30/0.46 => ( ( P @ X3 )
% 0.30/0.46 => ( Q @ X3 ) ) )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ R )
% 0.30/0.46 => ( P @ X3 ) )
% 0.30/0.46 => ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ R )
% 0.30/0.46 => ( Q @ X7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_reg
% 0.30/0.46 thf(fact_217_iterates__transfer,axiom,
% 0.30/0.46 ! [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 ) ) ).
% 0.30/0.46
% 0.30/0.46 % iterates_transfer
% 0.30/0.46 thf(fact_218_llist_Opred__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( coindu543516966_llist @ A )
% 0.30/0.46 @ ( coindu543516966_llist @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.pred_transfer
% 0.30/0.46 thf(fact_219_lfinite__iterates,axiom,
% 0.30/0.46 ! [A: $tType,F2: A > A,X: A] :
% 0.30/0.46 ~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F2 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % lfinite_iterates
% 0.30/0.46 thf(fact_220_llist_Opred__True,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( coindu543516966_llist @ A
% 0.30/0.46 @ ^ [Uu: A] : $true )
% 0.30/0.46 = ( ^ [Uu: coinductive_llist @ A] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.pred_True
% 0.30/0.46 thf(fact_221_llist_ODomainp__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( domainp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) )
% 0.30/0.46 = ( coindu543516966_llist @ A @ ( domainp @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.Domainp_rel
% 0.30/0.46 thf(fact_222_llist__all__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.30/0.46 @ ( coindu543516966_llist @ A )
% 0.30/0.46 @ ( coindu543516966_llist @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_all_transfer
% 0.30/0.46 thf(fact_223_stream__from__llist__setup_Ostream_Odomain__par,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,T2: C > B > $o,DR: C > $o,P22: ( coinductive_llist @ C ) > $o] :
% 0.30/0.46 ( ( ( domainp @ C @ B @ T2 )
% 0.30/0.46 = DR )
% 0.30/0.46 => ( ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ C @ B @ T2 )
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ P22
% 0.30/0.46 @ ^ [Xs2: coinductive_llist @ B] :
% 0.30/0.46 ~ ( coinductive_lfinite @ B @ Xs2 ) )
% 0.30/0.46 => ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
% 0.30/0.46 = ( inf_inf @ ( ( coinductive_llist @ C ) > $o ) @ ( coindu543516966_llist @ C @ DR ) @ P22 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream_from_llist_setup.stream.domain_par
% 0.30/0.46 thf(fact_224_case__llist__transfer,axiom,
% 0.30/0.46 ! [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 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_llist_transfer
% 0.30/0.46 thf(fact_225_llist__case__mono,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,Lnil: A > B,Lcons: A > C > ( coinductive_llist @ C ) > B,X: coinductive_llist @ C] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ Lnil )
% 0.30/0.46 => ( ! [X3: C,Xs3: coinductive_llist @ C] :
% 0.30/0.46 ( comple1396247847notone @ A @ B @ Orda @ Ordb
% 0.30/0.46 @ ^ [F: A] : ( Lcons @ F @ X3 @ Xs3 ) )
% 0.30/0.46 => ( comple1396247847notone @ A @ B @ Orda @ Ordb
% 0.30/0.46 @ ^ [F: A] : ( coindu1381640503_llist @ B @ C @ ( Lnil @ F ) @ ( Lcons @ F ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_case_mono
% 0.30/0.46 thf(fact_226_llist_Ocase__distrib,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 0.30/0.46 ( ( H2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ Llist ) )
% 0.30/0.46 = ( coindu1381640503_llist @ C @ A @ ( H2 @ F1 )
% 0.30/0.46 @ ^ [X14: A,X23: coinductive_llist @ A] : ( H2 @ ( F22 @ X14 @ X23 ) )
% 0.30/0.46 @ Llist ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.case_distrib
% 0.30/0.46 thf(fact_227_pcr__Domainp__par,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P23: A > $o,A2: C > A > $o,P1: C > $o,P22: C > $o] :
% 0.30/0.46 ( ( ( domainp @ A @ B @ B2 )
% 0.30/0.46 = P23 )
% 0.30/0.46 => ( ( ( domainp @ C @ A @ A2 )
% 0.30/0.46 = P1 )
% 0.30/0.46 => ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.30/0.46 @ P22
% 0.30/0.46 @ P23 )
% 0.30/0.46 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = ( inf_inf @ ( C > $o ) @ P1 @ P22 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pcr_Domainp_par
% 0.30/0.46 thf(fact_228_llist_Ocase__transfer,axiom,
% 0.30/0.46 ! [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 ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist.case_transfer
% 0.30/0.46 thf(fact_229_img__ord__def,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( partial_img_ord @ A @ C @ B )
% 0.30/0.46 = ( ^ [F: A > C,Ord2: C > C > B,X2: A,Y4: A] : ( Ord2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % img_ord_def
% 0.30/0.46 thf(fact_230_mk__less__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( partial_mk_less @ A )
% 0.30/0.46 = ( ^ [R5: A > A > $o,X2: A,Y4: A] :
% 0.30/0.46 ( ( R5 @ X2 @ Y4 )
% 0.30/0.46 & ~ ( R5 @ Y4 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mk_less_def
% 0.30/0.46 thf(fact_231_type__definition_ORep,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( member @ A @ ( Rep @ X ) @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Rep
% 0.30/0.46 thf(fact_232_type__definition__def,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( type_definition @ B @ A )
% 0.30/0.46 = ( ^ [Rep2: B > A,Abs2: A > B,A6: set @ A] :
% 0.30/0.46 ( ! [X2: B] : ( member @ A @ ( Rep2 @ X2 ) @ A6 )
% 0.30/0.46 & ! [X2: B] :
% 0.30/0.46 ( ( Abs2 @ ( Rep2 @ X2 ) )
% 0.30/0.46 = X2 )
% 0.30/0.46 & ! [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ A6 )
% 0.30/0.46 => ( ( Rep2 @ ( Abs2 @ Y4 ) )
% 0.30/0.46 = Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition_def
% 0.30/0.46 thf(fact_233_type__definition_ORep__inverse,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( Abs @ ( Rep @ X ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Rep_inverse
% 0.30/0.46 thf(fact_234_type__definition_OAbs__inverse,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( member @ A @ Y @ A2 )
% 0.30/0.46 => ( ( Rep @ ( Abs @ Y ) )
% 0.30/0.46 = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Abs_inverse
% 0.30/0.46 thf(fact_235_type__definition_ORep__inject,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B,Y: B] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( ( Rep @ X )
% 0.30/0.46 = ( Rep @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Rep_inject
% 0.30/0.46 thf(fact_236_type__definition_ORep__induct,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A,P: A > $o] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( member @ A @ Y @ A2 )
% 0.30/0.46 => ( ! [X3: B] : ( P @ ( Rep @ X3 ) )
% 0.30/0.46 => ( P @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Rep_induct
% 0.30/0.46 thf(fact_237_type__definition_OAbs__inject,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: A,Y: A] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( member @ A @ X @ A2 )
% 0.30/0.46 => ( ( member @ A @ Y @ A2 )
% 0.30/0.46 => ( ( ( Abs @ X )
% 0.30/0.46 = ( Abs @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Abs_inject
% 0.30/0.46 thf(fact_238_type__definition_OAbs__induct,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,P: B > $o,X: B] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ! [Y2: A] :
% 0.30/0.46 ( ( member @ A @ Y2 @ A2 )
% 0.30/0.46 => ( P @ ( Abs @ Y2 ) ) )
% 0.30/0.46 => ( P @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Abs_induct
% 0.30/0.46 thf(fact_239_type__definition_ORep__cases,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( member @ A @ Y @ A2 )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( Rep @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Rep_cases
% 0.30/0.46 thf(fact_240_type__definition_OAbs__cases,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ~ ! [Y2: A] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( Abs @ Y2 ) )
% 0.30/0.46 => ~ ( member @ A @ Y2 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Abs_cases
% 0.30/0.46 thf(fact_241_type__definition_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,A2: set @ A,Abs: A > B] :
% 0.30/0.46 ( ! [X3: B] : ( member @ A @ ( Rep @ X3 ) @ A2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( Abs @ ( Rep @ X3 ) )
% 0.30/0.46 = X3 )
% 0.30/0.46 => ( ! [Y2: A] :
% 0.30/0.46 ( ( member @ A @ Y2 @ A2 )
% 0.30/0.46 => ( ( Rep @ ( Abs @ Y2 ) )
% 0.30/0.46 = Y2 ) )
% 0.30/0.46 => ( type_definition @ B @ A @ Rep @ Abs @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.intro
% 0.30/0.46 thf(fact_242_llist__of__stream__siterates,axiom,
% 0.30/0.46 ! [A: $tType,F2: A > A,X: A] :
% 0.30/0.46 ( ( coindu1724414836stream @ A @ ( siterate @ A @ F2 @ X ) )
% 0.30/0.46 = ( coinductive_iterates @ A @ F2 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % llist_of_stream_siterates
% 0.30/0.46 thf(fact_243_nchotomy__relcomppE,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: B > A,R4: C > A > $o,S3: A > D > $o,A3: C,C3: D] :
% 0.30/0.46 ( ! [Y2: A] :
% 0.30/0.46 ? [X7: B] :
% 0.30/0.46 ( Y2
% 0.30/0.46 = ( F2 @ X7 ) )
% 0.30/0.46 => ( ( relcompp @ C @ A @ D @ R4 @ S3 @ A3 @ C3 )
% 0.30/0.46 => ~ ! [B5: B] :
% 0.30/0.46 ( ( R4 @ A3 @ ( F2 @ B5 ) )
% 0.30/0.46 => ~ ( S3 @ ( F2 @ B5 ) @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nchotomy_relcomppE
% 0.30/0.46 thf(fact_244_OO__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( relcompp @ A @ B @ B @ R
% 0.30/0.46 @ ^ [Y3: B,Z: B] : ( Y3 = Z ) )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % OO_eq
% 0.30/0.46 thf(fact_245_eq__OO,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( relcompp @ A @ A @ B
% 0.30/0.46 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.30/0.46 @ R )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % eq_OO
% 0.30/0.46 thf(fact_246_relcompp_OrelcompI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,A3: A,B3: B,S3: B > C > $o,C3: C] :
% 0.30/0.46 ( ( R4 @ A3 @ B3 )
% 0.30/0.46 => ( ( S3 @ B3 @ C3 )
% 0.30/0.46 => ( relcompp @ A @ B @ C @ R4 @ S3 @ A3 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.relcompI
% 0.30/0.46 thf(fact_247_relcompp_Oinducts,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,X1: A,X22: C,P: A > C > $o] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C @ R4 @ S3 @ X1 @ X22 )
% 0.30/0.46 => ( ! [A5: A,B5: B,C4: C] :
% 0.30/0.46 ( ( R4 @ A5 @ B5 )
% 0.30/0.46 => ( ( S3 @ B5 @ C4 )
% 0.30/0.46 => ( P @ A5 @ C4 ) ) )
% 0.30/0.46 => ( P @ X1 @ X22 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.inducts
% 0.30/0.46 thf(fact_248_relcompp__assoc,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType,B: $tType,C: $tType,R4: A > D > $o,S3: D > C > $o,T: C > B > $o] :
% 0.30/0.46 ( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R4 @ S3 ) @ T )
% 0.30/0.46 = ( relcompp @ A @ D @ B @ R4 @ ( relcompp @ D @ C @ B @ S3 @ T ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_assoc
% 0.30/0.46 thf(fact_249_relcompp__apply,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C )
% 0.30/0.46 = ( ^ [R5: A > B > $o,S5: B > C > $o,A4: A,C5: C] :
% 0.30/0.46 ? [B6: B] :
% 0.30/0.46 ( ( R5 @ A4 @ B6 )
% 0.30/0.46 & ( S5 @ B6 @ C5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_apply
% 0.30/0.46 thf(fact_250_relcompp_Osimps,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C )
% 0.30/0.46 = ( ^ [R6: A > B > $o,S2: B > C > $o,A1: A,A22: C] :
% 0.30/0.46 ? [A4: A,B6: B,C5: C] :
% 0.30/0.46 ( ( A1 = A4 )
% 0.30/0.46 & ( A22 = C5 )
% 0.30/0.46 & ( R6 @ A4 @ B6 )
% 0.30/0.46 & ( S2 @ B6 @ C5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.simps
% 0.30/0.46 thf(fact_251_relcompp_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C @ R4 @ S3 @ A12 @ A23 )
% 0.30/0.46 => ~ ! [B5: B] :
% 0.30/0.46 ( ( R4 @ A12 @ B5 )
% 0.30/0.46 => ~ ( S3 @ B5 @ A23 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.cases
% 0.30/0.46 thf(fact_252_relcomppE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,A3: A,C3: C] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C @ R4 @ S3 @ A3 @ C3 )
% 0.30/0.46 => ~ ! [B5: B] :
% 0.30/0.46 ( ( R4 @ A3 @ B5 )
% 0.30/0.46 => ~ ( S3 @ B5 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcomppE
% 0.30/0.46 thf(fact_253_Powp__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( powp @ A )
% 0.30/0.46 = ( ^ [A6: A > $o,B4: set @ A] :
% 0.30/0.46 ! [X2: A] :
% 0.30/0.46 ( ( member @ A @ X2 @ B4 )
% 0.30/0.46 => ( A6 @ X2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Powp_def
% 0.30/0.46 thf(fact_254_pos__fun__distr,axiom,
% 0.30/0.46 ! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F3: $tType,R: A > E > $o,S: B > F3 > $o,R8: E > C > $o,S6: F3 > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F3 ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ R @ S ) @ ( bNF_rel_fun @ E @ C @ F3 @ D @ R8 @ S6 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R @ R8 ) @ ( relcompp @ B @ F3 @ D @ S @ S6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pos_fun_distr
% 0.30/0.46 thf(fact_255_relcompp__mono,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R9: A > B > $o,R4: A > B > $o,S7: B > C > $o,S3: B > C > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ R9 @ R4 )
% 0.30/0.46 => ( ( ord_less_eq @ ( B > C > $o ) @ S7 @ S3 )
% 0.30/0.46 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R9 @ S7 ) @ ( relcompp @ A @ B @ C @ R4 @ S3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_mono
% 0.30/0.46
% 0.30/0.46 %----Helper facts (3)
% 0.30/0.46 thf(help_If_3_1_T,axiom,
% 0.30/0.46 ! [P: $o] :
% 0.30/0.46 ( ( P = $true )
% 0.30/0.46 | ( P = $false ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_2_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_1_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 %----Conjectures (2)
% 0.30/0.46 thf(conj_0,hypothesis,
% 0.30/0.46 ( bNF_rel_fun @ a @ a @ ( coinductive_llist @ b ) @ ( stream @ b )
% 0.30/0.46 @ ^ [Y3: a,Z: a] : ( Y3 = Z )
% 0.30/0.46 @ ^ [X2: coinductive_llist @ b,Y4: stream @ b] :
% 0.30/0.46 ( X2
% 0.30/0.46 = ( coindu1724414836stream @ b @ Y4 ) )
% 0.30/0.46 @ xb
% 0.30/0.46 @ yb ) ).
% 0.30/0.46
% 0.30/0.46 thf(conj_1,conjecture,
% 0.30/0.46 ( xb
% 0.30/0.46 = ( comp @ ( stream @ b ) @ ( coinductive_llist @ b ) @ a @ ( coindu1724414836stream @ b ) @ yb ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 0.31/1.80 ---- Embedded in THF ---
% 0.31/1.80 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.80 %%% Generated on Tue Feb 24 06:09:37 EST 2026
% 0.31/1.80 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.80 %%% Logic specification used:
% 0.31/1.80 %%% thf(spec, logic, $$dhol).
% 0.31/1.80
% 0.31/1.80 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.WjgrtYFCGm/DTF2DTF2534.p
% 0.31/1.80 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 0.31/1.80 thf(ty_t_Stream_Ostream, type, stream: ($tType > $tType)).
% 0.31/1.80 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.31/1.80 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 0.31/1.80 thf(ty_tf_b, type, b: $tType).
% 0.31/1.80 thf(ty_tf_a, type, a: $tType).
% 0.31/1.80 thf(sy_c_BNF__Def_Orel__fun, type, bNF_rel_fun: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > (C > $o)) > ((B > (D > $o)) > ((A > B) > ((C > D) > $o))))))).
% 0.31/1.80 thf(sy_c_Basic__BNFs_Opred__fun, type, basic_pred_fun: (!> [A:$tType,B:$tType]: (((A > $o) > ((B > $o) > ((A > B) > $o)))))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Oiterates, type, coinductive_iterates: (!> [A:$tType]: (((A > A) > (A > (coinductive_llist @ A)))))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Ollist_Ocase__llist, type, coindu1381640503_llist: (!> [B:$tType,A:$tType]: ((B > ((A > ((coinductive_llist @ A) > B)) > ((coinductive_llist @ A) > B)))))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist, type, coindu1259883913_llist: (!> [C:$tType,A:$tType]: (((C > $o) > ((C > A) > ((C > $o) > ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A)))))))))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Ollist_Ollist__all2, type, coindu1486289336t_all2: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((coinductive_llist @ A) > ((coinductive_llist @ B) > $o)))))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Ollist_Opred__llist, type, coindu543516966_llist: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > $o))))).
% 0.31/1.80 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 0.31/1.80 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream, type, coindu1724414836stream: (!> [A:$tType]: (((stream @ A) > (coinductive_llist @ A))))).
% 0.31/1.80 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream, type, coindu1183105481stream: (!> [A:$tType]: (((coinductive_llist @ A) > ((stream @ A) > $o))))).
% 0.31/1.80 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Opcr__stream, type, coindu773941317stream: (!> [C:$tType,B:$tType]: (((C > (B > $o)) > ((coinductive_llist @ C) > ((stream @ B) > $o)))))).
% 0.31/1.80 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist, type, coindu2010755910_llist: (!> [A:$tType]: (((coinductive_llist @ A) > (stream @ A))))).
% 0.31/1.80 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream, type, coindu139217191stream: (!> [A:$tType,B:$tType]: (((A > B) > ((A > A) > (A > (stream @ B))))))).
% 0.31/1.80 thf(sy_c_Complete__Partial__Order_Omonotone, type, comple1396247847notone: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > (B > $o)) > ((A > B) > $o)))))).
% 0.31/1.80 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 0.31/1.80 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))))))).
% 0.31/1.80 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 0.31/1.80 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 0.31/1.80 thf(sy_c_Lifting_Orel__pred__comp, type, rel_pred_comp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((B > $o) > (A > $o)))))).
% 0.31/1.80 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.31/1.80 thf(sy_c_Partial__Function_Ofun__ord, type, partial_fun_ord: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((C > A) > ((C > B) > $o)))))).
% 0.31/1.80 thf(sy_c_Partial__Function_Oimg__ord, type, partial_img_ord: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > ((C > (C > B)) > (A > (A > B))))))).
% 0.31/1.80 thf(sy_c_Partial__Function_Omk__less, type, partial_mk_less: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > $o)))))).
% 0.31/1.80 thf(sy_c_Pure_Otype, type, type: (!> [A:$tType]: ((itself @ A)))).
% 0.31/1.80 thf(sy_c_Quotient_OBex1__rel, type, bex1_rel: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > $o))))).
% 0.31/1.80 thf(sy_c_Quotient_OQuotient3, type, quotient3: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > $o)))))).
% 0.31/1.80 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 0.31/1.80 thf(sy_c_Relation_OPowp, type, powp: (!> [A:$tType]: (((A > $o) > ((set @ A) > $o))))).
% 0.31/1.80 thf(sy_c_Relation_Oreflp, type, reflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 0.31/1.80 thf(sy_c_Relation_Orelcompp, type, relcompp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (A > (C > $o))))))).
% 0.31/1.80 thf(sy_c_Set_OBall, type, ball: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 0.31/1.80 thf(sy_c_Set_OBex, type, bex: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 0.31/1.80 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.31/1.80 thf(sy_c_Stream_Ositerate, type, siterate: (!> [A:$tType]: (((A > A) > (A > (stream @ A)))))).
% 0.31/1.80 thf(sy_c_Stream_Ostream_Ocase__stream, type, case_stream: (!> [A:$tType,B:$tType]: (((A > ((stream @ A) > B)) > ((stream @ A) > B))))).
% 0.31/1.80 thf(sy_c_Stream_Ostream_Ocorec__stream, type, corec_stream: (!> [C:$tType,A:$tType]: (((C > A) > ((C > $o) > ((C > (stream @ A)) > ((C > C) > (C > (stream @ A))))))))).
% 0.31/1.80 thf(sy_c_Stream_Ostream_Opred__stream, type, pred_stream: (!> [A:$tType]: (((A > $o) > ((stream @ A) > $o))))).
% 0.31/1.80 thf(sy_c_Stream_Ostream_Ostream__all2, type, stream_all2: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((stream @ A) > ((stream @ B) > $o)))))).
% 0.31/1.80 thf(sy_c_Transfer_Obi__total, type, bi_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 0.31/1.80 thf(sy_c_Transfer_Oleft__total, type, left_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 0.31/1.80 thf(sy_c_Transfer_Orev__implies, type, rev_implies: ($o > ($o > $o))).
% 0.31/1.80 thf(sy_c_Transfer_Oright__total, type, right_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 0.31/1.80 thf(sy_c_Transfer_Otransfer__bforall, type, transfer_bforall: (!> [A:$tType]: (((A > $o) > ((A > $o) > $o))))).
% 0.31/1.80 thf(sy_c_Transfer_Otransfer__forall, type, transfer_forall: (!> [A:$tType]: (((A > $o) > $o)))).
% 0.31/1.80 thf(sy_c_Typedef_Otype__definition, type, type_definition: (!> [B:$tType,A:$tType]: (((B > A) > ((A > B) > ((set @ A) > $o)))))).
% 0.31/1.80 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.31/1.80 thf(sy_v_xb, type, xb: (a > (coinductive_llist @ b))).
% 0.31/1.80 thf(sy_v_yb, type, yb: (a > (stream @ b))).
% 0.31/1.80 thf(fact_0_stream_ORep__inject, axiom, (! [A:$tType,X:(stream @ A),Y:(stream @ A)]: (((((coindu1724414836stream @ A) @ X) = ((coindu1724414836stream @ A) @ Y)) = (X = Y))))).
% 0.31/1.80 thf(fact_1_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F:(B > A),G:(C > B),X2:C]: ((F @ (G @ X2)))))))).
% 0.31/1.80 thf(fact_2_rel__funI, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F2:(A > C),G2:(B > D)]: (((! [X3:A,Y2:B]: ((((A2 @ X3) @ Y2) => ((B2 @ (F2 @ X3)) @ (G2 @ Y2))))) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F2) @ G2))))).
% 0.31/1.80 thf(fact_3_fun_Orel__map_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,Sa:(A > (C > $o)),X:(D > A),G2:(B > C),Y:(D > B)]: ((((((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Sa) @ X) @ (((((comp @ B) @ C) @ D) @ G2) @ Y)) = ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ (^ [X2:A,Y4:B]: (((Sa @ X2) @ (G2 @ Y4))))) @ X) @ Y))))).
% 0.31/1.80 thf(fact_4_fun_Orel__map_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,Sb:(C > (B > $o)),I:(A > C),X:(D > A),Y:(D > B)]: ((((((((((bNF_rel_fun @ D) @ D) @ C) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Sb) @ (((((comp @ A) @ C) @ D) @ I) @ X)) @ Y) = ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ (^ [X2:A]: ((Sb @ (I @ X2))))) @ X) @ Y))))).
% 0.31/1.80 thf(fact_5_o__rsp_I2_J, axiom, (! [E:$tType,F3:$tType,H:$tType,G3:$tType,R1:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (G3 > H)) @ (G3 > H)) @ ((E > G3) > (E > H))) @ ((F3 > G3) > (F3 > H))) @ (^ [Y3:(G3 > H),Z:(G3 > H)]: ((Y3 = Z)))) @ ((((((bNF_rel_fun @ (E > G3)) @ (F3 > G3)) @ (E > H)) @ (F3 > H)) @ ((((((bNF_rel_fun @ E) @ F3) @ G3) @ G3) @ R1) @ (^ [Y3:G3,Z:G3]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ E) @ F3) @ H) @ H) @ R1) @ (^ [Y3:H,Z:H]: ((Y3 = Z)))))) @ (((comp @ G3) @ H) @ E)) @ (((comp @ G3) @ H) @ F3))))).
% 0.31/1.80 thf(fact_6_o__rsp_I1_J, axiom, (! [A:$tType,B:$tType,E:$tType,F3:$tType,D:$tType,C:$tType,R2:(A > (C > $o)),R3:(B > (D > $o)),R1:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > A) > (E > B))) @ ((F3 > C) > (F3 > D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ R2) @ R3)) @ ((((((bNF_rel_fun @ (E > A)) @ (F3 > C)) @ (E > B)) @ (F3 > D)) @ ((((((bNF_rel_fun @ E) @ F3) @ A) @ C) @ R1) @ R2)) @ ((((((bNF_rel_fun @ E) @ F3) @ B) @ D) @ R1) @ R3))) @ (((comp @ A) @ B) @ E)) @ (((comp @ C) @ D) @ F3))))).
% 0.31/1.80 thf(fact_7_fun_Omap__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,G3:$tType,F3:$tType,Rb:(A > (F3 > $o)),Sd:(B > (G3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (F3 > G3)) @ ((D > A) > (D > B))) @ ((D > F3) > (D > G3))) @ ((((((bNF_rel_fun @ A) @ F3) @ B) @ G3) @ Rb) @ Sd)) @ ((((((bNF_rel_fun @ (D > A)) @ (D > F3)) @ (D > B)) @ (D > G3)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ F3) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Rb)) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ G3) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Sd))) @ (((comp @ A) @ B) @ D)) @ (((comp @ F3) @ G3) @ D))))).
% 0.31/1.80 thf(fact_8_comp__transfer, axiom, (! [A:$tType,B:$tType,E:$tType,F3:$tType,D:$tType,C:$tType,B2:(A > (C > $o)),C2:(B > (D > $o)),A2:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > A) > (E > B))) @ ((F3 > C) > (F3 > D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ B2) @ C2)) @ ((((((bNF_rel_fun @ (E > A)) @ (F3 > C)) @ (E > B)) @ (F3 > D)) @ ((((((bNF_rel_fun @ E) @ F3) @ A) @ C) @ A2) @ B2)) @ ((((((bNF_rel_fun @ E) @ F3) @ B) @ D) @ A2) @ C2))) @ (((comp @ A) @ B) @ E)) @ (((comp @ C) @ D) @ F3))))).
% 0.31/1.80 thf(fact_9_K__record__comp, axiom, (! [C:$tType,B:$tType,A:$tType,C3:B,F2:(A > C)]: (((((((comp @ C) @ B) @ A) @ (^ [X2:C]: (C3))) @ F2) = (^ [X2:A]: (C3)))))).
% 0.31/1.80 thf(fact_10_fun_Omap__ident, axiom, (! [A:$tType,D:$tType,T:(D > A)]: (((((((comp @ A) @ A) @ D) @ (^ [X2:A]: (X2))) @ T) = T)))).
% 0.31/1.80 thf(fact_11_stream__from__llist__setup_Ocr__stream__def, axiom, (! [A:$tType]: (((coindu1183105481stream @ A) = (^ [X2:(coinductive_llist @ A),Y4:(stream @ A)]: ((X2 = ((coindu1724414836stream @ A) @ Y4)))))))).
% 0.31/1.80 thf(fact_12_llist__of__stream__corec__stream, axiom, (! [A:$tType,B:$tType,SHD:(B > A),EndORmore:(B > $o),STL_more:(B > (stream @ A)),STL_end:(B > B),X:B]: ((((coindu1724414836stream @ A) @ (((((((corec_stream @ B) @ A) @ SHD) @ EndORmore) @ STL_more) @ STL_end) @ X)) = ((((((((coindu1259883913_llist @ B) @ A) @ (^ [Uu:B]: ($false))) @ SHD) @ EndORmore) @ (((((comp @ (stream @ A)) @ (coinductive_llist @ A)) @ B) @ (coindu1724414836stream @ A)) @ STL_more)) @ STL_end) @ X))))).
% 0.31/1.80 thf(fact_13_rel__fun__def__butlast, axiom, (! [B:$tType,D:$tType,C:$tType,E:$tType,F3:$tType,A:$tType,R:(A > (B > $o)),S:(C > (E > $o)),T2:(D > (F3 > $o)),F2:(A > (C > D)),G2:(B > (E > F3))]: ((((((((((bNF_rel_fun @ A) @ B) @ (C > D)) @ (E > F3)) @ R) @ ((((((bNF_rel_fun @ C) @ E) @ D) @ F3) @ S) @ T2)) @ F2) @ G2) = (! [X2:A,Y4:B]: ((((R @ X2) @ Y4) => ((((((((bNF_rel_fun @ C) @ E) @ D) @ F3) @ S) @ T2) @ (F2 @ X2)) @ (G2 @ Y4))))))))).
% 0.31/1.80 thf(fact_14_If__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ $o) @ $o) @ (A > (A > A))) @ (B > (B > B))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2))) @ (if @ A)) @ (if @ B))))).
% 0.31/1.80 thf(fact_15_fun_Orel__transfer, axiom, (! [B:$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) @ $o) @ Sc) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (D > A)) @ (D > C)) @ ((D > B) > $o)) @ ((D > E) > $o)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Sa)) @ ((((((bNF_rel_fun @ (D > B)) @ (D > E)) @ $o) @ $o) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ E) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Sc)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ (((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z))))) @ (((((bNF_rel_fun @ D) @ D) @ C) @ E) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))))))).
% 0.31/1.80 thf(fact_16_fun_Orel__refl, axiom, (! [B:$tType,D:$tType,Ra:(B > (B > $o)),X:(D > B)]: (((! [X3:B]: (((Ra @ X3) @ X3))) => ((((((((bNF_rel_fun @ D) @ D) @ B) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ Ra) @ X) @ X))))).
% 0.31/1.80 thf(fact_17_fun_Orel__eq, axiom, (! [A:$tType,D:$tType]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ A) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) = (^ [Y3:(D > A),Z:(D > A)]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_18_rel__fun__mono_H, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,Y5:(A > (B > $o)),X4:(A > (B > $o)),A2:(C > (D > $o)),B2:(C > (D > $o)),F2:(A > C),G2:(B > D)]: (((! [X3:A,Y2:B]: ((((Y5 @ X3) @ Y2) => ((X4 @ X3) @ Y2)))) => ((! [X3:C,Y2:D]: ((((A2 @ X3) @ Y2) => ((B2 @ X3) @ Y2)))) => (((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ X4) @ A2) @ F2) @ G2) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ Y5) @ B2) @ F2) @ G2))))))).
% 0.31/1.80 thf(fact_19_rel__fun__mono, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,X4:(A > (B > $o)),A2:(C > (D > $o)),F2:(A > C),G2:(B > D),Y5:(A > (B > $o)),B2:(C > (D > $o))]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ X4) @ A2) @ F2) @ G2) => ((! [X3:A,Y2:B]: ((((Y5 @ X3) @ Y2) => ((X4 @ X3) @ Y2)))) => ((! [X3:C,Y2:D]: ((((A2 @ X3) @ Y2) => ((B2 @ X3) @ Y2)))) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ Y5) @ B2) @ F2) @ G2))))))).
% 0.31/1.80 thf(fact_20_let__rsp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (B > $o)),R2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D)) @ R1) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ C) @ D) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R1) @ R2)) @ R2)) @ (^ [S2:A,F:(A > C)]: ((F @ S2)))) @ (^ [S2:B,F:(B > D)]: ((F @ S2))))))).
% 0.31/1.80 thf(fact_21_rel__funD, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F2:(A > C),G2:(B > D),X:A,Y:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F2) @ G2) => (((A2 @ X) @ Y) => ((B2 @ (F2 @ X)) @ (G2 @ Y))))))).
% 0.31/1.80 thf(fact_22_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G2:(C > B),H2:(A > C),R12:(D > B),R22:(A > D),F2:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G2) @ H2) = (((((comp @ D) @ B) @ A) @ R12) @ R22)) => (((((((comp @ B) @ E) @ D) @ F2) @ R12) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F2) @ G2)) @ H2) = (((((comp @ D) @ E) @ A) @ L) @ R22))))))).
% 0.31/1.80 thf(fact_23_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F2:(C > B),G2:(A > C),L1:(D > B),L2:(A > D),H2:(E > A),R4:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F2) @ G2) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H2) = R4) => ((((((comp @ C) @ B) @ E) @ F2) @ (((((comp @ A) @ C) @ E) @ G2) @ H2)) = (((((comp @ D) @ B) @ E) @ L1) @ R4))))))).
% 0.31/1.80 thf(fact_24_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G2:(C > B),H2:(A > C),R4:(A > B),F2:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G2) @ H2) = R4) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F2) @ G2)) @ H2) = (((((comp @ B) @ D) @ A) @ F2) @ R4)))))).
% 0.31/1.80 thf(fact_25_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F2:(C > B),G2:(A > C),L:(A > B),H2:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F2) @ G2) = L) => ((((((comp @ C) @ B) @ D) @ F2) @ (((((comp @ A) @ C) @ D) @ G2) @ H2)) = (((((comp @ A) @ B) @ D) @ L) @ H2)))))).
% 0.31/1.80 thf(fact_26_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G2:(B > C),F2:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G2) @ (((((comp @ A) @ B) @ D) @ F2) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G2) @ F2)) @ V))))).
% 0.31/1.80 thf(fact_27_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F2:(B > A),G2:(C > B),X:C,H2:(D > A),K:(C > D)]: ((((F2 @ (G2 @ X)) = (H2 @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F2) @ G2) @ X) = ((((((comp @ D) @ A) @ C) @ H2) @ K) @ X)))))).
% 0.31/1.80 thf(fact_28_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A3:(C > B),B3:(A > C),C3:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = C3) => ((A3 @ (B3 @ V)) = (C3 @ V)))))).
% 0.31/1.80 thf(fact_29_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B3:(A > C),C3:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = (((((comp @ D) @ B) @ A) @ C3) @ D2)) => (! [V2:A]: (((A3 @ (B3 @ V2)) = (C3 @ (D2 @ V2))))))))).
% 0.31/1.80 thf(fact_30_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B3:(A > C),C3:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = (((((comp @ D) @ B) @ A) @ C3) @ D2)) => ((A3 @ (B3 @ V)) = (C3 @ (D2 @ V))))))).
% 0.31/1.80 thf(fact_31_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F2:(D > B),G2:(C > D),H2:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F2) @ G2)) @ H2) = (((((comp @ D) @ B) @ A) @ F2) @ (((((comp @ C) @ D) @ A) @ G2) @ H2)))))).
% 0.31/1.80 thf(fact_32_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F:(B > C),G:(A > B),X2:A]: ((F @ (G @ X2)))))))).
% 0.31/1.80 thf(fact_33_Let__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D)) @ A2) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ C) @ D) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ B2)) @ (^ [S2:A,F:(A > C)]: ((F @ S2)))) @ (^ [S2:B,F:(B > D)]: ((F @ S2))))))).
% 0.31/1.80 thf(fact_34_stream_Ocorec__disc, axiom, (! [A:$tType,C:$tType]: ((((corec_stream @ C) @ A) = ((corec_stream @ C) @ A))))).
% 0.31/1.80 thf(fact_35_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M:(B > A),G2:(C > B),X:C,N:(D > A),H2:(C > D),F2:(A > E)]: ((((M @ (G2 @ X)) = (N @ (H2 @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F2) @ M)) @ G2) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F2) @ N)) @ H2) @ X)))))).
% 0.31/1.80 thf(fact_36_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F2:(B > A),G2:(C > B),X:C,F4:(D > A),G4:(E > D),X5:E]: ((((F2 @ (G2 @ X)) = (F4 @ (G4 @ X5))) => (((((((comp @ B) @ A) @ C) @ F2) @ G2) @ X) = ((((((comp @ D) @ A) @ E) @ F4) @ G4) @ X5)))))).
% 0.31/1.80 thf(fact_37_rel__funD2, axiom, (! [B:$tType,C:$tType,A:$tType,A2:(A > (A > $o)),B2:(B > (C > $o)),F2:(A > B),G2:(A > C),X:A]: ((((((((((bNF_rel_fun @ A) @ A) @ B) @ C) @ A2) @ B2) @ F2) @ G2) => (((A2 @ X) @ X) => ((B2 @ (F2 @ X)) @ (G2 @ X))))))).
% 0.31/1.80 thf(fact_38_apply__rsp_H, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (B > $o)),R2:(C > (D > $o)),F2:(A > C),G2:(B > D),X:A,Y:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R1) @ R2) @ F2) @ G2) => (((R1 @ X) @ Y) => ((R2 @ (F2 @ X)) @ (G2 @ Y))))))).
% 0.31/1.80 thf(fact_39_rel__funE, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F2:(A > C),G2:(B > D),X:A,Y:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F2) @ G2) => (((A2 @ X) @ Y) => ((B2 @ (F2 @ X)) @ (G2 @ Y))))))).
% 0.31/1.80 thf(fact_40_stream__from__llist__setup_Ostream_Opcr__cr__eq, axiom, (! [D:$tType]: (((((coindu773941317stream @ D) @ D) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) = (coindu1183105481stream @ D))))).
% 0.31/1.80 thf(fact_41_stream__of__llist__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((((coindu2010755910_llist @ A) @ ((coindu1724414836stream @ A) @ Xs)) = Xs)))).
% 0.31/1.80 thf(fact_42_stream_ORep__inverse, axiom, (! [A:$tType,X:(stream @ A)]: ((((coindu2010755910_llist @ A) @ ((coindu1724414836stream @ A) @ X)) = X)))).
% 0.31/1.80 thf(fact_43_map__fun__parametric, axiom, (! [A:$tType,B:$tType,E:$tType,F3:$tType,H:$tType,G3:$tType,D:$tType,C:$tType,A2:(A > (C > $o)),B2:(B > (D > $o)),C2:(E > (G3 > $o)),D3:(F3 > (H > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > F3) > ((B > E) > (A > F3)))) @ ((G3 > H) > ((D > G3) > (C > H)))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (E > F3)) @ (G3 > H)) @ ((B > E) > (A > F3))) @ ((D > G3) > (C > H))) @ ((((((bNF_rel_fun @ E) @ G3) @ F3) @ H) @ C2) @ D3)) @ ((((((bNF_rel_fun @ (B > E)) @ (D > G3)) @ (A > F3)) @ (C > H)) @ ((((((bNF_rel_fun @ B) @ D) @ E) @ G3) @ B2) @ C2)) @ ((((((bNF_rel_fun @ A) @ C) @ F3) @ H) @ A2) @ D3)))) @ ((((map_fun @ A) @ B) @ E) @ F3)) @ ((((map_fun @ C) @ D) @ G3) @ H))))).
% 0.31/1.80 thf(fact_44_map__fun__apply, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType]: ((((((map_fun @ B) @ C) @ D) @ A) = (^ [F:(B > C),G:(D > A),H3:(C > D),X2:B]: ((G @ (H3 @ (F @ X2))))))))).
% 0.31/1.80 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 0.31/1.80 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 0.31/1.80 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.31/1.80 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X3:A]: (((F2 @ X3) = (G2 @ X3)))) => (F2 = G2))))).
% 0.31/1.80 thf(fact_49_map__fun__def, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType]: ((((((map_fun @ C) @ A) @ B) @ D) = (^ [F:(C > A),G:(B > D),H3:(A > B)]: ((((((comp @ A) @ D) @ C) @ (((((comp @ B) @ D) @ A) @ G) @ H3)) @ F))))))).
% 0.31/1.80 thf(fact_50_map__fun_Ocomp, axiom, (! [E:$tType,C:$tType,A:$tType,F3:$tType,D:$tType,B:$tType,F2:(E > C),G2:(D > F3),H2:(C > A),I:(B > D)]: (((((((comp @ (C > D)) @ (E > F3)) @ (A > B)) @ ((((((map_fun @ E) @ C) @ D) @ F3) @ F2) @ G2)) @ ((((((map_fun @ C) @ A) @ B) @ D) @ H2) @ I)) = ((((((map_fun @ E) @ A) @ B) @ F3) @ (((((comp @ C) @ A) @ E) @ H2) @ F2)) @ (((((comp @ D) @ F3) @ B) @ G2) @ I)))))).
% 0.31/1.80 thf(fact_51_map__fun_Ocompositionality, axiom, (! [D:$tType,F3:$tType,C:$tType,E:$tType,B:$tType,A:$tType,F2:(E > C),G2:(D > F3),H2:(C > A),I:(B > D),Fun:(A > B)]: (((((((((map_fun @ E) @ C) @ D) @ F3) @ F2) @ G2) @ (((((((map_fun @ C) @ A) @ B) @ D) @ H2) @ I) @ Fun)) = (((((((map_fun @ E) @ A) @ B) @ F3) @ (((((comp @ C) @ A) @ E) @ H2) @ F2)) @ (((((comp @ D) @ F3) @ B) @ G2) @ I)) @ Fun))))).
% 0.31/1.80 thf(fact_52_fun_Opred__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((D > A) > $o)) @ ((D > B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (D > A)) @ (D > B)) @ $o) @ $o) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ R)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true)))) @ (((basic_pred_fun @ D) @ B) @ (^ [Uu:D]: ($true))))))).
% 0.31/1.80 thf(fact_53_if__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((((((bNF_rel_fun @ $o) @ $o) @ (A > (A > A))) @ (A > (A > A))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((((((bNF_rel_fun @ A) @ A) @ (A > A)) @ (A > A)) @ R) @ ((((((bNF_rel_fun @ A) @ A) @ A) @ A) @ R) @ R))) @ (if @ A)) @ (if @ A)))))).
% 0.31/1.80 thf(fact_54_stream_Ocorec__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,S:(C > (D > $o)),R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (C > A)) @ (D > B)) @ ((C > $o) > ((C > (stream @ A)) > ((C > C) > (C > (stream @ A)))))) @ ((D > $o) > ((D > (stream @ B)) > ((D > D) > (D > (stream @ B)))))) @ ((((((bNF_rel_fun @ C) @ D) @ A) @ B) @ S) @ R)) @ ((((((bNF_rel_fun @ (C > $o)) @ (D > $o)) @ ((C > (stream @ A)) > ((C > C) > (C > (stream @ A))))) @ ((D > (stream @ B)) > ((D > D) > (D > (stream @ B))))) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ S) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (C > (stream @ A))) @ (D > (stream @ B))) @ ((C > C) > (C > (stream @ A)))) @ ((D > D) > (D > (stream @ B)))) @ ((((((bNF_rel_fun @ C) @ D) @ (stream @ A)) @ (stream @ B)) @ S) @ (((stream_all2 @ A) @ B) @ R))) @ ((((((bNF_rel_fun @ (C > C)) @ (D > D)) @ (C > (stream @ A))) @ (D > (stream @ B))) @ ((((((bNF_rel_fun @ C) @ D) @ C) @ D) @ S) @ S)) @ ((((((bNF_rel_fun @ C) @ D) @ (stream @ A)) @ (stream @ B)) @ S) @ (((stream_all2 @ A) @ B) @ R)))))) @ ((corec_stream @ C) @ A)) @ ((corec_stream @ D) @ B))))).
% 0.31/1.80 thf(fact_55_unfold__stream__transfer, axiom, (! [B:$tType,A:$tType]: (((((((((bNF_rel_fun @ (A > B)) @ (A > B)) @ ((A > A) > (A > (coinductive_llist @ B)))) @ ((A > A) > (A > (stream @ B)))) @ (^ [Y3:(A > B),Z:(A > B)]: ((Y3 = Z)))) @ ((((((bNF_rel_fun @ (A > A)) @ (A > A)) @ (A > (coinductive_llist @ B))) @ (A > (stream @ B))) @ (^ [Y3:(A > A),Z:(A > A)]: ((Y3 = Z)))) @ ((((((bNF_rel_fun @ A) @ A) @ (coinductive_llist @ B)) @ (stream @ B)) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ (((coindu773941317stream @ B) @ B) @ (^ [Y3:B,Z:B]: ((Y3 = Z))))))) @ (((coindu1441602521_llist @ A) @ B) @ (^ [Uu:A]: ($false)))) @ ((coindu139217191stream @ A) @ B))))).
% 0.31/1.80 thf(fact_56_fun__ord__parametric, axiom, (! [C:$tType,D:$tType,A:$tType,B:$tType,F3:$tType,E:$tType,C2:(A > (B > $o)),A2:(C > (E > $o)),B2:(D > (F3 > $o))]: (((((bi_total @ A) @ B) @ C2) => ((((((((bNF_rel_fun @ (C > (D > $o))) @ (E > (F3 > $o))) @ ((A > C) > ((A > D) > $o))) @ ((B > E) > ((B > F3) > $o))) @ ((((((bNF_rel_fun @ C) @ E) @ (D > $o)) @ (F3 > $o)) @ A2) @ ((((((bNF_rel_fun @ D) @ F3) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > E)) @ ((A > D) > $o)) @ ((B > F3) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ E) @ C2) @ A2)) @ ((((((bNF_rel_fun @ (A > D)) @ (B > F3)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ D) @ F3) @ C2) @ B2)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ (((partial_fun_ord @ C) @ D) @ A)) @ (((partial_fun_ord @ E) @ F3) @ B)))))).
% 0.31/1.80 thf(fact_57_llist__of__stream__stream__of__llist, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu1724414836stream @ A) @ ((coindu2010755910_llist @ A) @ Xs)) = Xs))))).
% 0.31/1.80 thf(fact_58_cr__streamI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coindu1183105481stream @ A) @ Xs) @ ((coindu2010755910_llist @ A) @ Xs)))))).
% 0.31/1.80 thf(fact_59_stream_OAbs__inverse, axiom, (! [A:$tType,Y:(coinductive_llist @ A)]: (((((member @ (coinductive_llist @ A)) @ Y) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))) => (((coindu1724414836stream @ A) @ ((coindu2010755910_llist @ A) @ Y)) = Y))))).
% 0.31/1.80 thf(fact_60_llist__of__stream__unfold__stream, axiom, (! [A:$tType,B:$tType,SHD:(B > A),STL:(B > B),X:B]: ((((coindu1724414836stream @ A) @ (((((coindu139217191stream @ B) @ A) @ SHD) @ STL) @ X)) = ((((((coindu1441602521_llist @ B) @ A) @ (^ [Uu:B]: ($false))) @ SHD) @ STL) @ X))))).
% 0.31/1.80 thf(fact_61_stream_Odisc__transfer, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (stream @ A)) @ (stream @ B)) @ $o) @ $o) @ (((stream_all2 @ A) @ B) @ R)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (^ [Stream:(stream @ A)]: ((Stream = Stream)))) @ (^ [Stream:(stream @ B)]: ((Stream = Stream))))))).
% 0.31/1.80 thf(fact_62_stream_Orel__refl, axiom, (! [B:$tType,Ra:(B > (B > $o)),X:(stream @ B)]: (((! [X3:B]: (((Ra @ X3) @ X3))) => (((((stream_all2 @ B) @ B) @ Ra) @ X) @ X))))).
% 0.31/1.80 thf(fact_63_stream_Orel__eq, axiom, (! [A:$tType]: (((((stream_all2 @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) = (^ [Y3:(stream @ A),Z:(stream @ A)]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_64_stream_Obi__total__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((bi_total @ A) @ B) @ R) => (((bi_total @ (stream @ A)) @ (stream @ B)) @ (((stream_all2 @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_65_bi__total__def, axiom, (! [B:$tType,A:$tType]: ((((bi_total @ A) @ B) = (^ [R5:(A > (B > $o))]: (((! [X2:A]: (((^ [P2:(B > $o)]: ((? [X6:B]: ((P2 @ X6))))) @ (R5 @ X2)))) & (! [Y4:B]: ((? [X2:A]: (((R5 @ X2) @ Y4)))))))))))).
% 0.31/1.80 thf(fact_66_bi__total__eq, axiom, (! [A:$tType]: ((((bi_total @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_67_Quotient3__rep__reflp, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((R @ (Rep @ A3)) @ (Rep @ A3)))))).
% 0.31/1.80 thf(fact_68_Quotient3__rep__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ R4) => ((R @ (Rep @ (Abs @ R4))) @ R4)))))).
% 0.31/1.80 thf(fact_69_Quotient3__rel__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A3:B,B3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ (Rep @ A3)) @ (Rep @ B3)) = (A3 = B3)))))).
% 0.31/1.80 thf(fact_70_Quotient3__rel__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ S3) => ((Abs @ R4) = (Abs @ S3))))))).
% 0.31/1.80 thf(fact_71_Quotient3__abs__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((Abs @ (Rep @ A3)) = A3))))).
% 0.31/1.80 thf(fact_72_rep__abs__rsp__left, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ (Rep @ (Abs @ X1))) @ X22)))))).
% 0.31/1.80 thf(fact_73_Quotient3__refl2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ S3) => ((R @ S3) @ S3)))))).
% 0.31/1.80 thf(fact_74_Quotient3__refl1, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ S3) => ((R @ R4) @ R4)))))).
% 0.31/1.80 thf(fact_75_Quotient3__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((R @ R4) @ R4) & (((R @ S3) @ S3) & ((Abs @ R4) = (Abs @ S3)))) = ((R @ R4) @ S3)))))).
% 0.31/1.80 thf(fact_76_Quotient3__def, axiom, (! [B:$tType,A:$tType]: ((((quotient3 @ A) @ B) = (^ [R5:(A > (A > $o)),Abs2:(A > B),Rep2:(B > A)]: (((! [A4:B]: (((Abs2 @ (Rep2 @ A4)) = A4))) & ((! [A4:B]: (((R5 @ (Rep2 @ A4)) @ (Rep2 @ A4)))) & (! [R6:A,S2:A]: ((((R5 @ R6) @ S2) = (((R5 @ R6) @ R6) & (((R5 @ S2) @ S2) & ((Abs2 @ R6) = (Abs2 @ S2))))))))))))))).
% 0.31/1.80 thf(fact_77_rep__abs__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ X1) @ (Rep @ (Abs @ X22)))))))).
% 0.31/1.80 thf(fact_78_pred__funI, axiom, (! [B:$tType,A:$tType,A2:(A > $o),B2:(B > $o),F2:(A > B)]: (((! [X3:A]: (((A2 @ X3) => (B2 @ (F2 @ X3))))) => (((((basic_pred_fun @ A) @ B) @ A2) @ B2) @ F2))))).
% 0.31/1.80 thf(fact_79_equals__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),Xa:A,Xb:A,Ya:A,Yb:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ Xa) @ Xb) => (((R @ Ya) @ Yb) => (((R @ Xa) @ Ya) = ((R @ Xb) @ Yb)))))))).
% 0.31/1.80 thf(fact_80_Quotient3I, axiom, (! [B:$tType,A:$tType,Abs:(B > A),Rep:(A > B),R:(B > (B > $o))]: (((! [A5:A]: (((Abs @ (Rep @ A5)) = A5))) => ((! [A5:A]: (((R @ (Rep @ A5)) @ (Rep @ A5)))) => ((! [R7:B,S4:B]: ((((R @ R7) @ S4) = (((R @ R7) @ R7) & (((R @ S4) @ S4) & ((Abs @ R7) = (Abs @ S4))))))) => (((((quotient3 @ B) @ A) @ R) @ Abs) @ Rep))))))).
% 0.31/1.80 thf(fact_81_cond__prs, axiom, (! [A:$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))))))).
% 0.31/1.80 thf(fact_82_fun_Opred__True, axiom, (! [A:$tType,D:$tType]: ((((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (^ [Uu:A]: ($true))) = (^ [Uu:(D > A)]: ($true)))))).
% 0.31/1.80 thf(fact_83_fun__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R2:(C > (C > $o)),Abs22:(C > D),Rep22:(D > C)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => ((((((quotient3 @ C) @ D) @ R2) @ Abs22) @ Rep22) => (((((quotient3 @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R1) @ R2)) @ ((((((map_fun @ B) @ A) @ C) @ D) @ Rep1) @ Abs22)) @ ((((((map_fun @ A) @ B) @ D) @ C) @ Abs1) @ Rep22))))))).
% 0.31/1.80 thf(fact_84_apply__rspQ3_H_H, axiom, (! [C:$tType,A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),S:(C > (C > $o)),F2:(A > C),X:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R) @ S) @ F2) @ F2) => ((S @ (F2 @ (Rep @ X))) @ (F2 @ (Rep @ X)))))))).
% 0.31/1.80 thf(fact_85_apply__rspQ3, axiom, (! [B:$tType,C:$tType,A:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),F2:(A > C),G2:(A > C),X:A,Y:A]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => (((((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R1) @ R2) @ F2) @ G2) => (((R1 @ X) @ Y) => ((R2 @ (F2 @ X)) @ (G2 @ Y)))))))).
% 0.31/1.80 thf(fact_86_let__prs, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),Abs23:(C > D),Rep23:(D > C)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ C) @ D) @ R2) @ Abs23) @ Rep23) => ((((((((map_fun @ D) @ C) @ ((C > A) > A)) @ ((D > B) > B)) @ Rep23) @ ((((((map_fun @ (D > B)) @ (C > A)) @ A) @ B) @ ((((((map_fun @ C) @ D) @ B) @ A) @ Abs23) @ Rep12)) @ Abs12)) @ (^ [S2:C,F:(C > A)]: ((F @ S2)))) = (^ [S2:D,F:(D > B)]: ((F @ S2))))))))).
% 0.31/1.80 thf(fact_87_lfinite__llist__of__stream, axiom, (! [A:$tType,Xs:(stream @ A)]: ((~ (((coinductive_lfinite @ A) @ ((coindu1724414836stream @ A) @ Xs))))))).
% 0.31/1.80 thf(fact_88_lambda__prs1, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),Abs23:(C > D),Rep23:(D > C),F2:(B > D)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ C) @ D) @ R2) @ Abs23) @ Rep23) => ((((((((map_fun @ B) @ A) @ C) @ D) @ Rep12) @ Abs23) @ (((((((map_fun @ A) @ B) @ D) @ C) @ Abs12) @ Rep23) @ F2)) = F2)))))).
% 0.31/1.80 thf(fact_89_lambda__prs, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),Abs23:(C > D),Rep23:(D > C),F2:(B > D)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ C) @ D) @ R2) @ Abs23) @ Rep23) => ((((((((map_fun @ B) @ A) @ C) @ D) @ Rep12) @ Abs23) @ (^ [X2:A]: ((Rep23 @ (F2 @ (Abs12 @ X2)))))) = F2)))))).
% 0.31/1.80 thf(fact_90_stream_ORep__induct, axiom, (! [A:$tType,Y:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ (coinductive_llist @ A)) @ Y) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))) => ((! [X3:(stream @ A)]: ((P @ ((coindu1724414836stream @ A) @ X3)))) => (P @ Y)))))).
% 0.31/1.80 thf(fact_91_stream_ORep__cases, axiom, (! [A:$tType,Y:(coinductive_llist @ A)]: (((((member @ (coinductive_llist @ A)) @ Y) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))) => (~ ((! [X3:(stream @ A)]: ((Y != ((coindu1724414836stream @ A) @ X3)))))))))).
% 0.31/1.80 thf(fact_92_stream_ORep, axiom, (! [A:$tType,X:(stream @ A)]: ((((member @ (coinductive_llist @ A)) @ ((coindu1724414836stream @ A) @ X)) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2)))))))))).
% 0.31/1.80 thf(fact_93_stream_OAbs__cases, axiom, (! [A:$tType,X:(stream @ A)]: ((~ ((! [Y2:(coinductive_llist @ A)]: (((X = ((coindu2010755910_llist @ A) @ Y2)) => (~ ((((member @ (coinductive_llist @ A)) @ Y2) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))))))))))))).
% 0.31/1.80 thf(fact_94_stream_OAbs__induct, axiom, (! [A:$tType,P:((stream @ A) > $o),X:(stream @ A)]: (((! [Y2:(coinductive_llist @ A)]: (((((member @ (coinductive_llist @ A)) @ Y2) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))) => (P @ ((coindu2010755910_llist @ A) @ Y2))))) => (P @ X))))).
% 0.31/1.80 thf(fact_95_stream_OAbs__inject, axiom, (! [A:$tType,X:(coinductive_llist @ A),Y:(coinductive_llist @ A)]: (((((member @ (coinductive_llist @ A)) @ X) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))) => ((((member @ (coinductive_llist @ A)) @ Y) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))) => ((((coindu2010755910_llist @ A) @ X) = ((coindu2010755910_llist @ A) @ Y)) = (X = Y))))))).
% 0.31/1.80 thf(fact_96_unfold__stream__ltl__unroll, axiom, (! [A:$tType,B:$tType,SHD:(B > A),STL:(B > B),B3:B]: (((((((coindu139217191stream @ B) @ A) @ SHD) @ STL) @ (STL @ B3)) = (((((coindu139217191stream @ B) @ A) @ (((((comp @ B) @ A) @ B) @ SHD) @ STL)) @ STL) @ B3))))).
% 0.31/1.80 thf(fact_97_stream_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))) @ ((stream @ A) > ((stream @ B) > $o))) @ ((stream @ C) > ((stream @ D) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ (B > $o)) @ (D > $o)) @ Sa) @ ((((((bNF_rel_fun @ B) @ D) @ $o) @ $o) @ Sc) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (stream @ A)) @ (stream @ C)) @ ((stream @ B) > $o)) @ ((stream @ D) > $o)) @ (((stream_all2 @ A) @ C) @ Sa)) @ ((((((bNF_rel_fun @ (stream @ B)) @ (stream @ D)) @ $o) @ $o) @ (((stream_all2 @ B) @ D) @ Sc)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((stream_all2 @ A) @ B)) @ ((stream_all2 @ C) @ D))))).
% 0.31/1.80 thf(fact_98_o__prs_I1_J, axiom, (! [C:$tType,E:$tType,A:$tType,B:$tType,F3:$tType,D:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),Abs23:(C > D),Rep23:(D > C),R3:(E > (E > $o)),Abs3:(E > F3),Rep3:(F3 > E)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ C) @ D) @ R2) @ Abs23) @ Rep23) => ((((((quotient3 @ E) @ F3) @ R3) @ Abs3) @ Rep3) => ((((((((map_fun @ (D > F3)) @ (C > E)) @ ((A > C) > (A > E))) @ ((B > D) > (B > F3))) @ ((((((map_fun @ C) @ D) @ F3) @ E) @ Abs23) @ Rep3)) @ ((((((map_fun @ (B > D)) @ (A > C)) @ (A > E)) @ (B > F3)) @ ((((((map_fun @ A) @ B) @ D) @ C) @ Abs12) @ Rep23)) @ ((((((map_fun @ B) @ A) @ E) @ F3) @ Rep12) @ Abs3))) @ (((comp @ C) @ E) @ A)) = (((comp @ D) @ F3) @ B)))))))).
% 0.31/1.80 thf(fact_99_fun_Omap__cong__pred, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Ya:(D > A),F2:(A > B),G2:(A > B)]: (((X = Ya) => ((((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (^ [Z2:A]: (((F2 @ Z2) = (G2 @ Z2))))) @ Ya) => ((((((comp @ A) @ B) @ D) @ F2) @ X) = (((((comp @ A) @ B) @ D) @ G2) @ Ya))))))).
% 0.31/1.80 thf(fact_100_fun_Opred__map, axiom, (! [B:$tType,A:$tType,D:$tType,Q:(B > $o),F2:(A > B),X:(D > A)]: (((((((basic_pred_fun @ D) @ B) @ (^ [Uu:D]: ($true))) @ Q) @ (((((comp @ A) @ B) @ D) @ F2) @ X)) = (((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (((((comp @ B) @ $o) @ A) @ Q) @ F2)) @ X))))).
% 0.31/1.80 thf(fact_101_quot__rel__rsp, axiom, (! [B:$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) @ $o) @ R) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ R) @ R))))).
% 0.31/1.80 thf(fact_102_corec__llist__never__stop, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),MORE:(B > (coinductive_llist @ A)),LTL:(B > B),X:B]: ((((((((((coindu1259883913_llist @ B) @ A) @ IS_LNIL) @ LHD) @ (^ [Uu:B]: ($false))) @ MORE) @ LTL) @ X) = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ X))))).
% 0.31/1.80 thf(fact_103_unfold__llist__ltl__unroll, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B3:B]: ((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B3)) = ((((((coindu1441602521_llist @ B) @ A) @ (((((comp @ B) @ $o) @ B) @ IS_LNIL) @ LTL)) @ (((((comp @ B) @ A) @ B) @ LHD) @ LTL)) @ LTL) @ B3))))).
% 0.31/1.80 thf(fact_104_monotone__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ ((C > (C > $o)) > ((A > C) > $o))) @ ((D > (D > $o)) > ((B > D) > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (C > (C > $o))) @ (D > (D > $o))) @ ((A > C) > $o)) @ ((B > D) > $o)) @ ((((((bNF_rel_fun @ C) @ D) @ (C > $o)) @ (D > $o)) @ B2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((comple1396247847notone @ A) @ C)) @ ((comple1396247847notone @ B) @ D)))))).
% 0.31/1.80 thf(fact_105_stream_Otype__definition__axioms, axiom, (! [A:$tType]: ((((((type_definition @ (stream @ A)) @ (coinductive_llist @ A)) @ (coindu1724414836stream @ A)) @ (coindu2010755910_llist @ A)) @ ((collect @ (coinductive_llist @ A)) @ (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2)))))))))).
% 0.31/1.80 thf(fact_106_relcompp__transfer, axiom, (! [C:$tType,A:$tType,E:$tType,F3:$tType,B:$tType,D:$tType,B2:(A > (B > $o)),A2:(C > (D > $o)),C2:(E > (F3 > $o))]: (((((bi_total @ A) @ B) @ B2) => ((((((((bNF_rel_fun @ (C > (A > $o))) @ (D > (B > $o))) @ ((A > (E > $o)) > (C > (E > $o)))) @ ((B > (F3 > $o)) > (D > (F3 > $o)))) @ ((((((bNF_rel_fun @ C) @ D) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (A > (E > $o))) @ (B > (F3 > $o))) @ (C > (E > $o))) @ (D > (F3 > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (E > $o)) @ (F3 > $o)) @ B2) @ ((((((bNF_rel_fun @ E) @ F3) @ $o) @ $o) @ C2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ C) @ D) @ (E > $o)) @ (F3 > $o)) @ A2) @ ((((((bNF_rel_fun @ E) @ F3) @ $o) @ $o) @ C2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))))) @ (((relcompp @ C) @ A) @ E)) @ (((relcompp @ D) @ B) @ F3)))))).
% 0.31/1.80 thf(fact_107_bex1__rel__rsp, axiom, (! [B:$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) @ $o) @ R) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((bex1_rel @ A) @ R)) @ ((bex1_rel @ A) @ R)))))).
% 0.31/1.80 thf(fact_108_left__total__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_total @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((left_total @ A) @ C)) @ ((left_total @ B) @ D))))))).
% 0.31/1.80 thf(fact_109_fun_Orel__compp, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S:(B > (C > $o))]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ (((((relcompp @ A) @ B) @ C) @ R) @ S)) = (((((relcompp @ (D > A)) @ (D > B)) @ (D > C)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ R)) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ C) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ S)))))).
% 0.31/1.80 thf(fact_110_conj__comp__iff, axiom, (! [B:$tType,A:$tType,P:(B > $o),Q:(B > $o),G2:(A > B)]: (((((((comp @ B) @ $o) @ A) @ (^ [X2:B]: (((P @ X2) & (Q @ X2))))) @ G2) = (^ [X2:A]: ((((((((comp @ B) @ $o) @ A) @ P) @ G2) @ X2) & ((((((comp @ B) @ $o) @ A) @ Q) @ G2) @ X2)))))))).
% 0.31/1.80 thf(fact_111_stream_Orel__compp, axiom, (! [A:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S:(B > (C > $o))]: (((((stream_all2 @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R) @ S)) = (((((relcompp @ (stream @ A)) @ (stream @ B)) @ (stream @ C)) @ (((stream_all2 @ A) @ B) @ R)) @ (((stream_all2 @ B) @ C) @ S)))))).
% 0.31/1.80 thf(fact_112_stream_Oleft__total__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((left_total @ A) @ B) @ R) => (((left_total @ (stream @ A)) @ (stream @ B)) @ (((stream_all2 @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_113_eq__comp__r, axiom, (! [A:$tType,R:(A > (A > $o))]: (((((((relcompp @ A) @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ (((((relcompp @ A) @ A) @ A) @ R) @ (^ [Y3:A,Z:A]: ((Y3 = Z))))) = R)))).
% 0.31/1.80 thf(fact_114_bex1__rel__aux, axiom, (! [A:$tType,R:(A > (A > $o)),X:(A > $o),Y:(A > $o)]: (((! [Xa2:A,Ya2:A]: ((((R @ Xa2) @ Ya2) => ((X @ Xa2) = (Y @ Ya2))))) => ((((bex1_rel @ A) @ R) @ X) => (((bex1_rel @ A) @ R) @ Y)))))).
% 0.31/1.80 thf(fact_115_bex1__rel__aux2, axiom, (! [A:$tType,R:(A > (A > $o)),X:(A > $o),Y:(A > $o)]: (((! [Xa2:A,Ya2:A]: ((((R @ Xa2) @ Ya2) => ((X @ Xa2) = (Y @ Ya2))))) => ((((bex1_rel @ A) @ R) @ Y) => (((bex1_rel @ A) @ R) @ X)))))).
% 0.31/1.80 thf(fact_116_left__totalE, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((left_total @ A) @ B) @ R) => (! [X7:A]: ((? [X12:B]: (((R @ X7) @ X12))))))))).
% 0.31/1.80 thf(fact_117_left__totalI, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((! [X3:A]: ((? [X13:B]: (((R @ X3) @ X13))))) => (((left_total @ A) @ B) @ R))))).
% 0.31/1.80 thf(fact_118_left__total__OO, axiom, (! [A:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S:(B > (C > $o))]: (((((left_total @ A) @ B) @ R) => ((((left_total @ B) @ C) @ S) => (((left_total @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R) @ S))))))).
% 0.31/1.80 thf(fact_119_left__total__eq, axiom, (! [A:$tType]: ((((left_total @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_120_left__total__def, axiom, (! [B:$tType,A:$tType]: ((((left_total @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [X2:A]: (((^ [P2:(B > $o)]: ((? [X6:B]: ((P2 @ X6))))) @ (R5 @ X2)))))))))).
% 0.31/1.80 thf(fact_121_bi__total__OO, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(A > (B > $o)),B2:(B > (C > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_total @ B) @ C) @ B2) => (((bi_total @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ A2) @ B2))))))).
% 0.31/1.80 thf(fact_122_typedef__rep__transfer, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),T2:(A > (B > $o))]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((T2 = (^ [X2:A,Y4:B]: ((X2 = (Rep @ Y4))))) => ((((((((bNF_rel_fun @ A) @ B) @ A) @ A) @ T2) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ (^ [X2:A]: (X2))) @ Rep)))))).
% 0.31/1.80 thf(fact_123_OOO__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(B > (B > $o)),Abs23:(B > C),Rep23:(C > B),R23:(A > (A > $o))]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ B) @ C) @ R2) @ Abs23) @ Rep23) => ((! [X3:A,Y2:A]: ((((R23 @ X3) @ Y2) => (((R1 @ X3) @ X3) => (((R1 @ Y2) @ Y2) => ((R2 @ (Abs12 @ X3)) @ (Abs12 @ Y2))))))) => ((! [X3:B,Y2:B]: ((((R2 @ X3) @ Y2) => ((R23 @ (Rep12 @ X3)) @ (Rep12 @ Y2))))) => (((((quotient3 @ A) @ C) @ (((((relcompp @ A) @ A) @ A) @ R1) @ (((((relcompp @ A) @ A) @ A) @ R23) @ R1))) @ (((((comp @ B) @ C) @ A) @ Abs23) @ Abs12)) @ (((((comp @ B) @ A) @ C) @ Rep12) @ Rep23))))))))).
% 0.31/1.80 thf(fact_124_OOO__eq__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),Abs23:(B > C),Rep23:(C > B)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ B) @ C) @ (^ [Y3:B,Z:B]: ((Y3 = Z)))) @ Abs23) @ Rep23) => (((((quotient3 @ A) @ C) @ (((((relcompp @ A) @ A) @ A) @ R1) @ (((((relcompp @ A) @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ R1))) @ (((((comp @ B) @ C) @ A) @ Abs23) @ Abs12)) @ (((((comp @ B) @ A) @ C) @ Rep12) @ Rep23))))))).
% 0.31/1.80 thf(fact_125_monotone__applyI, axiom, (! [B:$tType,A:$tType,C:$tType,Orda:(A > (A > $o)),Ordb:(B > (B > $o)),F5:(A > B),X:C]: (((((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ F5) => (((((comple1396247847notone @ (C > A)) @ B) @ ((((partial_fun_ord @ A) @ A) @ C) @ Orda)) @ Ordb) @ (^ [F:(C > A)]: ((F5 @ (F @ X))))))))).
% 0.31/1.80 thf(fact_126_monotone__fun__ord__apply, axiom, (! [A:$tType,C:$tType,B:$tType,Orda:(A > (A > $o)),Ordb:(C > (C > $o)),F2:(A > (B > C))]: (((((((comple1396247847notone @ A) @ (B > C)) @ Orda) @ ((((partial_fun_ord @ C) @ C) @ B) @ Ordb)) @ F2) = (! [X2:B]: ((((((comple1396247847notone @ A) @ C) @ Orda) @ Ordb) @ (^ [Y4:A]: (((F2 @ Y4) @ X2)))))))))).
% 0.31/1.80 thf(fact_127_stream_Ocase__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R:(A > (B > $o)),S:(C > (D > $o))]: (((((((((bNF_rel_fun @ (A > ((stream @ A) > C))) @ (B > ((stream @ B) > D))) @ ((stream @ A) > C)) @ ((stream @ B) > D)) @ ((((((bNF_rel_fun @ A) @ B) @ ((stream @ A) > C)) @ ((stream @ B) > D)) @ R) @ ((((((bNF_rel_fun @ (stream @ A)) @ (stream @ B)) @ C) @ D) @ (((stream_all2 @ A) @ B) @ R)) @ S))) @ ((((((bNF_rel_fun @ (stream @ A)) @ (stream @ B)) @ C) @ D) @ (((stream_all2 @ A) @ B) @ R)) @ S)) @ ((case_stream @ A) @ C)) @ ((case_stream @ B) @ D))))).
% 0.31/1.80 thf(fact_128_monotone__if__fun, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,Orda:(B > (B > $o)),Ordb:(D > (D > $o)),F5:((A > B) > (C > D)),G5:((A > B) > (C > D)),C3:(C > $o)]: (((((((comple1396247847notone @ (A > B)) @ (C > D)) @ ((((partial_fun_ord @ B) @ B) @ A) @ Orda)) @ ((((partial_fun_ord @ D) @ D) @ C) @ Ordb)) @ F5) => ((((((comple1396247847notone @ (A > B)) @ (C > D)) @ ((((partial_fun_ord @ B) @ B) @ A) @ Orda)) @ ((((partial_fun_ord @ D) @ D) @ C) @ Ordb)) @ G5) => (((((comple1396247847notone @ (A > B)) @ (C > D)) @ ((((partial_fun_ord @ B) @ B) @ A) @ Orda)) @ ((((partial_fun_ord @ D) @ D) @ C) @ Ordb)) @ (^ [F:(A > B),N2:C]: (((((if @ D) @ (C3 @ N2)) @ ((F5 @ F) @ N2)) @ ((G5 @ F) @ N2)))))))))).
% 0.31/1.80 thf(fact_129_monotone__fun__apply__fun, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,Ord:(C > (C > $o)),T:A,G2:(D > B)]: ((((((comple1396247847notone @ (A > (B > C))) @ (D > C)) @ ((((partial_fun_ord @ (B > C)) @ (B > C)) @ A) @ ((((partial_fun_ord @ C) @ C) @ B) @ Ord))) @ ((((partial_fun_ord @ C) @ C) @ D) @ Ord)) @ (^ [F:(A > (B > C)),N2:D]: (((F @ T) @ (G2 @ N2)))))))).
% 0.31/1.80 thf(fact_130_stream_Ocase__distrib, axiom, (! [B:$tType,C:$tType,A:$tType,H2:(B > C),F2:(A > ((stream @ A) > B)),Stream2:(stream @ A)]: (((H2 @ ((((case_stream @ A) @ B) @ F2) @ Stream2)) = ((((case_stream @ A) @ C) @ (^ [X14:A,X23:(stream @ A)]: ((H2 @ ((F2 @ X14) @ X23))))) @ Stream2))))).
% 0.31/1.80 thf(fact_131_monotone__id_H, axiom, (! [A:$tType,Ord:(A > (A > $o))]: ((((((comple1396247847notone @ A) @ A) @ Ord) @ Ord) @ (^ [X2:A]: (X2)))))).
% 0.31/1.80 thf(fact_132_call__mono, axiom, (! [B:$tType,A:$tType,Ord:(B > (B > $o)),T:A]: ((((((comple1396247847notone @ (A > B)) @ B) @ ((((partial_fun_ord @ B) @ B) @ A) @ Ord)) @ Ord) @ (^ [F:(A > B)]: ((F @ T))))))).
% 0.31/1.80 thf(fact_133_stream_Opred__transfer, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((stream @ A) > $o)) @ ((stream @ B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (stream @ A)) @ (stream @ B)) @ $o) @ $o) @ (((stream_all2 @ A) @ B) @ R)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (pred_stream @ A)) @ (pred_stream @ B))))).
% 0.31/1.80 thf(fact_134_transfer__forall__transfer_I1_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 0.31/1.80 thf(fact_135_Domainp__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,B2:(A > (B > $o)),A2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ B2) => ((((((((bNF_rel_fun @ (C > (A > $o))) @ (D > (B > $o))) @ (C > $o)) @ (D > $o)) @ ((((((bNF_rel_fun @ C) @ D) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((domainp @ C) @ A)) @ ((domainp @ D) @ B)))))).
% 0.31/1.80 thf(fact_136_stream_Opred__True, axiom, (! [A:$tType]: ((((pred_stream @ A) @ (^ [Uu:A]: ($true))) = (^ [Uu:(stream @ A)]: ($true)))))).
% 0.31/1.80 thf(fact_137_fun_ODomainp__rel, axiom, (! [C:$tType,B:$tType,A:$tType,R:(A > (B > $o))]: (((((domainp @ (C > A)) @ (C > B)) @ ((((((bNF_rel_fun @ C) @ C) @ A) @ B) @ (^ [Y3:C,Z:C]: ((Y3 = Z)))) @ R)) = ((((basic_pred_fun @ C) @ A) @ (^ [Uu:C]: ($true))) @ (((domainp @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_138_stream_ODomainp__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((domainp @ (stream @ A)) @ (stream @ B)) @ (((stream_all2 @ A) @ B) @ R)) = ((pred_stream @ A) @ (((domainp @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_139_Domainp__iff, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = (^ [T3:(A > (B > $o)),X2:A]: (((^ [P2:(B > $o)]: ((? [X6:B]: ((P2 @ X6))))) @ (T3 @ X2)))))))).
% 0.31/1.80 thf(fact_140_Domainp__refl, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = ((domainp @ A) @ B))))).
% 0.31/1.80 thf(fact_141_pcr__Domainp__total, axiom, (! [A:$tType,B:$tType,C:$tType,B2:(A > (B > $o)),A2:(C > (A > $o)),P:(C > $o)]: (((((left_total @ A) @ B) @ B2) => (((((domainp @ C) @ A) @ A2) = P) => ((((domainp @ C) @ B) @ (((((relcompp @ C) @ A) @ B) @ A2) @ B2)) = P)))))).
% 0.31/1.80 thf(fact_142_stream__from__llist__setup_Ostream_Odomain__eq, axiom, (! [A:$tType]: (((((domainp @ (coinductive_llist @ A)) @ (stream @ A)) @ (((coindu773941317stream @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z))))) = (^ [Xs2:(coinductive_llist @ A)]: ((~ (((coinductive_lfinite @ A) @ Xs2))))))))).
% 0.31/1.80 thf(fact_143_fun__ord__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((partial_fun_ord @ A) @ B) @ C) = (^ [Ord2:(A > (B > $o)),F:(C > A),G:(C > B)]: ((! [X2:C]: (((Ord2 @ (F @ X2)) @ (G @ X2)))))))))).
% 0.31/1.80 thf(fact_144_pcr__Domainp__par__left__total, axiom, (! [A:$tType,B:$tType,C:$tType,B2:(A > (B > $o)),P:(A > $o),A2:(C > (A > $o)),P3:(C > $o)]: ((((((domainp @ A) @ B) @ B2) = P) => ((((left_total @ C) @ A) @ A2) => (((((((((bNF_rel_fun @ C) @ A) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ P3) @ P) => ((((domainp @ C) @ B) @ (((((relcompp @ C) @ A) @ B) @ A2) @ B2)) = P3))))))).
% 0.31/1.80 thf(fact_145_let__mono, axiom, (! [A:$tType,C:$tType,B:$tType,Orda:(B > (B > $o)),Ordb:(C > (C > $o)),B3:(B > (A > C)),T:A]: (((! [X3:A]: ((((((comple1396247847notone @ B) @ C) @ Orda) @ Ordb) @ (^ [F:B]: (((B3 @ F) @ X3)))))) => (((((comple1396247847notone @ B) @ C) @ Orda) @ Ordb) @ (^ [F:B]: (((B3 @ F) @ T)))))))).
% 0.31/1.80 thf(fact_146_if__mono, axiom, (! [B:$tType,A:$tType,Orda:(A > (A > $o)),Ordb:(B > (B > $o)),F5:(A > B),G5:(A > B),C3:$o]: (((((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ F5) => ((((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ G5) => (((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ (^ [F:A]: (((((if @ B) @ C3) @ (F5 @ F)) @ (G5 @ F)))))))))).
% 0.31/1.80 thf(fact_147_transfer__forall__transfer_I4_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ rev_implies) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 0.31/1.80 thf(fact_148_transfer__forall__transfer_I5_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ rev_implies)) @ rev_implies) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 0.31/1.80 thf(fact_149_reflp__transfer_I1_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (reflp @ A)) @ (reflp @ B)))))).
% 0.31/1.80 thf(fact_150_right__total__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_total @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((right_total @ A) @ C)) @ ((right_total @ B) @ D))))))).
% 0.31/1.80 thf(fact_151_fun_Orel__reflp, axiom, (! [D:$tType,A:$tType,R:(A > (A > $o))]: ((((reflp @ A) @ R) => ((reflp @ (D > A)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ A) @ (^ [Y3:D,Z:D]: ((Y3 = Z)))) @ R)))))).
% 0.31/1.80 thf(fact_152_DEADID_Orel__reflp, axiom, (! [A:$tType]: (((reflp @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_153_right__total__def, axiom, (! [B:$tType,A:$tType]: ((((right_total @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [Y4:B]: ((? [X2:A]: (((R5 @ X2) @ Y4))))))))))).
% 0.31/1.80 thf(fact_154_rev__implies__def, axiom, (rev_implies = (^ [X2:$o,Y4:$o]: ((Y4 => X2))))).
% 0.31/1.80 thf(fact_155_right__total__eq, axiom, (! [A:$tType]: ((((right_total @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_156_right__totalI, axiom, (! [A:$tType,B:$tType,A2:(B > (A > $o))]: (((! [Y2:A]: ((? [X7:B]: (((A2 @ X7) @ Y2))))) => (((right_total @ B) @ A) @ A2))))).
% 0.31/1.80 thf(fact_157_right__totalE, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o)),Y:B]: (((((right_total @ A) @ B) @ A2) => (~ ((! [X3:A]: ((~ (((A2 @ X3) @ Y))))))))))).
% 0.31/1.80 thf(fact_158_stream__from__llist__setup_Ostream_Oright__total, axiom, (! [F3:$tType,E:$tType,T2:(E > (F3 > $o))]: (((((right_total @ E) @ F3) @ T2) => (((right_total @ (coinductive_llist @ E)) @ (stream @ F3)) @ (((coindu773941317stream @ E) @ F3) @ T2)))))).
% 0.31/1.80 thf(fact_159_stream_Oright__total__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((right_total @ A) @ B) @ R) => (((right_total @ (stream @ A)) @ (stream @ B)) @ (((stream_all2 @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_160_stream_Orel__reflp, axiom, (! [A:$tType,R:(A > (A > $o))]: ((((reflp @ A) @ R) => ((reflp @ (stream @ A)) @ (((stream_all2 @ A) @ A) @ R)))))).
% 0.31/1.80 thf(fact_161_right__total__OO, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(A > (B > $o)),B2:(B > (C > $o))]: (((((right_total @ A) @ B) @ A2) => ((((right_total @ B) @ C) @ B2) => (((right_total @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ A2) @ B2))))))).
% 0.31/1.80 thf(fact_162_typedef__right__total, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),T2:(A > (B > $o))]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((T2 = (^ [X2:A,Y4:B]: ((X2 = (Rep @ Y4))))) => (((right_total @ A) @ B) @ T2)))))).
% 0.31/1.80 thf(fact_163_reflp__transfer_I2_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (=>)))) @ (=>)) @ (reflp @ A)) @ (reflp @ B)))))).
% 0.31/1.80 thf(fact_164_reflp__transfer_I3_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ (=>)) @ (reflp @ A)) @ (reflp @ B)))))).
% 0.31/1.80 thf(fact_165_bi__totalI, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((left_total @ A) @ B) @ R) => ((((right_total @ A) @ B) @ R) => (((bi_total @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_166_bi__total__alt__def, axiom, (! [B:$tType,A:$tType]: ((((bi_total @ A) @ B) = (^ [A6:(A > (B > $o))]: (((((left_total @ A) @ B) @ A6) & (((right_total @ A) @ B) @ A6)))))))).
% 0.31/1.80 thf(fact_167_reflp__transfer_I4_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ rev_implies))) @ rev_implies) @ (reflp @ A)) @ (reflp @ B)))))).
% 0.31/1.80 thf(fact_168_reflp__transfer_I5_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ rev_implies) @ (reflp @ A)) @ (reflp @ B)))))).
% 0.31/1.80 thf(fact_169_transfer__forall__transfer_I3_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (=>))) @ (=>)) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 0.31/1.80 thf(fact_170_transfer__forall__transfer_I2_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (=>)) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 0.31/1.80 thf(fact_171_Domainp__forall__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((transfer_bforall @ A) @ (((domainp @ A) @ B) @ A2))) @ (transfer_forall @ B)))))).
% 0.31/1.80 thf(fact_172_right__total__relcompp__transfer, axiom, (! [C:$tType,A:$tType,E:$tType,F3:$tType,B:$tType,D:$tType,B2:(A > (B > $o)),A2:(C > (D > $o)),C2:(E > (F3 > $o))]: (((((right_total @ A) @ B) @ B2) => ((((((((bNF_rel_fun @ (C > (A > $o))) @ (D > (B > $o))) @ ((A > (E > $o)) > (C > (E > $o)))) @ ((B > (F3 > $o)) > (D > (F3 > $o)))) @ ((((((bNF_rel_fun @ C) @ D) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ (A > (E > $o))) @ (B > (F3 > $o))) @ (C > (E > $o))) @ (D > (F3 > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (E > $o)) @ (F3 > $o)) @ B2) @ ((((((bNF_rel_fun @ E) @ F3) @ $o) @ $o) @ C2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ C) @ D) @ (E > $o)) @ (F3 > $o)) @ A2) @ ((((((bNF_rel_fun @ E) @ F3) @ $o) @ $o) @ C2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))))) @ (^ [R5:(C > (A > $o)),S5:(A > (E > $o)),X2:C,Z2:E]: ((? [Y4:A]: (((((member @ A) @ Y4) @ ((collect @ A) @ (((domainp @ A) @ B) @ B2))) & (((R5 @ X2) @ Y4) & ((S5 @ Y4) @ Z2)))))))) @ (((relcompp @ D) @ B) @ F3)))))).
% 0.31/1.80 thf(fact_173_right__total__Domainp__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,B2:(A > (B > $o)),A2:(C > (D > $o))]: (((((right_total @ A) @ B) @ B2) => ((((((((bNF_rel_fun @ (C > (A > $o))) @ (D > (B > $o))) @ (C > $o)) @ (D > $o)) @ ((((((bNF_rel_fun @ C) @ D) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ B2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [T3:(C > (A > $o)),X2:C]: ((? [Y4:A]: (((((member @ A) @ Y4) @ ((collect @ A) @ (((domainp @ A) @ B) @ B2))) & ((T3 @ X2) @ Y4))))))) @ ((domainp @ D) @ B)))))).
% 0.31/1.80 thf(fact_174_bex__reg, axiom, (! [A:$tType,R:(set @ A),P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ R) => ((P @ X3) => (Q @ X3))))) => ((? [X7:A]: (((((member @ A) @ X7) @ R) & (P @ X7)))) => (? [X3:A]: (((((member @ A) @ X3) @ R) & (Q @ X3))))))))).
% 0.31/1.80 thf(fact_175_right__total__Ex__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((bex @ A) @ ((collect @ A) @ (((domainp @ A) @ B) @ A2)))) @ (^ [P2:(B > $o)]: ((? [X6:B]: ((P2 @ X6)))))))))).
% 0.31/1.80 thf(fact_176_Domainp__prod__fun__eq, axiom, (! [C:$tType,B:$tType,A:$tType,T2:(B > (C > $o))]: (((((domainp @ (A > B)) @ (A > C)) @ ((((((bNF_rel_fun @ A) @ A) @ B) @ C) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ T2)) = (^ [F:(A > B)]: ((! [X2:A]: (((((domainp @ B) @ C) @ T2) @ (F @ X2)))))))))).
% 0.31/1.80 thf(fact_177_stream__from__llist__setup_Ostream_Odomain__par__left__total, axiom, (! [B:$tType,C:$tType,T2:(C > (B > $o)),P3:((coinductive_llist @ C) > $o)]: (((((left_total @ (coinductive_llist @ C)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ C) @ B) @ T2)) => (((((((((bNF_rel_fun @ (coinductive_llist @ C)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ C) @ B) @ T2)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ P3) @ (^ [Xs2:(coinductive_llist @ B)]: ((~ (((coinductive_lfinite @ B) @ Xs2)))))) => ((((domainp @ (coinductive_llist @ C)) @ (stream @ B)) @ (((coindu773941317stream @ C) @ B) @ T2)) = P3)))))).
% 0.31/1.80 thf(fact_178_llist_Orel__reflp, axiom, (! [A:$tType,R:(A > (A > $o))]: ((((reflp @ A) @ R) => ((reflp @ (coinductive_llist @ A)) @ (((coindu1486289336t_all2 @ A) @ A) @ R)))))).
% 0.31/1.80 thf(fact_179_llist_Oright__total__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((right_total @ A) @ B) @ R) => (((right_total @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_180_bex__reg__left, axiom, (! [A:$tType,R:(set @ A),Q:(A > $o),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ R) => ((Q @ X3) => (P @ X3))))) => ((? [X7:A]: (((((member @ A) @ X7) @ R) & (Q @ X7)))) => (? [X12:A]: ((P @ X12)))))))).
% 0.31/1.80 thf(fact_181_rel__fun__eq__rel, axiom, (! [C:$tType,B:$tType,A:$tType,R:(B > (C > $o))]: ((((((((bNF_rel_fun @ A) @ A) @ B) @ C) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ R) = (^ [F:(A > B),G:(A > C)]: ((! [X2:A]: (((R @ (F @ X2)) @ (G @ X2)))))))))).
% 0.31/1.80 thf(fact_182_rel__fun__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((bNF_rel_fun @ A) @ C) @ B) @ D) = (^ [A6:(A > (C > $o)),B4:(B > (D > $o)),F:(A > B),G:(C > D)]: ((! [X2:A,Y4:C]: ((((A6 @ X2) @ Y4) => ((B4 @ (F @ X2)) @ (G @ Y4))))))))))).
% 0.31/1.80 thf(fact_183_stream__from__llist__setup_Ostream_Odomain, axiom, (! [B:$tType,C:$tType,T2:(C > (B > $o))]: (((((domainp @ (coinductive_llist @ C)) @ (stream @ B)) @ (((coindu773941317stream @ C) @ B) @ T2)) = (^ [X2:(coinductive_llist @ C)]: ((? [Y4:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ C) @ B) @ T2) @ X2) @ Y4) & (~ (((coinductive_lfinite @ B) @ Y4)))))))))))).
% 0.31/1.80 thf(fact_184_llist__all2__transfer, axiom, (! [A:$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) @ $o) @ R) @ (^ [Y3:$o,Z:$o]: ((Y3 = 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) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((coindu1486289336t_all2 @ A) @ A)) @ ((coindu1486289336t_all2 @ B) @ B))))).
% 0.31/1.80 thf(fact_185_llist_Orel__transfer, axiom, (! [A:$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) @ $o) @ Sc) @ (^ [Y3:$o,Z:$o]: ((Y3 = 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) @ $o) @ (((coindu1486289336t_all2 @ B) @ D) @ Sc)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))))) @ ((coindu1486289336t_all2 @ A) @ B)) @ ((coindu1486289336t_all2 @ C) @ D))))).
% 0.31/1.80 thf(fact_186_lfinite__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (coinductive_lfinite @ A)) @ (coinductive_lfinite @ B))))).
% 0.31/1.80 thf(fact_187_transfer__forall__def, axiom, (! [A:$tType]: (((transfer_forall @ A) = (^ [P2:(A > $o)]: ((! [X6:A]: ((P2 @ X6))))))))).
% 0.31/1.80 thf(fact_188_OO__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((relcompp @ A) @ C) @ B) = (^ [R5:(A > (C > $o)),S5:(C > (B > $o)),X2:A,Z2:B]: ((? [Y4:C]: ((((R5 @ X2) @ Y4) & ((S5 @ Y4) @ Z2)))))))))).
% 0.31/1.80 thf(fact_189_llist_Obi__total__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((bi_total @ A) @ B) @ R) => (((bi_total @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_190_llist__all2__lfiniteD, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => (((coinductive_lfinite @ A) @ Xs) = ((coinductive_lfinite @ B) @ Ys)))))).
% 0.31/1.80 thf(fact_191_llist_Orel__refl, axiom, (! [B:$tType,Ra:(B > (B > $o)),X:(coinductive_llist @ B)]: (((! [X3:B]: (((Ra @ X3) @ X3))) => (((((coindu1486289336t_all2 @ B) @ B) @ Ra) @ X) @ X))))).
% 0.31/1.80 thf(fact_192_llist_Orel__eq, axiom, (! [A:$tType]: (((((coindu1486289336t_all2 @ A) @ A) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) = (^ [Y3:(coinductive_llist @ A),Z:(coinductive_llist @ A)]: ((Y3 = Z))))))).
% 0.31/1.80 thf(fact_193_llist__all2__mono, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B),P3:(A > (B > $o))]: (((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) => ((! [X3:A,Y2:B]: ((((P @ X3) @ Y2) => ((P3 @ X3) @ Y2)))) => (((((coindu1486289336t_all2 @ A) @ B) @ P3) @ Xs) @ Ys)))))).
% 0.31/1.80 thf(fact_194_llist__all2__rsp, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),S:(A > (A > $o)),T2:(B > (B > $o)),X:(coinductive_llist @ A),Y:(coinductive_llist @ B),A3:(coinductive_llist @ A),B3:(coinductive_llist @ B)]: (((! [X3:A,Y2:B]: ((((R @ X3) @ Y2) => (! [A5:A,B5:B]: ((((R @ A5) @ B5) => (((S @ X3) @ A5) = ((T2 @ Y2) @ B5)))))))) => ((((((coindu1486289336t_all2 @ A) @ B) @ R) @ X) @ Y) => ((((((coindu1486289336t_all2 @ A) @ B) @ R) @ A3) @ B3) => ((((((coindu1486289336t_all2 @ A) @ A) @ S) @ X) @ A3) = (((((coindu1486289336t_all2 @ B) @ B) @ T2) @ Y) @ B3)))))))).
% 0.31/1.80 thf(fact_195_llist__all2__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((((((coindu1486289336t_all2 @ A) @ B) @ (^ [X2:A,Y4:B]: ((((P @ X2) @ Y4) & ((Q @ X2) @ Y4))))) @ Xs) @ Ys) = ((((((coindu1486289336t_all2 @ A) @ B) @ P) @ Xs) @ Ys) & (((((coindu1486289336t_all2 @ A) @ B) @ Q) @ Xs) @ Ys)))))).
% 0.31/1.80 thf(fact_196_llist_Oleft__total__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((left_total @ A) @ B) @ R) => (((left_total @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_197_llist_Orel__compp, axiom, (! [A:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S:(B > (C > $o))]: (((((coindu1486289336t_all2 @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R) @ S)) = (((((relcompp @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (coinductive_llist @ C)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ (((coindu1486289336t_all2 @ B) @ C) @ S)))))).
% 0.31/1.80 thf(fact_198_pred__fun__def, axiom, (! [B:$tType,A:$tType]: ((((basic_pred_fun @ A) @ B) = (^ [A6:(A > $o),B4:(B > $o),F:(A > B)]: ((! [X2:A]: (((A6 @ X2) => (B4 @ (F @ X2))))))))))).
% 0.31/1.80 thf(fact_199_pcr__Domainp, axiom, (! [B:$tType,A:$tType,C:$tType,B2:(A > (B > $o)),P:(A > $o),A2:(C > (A > $o))]: ((((((domainp @ A) @ B) @ B2) = P) => ((((domainp @ C) @ B) @ (((((relcompp @ C) @ A) @ B) @ A2) @ B2)) = (^ [X2:C]: ((? [Y4:A]: ((((A2 @ X2) @ Y4) & (P @ Y4))))))))))).
% 0.31/1.80 thf(fact_200_stream__from__llist__setup_Ostream_Orep__transfer, axiom, (! [E:$tType,F3:$tType,T2:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (coinductive_llist @ E)) @ (stream @ F3)) @ (coinductive_llist @ E)) @ (coinductive_llist @ F3)) @ (((coindu773941317stream @ E) @ F3) @ T2)) @ (((coindu1486289336t_all2 @ E) @ F3) @ T2)) @ (^ [X2:(coinductive_llist @ E)]: (X2))) @ (coindu1724414836stream @ F3))))).
% 0.31/1.80 thf(fact_201_stream__from__llist__setup_Opcr__stream__def, axiom, (! [B:$tType,C:$tType]: ((((coindu773941317stream @ C) @ B) = (^ [T3:(C > (B > $o))]: ((((((relcompp @ (coinductive_llist @ C)) @ (coinductive_llist @ B)) @ (stream @ B)) @ (((coindu1486289336t_all2 @ C) @ B) @ T3)) @ (coindu1183105481stream @ B)))))))).
% 0.31/1.80 thf(fact_202_transfer__bforall__def, axiom, (! [A:$tType]: (((transfer_bforall @ A) = (^ [P4:(A > $o),Q2:(A > $o)]: ((! [X2:A]: (((P4 @ X2) => (Q2 @ X2)))))))))).
% 0.31/1.80 thf(fact_203_unfold__llist__transfer, axiom, (! [A:$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) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ ((A > A) > (A > (coinductive_llist @ C)))) @ ((B > B) > (B > (coinductive_llist @ D)))) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (A > A)) @ (B > B)) @ (A > (coinductive_llist @ C))) @ (B > (coinductive_llist @ D))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2)) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A2) @ (((coindu1486289336t_all2 @ C) @ D) @ B2))))) @ ((coindu1441602521_llist @ A) @ C)) @ ((coindu1441602521_llist @ B) @ D))))).
% 0.31/1.80 thf(fact_204_llist_Ocorec__transfer, axiom, (! [C:$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) @ $o) @ S) @ (^ [Y3:$o,Z:$o]: ((Y3 = 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) @ $o) @ S) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (C > (coinductive_llist @ A))) @ (D > (coinductive_llist @ B))) @ ((C > C) > (C > (coinductive_llist @ A)))) @ ((D > D) > (D > (coinductive_llist @ B)))) @ ((((((bNF_rel_fun @ C) @ D) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ S) @ (((coindu1486289336t_all2 @ A) @ B) @ R))) @ ((((((bNF_rel_fun @ (C > C)) @ (D > D)) @ (C > (coinductive_llist @ A))) @ (D > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ C) @ D) @ C) @ D) @ S) @ S)) @ ((((((bNF_rel_fun @ C) @ D) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ S) @ (((coindu1486289336t_all2 @ A) @ B) @ R))))))) @ ((coindu1259883913_llist @ C) @ A)) @ ((coindu1259883913_llist @ D) @ B))))).
% 0.31/1.80 thf(fact_205_llist__corec__transfer, axiom, (! [A:$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) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = 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) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (A > (coinductive_llist @ C))) @ (B > (coinductive_llist @ D))) @ ((A > A) > (A > (coinductive_llist @ C)))) @ ((B > B) > (B > (coinductive_llist @ D)))) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A2) @ (((coindu1486289336t_all2 @ C) @ D) @ B2))) @ ((((((bNF_rel_fun @ (A > A)) @ (B > B)) @ (A > (coinductive_llist @ C))) @ (B > (coinductive_llist @ D))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2)) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A2) @ (((coindu1486289336t_all2 @ C) @ D) @ B2))))))) @ ((coindu1259883913_llist @ A) @ C)) @ ((coindu1259883913_llist @ B) @ D))))).
% 0.31/1.80 thf(fact_206_Ex__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (^ [P2:(A > $o)]: ((? [X6:A]: ((P2 @ X6)))))) @ (^ [P2:(B > $o)]: ((? [X6:B]: ((P2 @ X6)))))))))).
% 0.31/1.80 thf(fact_207_All__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (^ [P2:(A > $o)]: ((! [X6:A]: ((P2 @ X6)))))) @ (^ [P2:(B > $o)]: ((! [X6:B]: ((P2 @ X6)))))))))).
% 0.31/1.80 thf(fact_208_bi__total__alt__def2, axiom, (! [B:$tType,A:$tType]: ((((bi_total @ A) @ B) = (^ [R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R5) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ (^ [P2:(A > $o)]: ((! [X6:A]: ((P2 @ X6)))))) @ (^ [P2:(B > $o)]: ((! [X6:B]: ((P2 @ X6)))))))))))).
% 0.31/1.80 thf(fact_209_right__total__alt__def2, axiom, (! [B:$tType,A:$tType]: ((((right_total @ A) @ B) = (^ [R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R5) @ (=>))) @ (=>)) @ (^ [P2:(A > $o)]: ((! [X6:A]: ((P2 @ X6)))))) @ (^ [P2:(B > $o)]: ((! [X6:B]: ((P2 @ X6)))))))))))).
% 0.31/1.80 thf(fact_210_Bex__def__raw, axiom, (! [A:$tType]: (((bex @ A) = (^ [A6:(set @ A),P4:(A > $o)]: ((? [X2:A]: (((((member @ A) @ X2) @ A6) & (P4 @ X2)))))))))).
% 0.31/1.80 thf(fact_211_rel__pred__comp__def, axiom, (! [B:$tType,A:$tType]: ((((rel_pred_comp @ A) @ B) = (^ [R5:(A > (B > $o)),P4:(B > $o),X2:A]: ((? [Y4:B]: ((((R5 @ X2) @ Y4) & (P4 @ Y4)))))))))).
% 0.31/1.80 thf(fact_212_right__total__All__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ ((ball @ A) @ ((collect @ A) @ (((domainp @ A) @ B) @ A2)))) @ (^ [P2:(B > $o)]: ((! [X6:B]: ((P2 @ X6)))))))))).
% 0.31/1.80 thf(fact_213_ball__reg__right, axiom, (! [A:$tType,R:(set @ A),P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ R) => ((P @ X3) => (Q @ X3))))) => ((! [X12:A]: ((P @ X12))) => (! [X7:A]: (((((member @ A) @ X7) @ R) => (Q @ X7))))))))).
% 0.31/1.80 thf(fact_214_Ball__def__raw, axiom, (! [A:$tType]: (((ball @ A) = (^ [A6:(set @ A),P4:(A > $o)]: ((! [X2:A]: (((((member @ A) @ X2) @ A6) => (P4 @ X2)))))))))).
% 0.31/1.80 thf(fact_215_Ball__comp__iff, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(B > (set @ C)),F2:(C > $o),G2:(A > B)]: (((((((comp @ B) @ $o) @ A) @ (^ [X2:B]: ((! [Y4:C]: (((((member @ C) @ Y4) @ (A2 @ X2)) => (F2 @ Y4))))))) @ G2) = (^ [X2:A]: ((! [Y4:C]: (((((member @ C) @ Y4) @ ((((((comp @ B) @ (set @ C)) @ A) @ A2) @ G2) @ X2)) => (F2 @ Y4)))))))))).
% 0.31/1.80 thf(fact_216_ball__reg, axiom, (! [A:$tType,R:(set @ A),P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ R) => ((P @ X3) => (Q @ X3))))) => ((! [X3:A]: (((((member @ A) @ X3) @ R) => (P @ X3)))) => (! [X7:A]: (((((member @ A) @ X7) @ R) => (Q @ X7))))))))).
% 0.31/1.80 thf(fact_217_iterates__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > A)) @ (B > B)) @ (A > (coinductive_llist @ A))) @ (B > (coinductive_llist @ B))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2)) @ ((((((bNF_rel_fun @ A) @ B) @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ A2) @ (((coindu1486289336t_all2 @ A) @ B) @ A2))) @ (coinductive_iterates @ A)) @ (coinductive_iterates @ B))))).
% 0.31/1.80 thf(fact_218_llist_Opred__transfer, axiom, (! [A:$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) @ $o) @ R) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (coindu543516966_llist @ A)) @ (coindu543516966_llist @ B))))).
% 0.31/1.80 thf(fact_219_lfinite__iterates, axiom, (! [A:$tType,F2:(A > A),X:A]: ((~ (((coinductive_lfinite @ A) @ (((coinductive_iterates @ A) @ F2) @ X))))))).
% 0.31/1.80 thf(fact_220_llist_Opred__True, axiom, (! [A:$tType]: ((((coindu543516966_llist @ A) @ (^ [Uu:A]: ($true))) = (^ [Uu:(coinductive_llist @ A)]: ($true)))))).
% 0.31/1.80 thf(fact_221_llist_ODomainp__rel, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((domainp @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) = ((coindu543516966_llist @ A) @ (((domainp @ A) @ B) @ R)))))).
% 0.31/1.80 thf(fact_222_llist__all__transfer, axiom, (! [A:$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) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ A) @ B) @ A2)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z))))) @ (coindu543516966_llist @ A)) @ (coindu543516966_llist @ B))))).
% 0.31/1.80 thf(fact_223_stream__from__llist__setup_Ostream_Odomain__par, axiom, (! [B:$tType,C:$tType,T2:(C > (B > $o)),DR:(C > $o),P22:((coinductive_llist @ C) > $o)]: ((((((domainp @ C) @ B) @ T2) = DR) => (((((((((bNF_rel_fun @ (coinductive_llist @ C)) @ (coinductive_llist @ B)) @ $o) @ $o) @ (((coindu1486289336t_all2 @ C) @ B) @ T2)) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ P22) @ (^ [Xs2:(coinductive_llist @ B)]: ((~ (((coinductive_lfinite @ B) @ Xs2)))))) => ((((domainp @ (coinductive_llist @ C)) @ (stream @ B)) @ (((coindu773941317stream @ C) @ B) @ T2)) = (((inf_inf @ ((coinductive_llist @ C) > $o)) @ ((coindu543516966_llist @ C) @ DR)) @ P22))))))).
% 0.31/1.80 thf(fact_224_case__llist__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,B2:(A > (B > $o)),A2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((C > ((coinductive_llist @ C) > A)) > ((coinductive_llist @ C) > A))) @ ((D > ((coinductive_llist @ D) > B)) > ((coinductive_llist @ D) > B))) @ B2) @ ((((((bNF_rel_fun @ (C > ((coinductive_llist @ C) > A))) @ (D > ((coinductive_llist @ D) > B))) @ ((coinductive_llist @ C) > A)) @ ((coinductive_llist @ D) > B)) @ ((((((bNF_rel_fun @ C) @ D) @ ((coinductive_llist @ C) > A)) @ ((coinductive_llist @ D) > B)) @ A2) @ ((((((bNF_rel_fun @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A) @ B) @ (((coindu1486289336t_all2 @ C) @ D) @ A2)) @ B2))) @ ((((((bNF_rel_fun @ (coinductive_llist @ C)) @ (coinductive_llist @ D)) @ A) @ B) @ (((coindu1486289336t_all2 @ C) @ D) @ A2)) @ B2))) @ ((coindu1381640503_llist @ A) @ C)) @ ((coindu1381640503_llist @ B) @ D))))).
% 0.31/1.80 thf(fact_225_llist__case__mono, axiom, (! [C:$tType,B:$tType,A:$tType,Orda:(A > (A > $o)),Ordb:(B > (B > $o)),Lnil:(A > B),Lcons:(A > (C > ((coinductive_llist @ C) > B))),X:(coinductive_llist @ C)]: (((((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ Lnil) => ((! [X3:C,Xs3:(coinductive_llist @ C)]: ((((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ (^ [F:A]: ((((Lcons @ F) @ X3) @ Xs3)))))) => (((((comple1396247847notone @ A) @ B) @ Orda) @ Ordb) @ (^ [F:A]: ((((((coindu1381640503_llist @ B) @ C) @ (Lnil @ F)) @ (Lcons @ F)) @ X))))))))).
% 0.31/1.80 thf(fact_226_llist_Ocase__distrib, axiom, (! [B:$tType,C:$tType,A:$tType,H2:(B > C),F1:B,F22:(A > ((coinductive_llist @ A) > B)),Llist:(coinductive_llist @ A)]: (((H2 @ (((((coindu1381640503_llist @ B) @ A) @ F1) @ F22) @ Llist)) = (((((coindu1381640503_llist @ C) @ A) @ (H2 @ F1)) @ (^ [X14:A,X23:(coinductive_llist @ A)]: ((H2 @ ((F22 @ X14) @ X23))))) @ Llist))))).
% 0.31/1.80 thf(fact_227_pcr__Domainp__par, axiom, (! [A:$tType,B:$tType,C:$tType,B2:(A > (B > $o)),P23:(A > $o),A2:(C > (A > $o)),P1:(C > $o),P22:(C > $o)]: ((((((domainp @ A) @ B) @ B2) = P23) => (((((domainp @ C) @ A) @ A2) = P1) => (((((((((bNF_rel_fun @ C) @ A) @ $o) @ $o) @ A2) @ (^ [Y3:$o,Z:$o]: ((Y3 = Z)))) @ P22) @ P23) => ((((domainp @ C) @ B) @ (((((relcompp @ C) @ A) @ B) @ A2) @ B2)) = (((inf_inf @ (C > $o)) @ P1) @ P22)))))))).
% 0.31/1.80 thf(fact_228_llist_Ocase__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,S:(C > (D > $o)),R:(A > (B > $o))]: (((((((((bNF_rel_fun @ C) @ D) @ ((A > ((coinductive_llist @ A) > C)) > ((coinductive_llist @ A) > C))) @ ((B > ((coinductive_llist @ B) > D)) > ((coinductive_llist @ B) > D))) @ S) @ ((((((bNF_rel_fun @ (A > ((coinductive_llist @ A) > C))) @ (B > ((coinductive_llist @ B) > D))) @ ((coinductive_llist @ A) > C)) @ ((coinductive_llist @ B) > D)) @ ((((((bNF_rel_fun @ A) @ B) @ ((coinductive_llist @ A) > C)) @ ((coinductive_llist @ B) > D)) @ R) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ C) @ D) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ S))) @ ((((((bNF_rel_fun @ (coinductive_llist @ A)) @ (coinductive_llist @ B)) @ C) @ D) @ (((coindu1486289336t_all2 @ A) @ B) @ R)) @ S))) @ ((coindu1381640503_llist @ C) @ A)) @ ((coindu1381640503_llist @ D) @ B))))).
% 0.31/1.80 thf(fact_229_img__ord__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((partial_img_ord @ A) @ C) @ B) = (^ [F:(A > C),Ord2:(C > (C > B)),X2:A,Y4:A]: (((Ord2 @ (F @ X2)) @ (F @ Y4)))))))).
% 0.31/1.80 thf(fact_230_mk__less__def, axiom, (! [A:$tType]: (((partial_mk_less @ A) = (^ [R5:(A > (A > $o)),X2:A,Y4:A]: ((((R5 @ X2) @ Y4) & (~ (((R5 @ Y4) @ X2)))))))))).
% 0.31/1.80 thf(fact_231_type__definition_ORep, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (((member @ A) @ (Rep @ X)) @ A2))))).
% 0.31/1.80 thf(fact_232_type__definition__def, axiom, (! [A:$tType,B:$tType]: ((((type_definition @ B) @ A) = (^ [Rep2:(B > A),Abs2:(A > B),A6:(set @ A)]: (((! [X2:B]: ((((member @ A) @ (Rep2 @ X2)) @ A6))) & ((! [X2:B]: (((Abs2 @ (Rep2 @ X2)) = X2))) & (! [Y4:A]: (((((member @ A) @ Y4) @ A6) => ((Rep2 @ (Abs2 @ Y4)) = Y4)))))))))))).
% 0.31/1.80 thf(fact_233_type__definition_ORep__inverse, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((Abs @ (Rep @ X)) = X))))).
% 0.31/1.80 thf(fact_234_type__definition_OAbs__inverse, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),Y:A]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ Y) @ A2) => ((Rep @ (Abs @ Y)) = Y)))))).
% 0.31/1.80 thf(fact_235_type__definition_ORep__inject, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B,Y:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (((Rep @ X) = (Rep @ Y)) = (X = Y)))))).
% 0.31/1.80 thf(fact_236_type__definition_ORep__induct, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),Y:A,P:(A > $o)]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ Y) @ A2) => ((! [X3:B]: ((P @ (Rep @ X3)))) => (P @ Y))))))).
% 0.31/1.80 thf(fact_237_type__definition_OAbs__inject, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:A,Y:A]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ X) @ A2) => ((((member @ A) @ Y) @ A2) => (((Abs @ X) = (Abs @ Y)) = (X = Y)))))))).
% 0.31/1.80 thf(fact_238_type__definition_OAbs__induct, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),P:(B > $o),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((! [Y2:A]: (((((member @ A) @ Y2) @ A2) => (P @ (Abs @ Y2))))) => (P @ X)))))).
% 0.31/1.80 thf(fact_239_type__definition_ORep__cases, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),Y:A]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ Y) @ A2) => (~ ((! [X3:B]: ((Y != (Rep @ X3))))))))))).
% 0.31/1.80 thf(fact_240_type__definition_OAbs__cases, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (~ ((! [Y2:A]: (((X = (Abs @ Y2)) => (~ ((((member @ A) @ Y2) @ A2)))))))))))).
% 0.31/1.80 thf(fact_241_type__definition_Ointro, axiom, (! [B:$tType,A:$tType,Rep:(B > A),A2:(set @ A),Abs:(A > B)]: (((! [X3:B]: ((((member @ A) @ (Rep @ X3)) @ A2))) => ((! [X3:B]: (((Abs @ (Rep @ X3)) = X3))) => ((! [Y2:A]: (((((member @ A) @ Y2) @ A2) => ((Rep @ (Abs @ Y2)) = Y2)))) => (((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2))))))).
% 0.31/1.80 thf(fact_242_llist__of__stream__siterates, axiom, (! [A:$tType,F2:(A > A),X:A]: ((((coindu1724414836stream @ A) @ (((siterate @ A) @ F2) @ X)) = (((coinductive_iterates @ A) @ F2) @ X))))).
% 0.31/1.80 thf(fact_243_nchotomy__relcomppE, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F2:(B > A),R4:(C > (A > $o)),S3:(A > (D > $o)),A3:C,C3:D]: (((! [Y2:A]: ((? [X7:B]: ((Y2 = (F2 @ X7)))))) => ((((((((relcompp @ C) @ A) @ D) @ R4) @ S3) @ A3) @ C3) => (~ ((! [B5:B]: ((((R4 @ A3) @ (F2 @ B5)) => (~ (((S3 @ (F2 @ B5)) @ C3))))))))))))).
% 0.31/1.80 thf(fact_244_OO__eq, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((((relcompp @ A) @ B) @ B) @ R) @ (^ [Y3:B,Z:B]: ((Y3 = Z)))) = R)))).
% 0.31/1.80 thf(fact_245_eq__OO, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((((relcompp @ A) @ A) @ B) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ R) = R)))).
% 0.31/1.80 thf(fact_246_relcompp_OrelcompI, axiom, (! [A:$tType,B:$tType,C:$tType,R4:(A > (B > $o)),A3:A,B3:B,S3:(B > (C > $o)),C3:C]: ((((R4 @ A3) @ B3) => (((S3 @ B3) @ C3) => (((((((relcompp @ A) @ B) @ C) @ R4) @ S3) @ A3) @ C3)))))).
% 0.31/1.80 thf(fact_247_relcompp_Oinducts, axiom, (! [B:$tType,A:$tType,C:$tType,R4:(A > (B > $o)),S3:(B > (C > $o)),X1:A,X22:C,P:(A > (C > $o))]: (((((((((relcompp @ A) @ B) @ C) @ R4) @ S3) @ X1) @ X22) => ((! [A5:A,B5:B,C4:C]: ((((R4 @ A5) @ B5) => (((S3 @ B5) @ C4) => ((P @ A5) @ C4))))) => ((P @ X1) @ X22)))))).
% 0.31/1.80 thf(fact_248_relcompp__assoc, axiom, (! [A:$tType,D:$tType,B:$tType,C:$tType,R4:(A > (D > $o)),S3:(D > (C > $o)),T:(C > (B > $o))]: (((((((relcompp @ A) @ C) @ B) @ (((((relcompp @ A) @ D) @ C) @ R4) @ S3)) @ T) = (((((relcompp @ A) @ D) @ B) @ R4) @ (((((relcompp @ D) @ C) @ B) @ S3) @ T)))))).
% 0.31/1.80 thf(fact_249_relcompp__apply, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcompp @ A) @ B) @ C) = (^ [R5:(A > (B > $o)),S5:(B > (C > $o)),A4:A,C5:C]: ((? [B6:B]: ((((R5 @ A4) @ B6) & ((S5 @ B6) @ C5)))))))))).
% 0.31/1.80 thf(fact_250_relcompp_Osimps, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcompp @ A) @ B) @ C) = (^ [R6:(A > (B > $o)),S2:(B > (C > $o)),A1:A,A22:C]: ((? [A4:A,B6:B,C5:C]: (((A1 = A4) & ((A22 = C5) & (((R6 @ A4) @ B6) & ((S2 @ B6) @ C5)))))))))))).
% 0.31/1.80 thf(fact_251_relcompp_Ocases, axiom, (! [A:$tType,B:$tType,C:$tType,R4:(A > (B > $o)),S3:(B > (C > $o)),A12:A,A23:C]: (((((((((relcompp @ A) @ B) @ C) @ R4) @ S3) @ A12) @ A23) => (~ ((! [B5:B]: ((((R4 @ A12) @ B5) => (~ (((S3 @ B5) @ A23)))))))))))).
% 0.31/1.80 thf(fact_252_relcomppE, axiom, (! [A:$tType,B:$tType,C:$tType,R4:(A > (B > $o)),S3:(B > (C > $o)),A3:A,C3:C]: (((((((((relcompp @ A) @ B) @ C) @ R4) @ S3) @ A3) @ C3) => (~ ((! [B5:B]: ((((R4 @ A3) @ B5) => (~ (((S3 @ B5) @ C3)))))))))))).
% 0.31/1.80 thf(fact_253_Powp__def, axiom, (! [A:$tType]: (((powp @ A) = (^ [A6:(A > $o),B4:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ B4) => (A6 @ X2)))))))))).
% 0.31/1.80 thf(fact_254_pos__fun__distr, axiom, (! [E:$tType,C:$tType,A:$tType,B:$tType,D:$tType,F3:$tType,R:(A > (E > $o)),S:(B > (F3 > $o)),R8:(E > (C > $o)),S6:(F3 > (D > $o))]: ((((ord_less_eq @ ((A > B) > ((C > D) > $o))) @ (((((relcompp @ (A > B)) @ (E > F3)) @ (C > D)) @ ((((((bNF_rel_fun @ A) @ E) @ B) @ F3) @ R) @ S)) @ ((((((bNF_rel_fun @ E) @ C) @ F3) @ D) @ R8) @ S6))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ (((((relcompp @ A) @ E) @ C) @ R) @ R8)) @ (((((relcompp @ B) @ F3) @ D) @ S) @ S6)))))).
% 0.31/1.80 thf(fact_255_relcompp__mono, axiom, (! [A:$tType,C:$tType,B:$tType,R9:(A > (B > $o)),R4:(A > (B > $o)),S7:(B > (C > $o)),S3:(B > (C > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R9) @ R4) => ((((ord_less_eq @ (B > (C > $o))) @ S7) @ S3) => (((ord_less_eq @ (A > (C > $o))) @ (((((relcompp @ A) @ B) @ C) @ R9) @ S7)) @ (((((relcompp @ A) @ B) @ C) @ R4) @ S3))))))).
% 0.31/1.80 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 0.31/1.80 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 0.31/1.80 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 0.31/1.80 thf(conj_0, hypothesis, ((((((((bNF_rel_fun @ a) @ a) @ (coinductive_llist @ b)) @ (stream @ b)) @ (^ [Y3:a,Z:a]: ((Y3 = Z)))) @ (^ [X2:(coinductive_llist @ b),Y4:(stream @ b)]: ((X2 = ((coindu1724414836stream @ b) @ Y4))))) @ xb) @ yb)).
% 0.31/1.80 thf(conj_1, conjecture, (xb = (((((comp @ (stream @ b)) @ (coinductive_llist @ b)) @ a) @ (coindu1724414836stream @ b)) @ yb))).
% 0.31/1.80 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.WjgrtYFCGm/DTF2DTF2534.p
% 0.31/1.80 ------------------------
% 0.31/1.87 ---- Cleaned THF ---
% 0.31/1.87 thf(ty_t_Coinductive__List_Ollist,type,
% 0.31/1.87 coinductive_llist: $tType > $tType ).
% 0.31/1.87
% 0.31/1.87 thf(ty_t_Stream_Ostream,type,
% 0.31/1.87 stream: $tType > $tType ).
% 0.31/1.87
% 0.31/1.87 thf(ty_t_Set_Oset,type,
% 0.31/1.87 set: $tType > $tType ).
% 0.31/1.87
% 0.31/1.87 thf(ty_t_itself,type,
% 0.31/1.87 itself: $tType > $tType ).
% 0.31/1.87
% 0.31/1.87 thf(ty_tf_b,type,
% 0.31/1.87 b: $tType ).
% 0.31/1.87
% 0.31/1.87 thf(ty_tf_a,type,
% 0.31/1.87 a: $tType ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.31/1.87 bNF_rel_fun:
% 0.31/1.87 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Basic__BNFs_Opred__fun,type,
% 0.31/1.87 basic_pred_fun:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Oiterates,type,
% 0.31/1.87 coinductive_iterates:
% 0.31/1.87 !>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.31/1.87 coinductive_lfinite:
% 0.31/1.87 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
% 0.31/1.87 coindu1381640503_llist:
% 0.31/1.87 !>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 0.31/1.87 coindu1259883913_llist:
% 0.31/1.87 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Ollist_Ollist__all2,type,
% 0.31/1.87 coindu1486289336t_all2:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
% 0.31/1.87 coindu543516966_llist:
% 0.31/1.87 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.31/1.87 coindu1441602521_llist:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
% 0.31/1.87 coindu1724414836stream:
% 0.31/1.87 !>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
% 0.31/1.87 coindu1183105481stream:
% 0.31/1.87 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Opcr__stream,type,
% 0.31/1.87 coindu773941317stream:
% 0.31/1.87 !>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( coinductive_llist @ C ) > ( stream @ B ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
% 0.31/1.87 coindu2010755910_llist:
% 0.31/1.87 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
% 0.31/1.87 coindu139217191stream:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 0.31/1.87 comple1396247847notone:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Fun_Ocomp,type,
% 0.31/1.87 comp:
% 0.31/1.87 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Fun_Omap__fun,type,
% 0.31/1.87 map_fun:
% 0.31/1.87 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_If,type,
% 0.31/1.87 if:
% 0.31/1.87 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.31/1.87 inf_inf:
% 0.31/1.87 !>[A: $tType] : ( A > A > A ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Lifting_Orel__pred__comp,type,
% 0.31/1.87 rel_pred_comp:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/1.87 ord_less_eq:
% 0.31/1.87 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Partial__Function_Ofun__ord,type,
% 0.31/1.87 partial_fun_ord:
% 0.31/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Partial__Function_Oimg__ord,type,
% 0.31/1.87 partial_img_ord:
% 0.31/1.87 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Partial__Function_Omk__less,type,
% 0.31/1.87 partial_mk_less:
% 0.31/1.87 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Pure_Otype,type,
% 0.31/1.87 type:
% 0.31/1.87 !>[A: $tType] : ( itself @ A ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Quotient_OBex1__rel,type,
% 0.31/1.87 bex1_rel:
% 0.31/1.87 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Quotient_OQuotient3,type,
% 0.31/1.87 quotient3:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Relation_ODomainp,type,
% 0.31/1.87 domainp:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Relation_OPowp,type,
% 0.31/1.87 powp:
% 0.31/1.87 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Relation_Oreflp,type,
% 0.31/1.87 reflp:
% 0.31/1.87 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Relation_Orelcompp,type,
% 0.31/1.87 relcompp:
% 0.31/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Set_OBall,type,
% 0.31/1.87 ball:
% 0.31/1.87 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Set_OBex,type,
% 0.31/1.87 bex:
% 0.31/1.87 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Set_OCollect,type,
% 0.31/1.87 collect:
% 0.31/1.87 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Stream_Ositerate,type,
% 0.31/1.87 siterate:
% 0.31/1.87 !>[A: $tType] : ( ( A > A ) > A > ( stream @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Stream_Ostream_Ocase__stream,type,
% 0.31/1.87 case_stream:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Stream_Ostream_Ocorec__stream,type,
% 0.31/1.87 corec_stream:
% 0.31/1.87 !>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Stream_Ostream_Opred__stream,type,
% 0.31/1.87 pred_stream:
% 0.31/1.87 !>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Stream_Ostream_Ostream__all2,type,
% 0.31/1.87 stream_all2:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream @ A ) > ( stream @ B ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Transfer_Obi__total,type,
% 0.31/1.87 bi_total:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Transfer_Oleft__total,type,
% 0.31/1.87 left_total:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Transfer_Orev__implies,type,
% 0.31/1.87 rev_implies: $o > $o > $o ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Transfer_Oright__total,type,
% 0.31/1.87 right_total:
% 0.31/1.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Transfer_Otransfer__bforall,type,
% 0.31/1.87 transfer_bforall:
% 0.31/1.87 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Transfer_Otransfer__forall,type,
% 0.31/1.87 transfer_forall:
% 0.31/1.87 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_Typedef_Otype__definition,type,
% 0.31/1.87 type_definition:
% 0.31/1.87 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_c_member,type,
% 0.31/1.87 member:
% 0.31/1.87 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_v_xb,type,
% 0.31/1.87 xb: a > ( coinductive_llist @ b ) ).
% 0.31/1.87
% 0.31/1.87 thf(sy_v_yb,type,
% 0.31/1.87 yb: a > ( stream @ b ) ).
% 0.31/1.87
% 0.31/1.87 thf(fact_0_stream_ORep__inject,axiom,
% 0.31/1.87 ! [A: $tType,X: stream @ A,Y: stream @ A] :
% 0.31/1.87 ( ( ( coindu1724414836stream @ A @ X )
% 0.31/1.87 = ( coindu1724414836stream @ A @ Y ) )
% 0.31/1.87 = ( X = Y ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(fact_1_comp__apply,axiom,
% 0.31/1.87 ! [C: $tType,A: $tType,B: $tType] :
% 0.31/1.88 ( ( comp @ B @ A @ C )
% 0.31/1.88 = ( ^ [F: B > A,G: C > B,X2: C] : ( F @ ( G @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_2_rel__funI,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D] :
% 0.31/1.88 ( ! [X3: A,Y2: B] :
% 0.31/1.88 ( ( A2 @ X3 @ Y2 )
% 0.31/1.88 => ( B2 @ ( F2 @ X3 ) @ ( G2 @ Y2 ) ) )
% 0.31/1.88 => ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_3_fun_Orel__map_I2_J,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G2: B > C,Y: D > B] :
% 0.31/1.88 ( ( bNF_rel_fun @ D @ D @ A @ C
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Sa
% 0.31/1.88 @ X
% 0.31/1.88 @ ( comp @ B @ C @ D @ G2 @ Y ) )
% 0.31/1.88 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ ^ [X2: A,Y4: B] : ( Sa @ X2 @ ( G2 @ Y4 ) )
% 0.31/1.88 @ X
% 0.31/1.88 @ Y ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_4_fun_Orel__map_I1_J,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
% 0.31/1.88 ( ( bNF_rel_fun @ D @ D @ C @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Sb
% 0.31/1.88 @ ( comp @ A @ C @ D @ I @ X )
% 0.31/1.88 @ Y )
% 0.31/1.88 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
% 0.31/1.88 @ X
% 0.31/1.88 @ Y ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_5_o__rsp_I2_J,axiom,
% 0.31/1.88 ! [E: $tType,F3: $tType,H: $tType,G3: $tType,R1: E > F3 > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( G3 > H ) @ ( G3 > H ) @ ( ( E > G3 ) > E > H ) @ ( ( F3 > G3 ) > F3 > H )
% 0.31/1.88 @ ^ [Y3: G3 > H,Z: G3 > H] : ( Y3 = Z )
% 0.31/1.88 @ ( bNF_rel_fun @ ( E > G3 ) @ ( F3 > G3 ) @ ( E > H ) @ ( F3 > H )
% 0.31/1.88 @ ( bNF_rel_fun @ E @ F3 @ G3 @ G3 @ R1
% 0.31/1.88 @ ^ [Y3: G3,Z: G3] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ E @ F3 @ H @ H @ R1
% 0.31/1.88 @ ^ [Y3: H,Z: H] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( comp @ G3 @ H @ E )
% 0.31/1.88 @ ( comp @ G3 @ H @ F3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_6_o__rsp_I1_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R3: B > D > $o,R1: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R2 @ R3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R1 @ R3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_7_fun_Omap__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,D: $tType,G3: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G3 > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > B ) @ ( F3 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G3 @ Rb @ Sd )
% 0.31/1.88 @ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G3 )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ A @ F3
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Rb )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ B @ G3
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Sd ) )
% 0.31/1.88 @ ( comp @ A @ B @ D )
% 0.31/1.88 @ ( comp @ F3 @ G3 @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_8_comp__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B2: A > C > $o,C2: B > D > $o,A2: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B2 @ C2 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A2 @ B2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A2 @ C2 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_9_K__record__comp,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,C3: B,F2: A > C] :
% 0.31/1.88 ( ( comp @ C @ B @ A
% 0.31/1.88 @ ^ [X2: C] : C3
% 0.31/1.88 @ F2 )
% 0.31/1.88 = ( ^ [X2: A] : C3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_10_fun_Omap__ident,axiom,
% 0.31/1.88 ! [A: $tType,D: $tType,T: D > A] :
% 0.31/1.88 ( ( comp @ A @ A @ D
% 0.31/1.88 @ ^ [X2: A] : X2
% 0.31/1.88 @ T )
% 0.31/1.88 = T ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_11_stream__from__llist__setup_Ocr__stream__def,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( coindu1183105481stream @ A )
% 0.31/1.88 = ( ^ [X2: coinductive_llist @ A,Y4: stream @ A] :
% 0.31/1.88 ( X2
% 0.31/1.88 = ( coindu1724414836stream @ A @ Y4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_12_llist__of__stream__corec__stream,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,SHD: B > A,EndORmore: B > $o,STL_more: B > ( stream @ A ),STL_end: B > B,X: B] :
% 0.31/1.88 ( ( coindu1724414836stream @ A @ ( corec_stream @ B @ A @ SHD @ EndORmore @ STL_more @ STL_end @ X ) )
% 0.31/1.88 = ( coindu1259883913_llist @ B @ A
% 0.31/1.88 @ ^ [Uu: B] : $false
% 0.31/1.88 @ SHD
% 0.31/1.88 @ EndORmore
% 0.31/1.88 @ ( comp @ ( stream @ A ) @ ( coinductive_llist @ A ) @ B @ ( coindu1724414836stream @ A ) @ STL_more )
% 0.31/1.88 @ STL_end
% 0.31/1.88 @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_13_rel__fun__def__butlast,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,C: $tType,E: $tType,F3: $tType,A: $tType,R: A > B > $o,S: C > E > $o,T2: D > F3 > $o,F2: A > C > D,G2: B > E > F3] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T2 ) @ F2 @ G2 )
% 0.31/1.88 = ( ! [X2: A,Y4: B] :
% 0.31/1.88 ( ( R @ X2 @ Y4 )
% 0.31/1.88 => ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T2 @ ( F2 @ X2 ) @ ( G2 @ Y4 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_14_If__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 0.31/1.88 @ ( if @ A )
% 0.31/1.88 @ ( if @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_15_fun_Orel__transfer,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
% 0.31/1.88 @ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ A @ C
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Sa )
% 0.31/1.88 @ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ B @ E
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Sc )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ C @ E
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_16_fun_Orel__refl,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,Ra: B > B > $o,X: D > B] :
% 0.31/1.88 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.31/1.88 => ( bNF_rel_fun @ D @ D @ B @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ Ra
% 0.31/1.88 @ X
% 0.31/1.88 @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_17_fun_Orel__eq,axiom,
% 0.31/1.88 ! [A: $tType,D: $tType] :
% 0.31/1.88 ( ( bNF_rel_fun @ D @ D @ A @ A
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.31/1.88 = ( ^ [Y3: D > A,Z: D > A] : ( Y3 = Z ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_18_rel__fun__mono_H,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F2: A > C,G2: B > D] :
% 0.31/1.88 ( ! [X3: A,Y2: B] :
% 0.31/1.88 ( ( Y5 @ X3 @ Y2 )
% 0.31/1.88 => ( X4 @ X3 @ Y2 ) )
% 0.31/1.88 => ( ! [X3: C,Y2: D] :
% 0.31/1.88 ( ( A2 @ X3 @ Y2 )
% 0.31/1.88 => ( B2 @ X3 @ Y2 ) )
% 0.31/1.88 => ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F2 @ G2 )
% 0.31/1.88 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F2 @ G2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_19_rel__fun__mono,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F2: A > C,G2: B > D,Y5: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F2 @ G2 )
% 0.31/1.88 => ( ! [X3: A,Y2: B] :
% 0.31/1.88 ( ( Y5 @ X3 @ Y2 )
% 0.31/1.88 => ( X4 @ X3 @ Y2 ) )
% 0.31/1.88 => ( ! [X3: C,Y2: D] :
% 0.31/1.88 ( ( A2 @ X3 @ Y2 )
% 0.31/1.88 => ( B2 @ X3 @ Y2 ) )
% 0.31/1.88 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F2 @ G2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_20_let__rsp,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
% 0.31/1.88 ( 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 )
% 0.31/1.88 @ ^ [S2: A,F: A > C] : ( F @ S2 )
% 0.31/1.88 @ ^ [S2: B,F: B > D] : ( F @ S2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_21_rel__funD,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 )
% 0.31/1.88 => ( ( A2 @ X @ Y )
% 0.31/1.88 => ( B2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_22_rewriteR__comp__comp2,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H2: A > C,R12: D > B,R22: A > D,F2: B > E,L: D > E] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ G2 @ H2 )
% 0.31/1.88 = ( comp @ D @ B @ A @ R12 @ R22 ) )
% 0.31/1.88 => ( ( ( comp @ B @ E @ D @ F2 @ R12 )
% 0.31/1.88 = L )
% 0.31/1.88 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H2 )
% 0.31/1.88 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_23_rewriteL__comp__comp2,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H2: E > A,R4: E > D] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 0.31/1.88 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.31/1.88 => ( ( ( comp @ A @ D @ E @ L2 @ H2 )
% 0.31/1.88 = R4 )
% 0.31/1.88 => ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H2 ) )
% 0.31/1.88 = ( comp @ D @ B @ E @ L1 @ R4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_24_rewriteR__comp__comp,axiom,
% 0.31/1.88 ! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H2: A > C,R4: A > B,F2: B > D] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ G2 @ H2 )
% 0.31/1.88 = R4 )
% 0.31/1.88 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H2 )
% 0.31/1.88 = ( comp @ B @ D @ A @ F2 @ R4 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_25_rewriteL__comp__comp,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H2: D > A] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ F2 @ G2 )
% 0.31/1.88 = L )
% 0.31/1.88 => ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H2 ) )
% 0.31/1.88 = ( comp @ A @ B @ D @ L @ H2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_26_fun_Omap__comp,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
% 0.31/1.88 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
% 0.31/1.88 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_27_comp__apply__eq,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X: C,H2: D > A,K: C > D] :
% 0.31/1.88 ( ( ( F2 @ ( G2 @ X ) )
% 0.31/1.88 = ( H2 @ ( K @ X ) ) )
% 0.31/1.88 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X )
% 0.31/1.88 = ( comp @ D @ A @ C @ H2 @ K @ X ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_28_comp__eq__dest__lhs,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.31/1.88 = C3 )
% 0.31/1.88 => ( ( A3 @ ( B3 @ V ) )
% 0.31/1.88 = ( C3 @ V ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_29_comp__eq__elim,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D2: A > D] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.31/1.88 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.31/1.88 => ! [V2: A] :
% 0.31/1.88 ( ( A3 @ ( B3 @ V2 ) )
% 0.31/1.88 = ( C3 @ ( D2 @ V2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_30_comp__eq__dest,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D2: A > D,V: A] :
% 0.31/1.88 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.31/1.88 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.31/1.88 => ( ( A3 @ ( B3 @ V ) )
% 0.31/1.88 = ( C3 @ ( D2 @ V ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_31_comp__assoc,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H2: A > C] :
% 0.31/1.88 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H2 )
% 0.31/1.88 = ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_32_comp__def,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType] :
% 0.31/1.88 ( ( comp @ B @ C @ A )
% 0.31/1.88 = ( ^ [F: B > C,G: A > B,X2: A] : ( F @ ( G @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_33_Let__transfer,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( 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 )
% 0.31/1.88 @ ^ [S2: A,F: A > C] : ( F @ S2 )
% 0.31/1.88 @ ^ [S2: B,F: B > D] : ( F @ S2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_34_stream_Ocorec__disc,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType] :
% 0.31/1.88 ( ( corec_stream @ C @ A )
% 0.31/1.88 = ( corec_stream @ C @ A ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_35_type__copy__map__cong0,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X: C,N: D > A,H2: C > D,F2: A > E] :
% 0.31/1.88 ( ( ( M @ ( G2 @ X ) )
% 0.31/1.88 = ( N @ ( H2 @ X ) ) )
% 0.31/1.88 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X )
% 0.31/1.88 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H2 @ X ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_36_comp__cong,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X: C,F4: D > A,G4: E > D,X5: E] :
% 0.31/1.88 ( ( ( F2 @ ( G2 @ X ) )
% 0.31/1.88 = ( F4 @ ( G4 @ X5 ) ) )
% 0.31/1.88 => ( ( comp @ B @ A @ C @ F2 @ G2 @ X )
% 0.31/1.88 = ( comp @ D @ A @ E @ F4 @ G4 @ X5 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_37_rel__funD2,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F2: A > B,G2: A > C,X: A] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F2 @ G2 )
% 0.31/1.88 => ( ( A2 @ X @ X )
% 0.31/1.88 => ( B2 @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_38_apply__rsp_H,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F2 @ G2 )
% 0.31/1.88 => ( ( R1 @ X @ Y )
% 0.31/1.88 => ( R2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_39_rel__funE,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 )
% 0.31/1.88 => ( ( A2 @ X @ Y )
% 0.31/1.88 => ( B2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_40_stream__from__llist__setup_Ostream_Opcr__cr__eq,axiom,
% 0.31/1.88 ! [D: $tType] :
% 0.31/1.88 ( ( coindu773941317stream @ D @ D
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z ) )
% 0.31/1.88 = ( coindu1183105481stream @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_41_stream__of__llist__llist__of__stream,axiom,
% 0.31/1.88 ! [A: $tType,Xs: stream @ A] :
% 0.31/1.88 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
% 0.31/1.88 = Xs ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_42_stream_ORep__inverse,axiom,
% 0.31/1.88 ! [A: $tType,X: stream @ A] :
% 0.31/1.88 ( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
% 0.31/1.88 = X ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_43_map__fun__parametric,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,E: $tType,F3: $tType,H: $tType,G3: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G3 > $o,D3: F3 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F3 ) > ( B > E ) > A > F3 ) @ ( ( G3 > H ) > ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F3 ) @ ( G3 > H ) @ ( ( B > E ) > A > F3 ) @ ( ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ E @ G3 @ F3 @ H @ C2 @ D3 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G3 ) @ ( A > F3 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G3 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F3 @ H @ A2 @ D3 ) ) ) @ ( map_fun @ A @ B @ E @ F3 ) @ ( map_fun @ C @ D @ G3 @ H ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_44_map__fun__apply,axiom,
% 0.31/1.88 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 0.31/1.88 ( ( map_fun @ B @ C @ D @ A )
% 0.31/1.88 = ( ^ [F: B > C,G: D > A,H3: C > D,X2: B] : ( G @ ( H3 @ ( F @ X2 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/1.88 ! [A: $tType,A3: A,P: A > $o] :
% 0.31/1.88 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.31/1.88 = ( P @ A3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/1.88 ! [A: $tType,A2: set @ A] :
% 0.31/1.88 ( ( collect @ A
% 0.31/1.88 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.31/1.88 = A2 ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_47_Collect__cong,axiom,
% 0.31/1.88 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( P @ X3 )
% 0.31/1.88 = ( Q @ X3 ) )
% 0.31/1.88 => ( ( collect @ A @ P )
% 0.31/1.88 = ( collect @ A @ Q ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_48_ext,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( F2 @ X3 )
% 0.31/1.88 = ( G2 @ X3 ) )
% 0.31/1.88 => ( F2 = G2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_49_map__fun__def,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,A: $tType,C: $tType] :
% 0.31/1.88 ( ( map_fun @ C @ A @ B @ D )
% 0.31/1.88 = ( ^ [F: C > A,G: B > D,H3: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G @ H3 ) @ F ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_50_map__fun_Ocomp,axiom,
% 0.31/1.88 ! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F2: E > C,G2: D > F3,H2: C > A,I: B > D] :
% 0.31/1.88 ( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F2 @ G2 ) @ ( map_fun @ C @ A @ B @ D @ H2 @ I ) )
% 0.31/1.88 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F2 ) @ ( comp @ D @ F3 @ B @ G2 @ I ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_51_map__fun_Ocompositionality,axiom,
% 0.31/1.88 ! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F2: E > C,G2: D > F3,H2: C > A,I: B > D,Fun: A > B] :
% 0.31/1.88 ( ( map_fun @ E @ C @ D @ F3 @ F2 @ G2 @ ( map_fun @ C @ A @ B @ D @ H2 @ I @ Fun ) )
% 0.31/1.88 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F2 ) @ ( comp @ D @ F3 @ B @ G2 @ I ) @ Fun ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_52_fun_Opred__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,D: $tType,R: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ R )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( basic_pred_fun @ D @ A
% 0.31/1.88 @ ^ [Uu: D] : $true )
% 0.31/1.88 @ ( basic_pred_fun @ D @ B
% 0.31/1.88 @ ^ [Uu: D] : $true ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_53_if__rsp,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
% 0.31/1.88 @ ( if @ A )
% 0.31/1.88 @ ( if @ A ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_54_stream_Ocorec__transfer,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > $o ) > ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S @ R )
% 0.31/1.88 @ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( ( C > C ) > C > ( stream @ A ) ) @ ( ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S @ ( stream_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S @ ( stream_all2 @ A @ B @ R ) ) ) ) )
% 0.31/1.88 @ ( corec_stream @ C @ A )
% 0.31/1.88 @ ( corec_stream @ D @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_55_unfold__stream__transfer,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > B ) @ ( A > B ) @ ( ( A > A ) > A > ( coinductive_llist @ B ) ) @ ( ( A > A ) > A > ( stream @ B ) )
% 0.31/1.88 @ ^ [Y3: A > B,Z: A > B] : ( Y3 = Z )
% 0.31/1.88 @ ( bNF_rel_fun @ ( A > A ) @ ( A > A ) @ ( A > ( coinductive_llist @ B ) ) @ ( A > ( stream @ B ) )
% 0.31/1.88 @ ^ [Y3: A > A,Z: A > A] : ( Y3 = Z )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ A @ ( coinductive_llist @ B ) @ ( stream @ B )
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ ( coindu773941317stream @ B @ B
% 0.31/1.88 @ ^ [Y3: B,Z: B] : ( Y3 = Z ) ) ) )
% 0.31/1.88 @ ( coindu1441602521_llist @ A @ B
% 0.31/1.88 @ ^ [Uu: A] : $false )
% 0.31/1.88 @ ( coindu139217191stream @ A @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_56_fun__ord__parametric,axiom,
% 0.31/1.88 ! [C: $tType,D: $tType,A: $tType,B: $tType,F3: $tType,E: $tType,C2: A > B > $o,A2: C > E > $o,B2: D > F3 > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ C2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F3 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F3 ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F3 > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ D @ F3 @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F3 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C2 @ A2 )
% 0.31/1.88 @ ( bNF_rel_fun @ ( A > D ) @ ( B > F3 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F3 @ C2 @ B2 )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( partial_fun_ord @ C @ D @ A )
% 0.31/1.88 @ ( partial_fun_ord @ E @ F3 @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_57_llist__of__stream__stream__of__llist,axiom,
% 0.31/1.88 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.31/1.88 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.31/1.88 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
% 0.31/1.88 = Xs ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_58_cr__streamI,axiom,
% 0.31/1.88 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.31/1.88 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.31/1.88 => ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_59_stream_OAbs__inverse,axiom,
% 0.31/1.88 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.31/1.88 ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.31/1.88 => ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Y ) )
% 0.31/1.88 = Y ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_60_llist__of__stream__unfold__stream,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,SHD: B > A,STL: B > B,X: B] :
% 0.31/1.88 ( ( coindu1724414836stream @ A @ ( coindu139217191stream @ B @ A @ SHD @ STL @ X ) )
% 0.31/1.88 = ( coindu1441602521_llist @ B @ A
% 0.31/1.88 @ ^ [Uu: B] : $false
% 0.31/1.88 @ SHD
% 0.31/1.88 @ STL
% 0.31/1.88 @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_61_stream_Odisc__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ^ [Stream: stream @ A] : ( Stream = Stream )
% 0.31/1.88 @ ^ [Stream: stream @ B] : ( Stream = Stream ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_62_stream_Orel__refl,axiom,
% 0.31/1.88 ! [B: $tType,Ra: B > B > $o,X: stream @ B] :
% 0.31/1.88 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.31/1.88 => ( stream_all2 @ B @ B @ Ra @ X @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_63_stream_Orel__eq,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( stream_all2 @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.31/1.88 = ( ^ [Y3: stream @ A,Z: stream @ A] : ( Y3 = Z ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_64_stream_Obi__total__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ R )
% 0.31/1.88 => ( bi_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_65_bi__total__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( bi_total @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > B > $o] :
% 0.31/1.88 ( ! [X2: A] :
% 0.31/1.88 ( ^ [P2: B > $o] :
% 0.31/1.88 ? [X6: B] : ( P2 @ X6 )
% 0.31/1.88 @ ( R5 @ X2 ) )
% 0.31/1.88 & ! [Y4: B] :
% 0.31/1.88 ? [X2: A] : ( R5 @ X2 @ Y4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_66_bi__total__eq,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( bi_total @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_67_Quotient3__rep__reflp,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_68_Quotient3__rep__abs,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ R4 @ R4 )
% 0.31/1.88 => ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_69_Quotient3__rel__rep,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B,B3: B] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 0.31/1.88 = ( A3 = B3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_70_Quotient3__rel__abs,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ R4 @ S3 )
% 0.31/1.88 => ( ( Abs @ R4 )
% 0.31/1.88 = ( Abs @ S3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_71_Quotient3__abs__rep,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( Abs @ ( Rep @ A3 ) )
% 0.31/1.88 = A3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_72_rep__abs__rsp__left,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ X1 @ X22 )
% 0.31/1.88 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_73_Quotient3__refl2,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ R4 @ S3 )
% 0.31/1.88 => ( R @ S3 @ S3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_74_Quotient3__refl1,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ R4 @ S3 )
% 0.31/1.88 => ( R @ R4 @ R4 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_75_Quotient3__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( ( R @ R4 @ R4 )
% 0.31/1.88 & ( R @ S3 @ S3 )
% 0.31/1.88 & ( ( Abs @ R4 )
% 0.31/1.88 = ( Abs @ S3 ) ) )
% 0.31/1.88 = ( R @ R4 @ S3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_76_Quotient3__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( quotient3 @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > A > $o,Abs2: A > B,Rep2: B > A] :
% 0.31/1.88 ( ! [A4: B] :
% 0.31/1.88 ( ( Abs2 @ ( Rep2 @ A4 ) )
% 0.31/1.88 = A4 )
% 0.31/1.88 & ! [A4: B] : ( R5 @ ( Rep2 @ A4 ) @ ( Rep2 @ A4 ) )
% 0.31/1.88 & ! [R6: A,S2: A] :
% 0.31/1.88 ( ( R5 @ R6 @ S2 )
% 0.31/1.88 = ( ( R5 @ R6 @ R6 )
% 0.31/1.88 & ( R5 @ S2 @ S2 )
% 0.31/1.88 & ( ( Abs2 @ R6 )
% 0.31/1.88 = ( Abs2 @ S2 ) ) ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_77_rep__abs__rsp,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ X1 @ X22 )
% 0.31/1.88 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_78_pred__funI,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,A2: A > $o,B2: B > $o,F2: A > B] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( A2 @ X3 )
% 0.31/1.88 => ( B2 @ ( F2 @ X3 ) ) )
% 0.31/1.88 => ( basic_pred_fun @ A @ B @ A2 @ B2 @ F2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_79_equals__rsp,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( R @ Xa @ Xb )
% 0.31/1.88 => ( ( R @ Ya @ Yb )
% 0.31/1.88 => ( ( R @ Xa @ Ya )
% 0.31/1.88 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_80_Quotient3I,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 0.31/1.88 ( ! [A5: A] :
% 0.31/1.88 ( ( Abs @ ( Rep @ A5 ) )
% 0.31/1.88 = A5 )
% 0.31/1.88 => ( ! [A5: A] : ( R @ ( Rep @ A5 ) @ ( Rep @ A5 ) )
% 0.31/1.88 => ( ! [R7: B,S4: B] :
% 0.31/1.88 ( ( R @ R7 @ S4 )
% 0.31/1.88 = ( ( R @ R7 @ R7 )
% 0.31/1.88 & ( R @ S4 @ S4 )
% 0.31/1.88 & ( ( Abs @ R7 )
% 0.31/1.88 = ( Abs @ S4 ) ) ) )
% 0.31/1.88 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_81_cond__prs,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A3: $o,B3: B,C3: B] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.31/1.88 => ( ( A3
% 0.31/1.88 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.31/1.88 = B3 ) )
% 0.31/1.88 & ( ~ A3
% 0.31/1.88 => ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.31/1.88 = C3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_82_fun_Opred__True,axiom,
% 0.31/1.88 ! [A: $tType,D: $tType] :
% 0.31/1.88 ( ( basic_pred_fun @ D @ A
% 0.31/1.88 @ ^ [Uu: D] : $true
% 0.31/1.88 @ ^ [Uu: A] : $true )
% 0.31/1.88 = ( ^ [Uu: D > A] : $true ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_83_fun__quotient3,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.31/1.88 => ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
% 0.31/1.88 => ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs22 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_84_apply__rspQ3_H_H,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S: C > C > $o,F2: A > C,X: B] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S @ F2 @ F2 )
% 0.31/1.88 => ( S @ ( F2 @ ( Rep @ X ) ) @ ( F2 @ ( Rep @ X ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_85_apply__rspQ3,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F2: A > C,G2: A > C,X: A,Y: A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F2 @ G2 )
% 0.31/1.88 => ( ( R1 @ X @ Y )
% 0.31/1.88 => ( R2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_86_let__prs,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.31/1.88 => ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep23 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs23 @ Rep12 ) @ Abs12 )
% 0.31/1.88 @ ^ [S2: C,F: C > A] : ( F @ S2 ) )
% 0.31/1.88 = ( ^ [S2: D,F: D > B] : ( F @ S2 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_87_lfinite__llist__of__stream,axiom,
% 0.31/1.88 ! [A: $tType,Xs: stream @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_88_lambda__prs1,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,F2: B > D] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.31/1.88 => ( ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs23 @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep23 @ F2 ) )
% 0.31/1.88 = F2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_89_lambda__prs,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,F2: B > D] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.31/1.88 => ( ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs23
% 0.31/1.88 @ ^ [X2: A] : ( Rep23 @ ( F2 @ ( Abs12 @ X2 ) ) ) )
% 0.31/1.88 = F2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_90_stream_ORep__induct,axiom,
% 0.31/1.88 ! [A: $tType,Y: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.31/1.88 ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.31/1.88 => ( ! [X3: stream @ A] : ( P @ ( coindu1724414836stream @ A @ X3 ) )
% 0.31/1.88 => ( P @ Y ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_91_stream_ORep__cases,axiom,
% 0.31/1.88 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.31/1.88 ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.31/1.88 => ~ ! [X3: stream @ A] :
% 0.31/1.88 ( Y
% 0.31/1.88 != ( coindu1724414836stream @ A @ X3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_92_stream_ORep,axiom,
% 0.31/1.88 ! [A: $tType,X: stream @ A] :
% 0.31/1.88 ( member @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A @ X )
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_93_stream_OAbs__cases,axiom,
% 0.31/1.88 ! [A: $tType,X: stream @ A] :
% 0.31/1.88 ~ ! [Y2: coinductive_llist @ A] :
% 0.31/1.88 ( ( X
% 0.31/1.88 = ( coindu2010755910_llist @ A @ Y2 ) )
% 0.31/1.88 => ~ ( member @ ( coinductive_llist @ A ) @ Y2
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_94_stream_OAbs__induct,axiom,
% 0.31/1.88 ! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
% 0.31/1.88 ( ! [Y2: coinductive_llist @ A] :
% 0.31/1.88 ( ( member @ ( coinductive_llist @ A ) @ Y2
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.31/1.88 => ( P @ ( coindu2010755910_llist @ A @ Y2 ) ) )
% 0.31/1.88 => ( P @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_95_stream_OAbs__inject,axiom,
% 0.31/1.88 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
% 0.31/1.88 ( ( member @ ( coinductive_llist @ A ) @ X
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.31/1.88 => ( ( member @ ( coinductive_llist @ A ) @ Y
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) )
% 0.31/1.88 => ( ( ( coindu2010755910_llist @ A @ X )
% 0.31/1.88 = ( coindu2010755910_llist @ A @ Y ) )
% 0.31/1.88 = ( X = Y ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_96_unfold__stream__ltl__unroll,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B3: B] :
% 0.31/1.88 ( ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B3 ) )
% 0.31/1.88 = ( coindu139217191stream @ B @ A @ ( comp @ B @ A @ B @ SHD @ STL ) @ STL @ B3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_97_stream_Orel__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( stream @ A ) > ( stream @ B ) > $o ) @ ( ( stream @ C ) > ( stream @ D ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.31/1.88 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ C ) @ ( ( stream @ B ) > $o ) @ ( ( stream @ D ) > $o ) @ ( stream_all2 @ A @ C @ Sa )
% 0.31/1.88 @ ( bNF_rel_fun @ ( stream @ B ) @ ( stream @ D ) @ $o @ $o @ ( stream_all2 @ B @ D @ Sc )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( stream_all2 @ A @ B )
% 0.31/1.88 @ ( stream_all2 @ C @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_98_o__prs_I1_J,axiom,
% 0.31/1.88 ! [C: $tType,E: $tType,A: $tType,B: $tType,F3: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,R3: E > E > $o,Abs3: E > F3,Rep3: F3 > E] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.31/1.88 => ( ( quotient3 @ E @ F3 @ R3 @ Abs3 @ Rep3 )
% 0.31/1.88 => ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs23 @ Rep3 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep23 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep12 @ Abs3 ) ) @ ( comp @ C @ E @ A ) )
% 0.31/1.88 = ( comp @ D @ F3 @ B ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_99_fun_Omap__cong__pred,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F2: A > B,G2: A > B] :
% 0.31/1.88 ( ( X = Ya )
% 0.31/1.88 => ( ( basic_pred_fun @ D @ A
% 0.31/1.88 @ ^ [Uu: D] : $true
% 0.31/1.88 @ ^ [Z2: A] :
% 0.31/1.88 ( ( F2 @ Z2 )
% 0.31/1.88 = ( G2 @ Z2 ) )
% 0.31/1.88 @ Ya )
% 0.31/1.88 => ( ( comp @ A @ B @ D @ F2 @ X )
% 0.31/1.88 = ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_100_fun_Opred__map,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,D: $tType,Q: B > $o,F2: A > B,X: D > A] :
% 0.31/1.88 ( ( basic_pred_fun @ D @ B
% 0.31/1.88 @ ^ [Uu: D] : $true
% 0.31/1.88 @ Q
% 0.31/1.88 @ ( comp @ A @ B @ D @ F2 @ X ) )
% 0.31/1.88 = ( basic_pred_fun @ D @ A
% 0.31/1.88 @ ^ [Uu: D] : $true
% 0.31/1.88 @ ( comp @ B @ $o @ A @ Q @ F2 )
% 0.31/1.88 @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_101_quot__rel__rsp,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.31/1.88 => ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
% 0.31/1.88 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ R
% 0.31/1.88 @ R ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_102_corec__llist__never__stop,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coinductive_llist @ A ),LTL: B > B,X: B] :
% 0.31/1.88 ( ( coindu1259883913_llist @ B @ A @ IS_LNIL @ LHD
% 0.31/1.88 @ ^ [Uu: B] : $false
% 0.31/1.88 @ MORE
% 0.31/1.88 @ LTL
% 0.31/1.88 @ X )
% 0.31/1.88 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_103_unfold__llist__ltl__unroll,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B] :
% 0.31/1.88 ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) )
% 0.31/1.88 = ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_104_monotone__parametric,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( comple1396247847notone @ A @ C )
% 0.31/1.88 @ ( comple1396247847notone @ B @ D ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_105_stream_Otype__definition__axioms,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( type_definition @ ( stream @ A ) @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A ) @ ( coindu2010755910_llist @ A )
% 0.31/1.88 @ ( collect @ ( coinductive_llist @ A )
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_106_relcompp__transfer,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ B2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
% 0.31/1.88 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) ) )
% 0.31/1.88 @ ( relcompp @ C @ A @ E )
% 0.31/1.88 @ ( relcompp @ D @ B @ F3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_107_bex1__rel__rsp,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( bex1_rel @ A @ R )
% 0.31/1.88 @ ( bex1_rel @ A @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_108_left__total__parametric,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( ( bi_total @ C @ D @ B2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( left_total @ A @ C )
% 0.31/1.88 @ ( left_total @ B @ D ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_109_fun_Orel__compp,axiom,
% 0.31/1.88 ! [A: $tType,D: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.31/1.88 ( ( bNF_rel_fun @ D @ D @ A @ C
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ ( relcompp @ A @ B @ C @ R @ S ) )
% 0.31/1.88 = ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ R )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ B @ C
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ S ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_110_conj__comp__iff,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G2: A > B] :
% 0.31/1.88 ( ( comp @ B @ $o @ A
% 0.31/1.88 @ ^ [X2: B] :
% 0.31/1.88 ( ( P @ X2 )
% 0.31/1.88 & ( Q @ X2 ) )
% 0.31/1.88 @ G2 )
% 0.31/1.88 = ( ^ [X2: A] :
% 0.31/1.88 ( ( comp @ B @ $o @ A @ P @ G2 @ X2 )
% 0.31/1.88 & ( comp @ B @ $o @ A @ Q @ G2 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_111_stream_Orel__compp,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.31/1.88 ( ( stream_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
% 0.31/1.88 = ( relcompp @ ( stream @ A ) @ ( stream @ B ) @ ( stream @ C ) @ ( stream_all2 @ A @ B @ R ) @ ( stream_all2 @ B @ C @ S ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_112_stream_Oleft__total__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ R )
% 0.31/1.88 => ( left_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_113_eq__comp__r,axiom,
% 0.31/1.88 ! [A: $tType,R: A > A > $o] :
% 0.31/1.88 ( ( relcompp @ A @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ ( relcompp @ A @ A @ A @ R
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) )
% 0.31/1.88 = R ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_114_bex1__rel__aux,axiom,
% 0.31/1.88 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 0.31/1.88 ( ! [Xa2: A,Ya2: A] :
% 0.31/1.88 ( ( R @ Xa2 @ Ya2 )
% 0.31/1.88 => ( ( X @ Xa2 )
% 0.31/1.88 = ( Y @ Ya2 ) ) )
% 0.31/1.88 => ( ( bex1_rel @ A @ R @ X )
% 0.31/1.88 => ( bex1_rel @ A @ R @ Y ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_115_bex1__rel__aux2,axiom,
% 0.31/1.88 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 0.31/1.88 ( ! [Xa2: A,Ya2: A] :
% 0.31/1.88 ( ( R @ Xa2 @ Ya2 )
% 0.31/1.88 => ( ( X @ Xa2 )
% 0.31/1.88 = ( Y @ Ya2 ) ) )
% 0.31/1.88 => ( ( bex1_rel @ A @ R @ Y )
% 0.31/1.88 => ( bex1_rel @ A @ R @ X ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_116_left__totalE,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ R )
% 0.31/1.88 => ! [X7: A] :
% 0.31/1.88 ? [X12: B] : ( R @ X7 @ X12 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_117_left__totalI,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ? [X13: B] : ( R @ X3 @ X13 )
% 0.31/1.88 => ( left_total @ A @ B @ R ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_118_left__total__OO,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ R )
% 0.31/1.88 => ( ( left_total @ B @ C @ S )
% 0.31/1.88 => ( left_total @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_119_left__total__eq,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( left_total @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_120_left__total__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( left_total @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > B > $o] :
% 0.31/1.88 ! [X2: A] :
% 0.31/1.88 ( ^ [P2: B > $o] :
% 0.31/1.88 ? [X6: B] : ( P2 @ X6 )
% 0.31/1.88 @ ( R5 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_121_bi__total__OO,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( ( bi_total @ B @ C @ B2 )
% 0.31/1.88 => ( bi_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_122_typedef__rep__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T2: A > B > $o] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( T2
% 0.31/1.88 = ( ^ [X2: A,Y4: B] :
% 0.31/1.88 ( X2
% 0.31/1.88 = ( Rep @ Y4 ) ) ) )
% 0.31/1.88 => ( bNF_rel_fun @ A @ B @ A @ A @ T2
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ ^ [X2: A] : X2
% 0.31/1.88 @ Rep ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_123_OOO__quotient3,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: B > B > $o,Abs23: B > C,Rep23: C > B,R23: A > A > $o] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( quotient3 @ B @ C @ R2 @ Abs23 @ Rep23 )
% 0.31/1.88 => ( ! [X3: A,Y2: A] :
% 0.31/1.88 ( ( R23 @ X3 @ Y2 )
% 0.31/1.88 => ( ( R1 @ X3 @ X3 )
% 0.31/1.88 => ( ( R1 @ Y2 @ Y2 )
% 0.31/1.88 => ( R2 @ ( Abs12 @ X3 ) @ ( Abs12 @ Y2 ) ) ) ) )
% 0.31/1.88 => ( ! [X3: B,Y2: B] :
% 0.31/1.88 ( ( R2 @ X3 @ Y2 )
% 0.31/1.88 => ( R23 @ ( Rep12 @ X3 ) @ ( Rep12 @ Y2 ) ) )
% 0.31/1.88 => ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R23 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs23 @ Abs12 ) @ ( comp @ B @ A @ C @ Rep12 @ Rep23 ) ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_124_OOO__eq__quotient3,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,Abs23: B > C,Rep23: C > B] :
% 0.31/1.88 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.31/1.88 => ( ( quotient3 @ B @ C
% 0.31/1.88 @ ^ [Y3: B,Z: B] : ( Y3 = Z )
% 0.31/1.88 @ Abs23
% 0.31/1.88 @ Rep23 )
% 0.31/1.88 => ( quotient3 @ A @ C
% 0.31/1.88 @ ( relcompp @ A @ A @ A @ R1
% 0.31/1.88 @ ( relcompp @ A @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ R1 ) )
% 0.31/1.88 @ ( comp @ B @ C @ A @ Abs23 @ Abs12 )
% 0.31/1.88 @ ( comp @ B @ A @ C @ Rep12 @ Rep23 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_125_monotone__applyI,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,C: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,X: C] :
% 0.31/1.88 ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F5 )
% 0.31/1.88 => ( comple1396247847notone @ ( C > A ) @ B @ ( partial_fun_ord @ A @ A @ C @ Orda ) @ Ordb
% 0.31/1.88 @ ^ [F: C > A] : ( F5 @ ( F @ X ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_126_monotone__fun__ord__apply,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,Orda: A > A > $o,Ordb: C > C > $o,F2: A > B > C] :
% 0.31/1.88 ( ( comple1396247847notone @ A @ ( B > C ) @ Orda @ ( partial_fun_ord @ C @ C @ B @ Ordb ) @ F2 )
% 0.31/1.88 = ( ! [X2: B] :
% 0.31/1.88 ( comple1396247847notone @ A @ C @ Orda @ Ordb
% 0.31/1.88 @ ^ [Y4: A] : ( F2 @ Y4 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_127_stream_Ocase__transfer,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,S: C > D > $o] : ( bNF_rel_fun @ ( A > ( stream @ A ) > C ) @ ( B > ( stream @ B ) > D ) @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ R @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S ) @ ( case_stream @ A @ C ) @ ( case_stream @ B @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_128_monotone__if__fun,axiom,
% 0.31/1.88 ! [C: $tType,D: $tType,B: $tType,A: $tType,Orda: B > B > $o,Ordb: D > D > $o,F5: ( A > B ) > C > D,G5: ( A > B ) > C > D,C3: C > $o] :
% 0.31/1.88 ( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ F5 )
% 0.31/1.88 => ( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ G5 )
% 0.31/1.88 => ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb )
% 0.31/1.88 @ ^ [F: A > B,N2: C] : ( if @ D @ ( C3 @ N2 ) @ ( F5 @ F @ N2 ) @ ( G5 @ F @ N2 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_129_monotone__fun__apply__fun,axiom,
% 0.31/1.88 ! [D: $tType,C: $tType,B: $tType,A: $tType,Ord: C > C > $o,T: A,G2: D > B] :
% 0.31/1.88 ( comple1396247847notone @ ( A > B > C ) @ ( D > C ) @ ( partial_fun_ord @ ( B > C ) @ ( B > C ) @ A @ ( partial_fun_ord @ C @ C @ B @ Ord ) ) @ ( partial_fun_ord @ C @ C @ D @ Ord )
% 0.31/1.88 @ ^ [F: A > B > C,N2: D] : ( F @ T @ ( G2 @ N2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_130_stream_Ocase__distrib,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType,H2: B > C,F2: A > ( stream @ A ) > B,Stream2: stream @ A] :
% 0.31/1.88 ( ( H2 @ ( case_stream @ A @ B @ F2 @ Stream2 ) )
% 0.31/1.88 = ( case_stream @ A @ C
% 0.31/1.88 @ ^ [X14: A,X23: stream @ A] : ( H2 @ ( F2 @ X14 @ X23 ) )
% 0.31/1.88 @ Stream2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_131_monotone__id_H,axiom,
% 0.31/1.88 ! [A: $tType,Ord: A > A > $o] :
% 0.31/1.88 ( comple1396247847notone @ A @ A @ Ord @ Ord
% 0.31/1.88 @ ^ [X2: A] : X2 ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_132_call__mono,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Ord: B > B > $o,T: A] :
% 0.31/1.88 ( comple1396247847notone @ ( A > B ) @ B @ ( partial_fun_ord @ B @ B @ A @ Ord ) @ Ord
% 0.31/1.88 @ ^ [F: A > B] : ( F @ T ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_133_stream_Opred__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( stream @ A ) > $o ) @ ( ( stream @ B ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( pred_stream @ A )
% 0.31/1.88 @ ( pred_stream @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_134_transfer__forall__transfer_I1_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( transfer_forall @ A )
% 0.31/1.88 @ ( transfer_forall @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_135_Domainp__transfer,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ B2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( domainp @ C @ A )
% 0.31/1.88 @ ( domainp @ D @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_136_stream_Opred__True,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( pred_stream @ A
% 0.31/1.88 @ ^ [Uu: A] : $true )
% 0.31/1.88 = ( ^ [Uu: stream @ A] : $true ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_137_fun_ODomainp__rel,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( domainp @ ( C > A ) @ ( C > B )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ C @ A @ B
% 0.31/1.88 @ ^ [Y3: C,Z: C] : ( Y3 = Z )
% 0.31/1.88 @ R ) )
% 0.31/1.88 = ( basic_pred_fun @ C @ A
% 0.31/1.88 @ ^ [Uu: C] : $true
% 0.31/1.88 @ ( domainp @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_138_stream_ODomainp__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( domainp @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) )
% 0.31/1.88 = ( pred_stream @ A @ ( domainp @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_139_Domainp__iff,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( domainp @ A @ B )
% 0.31/1.88 = ( ^ [T3: A > B > $o,X2: A] :
% 0.31/1.88 ( ^ [P2: B > $o] :
% 0.31/1.88 ? [X6: B] : ( P2 @ X6 )
% 0.31/1.88 @ ( T3 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_140_Domainp__refl,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( domainp @ A @ B )
% 0.31/1.88 = ( domainp @ A @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_141_pcr__Domainp__total,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,A2: C > A > $o,P: C > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ B2 )
% 0.31/1.88 => ( ( ( domainp @ C @ A @ A2 )
% 0.31/1.88 = P )
% 0.31/1.88 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.31/1.88 = P ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_142_stream__from__llist__setup_Ostream_Odomain__eq,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( domainp @ ( coinductive_llist @ A ) @ ( stream @ A )
% 0.31/1.88 @ ( coindu773941317stream @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) )
% 0.31/1.88 = ( ^ [Xs2: coinductive_llist @ A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_143_fun__ord__def,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/1.88 ( ( partial_fun_ord @ A @ B @ C )
% 0.31/1.88 = ( ^ [Ord2: A > B > $o,F: C > A,G: C > B] :
% 0.31/1.88 ! [X2: C] : ( Ord2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_144_pcr__Domainp__par__left__total,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o,P3: C > $o] :
% 0.31/1.88 ( ( ( domainp @ A @ B @ B2 )
% 0.31/1.88 = P )
% 0.31/1.88 => ( ( left_total @ C @ A @ A2 )
% 0.31/1.88 => ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ P3
% 0.31/1.88 @ P )
% 0.31/1.88 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.31/1.88 = P3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_145_let__mono,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,Orda: B > B > $o,Ordb: C > C > $o,B3: B > A > C,T: A] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( comple1396247847notone @ B @ C @ Orda @ Ordb
% 0.31/1.88 @ ^ [F: B] : ( B3 @ F @ X3 ) )
% 0.31/1.88 => ( comple1396247847notone @ B @ C @ Orda @ Ordb
% 0.31/1.88 @ ^ [F: B] : ( B3 @ F @ T ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_146_if__mono,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,G5: A > B,C3: $o] :
% 0.31/1.88 ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F5 )
% 0.31/1.88 => ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ G5 )
% 0.31/1.88 => ( comple1396247847notone @ A @ B @ Orda @ Ordb
% 0.31/1.88 @ ^ [F: A] : ( if @ B @ C3 @ ( F5 @ F ) @ ( G5 @ F ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_147_transfer__forall__transfer_I4_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ rev_implies
% 0.31/1.88 @ ( transfer_forall @ A )
% 0.31/1.88 @ ( transfer_forall @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_148_transfer__forall__transfer_I5_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) @ rev_implies @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_149_reflp__transfer_I1_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( reflp @ A )
% 0.31/1.88 @ ( reflp @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_150_right__total__parametric,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( ( bi_total @ C @ D @ B2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( right_total @ A @ C )
% 0.31/1.88 @ ( right_total @ B @ D ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_151_fun_Orel__reflp,axiom,
% 0.31/1.88 ! [D: $tType,A: $tType,R: A > A > $o] :
% 0.31/1.88 ( ( reflp @ A @ R )
% 0.31/1.88 => ( reflp @ ( D > A )
% 0.31/1.88 @ ( bNF_rel_fun @ D @ D @ A @ A
% 0.31/1.88 @ ^ [Y3: D,Z: D] : ( Y3 = Z )
% 0.31/1.88 @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_152_DEADID_Orel__reflp,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( reflp @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_153_right__total__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( right_total @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > B > $o] :
% 0.31/1.88 ! [Y4: B] :
% 0.31/1.88 ? [X2: A] : ( R5 @ X2 @ Y4 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_154_rev__implies__def,axiom,
% 0.31/1.88 ( rev_implies
% 0.31/1.88 = ( ^ [X2: $o,Y4: $o] :
% 0.31/1.88 ( Y4
% 0.31/1.88 => X2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_155_right__total__eq,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( right_total @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_156_right__totalI,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: B > A > $o] :
% 0.31/1.88 ( ! [Y2: A] :
% 0.31/1.88 ? [X7: B] : ( A2 @ X7 @ Y2 )
% 0.31/1.88 => ( right_total @ B @ A @ A2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_157_right__totalE,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o,Y: B] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ~ ! [X3: A] :
% 0.31/1.88 ~ ( A2 @ X3 @ Y ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_158_stream__from__llist__setup_Ostream_Oright__total,axiom,
% 0.31/1.88 ! [F3: $tType,E: $tType,T2: E > F3 > $o] :
% 0.31/1.88 ( ( right_total @ E @ F3 @ T2 )
% 0.31/1.88 => ( right_total @ ( coinductive_llist @ E ) @ ( stream @ F3 ) @ ( coindu773941317stream @ E @ F3 @ T2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_159_stream_Oright__total__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ R )
% 0.31/1.88 => ( right_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_160_stream_Orel__reflp,axiom,
% 0.31/1.88 ! [A: $tType,R: A > A > $o] :
% 0.31/1.88 ( ( reflp @ A @ R )
% 0.31/1.88 => ( reflp @ ( stream @ A ) @ ( stream_all2 @ A @ A @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_161_right__total__OO,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( ( right_total @ B @ C @ B2 )
% 0.31/1.88 => ( right_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_162_typedef__right__total,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T2: A > B > $o] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( T2
% 0.31/1.88 = ( ^ [X2: A,Y4: B] :
% 0.31/1.88 ( X2
% 0.31/1.88 = ( Rep @ Y4 ) ) ) )
% 0.31/1.88 => ( right_total @ A @ B @ T2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_163_reflp__transfer_I2_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) ) @ (=>) @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_164_reflp__transfer_I3_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ (=>)
% 0.31/1.88 @ ( reflp @ A )
% 0.31/1.88 @ ( reflp @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_165_bi__totalI,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ R )
% 0.31/1.88 => ( ( right_total @ A @ B @ R )
% 0.31/1.88 => ( bi_total @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_166_bi__total__alt__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( bi_total @ A @ B )
% 0.31/1.88 = ( ^ [A6: A > B > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ A6 )
% 0.31/1.88 & ( right_total @ A @ B @ A6 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_167_reflp__transfer_I4_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) ) @ rev_implies @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_168_reflp__transfer_I5_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ rev_implies
% 0.31/1.88 @ ( reflp @ A )
% 0.31/1.88 @ ( reflp @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_169_transfer__forall__transfer_I3_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) @ (=>) @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_170_transfer__forall__transfer_I2_J,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ (=>)
% 0.31/1.88 @ ( transfer_forall @ A )
% 0.31/1.88 @ ( transfer_forall @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_171_Domainp__forall__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( transfer_bforall @ A @ ( domainp @ A @ B @ A2 ) )
% 0.31/1.88 @ ( transfer_forall @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_172_right__total__relcompp__transfer,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ B2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
% 0.31/1.88 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) ) )
% 0.31/1.88 @ ^ [R5: C > A > $o,S5: A > E > $o,X2: C,Z2: E] :
% 0.31/1.88 ? [Y4: A] :
% 0.31/1.88 ( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B2 ) ) )
% 0.31/1.88 & ( R5 @ X2 @ Y4 )
% 0.31/1.88 & ( S5 @ Y4 @ Z2 ) )
% 0.31/1.88 @ ( relcompp @ D @ B @ F3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_173_right__total__Domainp__transfer,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ B2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [T3: C > A > $o,X2: C] :
% 0.31/1.88 ? [Y4: A] :
% 0.31/1.88 ( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B2 ) ) )
% 0.31/1.88 & ( T3 @ X2 @ Y4 ) )
% 0.31/1.88 @ ( domainp @ D @ B ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_174_bex__reg,axiom,
% 0.31/1.88 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( member @ A @ X3 @ R )
% 0.31/1.88 => ( ( P @ X3 )
% 0.31/1.88 => ( Q @ X3 ) ) )
% 0.31/1.88 => ( ? [X7: A] :
% 0.31/1.88 ( ( member @ A @ X7 @ R )
% 0.31/1.88 & ( P @ X7 ) )
% 0.31/1.88 => ? [X3: A] :
% 0.31/1.88 ( ( member @ A @ X3 @ R )
% 0.31/1.88 & ( Q @ X3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_175_right__total__Ex__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( bex @ A @ ( collect @ A @ ( domainp @ A @ B @ A2 ) ) )
% 0.31/1.88 @ ^ [P2: B > $o] :
% 0.31/1.88 ? [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_176_Domainp__prod__fun__eq,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,T2: B > C > $o] :
% 0.31/1.88 ( ( domainp @ ( A > B ) @ ( A > C )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ A @ B @ C
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ T2 ) )
% 0.31/1.88 = ( ^ [F: A > B] :
% 0.31/1.88 ! [X2: A] : ( domainp @ B @ C @ T2 @ ( F @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_177_stream__from__llist__setup_Ostream_Odomain__par__left__total,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,T2: C > B > $o,P3: ( coinductive_llist @ C ) > $o] :
% 0.31/1.88 ( ( left_total @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ C @ B @ T2 ) )
% 0.31/1.88 => ( ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ C @ B @ T2 )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ P3
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ B] :
% 0.31/1.88 ~ ( coinductive_lfinite @ B @ Xs2 ) )
% 0.31/1.88 => ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
% 0.31/1.88 = P3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_178_llist_Orel__reflp,axiom,
% 0.31/1.88 ! [A: $tType,R: A > A > $o] :
% 0.31/1.88 ( ( reflp @ A @ R )
% 0.31/1.88 => ( reflp @ ( coinductive_llist @ A ) @ ( coindu1486289336t_all2 @ A @ A @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_179_llist_Oright__total__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ R )
% 0.31/1.88 => ( right_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_180_bex__reg__left,axiom,
% 0.31/1.88 ! [A: $tType,R: set @ A,Q: A > $o,P: A > $o] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( member @ A @ X3 @ R )
% 0.31/1.88 => ( ( Q @ X3 )
% 0.31/1.88 => ( P @ X3 ) ) )
% 0.31/1.88 => ( ? [X7: A] :
% 0.31/1.88 ( ( member @ A @ X7 @ R )
% 0.31/1.88 & ( Q @ X7 ) )
% 0.31/1.88 => ? [X12: A] : ( P @ X12 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_181_rel__fun__eq__rel,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,R: B > C > $o] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ A @ B @ C
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ R )
% 0.31/1.88 = ( ^ [F: A > B,G: A > C] :
% 0.31/1.88 ! [X2: A] : ( R @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_182_rel__fun__def,axiom,
% 0.31/1.88 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.31/1.88 ( ( bNF_rel_fun @ A @ C @ B @ D )
% 0.31/1.88 = ( ^ [A6: A > C > $o,B4: B > D > $o,F: A > B,G: C > D] :
% 0.31/1.88 ! [X2: A,Y4: C] :
% 0.31/1.88 ( ( A6 @ X2 @ Y4 )
% 0.31/1.88 => ( B4 @ ( F @ X2 ) @ ( G @ Y4 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_183_stream__from__llist__setup_Ostream_Odomain,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,T2: C > B > $o] :
% 0.31/1.88 ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
% 0.31/1.88 = ( ^ [X2: coinductive_llist @ C] :
% 0.31/1.88 ? [Y4: coinductive_llist @ B] :
% 0.31/1.88 ( ( coindu1486289336t_all2 @ C @ B @ T2 @ X2 @ Y4 )
% 0.31/1.88 & ~ ( coinductive_lfinite @ B @ Y4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_184_llist__all2__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o ) @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.31/1.88 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( coindu1486289336t_all2 @ A @ A )
% 0.31/1.88 @ ( coindu1486289336t_all2 @ B @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_185_llist_Orel__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.31/1.88 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ D ) > $o ) @ ( coindu1486289336t_all2 @ A @ C @ Sa )
% 0.31/1.88 @ ( bNF_rel_fun @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ $o @ $o @ ( coindu1486289336t_all2 @ B @ D @ Sc )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) ) )
% 0.31/1.88 @ ( coindu1486289336t_all2 @ A @ B )
% 0.31/1.88 @ ( coindu1486289336t_all2 @ C @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_186_lfinite__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( coinductive_lfinite @ A )
% 0.31/1.88 @ ( coinductive_lfinite @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_187_transfer__forall__def,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( transfer_forall @ A )
% 0.31/1.88 = ( ^ [P2: A > $o] :
% 0.31/1.88 ! [X6: A] : ( P2 @ X6 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_188_OO__def,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType] :
% 0.31/1.88 ( ( relcompp @ A @ C @ B )
% 0.31/1.88 = ( ^ [R5: A > C > $o,S5: C > B > $o,X2: A,Z2: B] :
% 0.31/1.88 ? [Y4: C] :
% 0.31/1.88 ( ( R5 @ X2 @ Y4 )
% 0.31/1.88 & ( S5 @ Y4 @ Z2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_189_llist_Obi__total__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ R )
% 0.31/1.88 => ( bi_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_190_llist__all2__lfiniteD,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.31/1.88 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.31/1.88 => ( ( coinductive_lfinite @ A @ Xs )
% 0.31/1.88 = ( coinductive_lfinite @ B @ Ys ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_191_llist_Orel__refl,axiom,
% 0.31/1.88 ! [B: $tType,Ra: B > B > $o,X: coinductive_llist @ B] :
% 0.31/1.88 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.31/1.88 => ( coindu1486289336t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_192_llist_Orel__eq,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( coindu1486289336t_all2 @ A @ A
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.31/1.88 = ( ^ [Y3: coinductive_llist @ A,Z: coinductive_llist @ A] : ( Y3 = Z ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_193_llist__all2__mono,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P3: A > B > $o] :
% 0.31/1.88 ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.31/1.88 => ( ! [X3: A,Y2: B] :
% 0.31/1.88 ( ( P @ X3 @ Y2 )
% 0.31/1.88 => ( P3 @ X3 @ Y2 ) )
% 0.31/1.88 => ( coindu1486289336t_all2 @ A @ B @ P3 @ Xs @ Ys ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_194_llist__all2__rsp,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T2: B > B > $o,X: coinductive_llist @ A,Y: coinductive_llist @ B,A3: coinductive_llist @ A,B3: coinductive_llist @ B] :
% 0.31/1.88 ( ! [X3: A,Y2: B] :
% 0.31/1.88 ( ( R @ X3 @ Y2 )
% 0.31/1.88 => ! [A5: A,B5: B] :
% 0.31/1.88 ( ( R @ A5 @ B5 )
% 0.31/1.88 => ( ( S @ X3 @ A5 )
% 0.31/1.88 = ( T2 @ Y2 @ B5 ) ) ) )
% 0.31/1.88 => ( ( coindu1486289336t_all2 @ A @ B @ R @ X @ Y )
% 0.31/1.88 => ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
% 0.31/1.88 => ( ( coindu1486289336t_all2 @ A @ A @ S @ X @ A3 )
% 0.31/1.88 = ( coindu1486289336t_all2 @ B @ B @ T2 @ Y @ B3 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_195_llist__all2__conj,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.31/1.88 ( ( coindu1486289336t_all2 @ A @ B
% 0.31/1.88 @ ^ [X2: A,Y4: B] :
% 0.31/1.88 ( ( P @ X2 @ Y4 )
% 0.31/1.88 & ( Q @ X2 @ Y4 ) )
% 0.31/1.88 @ Xs
% 0.31/1.88 @ Ys )
% 0.31/1.88 = ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
% 0.31/1.88 & ( coindu1486289336t_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_196_llist_Oleft__total__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( left_total @ A @ B @ R )
% 0.31/1.88 => ( left_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_197_llist_Orel__compp,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
% 0.31/1.88 ( ( coindu1486289336t_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
% 0.31/1.88 = ( relcompp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ C ) @ ( coindu1486289336t_all2 @ A @ B @ R ) @ ( coindu1486289336t_all2 @ B @ C @ S ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_198_pred__fun__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( basic_pred_fun @ A @ B )
% 0.31/1.88 = ( ^ [A6: A > $o,B4: B > $o,F: A > B] :
% 0.31/1.88 ! [X2: A] :
% 0.31/1.88 ( ( A6 @ X2 )
% 0.31/1.88 => ( B4 @ ( F @ X2 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_199_pcr__Domainp,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o] :
% 0.31/1.88 ( ( ( domainp @ A @ B @ B2 )
% 0.31/1.88 = P )
% 0.31/1.88 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.31/1.88 = ( ^ [X2: C] :
% 0.31/1.88 ? [Y4: A] :
% 0.31/1.88 ( ( A2 @ X2 @ Y4 )
% 0.31/1.88 & ( P @ Y4 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_200_stream__from__llist__setup_Ostream_Orep__transfer,axiom,
% 0.31/1.88 ! [E: $tType,F3: $tType,T2: E > F3 > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( coinductive_llist @ E ) @ ( stream @ F3 ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ F3 ) @ ( coindu773941317stream @ E @ F3 @ T2 ) @ ( coindu1486289336t_all2 @ E @ F3 @ T2 )
% 0.31/1.88 @ ^ [X2: coinductive_llist @ E] : X2
% 0.31/1.88 @ ( coindu1724414836stream @ F3 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_201_stream__from__llist__setup_Opcr__stream__def,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType] :
% 0.31/1.88 ( ( coindu773941317stream @ C @ B )
% 0.31/1.88 = ( ^ [T3: C > B > $o] : ( relcompp @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( stream @ B ) @ ( coindu1486289336t_all2 @ C @ B @ T3 ) @ ( coindu1183105481stream @ B ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_202_transfer__bforall__def,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( transfer_bforall @ A )
% 0.31/1.88 = ( ^ [P4: A > $o,Q2: A > $o] :
% 0.31/1.88 ! [X2: A] :
% 0.31/1.88 ( ( P4 @ X2 )
% 0.31/1.88 => ( Q2 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_203_unfold__llist__transfer,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > B ) > B > ( coinductive_llist @ D ) )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( 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 ) ) ) )
% 0.31/1.88 @ ( coindu1441602521_llist @ A @ C )
% 0.31/1.88 @ ( coindu1441602521_llist @ B @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_204_llist_Ocorec__transfer,axiom,
% 0.31/1.88 ! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
% 0.31/1.88 ( 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 ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( 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 )
% 0.31/1.88 @ ( 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 ) )
% 0.31/1.88 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( 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 ) ) ) ) ) )
% 0.31/1.88 @ ( coindu1259883913_llist @ C @ A )
% 0.31/1.88 @ ( coindu1259883913_llist @ D @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_205_llist__corec__transfer,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.31/1.88 ( 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 ) )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( 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 )
% 0.31/1.88 @ ( 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 ) )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( 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 ) ) ) ) ) )
% 0.31/1.88 @ ( coindu1259883913_llist @ A @ C )
% 0.31/1.88 @ ( coindu1259883913_llist @ B @ D ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_206_Ex__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ^ [P2: A > $o] :
% 0.31/1.88 ? [X6: A] : ( P2 @ X6 )
% 0.31/1.88 @ ^ [P2: B > $o] :
% 0.31/1.88 ? [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_207_All__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( bi_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ^ [P2: A > $o] :
% 0.31/1.88 ! [X6: A] : ( P2 @ X6 )
% 0.31/1.88 @ ^ [P2: B > $o] :
% 0.31/1.88 ! [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_208_bi__total__alt__def2,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( bi_total @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ^ [P2: A > $o] :
% 0.31/1.88 ! [X6: A] : ( P2 @ X6 )
% 0.31/1.88 @ ^ [P2: B > $o] :
% 0.31/1.88 ! [X6: B] : ( P2 @ X6 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_209_right__total__alt__def2,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( right_total @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5 @ (=>) ) @ (=>)
% 0.31/1.88 @ ^ [P2: A > $o] :
% 0.31/1.88 ! [X6: A] : ( P2 @ X6 )
% 0.31/1.88 @ ^ [P2: B > $o] :
% 0.31/1.88 ! [X6: B] : ( P2 @ X6 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_210_Bex__def__raw,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( bex @ A )
% 0.31/1.88 = ( ^ [A6: set @ A,P4: A > $o] :
% 0.31/1.88 ? [X2: A] :
% 0.31/1.88 ( ( member @ A @ X2 @ A6 )
% 0.31/1.88 & ( P4 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_211_rel__pred__comp__def,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType] :
% 0.31/1.88 ( ( rel_pred_comp @ A @ B )
% 0.31/1.88 = ( ^ [R5: A > B > $o,P4: B > $o,X2: A] :
% 0.31/1.88 ? [Y4: B] :
% 0.31/1.88 ( ( R5 @ X2 @ Y4 )
% 0.31/1.88 & ( P4 @ Y4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_212_right__total__All__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( ( right_total @ A @ B @ A2 )
% 0.31/1.88 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ ( ball @ A @ ( collect @ A @ ( domainp @ A @ B @ A2 ) ) )
% 0.31/1.88 @ ^ [P2: B > $o] :
% 0.31/1.88 ! [X6: B] : ( P2 @ X6 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_213_ball__reg__right,axiom,
% 0.31/1.88 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( member @ A @ X3 @ R )
% 0.31/1.88 => ( ( P @ X3 )
% 0.31/1.88 => ( Q @ X3 ) ) )
% 0.31/1.88 => ( ! [X12: A] : ( P @ X12 )
% 0.31/1.88 => ! [X7: A] :
% 0.31/1.88 ( ( member @ A @ X7 @ R )
% 0.31/1.88 => ( Q @ X7 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_214_Ball__def__raw,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( ball @ A )
% 0.31/1.88 = ( ^ [A6: set @ A,P4: A > $o] :
% 0.31/1.88 ! [X2: A] :
% 0.31/1.88 ( ( member @ A @ X2 @ A6 )
% 0.31/1.88 => ( P4 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_215_Ball__comp__iff,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,A2: B > ( set @ C ),F2: C > $o,G2: A > B] :
% 0.31/1.88 ( ( comp @ B @ $o @ A
% 0.31/1.88 @ ^ [X2: B] :
% 0.31/1.88 ! [Y4: C] :
% 0.31/1.88 ( ( member @ C @ Y4 @ ( A2 @ X2 ) )
% 0.31/1.88 => ( F2 @ Y4 ) )
% 0.31/1.88 @ G2 )
% 0.31/1.88 = ( ^ [X2: A] :
% 0.31/1.88 ! [Y4: C] :
% 0.31/1.88 ( ( member @ C @ Y4 @ ( comp @ B @ ( set @ C ) @ A @ A2 @ G2 @ X2 ) )
% 0.31/1.88 => ( F2 @ Y4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_216_ball__reg,axiom,
% 0.31/1.88 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.31/1.88 ( ! [X3: A] :
% 0.31/1.88 ( ( member @ A @ X3 @ R )
% 0.31/1.88 => ( ( P @ X3 )
% 0.31/1.88 => ( Q @ X3 ) ) )
% 0.31/1.88 => ( ! [X3: A] :
% 0.31/1.88 ( ( member @ A @ X3 @ R )
% 0.31/1.88 => ( P @ X3 ) )
% 0.31/1.88 => ! [X7: A] :
% 0.31/1.88 ( ( member @ A @ X7 @ R )
% 0.31/1.88 => ( Q @ X7 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_217_iterates__transfer,axiom,
% 0.31/1.88 ! [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 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_218_llist_Opred__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( coindu543516966_llist @ A )
% 0.31/1.88 @ ( coindu543516966_llist @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_219_lfinite__iterates,axiom,
% 0.31/1.88 ! [A: $tType,F2: A > A,X: A] :
% 0.31/1.88 ~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F2 @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_220_llist_Opred__True,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( coindu543516966_llist @ A
% 0.31/1.88 @ ^ [Uu: A] : $true )
% 0.31/1.88 = ( ^ [Uu: coinductive_llist @ A] : $true ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_221_llist_ODomainp__rel,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( domainp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) )
% 0.31/1.88 = ( coindu543516966_llist @ A @ ( domainp @ A @ B @ R ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_222_llist__all__transfer,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.31/1.88 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
% 0.31/1.88 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
% 0.31/1.88 @ ( coindu543516966_llist @ A )
% 0.31/1.88 @ ( coindu543516966_llist @ B ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_223_stream__from__llist__setup_Ostream_Odomain__par,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,T2: C > B > $o,DR: C > $o,P22: ( coinductive_llist @ C ) > $o] :
% 0.31/1.88 ( ( ( domainp @ C @ B @ T2 )
% 0.31/1.88 = DR )
% 0.31/1.88 => ( ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ C @ B @ T2 )
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ P22
% 0.31/1.88 @ ^ [Xs2: coinductive_llist @ B] :
% 0.31/1.88 ~ ( coinductive_lfinite @ B @ Xs2 ) )
% 0.31/1.88 => ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
% 0.31/1.88 = ( inf_inf @ ( ( coinductive_llist @ C ) > $o ) @ ( coindu543516966_llist @ C @ DR ) @ P22 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_224_case__llist__transfer,axiom,
% 0.31/1.88 ! [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 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_225_llist__case__mono,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,Lnil: A > B,Lcons: A > C > ( coinductive_llist @ C ) > B,X: coinductive_llist @ C] :
% 0.31/1.88 ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ Lnil )
% 0.31/1.88 => ( ! [X3: C,Xs3: coinductive_llist @ C] :
% 0.31/1.88 ( comple1396247847notone @ A @ B @ Orda @ Ordb
% 0.31/1.88 @ ^ [F: A] : ( Lcons @ F @ X3 @ Xs3 ) )
% 0.31/1.88 => ( comple1396247847notone @ A @ B @ Orda @ Ordb
% 0.31/1.88 @ ^ [F: A] : ( coindu1381640503_llist @ B @ C @ ( Lnil @ F ) @ ( Lcons @ F ) @ X ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_226_llist_Ocase__distrib,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
% 0.31/1.88 ( ( H2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ Llist ) )
% 0.31/1.88 = ( coindu1381640503_llist @ C @ A @ ( H2 @ F1 )
% 0.31/1.88 @ ^ [X14: A,X23: coinductive_llist @ A] : ( H2 @ ( F22 @ X14 @ X23 ) )
% 0.31/1.88 @ Llist ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_227_pcr__Domainp__par,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P23: A > $o,A2: C > A > $o,P1: C > $o,P22: C > $o] :
% 0.31/1.88 ( ( ( domainp @ A @ B @ B2 )
% 0.31/1.88 = P23 )
% 0.31/1.88 => ( ( ( domainp @ C @ A @ A2 )
% 0.31/1.88 = P1 )
% 0.31/1.88 => ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
% 0.31/1.88 @ ^ [Y3: $o,Z: $o] : ( Y3 = Z )
% 0.31/1.88 @ P22
% 0.31/1.88 @ P23 )
% 0.31/1.88 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.31/1.88 = ( inf_inf @ ( C > $o ) @ P1 @ P22 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_228_llist_Ocase__transfer,axiom,
% 0.31/1.88 ! [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 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_229_img__ord__def,axiom,
% 0.31/1.88 ! [B: $tType,C: $tType,A: $tType] :
% 0.31/1.88 ( ( partial_img_ord @ A @ C @ B )
% 0.31/1.88 = ( ^ [F: A > C,Ord2: C > C > B,X2: A,Y4: A] : ( Ord2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_230_mk__less__def,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( partial_mk_less @ A )
% 0.31/1.88 = ( ^ [R5: A > A > $o,X2: A,Y4: A] :
% 0.31/1.88 ( ( R5 @ X2 @ Y4 )
% 0.31/1.88 & ~ ( R5 @ Y4 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_231_type__definition_ORep,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( member @ A @ ( Rep @ X ) @ A2 ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_232_type__definition__def,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType] :
% 0.31/1.88 ( ( type_definition @ B @ A )
% 0.31/1.88 = ( ^ [Rep2: B > A,Abs2: A > B,A6: set @ A] :
% 0.31/1.88 ( ! [X2: B] : ( member @ A @ ( Rep2 @ X2 ) @ A6 )
% 0.31/1.88 & ! [X2: B] :
% 0.31/1.88 ( ( Abs2 @ ( Rep2 @ X2 ) )
% 0.31/1.88 = X2 )
% 0.31/1.88 & ! [Y4: A] :
% 0.31/1.88 ( ( member @ A @ Y4 @ A6 )
% 0.31/1.88 => ( ( Rep2 @ ( Abs2 @ Y4 ) )
% 0.31/1.88 = Y4 ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_233_type__definition_ORep__inverse,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( Abs @ ( Rep @ X ) )
% 0.31/1.88 = X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_234_type__definition_OAbs__inverse,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( member @ A @ Y @ A2 )
% 0.31/1.88 => ( ( Rep @ ( Abs @ Y ) )
% 0.31/1.88 = Y ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_235_type__definition_ORep__inject,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B,Y: B] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( ( Rep @ X )
% 0.31/1.88 = ( Rep @ Y ) )
% 0.31/1.88 = ( X = Y ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_236_type__definition_ORep__induct,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A,P: A > $o] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( member @ A @ Y @ A2 )
% 0.31/1.88 => ( ! [X3: B] : ( P @ ( Rep @ X3 ) )
% 0.31/1.88 => ( P @ Y ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_237_type__definition_OAbs__inject,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: A,Y: A] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( member @ A @ X @ A2 )
% 0.31/1.88 => ( ( member @ A @ Y @ A2 )
% 0.31/1.88 => ( ( ( Abs @ X )
% 0.31/1.88 = ( Abs @ Y ) )
% 0.31/1.88 = ( X = Y ) ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_238_type__definition_OAbs__induct,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,P: B > $o,X: B] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ! [Y2: A] :
% 0.31/1.88 ( ( member @ A @ Y2 @ A2 )
% 0.31/1.88 => ( P @ ( Abs @ Y2 ) ) )
% 0.31/1.88 => ( P @ X ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_239_type__definition_ORep__cases,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ( ( member @ A @ Y @ A2 )
% 0.31/1.88 => ~ ! [X3: B] :
% 0.31/1.88 ( Y
% 0.31/1.88 != ( Rep @ X3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_240_type__definition_OAbs__cases,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.31/1.88 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.88 => ~ ! [Y2: A] :
% 0.31/1.88 ( ( X
% 0.31/1.88 = ( Abs @ Y2 ) )
% 0.31/1.88 => ~ ( member @ A @ Y2 @ A2 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_241_type__definition_Ointro,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,Rep: B > A,A2: set @ A,Abs: A > B] :
% 0.31/1.88 ( ! [X3: B] : ( member @ A @ ( Rep @ X3 ) @ A2 )
% 0.31/1.88 => ( ! [X3: B] :
% 0.31/1.88 ( ( Abs @ ( Rep @ X3 ) )
% 0.31/1.88 = X3 )
% 0.31/1.88 => ( ! [Y2: A] :
% 0.31/1.88 ( ( member @ A @ Y2 @ A2 )
% 0.31/1.88 => ( ( Rep @ ( Abs @ Y2 ) )
% 0.31/1.88 = Y2 ) )
% 0.31/1.88 => ( type_definition @ B @ A @ Rep @ Abs @ A2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_242_llist__of__stream__siterates,axiom,
% 0.31/1.88 ! [A: $tType,F2: A > A,X: A] :
% 0.31/1.88 ( ( coindu1724414836stream @ A @ ( siterate @ A @ F2 @ X ) )
% 0.31/1.88 = ( coinductive_iterates @ A @ F2 @ X ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_243_nchotomy__relcomppE,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: B > A,R4: C > A > $o,S3: A > D > $o,A3: C,C3: D] :
% 0.31/1.88 ( ! [Y2: A] :
% 0.31/1.88 ? [X7: B] :
% 0.31/1.88 ( Y2
% 0.31/1.88 = ( F2 @ X7 ) )
% 0.31/1.88 => ( ( relcompp @ C @ A @ D @ R4 @ S3 @ A3 @ C3 )
% 0.31/1.88 => ~ ! [B5: B] :
% 0.31/1.88 ( ( R4 @ A3 @ ( F2 @ B5 ) )
% 0.31/1.88 => ~ ( S3 @ ( F2 @ B5 ) @ C3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_244_OO__eq,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( relcompp @ A @ B @ B @ R
% 0.31/1.88 @ ^ [Y3: B,Z: B] : ( Y3 = Z ) )
% 0.31/1.88 = R ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_245_eq__OO,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.31/1.88 ( ( relcompp @ A @ A @ B
% 0.31/1.88 @ ^ [Y3: A,Z: A] : ( Y3 = Z )
% 0.31/1.88 @ R )
% 0.31/1.88 = R ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_246_relcompp_OrelcompI,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,A3: A,B3: B,S3: B > C > $o,C3: C] :
% 0.31/1.88 ( ( R4 @ A3 @ B3 )
% 0.31/1.88 => ( ( S3 @ B3 @ C3 )
% 0.31/1.88 => ( relcompp @ A @ B @ C @ R4 @ S3 @ A3 @ C3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_247_relcompp_Oinducts,axiom,
% 0.31/1.88 ! [B: $tType,A: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,X1: A,X22: C,P: A > C > $o] :
% 0.31/1.88 ( ( relcompp @ A @ B @ C @ R4 @ S3 @ X1 @ X22 )
% 0.31/1.88 => ( ! [A5: A,B5: B,C4: C] :
% 0.31/1.88 ( ( R4 @ A5 @ B5 )
% 0.31/1.88 => ( ( S3 @ B5 @ C4 )
% 0.31/1.88 => ( P @ A5 @ C4 ) ) )
% 0.31/1.88 => ( P @ X1 @ X22 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_248_relcompp__assoc,axiom,
% 0.31/1.88 ! [A: $tType,D: $tType,B: $tType,C: $tType,R4: A > D > $o,S3: D > C > $o,T: C > B > $o] :
% 0.31/1.88 ( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R4 @ S3 ) @ T )
% 0.31/1.88 = ( relcompp @ A @ D @ B @ R4 @ ( relcompp @ D @ C @ B @ S3 @ T ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_249_relcompp__apply,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/1.88 ( ( relcompp @ A @ B @ C )
% 0.31/1.88 = ( ^ [R5: A > B > $o,S5: B > C > $o,A4: A,C5: C] :
% 0.31/1.88 ? [B6: B] :
% 0.31/1.88 ( ( R5 @ A4 @ B6 )
% 0.31/1.88 & ( S5 @ B6 @ C5 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_250_relcompp_Osimps,axiom,
% 0.31/1.88 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/1.88 ( ( relcompp @ A @ B @ C )
% 0.31/1.88 = ( ^ [R6: A > B > $o,S2: B > C > $o,A1: A,A22: C] :
% 0.31/1.88 ? [A4: A,B6: B,C5: C] :
% 0.31/1.88 ( ( A1 = A4 )
% 0.31/1.88 & ( A22 = C5 )
% 0.31/1.88 & ( R6 @ A4 @ B6 )
% 0.31/1.88 & ( S2 @ B6 @ C5 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_251_relcompp_Ocases,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
% 0.31/1.88 ( ( relcompp @ A @ B @ C @ R4 @ S3 @ A12 @ A23 )
% 0.31/1.88 => ~ ! [B5: B] :
% 0.31/1.88 ( ( R4 @ A12 @ B5 )
% 0.31/1.88 => ~ ( S3 @ B5 @ A23 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_252_relcomppE,axiom,
% 0.31/1.88 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,A3: A,C3: C] :
% 0.31/1.88 ( ( relcompp @ A @ B @ C @ R4 @ S3 @ A3 @ C3 )
% 0.31/1.88 => ~ ! [B5: B] :
% 0.31/1.88 ( ( R4 @ A3 @ B5 )
% 0.31/1.88 => ~ ( S3 @ B5 @ C3 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_253_Powp__def,axiom,
% 0.31/1.88 ! [A: $tType] :
% 0.31/1.88 ( ( powp @ A )
% 0.31/1.88 = ( ^ [A6: A > $o,B4: set @ A] :
% 0.31/1.88 ! [X2: A] :
% 0.31/1.88 ( ( member @ A @ X2 @ B4 )
% 0.31/1.88 => ( A6 @ X2 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_254_pos__fun__distr,axiom,
% 0.31/1.88 ! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F3: $tType,R: A > E > $o,S: B > F3 > $o,R8: E > C > $o,S6: F3 > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F3 ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ R @ S ) @ ( bNF_rel_fun @ E @ C @ F3 @ D @ R8 @ S6 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R @ R8 ) @ ( relcompp @ B @ F3 @ D @ S @ S6 ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(fact_255_relcompp__mono,axiom,
% 0.31/1.88 ! [A: $tType,C: $tType,B: $tType,R9: A > B > $o,R4: A > B > $o,S7: B > C > $o,S3: B > C > $o] :
% 0.31/1.88 ( ( ord_less_eq @ ( A > B > $o ) @ R9 @ R4 )
% 0.31/1.88 => ( ( ord_less_eq @ ( B > C > $o ) @ S7 @ S3 )
% 0.31/1.88 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R9 @ S7 ) @ ( relcompp @ A @ B @ C @ R4 @ S3 ) ) ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(help_If_3_1_T,axiom,
% 0.31/1.88 ! [P: $o] :
% 0.31/1.88 ( ( P = $true )
% 0.31/1.88 | ( P = $false ) ) ).
% 0.31/1.88
% 0.31/1.88 thf(help_If_2_1_T,axiom,
% 0.31/1.88 ! [A: $tType,X: A,Y: A] :
% 0.31/1.88 ( ( if @ A @ $false @ X @ Y )
% 0.31/1.88 = Y ) ).
% 0.31/1.88
% 0.31/1.88 thf(help_If_1_1_T,axiom,
% 0.31/1.88 ! [A: $tType,X: A,Y: A] :
% 0.31/1.88 ( ( if @ A @ $true @ X @ Y )
% 0.31/1.88 = X ) ).
% 0.31/1.88
% 0.31/1.88 thf(conj_0,hypothesis,
% 0.31/1.88 ( bNF_rel_fun @ a @ a @ ( coinductive_llist @ b ) @ ( stream @ b )
% 0.31/1.88 @ ^ [Y3: a,Z: a] : ( Y3 = Z )
% 0.31/1.88 @ ^ [X2: coinductive_llist @ b,Y4: stream @ b] :
% 0.31/1.88 ( X2
% 0.31/1.88 = ( coindu1724414836stream @ b @ Y4 ) )
% 0.31/1.88 @ xb
% 0.31/1.88 @ yb ) ).
% 0.31/1.88
% 0.31/1.88 thf(conj_1,conjecture,
% 0.31/1.88 ( xb
% 0.31/1.88 = ( comp @ ( stream @ b ) @ ( coinductive_llist @ b ) @ a @ ( coindu1724414836stream @ b ) @ yb ) ) ).
% 0.31/1.88 ------------------------
% 3.65/1.92 % (2635)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_2534 for (2999ds/27Mi)
% 3.65/1.92 % (2634)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_2534 for (2999ds/4Mi)
% 3.65/1.92 % (2636)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/2Mi)
% 3.65/1.92 % (2637)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_2534 for (2999ds/2Mi)
% 3.65/1.92 % (2633)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/183Mi)
% 3.65/1.92 % (2638)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_2534 for (2999ds/275Mi)
% 3.65/1.92 % (2639)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/18Mi)
% 3.65/1.92 % (2636)Instruction limit reached!
% 3.65/1.92 % (2636)------------------------------
% 3.65/1.92 % (2636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.92 % (2636)Termination reason: Unknown
% 3.65/1.92 % (2636)Termination phase: shuffling
% 3.65/1.92 % (2637)Instruction limit reached!
% 3.65/1.92 % (2637)------------------------------
% 3.65/1.92 % (2637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.92
% 3.65/1.92 % (2636)Memory used [KB]: 1663
% 3.65/1.92 % (2636)Time elapsed: 0.003 s
% 3.65/1.92 % (2636)Instructions burned: 2 (million)
% 3.65/1.92 % (2636)------------------------------
% 3.65/1.92 % (2636)------------------------------
% 3.65/1.92 % (2637)Termination reason: Unknown
% 3.65/1.92 % (2637)Termination phase: shuffling
% 3.65/1.92
% 3.65/1.92 % (2637)Memory used [KB]: 1663
% 3.65/1.92 % (2637)Time elapsed: 0.003 s
% 3.65/1.92 % (2637)Instructions burned: 3 (million)
% 3.65/1.92 % (2637)------------------------------
% 3.65/1.92 % (2637)------------------------------
% 3.65/1.93 % (2634)Instruction limit reached!
% 3.65/1.93 % (2634)------------------------------
% 3.65/1.93 % (2634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.93 % (2634)Termination reason: Unknown
% 3.65/1.93 % (2634)Termination phase: shuffling
% 3.65/1.93
% 3.65/1.93 % (2634)Memory used [KB]: 1663
% 3.65/1.93 % (2634)Time elapsed: 0.004 s
% 3.65/1.93 % (2634)Instructions burned: 4 (million)
% 3.65/1.93 % (2634)------------------------------
% 3.65/1.93 % (2634)------------------------------
% 3.65/1.93 % (2639)Instruction limit reached!
% 3.65/1.93 % (2639)------------------------------
% 3.65/1.93 % (2639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.93 % (2639)Termination reason: Unknown
% 3.65/1.93 % (2639)Termination phase: shuffling
% 3.65/1.93
% 3.65/1.93 % (2639)Memory used [KB]: 1918
% 3.65/1.93 % (2639)Time elapsed: 0.011 s
% 3.65/1.93 % (2639)Instructions burned: 19 (million)
% 3.65/1.93 % (2639)------------------------------
% 3.65/1.93 % (2639)------------------------------
% 3.65/1.94 % (2635)Instruction limit reached!
% 3.65/1.94 % (2635)------------------------------
% 3.65/1.94 % (2635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.94 % (2635)Termination reason: Unknown
% 3.65/1.94 % (2635)Termination phase: shuffling
% 3.65/1.94
% 3.65/1.94 % (2635)Memory used [KB]: 2046
% 3.65/1.94 % (2635)Time elapsed: 0.015 s
% 3.65/1.94 % (2635)Instructions burned: 27 (million)
% 3.65/1.94 % (2635)------------------------------
% 3.65/1.94 % (2635)------------------------------
% 3.65/1.94 % (2641)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_2534 for (2999ds/37Mi)
% 3.65/1.94 % (2640)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/3Mi)
% 3.65/1.94 % (2642)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_2534 for (2999ds/15Mi)
% 3.65/1.94 % (2640)Instruction limit reached!
% 3.65/1.94 % (2640)------------------------------
% 3.65/1.94 % (2640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.94 % (2640)Termination reason: Unknown
% 3.65/1.94 % (2640)Termination phase: shuffling
% 3.65/1.94
% 3.65/1.94 % (2640)Memory used [KB]: 1663
% 3.65/1.94 % (2640)Time elapsed: 0.003 s
% 3.65/1.94 % (2640)Instructions burned: 3 (million)
% 3.65/1.94 % (2640)------------------------------
% 3.65/1.94 % (2640)------------------------------
% 3.65/1.95 % (2642)Instruction limit reached!
% 3.65/1.95 % (2642)------------------------------
% 3.65/1.95 % (2642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.95 % (2642)Termination reason: Unknown
% 3.65/1.95 % (2643)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_2534 for (2999ds/3Mi)
% 3.65/1.95 % (2642)Termination phase: shuffling
% 3.65/1.95
% 3.65/1.95 % (2642)Memory used [KB]: 1918
% 3.65/1.95 % (2642)Time elapsed: 0.009 s
% 3.65/1.95 % (2642)Instructions burned: 15 (million)
% 3.65/1.95 % (2642)------------------------------
% 3.65/1.95 % (2642)------------------------------
% 3.65/1.95 % (2643)Instruction limit reached!
% 3.65/1.95 % (2643)------------------------------
% 3.65/1.95 % (2643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.95 % (2643)Termination reason: Unknown
% 3.65/1.95 % (2643)Termination phase: shuffling
% 3.65/1.95
% 3.65/1.95 % (2643)Memory used [KB]: 1663
% 3.65/1.95 % (2643)Time elapsed: 0.003 s
% 3.65/1.95 % (2643)Instructions burned: 3 (million)
% 3.65/1.95 % (2643)------------------------------
% 3.65/1.95 % (2643)------------------------------
% 3.65/1.95 % (2644)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_2534 for (2999ds/1041Mi)
% 3.65/1.96 % (2645)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/7Mi)
% 3.65/1.96 % (2641)Instruction limit reached!
% 3.65/1.96 % (2641)------------------------------
% 3.65/1.96 % (2641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.96 % (2641)Termination reason: Unknown
% 3.65/1.96 % (2641)Termination phase: shuffling
% 3.65/1.96
% 3.65/1.96 % (2641)Memory used [KB]: 2302
% 3.65/1.96 % (2641)Time elapsed: 0.019 s
% 3.65/1.96 % (2641)Instructions burned: 37 (million)
% 3.65/1.96 % (2641)------------------------------
% 3.65/1.96 % (2641)------------------------------
% 3.65/1.96 % (2645)Instruction limit reached!
% 3.65/1.96 % (2645)------------------------------
% 3.65/1.96 % (2645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.96 % (2645)Termination reason: Unknown
% 3.65/1.96 % (2645)Termination phase: shuffling
% 3.65/1.96
% 3.65/1.96 % (2645)Memory used [KB]: 1791
% 3.65/1.96 % (2645)Time elapsed: 0.005 s
% 3.65/1.96 % (2645)Instructions burned: 7 (million)
% 3.65/1.96 % (2645)------------------------------
% 3.65/1.96 % (2645)------------------------------
% 3.65/1.97 % (2646)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/16Mi)
% 3.65/1.97 % (2647)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_2534 for (2999ds/3Mi)
% 3.65/1.97 % (2647)Instruction limit reached!
% 3.65/1.97 % (2647)------------------------------
% 3.65/1.97 % (2647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.97 % (2647)Termination reason: Unknown
% 3.65/1.97 % (2647)Termination phase: shuffling
% 3.65/1.97
% 3.65/1.97 % (2647)Memory used [KB]: 1663
% 3.65/1.97 % (2647)Time elapsed: 0.003 s
% 3.65/1.97 % (2647)Instructions burned: 3 (million)
% 3.65/1.97 % (2647)------------------------------
% 3.65/1.97 % (2647)------------------------------
% 3.65/1.97 % (2646)Instruction limit reached!
% 3.65/1.97 % (2646)------------------------------
% 3.65/1.97 % (2646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.97 % (2646)Termination reason: Unknown
% 3.65/1.97 % (2646)Termination phase: shuffling
% 3.65/1.97
% 3.65/1.97 % (2646)Memory used [KB]: 1918
% 3.65/1.97 % (2646)Time elapsed: 0.010 s
% 3.65/1.97 % (2646)Instructions burned: 18 (million)
% 3.65/1.97 % (2646)------------------------------
% 3.65/1.97 % (2646)------------------------------
% 3.65/1.97 % (2648)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_2534 for (2999ds/3Mi)
% 3.65/1.98 % (2648)Instruction limit reached!
% 3.65/1.98 % (2648)------------------------------
% 3.65/1.98 % (2648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.98 % (2648)Termination reason: Unknown
% 3.65/1.98 % (2648)Termination phase: shuffling
% 3.65/1.98
% 3.65/1.98 % (2648)Memory used [KB]: 1663
% 3.65/1.98 % (2648)Time elapsed: 0.003 s
% 3.65/1.98 % (2648)Instructions burned: 3 (million)
% 3.65/1.98 % (2648)------------------------------
% 3.65/1.98 % (2648)------------------------------
% 3.65/1.98 % (2649)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/7Mi)
% 3.65/1.98 % (2649)Instruction limit reached!
% 3.65/1.98 % (2649)------------------------------
% 3.65/1.98 % (2649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.98 % (2649)Termination reason: Unknown
% 3.65/1.98 % (2649)Termination phase: shuffling
% 3.65/1.98
% 3.65/1.98 % (2649)Memory used [KB]: 1791
% 3.65/1.98 % (2649)Time elapsed: 0.005 s
% 3.65/1.98 % (2649)Instructions burned: 7 (million)
% 3.65/1.98 % (2649)------------------------------
% 3.65/1.98 % (2649)------------------------------
% 3.65/1.98 % (2650)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/3Mi)
% 3.65/1.99 % (2650)Instruction limit reached!
% 3.65/1.99 % (2650)------------------------------
% 3.65/1.99 % (2650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.99 % (2650)Termination reason: Unknown
% 3.65/1.99 % (2650)Termination phase: shuffling
% 3.65/1.99
% 3.65/1.99 % (2650)Memory used [KB]: 1663
% 3.65/1.99 % (2650)Time elapsed: 0.003 s
% 3.65/1.99 % (2650)Instructions burned: 3 (million)
% 3.65/1.99 % (2650)------------------------------
% 3.65/1.99 % (2650)------------------------------
% 3.65/1.99 % (2651)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/4Mi)
% 3.65/1.99 % (2651)Instruction limit reached!
% 3.65/1.99 % (2651)------------------------------
% 3.65/1.99 % (2651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/1.99 % (2651)Termination reason: Unknown
% 3.65/1.99 % (2651)Termination phase: shuffling
% 3.65/1.99
% 3.65/1.99 % (2651)Memory used [KB]: 1663
% 3.65/1.99 % (2651)Time elapsed: 0.004 s
% 3.65/1.99 % (2651)Instructions burned: 5 (million)
% 3.65/1.99 % (2651)------------------------------
% 3.65/1.99 % (2651)------------------------------
% 3.65/1.99 % (2652)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2534 for (2999ds/18Mi)
% 3.65/1.99 % (2638)First to succeed.
% 3.65/2.00 % (2638)Refutation found. Thanks to Tanya!
% 3.65/2.00 % SZS status Theorem for DTF2THF_2534
% 3.65/2.00 % SZS output start Proof for DTF2THF_2534
% 3.65/2.00 thf(type_def_6, type, coinductive_llist: ($tType) > $tType).
% 3.65/2.00 thf(type_def_7, type, stream: ($tType) > $tType).
% 3.65/2.00 thf(type_def_8, type, itself: ($tType) > $tType).
% 3.65/2.00 thf(type_def_9, type, set: ($tType) > $tType).
% 3.65/2.00 thf(type_def_10, type, a: $tType).
% 3.65/2.00 thf(type_def_11, type, b: $tType).
% 3.65/2.00 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 3.65/2.00 thf(func_def_1, type, stream: ($tType) > $tType).
% 3.65/2.00 thf(func_def_2, type, set: ($tType) > $tType).
% 3.65/2.00 thf(func_def_3, type, itself: ($tType) > $tType).
% 3.65/2.00 thf(func_def_4, type, b: $tType).
% 3.65/2.00 thf(func_def_5, type, a: $tType).
% 3.65/2.00 thf(func_def_6, type, bNF_rel_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > (X0 > X2) > (X1 > X3) > $o)).
% 3.65/2.00 thf(func_def_7, type, basic_pred_fun: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > $o) > (X0 > X1) > $o)).
% 3.65/2.00 thf(func_def_8, type, coinductive_iterates: !>[X0: $tType]:((X0 > X0) > X0 > coinductive_llist @ X0)).
% 3.65/2.00 thf(func_def_9, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.65/2.00 thf(func_def_10, type, coindu1381640503_llist: !>[X0: $tType, X1: $tType]:(X0 > (X1 > coinductive_llist @ X1 > X0) > coinductive_llist @ X1 > X0)).
% 3.65/2.00 thf(func_def_11, type, coindu1259883913_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > $o) > (X0 > coinductive_llist @ X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 3.65/2.00 thf(func_def_12, type, coindu1486289336t_all2: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > coinductive_llist @ X0 > coinductive_llist @ X1 > $o)).
% 3.65/2.00 thf(func_def_13, type, coindu543516966_llist: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > $o)).
% 3.65/2.00 thf(func_def_14, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 3.65/2.00 thf(func_def_15, type, coindu1724414836stream: !>[X0: $tType]:(stream @ X0 > coinductive_llist @ X0)).
% 3.65/2.00 thf(func_def_16, type, coindu1183105481stream: !>[X0: $tType]:(coinductive_llist @ X0 > stream @ X0 > $o)).
% 3.65/2.00 thf(func_def_17, type, coindu773941317stream: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > coinductive_llist @ X0 > stream @ X1 > $o)).
% 3.65/2.00 thf(func_def_18, type, coindu2010755910_llist: !>[X0: $tType]:(coinductive_llist @ X0 > stream @ X0)).
% 3.65/2.00 thf(func_def_19, type, coindu139217191stream: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X0) > X0 > stream @ X1)).
% 3.65/2.00 thf(func_def_20, type, comple1396247847notone: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X1 > $o) > (X0 > X1) > $o)).
% 3.65/2.00 thf(func_def_21, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 3.65/2.00 thf(func_def_22, type, map_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (X1 > X2) > X0 > X3)).
% 3.65/2.00 thf(func_def_23, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.65/2.00 thf(func_def_24, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.65/2.00 thf(func_def_25, type, rel_pred_comp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > $o) > X0 > $o)).
% 3.65/2.00 thf(func_def_26, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.65/2.00 thf(func_def_27, type, partial_fun_ord: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X2 > X0) > (X2 > X1) > $o)).
% 3.65/2.00 thf(func_def_28, type, partial_img_ord: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X1 > X1 > X2) > X0 > X0 > X2)).
% 3.65/2.00 thf(func_def_29, type, partial_mk_less: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > $o)).
% 3.65/2.00 thf(func_def_30, type, type: !>[X0: $tType]:(itself @ X0)).
% 3.65/2.00 thf(func_def_31, type, bex1_rel: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > $o)).
% 3.65/2.00 thf(func_def_32, type, quotient3: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > $o)).
% 3.65/2.00 thf(func_def_33, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 3.65/2.00 thf(func_def_34, type, powp: !>[X0: $tType]:((X0 > $o) > set @ X0 > $o)).
% 3.65/2.00 thf(func_def_35, type, reflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 3.65/2.00 thf(func_def_36, type, relcompp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > $o)).
% 3.65/2.00 thf(func_def_37, type, ball: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 3.65/2.00 thf(func_def_38, type, bex: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 3.65/2.00 thf(func_def_39, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.65/2.00 thf(func_def_40, type, siterate: !>[X0: $tType]:((X0 > X0) > X0 > stream @ X0)).
% 3.65/2.00 thf(func_def_41, type, case_stream: !>[X0: $tType, X1: $tType]:((X0 > stream @ X0 > X1) > stream @ X0 > X1)).
% 3.65/2.00 thf(func_def_42, type, corec_stream: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > (X0 > stream @ X1) > (X0 > X0) > X0 > stream @ X1)).
% 3.65/2.00 thf(func_def_43, type, pred_stream: !>[X0: $tType]:((X0 > $o) > stream @ X0 > $o)).
% 3.65/2.00 thf(func_def_44, type, stream_all2: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > stream @ X0 > stream @ X1 > $o)).
% 3.65/2.00 thf(func_def_45, type, bi_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 3.65/2.00 thf(func_def_46, type, left_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 3.65/2.00 thf(func_def_47, type, rev_implies: $o > $o > $o).
% 3.65/2.00 thf(func_def_48, type, right_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 3.65/2.00 thf(func_def_49, type, transfer_bforall: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > $o)).
% 3.65/2.00 thf(func_def_50, type, transfer_forall: !>[X0: $tType]:((X0 > $o) > $o)).
% 3.65/2.00 thf(func_def_51, type, type_definition: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X1 > $o)).
% 3.65/2.00 thf(func_def_52, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.65/2.00 thf(func_def_53, type, xb: a > coinductive_llist @ b).
% 3.65/2.00 thf(func_def_54, type, yb: a > stream @ b).
% 3.65/2.00 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X3 > X2) > (X0 > X1) > (X1 > X2 > $o) > (X0 > X3 > $o) > X0)).
% 3.65/2.00 thf(func_def_73, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X3 > X2) > (X0 > X1) > (X1 > X2 > $o) > (X0 > X3 > $o) > X3)).
% 3.65/2.00 thf(func_def_74, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((X0 > X3 > X4) > (X5 > X2 > X1) > (X1 > X4 > $o) > (X2 > X3 > $o) > (X5 > X0 > $o) > X5)).
% 3.65/2.00 thf(func_def_75, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((X0 > X3 > X4) > (X5 > X2 > X1) > (X1 > X4 > $o) > (X2 > X3 > $o) > (X5 > X0 > $o) > X0)).
% 3.65/2.00 thf(func_def_76, type, sK4: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X0)).
% 3.65/2.00 thf(func_def_77, type, sK5: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X1)).
% 3.65/2.00 thf(func_def_78, type, sK6: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X0)).
% 3.65/2.00 thf(func_def_79, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X1)).
% 3.65/2.00 thf(func_def_80, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.65/2.00 thf(func_def_81, type, sK9: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X0)).
% 3.65/2.00 thf(func_def_82, type, sK10: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X1)).
% 3.65/2.00 thf(func_def_83, type, sK11: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X0)).
% 3.65/2.00 thf(func_def_84, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > X1)).
% 3.65/2.00 thf(func_def_85, type, sK13: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.65/2.00 thf(func_def_87, type, ph15: !>[X0: $tType]:(X0)).
% 3.65/2.00 thf(func_def_88, type, sK16: a).
% 3.65/2.00 thf(f900,plain,(
% 3.65/2.00 $false),
% 3.65/2.00 inference(subsumption_resolution,[status(thm)],[f898,f870])).
% 3.65/2.00 thf(f870,plain,(
% 3.65/2.00 ( ! [X0 : a] : (((coindu1724414836stream @ b @ (yb @ X0)) = (xb @ X0))) )),
% 3.65/2.00 inference(equality_proxy_clausification,[status(thm)],[f869])).
% 3.65/2.00 thf(f869,plain,(
% 3.65/2.00 ( ! [X0 : a] : ((((coindu1724414836stream @ b @ (yb @ X0)) = (xb @ X0)) = $true)) )),
% 3.65/2.00 inference(trivial_inequality_removal,[status(thm)],[f868])).
% 3.65/2.00 thf(f868,plain,(
% 3.65/2.00 ( ! [X0 : a] : ((((coindu1724414836stream @ b @ (yb @ X0)) = (xb @ X0)) = $true) | ($true != $true)) )),
% 3.65/2.00 inference(boolean_simplification,[status(thm)],[f867])).
% 3.65/2.00 thf(f867,plain,(
% 3.65/2.00 ( ! [X0 : a] : ((((coindu1724414836stream @ b @ (yb @ X0)) = (xb @ X0)) = $true) | ((X0 = X0) != $true)) )),
% 3.65/2.00 inference(beta_eta_normalization,[status(thm)],[f866])).
% 3.65/2.00 thf(f866,plain,(
% 3.65/2.00 ( ! [X0 : a] : (((X0 = X0) != $true) | (((^[Y0 : coinductive_llist @ b]: ((^[Y1 : stream @ b]: ((coindu1724414836stream @ b @ Y1) = Y0)))) @ (xb @ X0) @ (yb @ X0)) = $true)) )),
% 3.65/2.00 inference(trivial_inequality_removal,[status(thm)],[f861])).
% 3.65/2.00 thf(f861,plain,(
% 3.65/2.00 ( ! [X0 : a] : ((((^[Y0 : coinductive_llist @ b]: ((^[Y1 : stream @ b]: ((coindu1724414836stream @ b @ Y1) = Y0)))) @ (xb @ X0) @ (yb @ X0)) = $true) | ((X0 = X0) != $true) | ($true != $true)) )),
% 3.65/2.00 inference(superposition,[status(thm)],[f800,f856])).
% 3.65/2.00 thf(f856,plain,(
% 3.65/2.00 ((bNF_rel_fun @ a @ a @ coinductive_llist @ b @ stream @ b @ = @ (^[Y0 : coinductive_llist @ b]: ((^[Y1 : stream @ b]: ((coindu1724414836stream @ b @ Y1) = Y0)))) @ xb @ yb) = $true)),
% 3.65/2.00 inference(beta_eta_normalization,[status(thm)],[f819])).
% 3.65/2.00 thf(f819,plain,(
% 3.65/2.00 ((bNF_rel_fun @ a @ a @ coinductive_llist @ b @ stream @ b @ (^[Y0 : a]: ((^[Y1 : a]: (Y0 = Y1)))) @ (^[Y0 : coinductive_llist @ b]: ((^[Y1 : stream @ b]: ((coindu1724414836stream @ b @ Y1) = Y0)))) @ xb @ yb) = $true)),
% 3.65/2.00 inference(cnf_transformation,[status(thm)],[f611])).
% 3.65/2.00 thf(f611,plain,(
% 3.65/2.00 ((bNF_rel_fun @ a @ a @ coinductive_llist @ b @ stream @ b @ (^[Y0 : a]: ((^[Y1 : a]: (Y0 = Y1)))) @ (^[Y0 : coinductive_llist @ b]: ((^[Y1 : stream @ b]: ((coindu1724414836stream @ b @ Y1) = Y0)))) @ xb @ yb) = $true)),
% 3.65/2.00 inference(fool_elimination,[status(thm)],[f610])).
% 3.65/2.00 thf(f610,plain,(
% 3.65/2.00 (bNF_rel_fun @ a @ a @ coinductive_llist @ b @ stream @ b @ (^[X0 : a, X1 : a] : ((X0 = X1))) @ (^[X2 : coinductive_llist @ b, X3 : stream @ b] : (((coindu1724414836stream @ b @ X3) = X2))) @ xb @ yb)),
% 3.65/2.00 inference(rectify,[status(thm)],[f260])).
% 3.65/2.00 thf(f260,axiom,(
% 3.65/2.00 (bNF_rel_fun @ a @ a @ coinductive_llist @ b @ stream @ b @ (^[X16 : a, X17 : a] : ((X16 = X17))) @ (^[X8 : coinductive_llist @ b, X18 : stream @ b] : (((coindu1724414836stream @ b @ X18) = X8))) @ xb @ yb)),
% 3.65/2.00 file('/export/starexec/sandbox/tmp/tmp.WjgrtYFCGm/DTF2THF_2534.p',conj_0)).
% 3.65/2.00 thf(f800,plain,(
% 3.65/2.00 ( ! [X1 : $tType,X0 : $tType,X2 : $tType,X3 : X2 > X2 > $o,X6 : X2 > X1,X7 : X2,X4 : X0 > X1 > $o,X5 : X2 > X0] : (($true != (bNF_rel_fun @ X2 @ X2 @ X0 @ X1 @ X3 @ X4 @ X5 @ X6)) | ((X4 @ (X5 @ X7) @ (X6 @ X7)) = $true) | ((X3 @ X7 @ X7) != $true)) )),
% 3.65/2.00 inference(cnf_transformation,[status(thm)],[f771])).
% 3.65/2.00 thf(f771,plain,(
% 3.65/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X2 > X2 > $o,X4 : X0 > X1 > $o,X5 : X2 > X0,X6 : X2 > X1,X7 : X2] : (((X4 @ (X5 @ X7) @ (X6 @ X7)) = $true) | ($true != (bNF_rel_fun @ X2 @ X2 @ X0 @ X1 @ X3 @ X4 @ X5 @ X6)) | ((X3 @ X7 @ X7) != $true))),
% 3.65/2.00 inference(flattening,[status(thm)],[f770])).
% 3.65/2.00 thf(f770,plain,(
% 3.65/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X2 > X2 > $o,X4 : X0 > X1 > $o,X5 : X2 > X0,X6 : X2 > X1,X7 : X2] : ((((X4 @ (X5 @ X7) @ (X6 @ X7)) = $true) | ((X3 @ X7 @ X7) != $true)) | ($true != (bNF_rel_fun @ X2 @ X2 @ X0 @ X1 @ X3 @ X4 @ X5 @ X6)))),
% 3.65/2.00 inference(ennf_transformation,[status(thm)],[f633])).
% 3.65/2.00 thf(f633,plain,(
% 3.65/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X2 > X2 > $o,X4 : X0 > X1 > $o,X5 : X2 > X0,X6 : X2 > X1,X7 : X2] : (($true = (bNF_rel_fun @ X2 @ X2 @ X0 @ X1 @ X3 @ X4 @ X5 @ X6)) => (((X3 @ X7 @ X7) = $true) => ((X4 @ (X5 @ X7) @ (X6 @ X7)) = $true)))),
% 3.65/2.00 inference(fool_elimination,[status(thm)],[f632])).
% 3.65/2.00 thf(f632,plain,(
% 3.65/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X2 > X2 > $o,X4 : X0 > X1 > $o,X5 : X2 > X0,X6 : X2 > X1,X7 : X2] : ((bNF_rel_fun @ X2 @ X2 @ X0 @ X1 @ X3 @ X4 @ X5 @ X6) => ((X3 @ X7 @ X7) => (X4 @ (X5 @ X7) @ (X6 @ X7))))),
% 3.65/2.00 inference(rectify,[status(thm)],[f38])).
% 3.65/2.00 thf(f38,axiom,(
% 3.65/2.00 ! [X2 : $tType,X1 : $tType,X0 : $tType,X9 : X0 > X0 > $o,X10 : X2 > X1 > $o,X11 : X0 > X2,X12 : X0 > X1,X4 : X0] : ((bNF_rel_fun @ X0 @ X0 @ X2 @ X1 @ X9 @ X10 @ X11 @ X12) => ((X9 @ X4 @ X4) => (X10 @ (X11 @ X4) @ (X12 @ X4))))),
% 3.65/2.00 file('/export/starexec/sandbox/tmp/tmp.WjgrtYFCGm/DTF2THF_2534.p',fact_37_rel__funD2)).
% 3.65/2.00 thf(f898,plain,(
% 3.65/2.00 ((coindu1724414836stream @ b @ (yb @ sK16)) != (xb @ sK16))),
% 3.65/2.00 inference(superposition,[status(thm)],[f857,f847])).
% 3.65/2.00 thf(f847,plain,(
% 3.65/2.00 ( ! [X1 : $tType,X0 : $tType,X2 : $tType,X3 : X0 > X1,X6 : X2,X4 : X2 > X0] : (((comp @ X0 @ X1 @ X2 @ X3 @ X4 @ X6) = (X3 @ (X4 @ X6)))) )),
% 3.65/2.00 inference(equality_resolution,[status(thm)],[f835])).
% 3.65/2.00 thf(f835,plain,(
% 3.65/2.00 ( ! [X1 : $tType,X0 : $tType,X2 : $tType,X3 : X0 > X1,X6 : X2,X4 : X2 > X0,X5 : X2 > X1] : (((X3 @ (X4 @ X6)) = (X5 @ X6)) | ((comp @ X0 @ X1 @ X2 @ X3 @ X4) != X5)) )),
% 3.65/2.00 inference(cnf_transformation,[status(thm)],[f757])).
% 3.65/2.00 thf(f757,plain,(
% 3.65/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X0 > X1,X4 : X2 > X0,X5 : X2 > X1,X6 : X2] : (((X3 @ (X4 @ X6)) = (X5 @ X6)) | ((comp @ X0 @ X1 @ X2 @ X3 @ X4) != X5))),
% 3.65/2.00 inference(ennf_transformation,[status(thm)],[f742])).
% 3.65/2.00 thf(f742,plain,(
% 3.65/2.00 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X0 > X1,X4 : X2 > X0,X5 : X2 > X1,X6 : X2] : (((comp @ X0 @ X1 @ X2 @ X3 @ X4) = X5) => ((X3 @ (X4 @ X6)) = (X5 @ X6)))),
% 3.65/2.00 inference(rectify,[status(thm)],[f29])).
% 3.65/2.00 thf(f29,axiom,(
% 3.65/2.00 ! [X1 : $tType,X2 : $tType,X0 : $tType,X55 : X1 > X2,X56 : X0 > X1,X31 : X0 > X2,X53 : X0] : (((comp @ X1 @ X2 @ X0 @ X55 @ X56) = X31) => ((X55 @ (X56 @ X53)) = (X31 @ X53)))),
% 3.65/2.00 file('/export/starexec/sandbox/tmp/tmp.WjgrtYFCGm/DTF2THF_2534.p',fact_28_comp__eq__dest__lhs)).
% 3.65/2.00 thf(f857,plain,(
% 3.65/2.00 ((comp @ stream @ b @ coinductive_llist @ b @ a @ coindu1724414836stream @ b @ yb @ sK16) != (xb @ sK16))),
% 3.65/2.00 inference(negative_extensionality,[status(thm)],[f825])).
% 3.65/2.00 thf(f825,plain,(
% 3.65/2.00 (xb != (comp @ stream @ b @ coinductive_llist @ b @ a @ coindu1724414836stream @ b @ yb))),
% 3.65/2.00 inference(cnf_transformation,[status(thm)],[f726])).
% 3.65/2.00 thf(f726,plain,(
% 3.65/2.00 (xb != (comp @ stream @ b @ coinductive_llist @ b @ a @ coindu1724414836stream @ b @ yb))),
% 3.65/2.00 inference(flattening,[status(thm)],[f262])).
% 3.65/2.00 thf(f262,negated_conjecture,(
% 3.65/2.00 ~(xb = (comp @ stream @ b @ coinductive_llist @ b @ a @ coindu1724414836stream @ b @ yb))),
% 3.65/2.00 inference(negated_conjecture,[status(cth)],[f261])).
% 3.65/2.00 thf(f261,conjecture,(
% 3.65/2.00 (xb = (comp @ stream @ b @ coinductive_llist @ b @ a @ coindu1724414836stream @ b @ yb))),
% 3.65/2.00 file('/export/starexec/sandbox/tmp/tmp.WjgrtYFCGm/DTF2THF_2534.p',conj_1)).
% 3.65/2.00 % SZS output end Proof for DTF2THF_2534
% 3.65/2.00 % (2638)------------------------------
% 3.65/2.00 % (2638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.65/2.00 % (2638)Termination reason: Refutation
% 3.65/2.00
% 3.65/2.00 % (2638)Memory used [KB]: 7803
% 3.65/2.00 % (2638)Time elapsed: 0.074 s
% 3.65/2.00 % (2638)Instructions burned: 144 (million)
% 3.65/2.00 % (2638)------------------------------
% 3.65/2.00 % (2638)------------------------------
% 3.65/2.00 % (2632)Success in time 0.109 s
%------------------------------------------------------------------------------