%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT141^1 : 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 : n029.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 08:44:54 AM UTC 2026
% Result : Theorem 2.56s 1.53s
% Output : Refutation 2.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT141^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n029.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 06:11:14 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.42 ---- Original DTF file ---
% 0.16/0.43 thf(spec,logic,$$dhol).
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 % File : DAT141^1 : TPTP v9.2.1. Released v7.0.0.
% 0.16/0.43 % Domain : Data Structures
% 0.16/0.43 % Problem : Coinductive list prefix 70
% 0.16/0.43 % Version : [Bla16] axioms : Especial.
% 0.16/0.43 % English :
% 0.16/0.43
% 0.16/0.43 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.16/0.43 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.16/0.43 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.16/0.43 % Source : [Bla16]
% 0.16/0.43 % Names : coinductive_list_prefix__70.p [Bla16]
% 0.16/0.43
% 0.16/0.43 % Status : Theorem
% 0.16/0.43 % Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% 0.16/0.43 % Syntax : Number of formulae : 325 ( 181 unt; 55 typ; 0 def)
% 0.16/0.43 % Number of atoms : 445 ( 329 equ; 0 cnn)
% 0.16/0.43 % Maximal formula atoms : 6 ( 1 avg)
% 0.16/0.43 % Number of connectives : 4668 ( 43 ~; 1 |; 23 &;4439 @)
% 0.16/0.43 % ( 0 <=>; 162 =>; 0 <=; 0 <~>)
% 0.16/0.43 % Maximal formula depth : 28 ( 8 avg)
% 0.16/0.43 % Number of types : 4 ( 3 usr)
% 0.16/0.43 % Number of type conns : 629 ( 629 >; 0 *; 0 +; 0 <<)
% 0.16/0.43 % Number of symbols : 55 ( 52 usr; 3 con; 0-9 aty)
% 0.16/0.43 % Number of variables : 1597 ( 181 ^;1314 !; 7 ?;1597 :)
% 0.16/0.43 % ( 95 !>; 0 ?*; 0 @-; 0 @+)
% 0.16/0.43 % SPC : TH1_THM_EQU_NAR
% 0.16/0.43
% 0.16/0.43 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.16/0.43 % 2016-07-13 14:40:51.031
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 %----Could-be-implicit typings (7)
% 0.16/0.43 thf(ty_t_Coinductive__List_Ollist,type,
% 0.16/0.43 coinductive_llist: $tType > $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Product__Type_Ounit,type,
% 0.16/0.43 product_unit: $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Product__Type_Oprod,type,
% 0.16/0.43 product_prod: $tType > $tType > $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Set_Oset,type,
% 0.16/0.43 set: $tType > $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Nat_Onat,type,
% 0.16/0.43 nat: $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_itself,type,
% 0.16/0.43 itself: $tType > $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_tf_a,type,
% 0.16/0.43 a: $tType ).
% 0.16/0.43
% 0.16/0.43 %----Explicit typings (48)
% 0.16/0.43 thf(sy_cl_HOL_Otype,type,
% 0.16/0.43 type:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Lattices_Olattice,type,
% 0.16/0.43 lattice:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.16/0.43 semilattice_inf:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Def_OGrp,type,
% 0.16/0.43 bNF_Grp:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Def_Oconvol,type,
% 0.16/0.43 bNF_convol:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Def_Ocsquare,type,
% 0.16/0.43 bNF_csquare:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Def_OfstOp,type,
% 0.16/0.43 bNF_fstOp:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.16/0.43 bNF_pick_middlep:
% 0.16/0.43 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Def_OsndOp,type,
% 0.16/0.43 bNF_sndOp:
% 0.16/0.43 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.16/0.43 bNF_Greatest_image2:
% 0.16/0.43 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
% 0.16/0.43 basic_BNF_size_prod:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Basic__BNFs_Opred__fun,type,
% 0.16/0.43 basic_pred_fun:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.16/0.43 coinductive_LNil:
% 0.16/0.43 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 0.16/0.43 coinductive_lhd:
% 0.16/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Coinductive__List_Ollist_Oltl,type,
% 0.16/0.43 coinductive_ltl:
% 0.16/0.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.16/0.43 coindu1441602521_llist:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Fun_Ocomp,type,
% 0.16/0.43 comp:
% 0.16/0.43 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Fun_Oid,type,
% 0.16/0.43 id:
% 0.16/0.43 !>[A: $tType] : ( A > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.16/0.43 fun_reduction_pair:
% 0.16/0.43 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.16/0.43 fun_rp_inv_image:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_HOL_OThe,type,
% 0.16/0.43 the:
% 0.16/0.43 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_If,type,
% 0.16/0.43 if:
% 0.16/0.43 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.16/0.43 inf_inf:
% 0.16/0.43 !>[A: $tType] : ( A > A > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_OPair,type,
% 0.16/0.43 product_Pair:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oapfst,type,
% 0.16/0.43 product_apfst:
% 0.16/0.43 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oapsnd,type,
% 0.16/0.43 product_apsnd:
% 0.16/0.43 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.16/0.43 produc2004651681e_prod:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Omap__prod,type,
% 0.16/0.43 product_map_prod:
% 0.16/0.43 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.16/0.43 product_rec_bool:
% 0.16/0.43 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 0.16/0.43 product_rec_set_bool:
% 0.16/0.43 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.16/0.43 product_rec_prod:
% 0.16/0.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.16/0.43 product_rec_set_prod:
% 0.16/0.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.16/0.43 product_rec_set_unit:
% 0.16/0.43 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.16/0.43 product_rec_unit:
% 0.16/0.43 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.16/0.43 product_case_prod:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.16/0.43 product_fst:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.16/0.43 product_snd:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.16/0.43 product_swap:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Product__Type_Oscomp,type,
% 0.16/0.43 product_scomp:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Pure_Otype,type,
% 0.16/0.43 type2:
% 0.16/0.43 !>[A: $tType] : ( itself @ A ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Record_Oiso__tuple__update__accessor__cong__assist,type,
% 0.16/0.43 iso_tu2017585022assist:
% 0.16/0.43 !>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
% 0.16/0.43 iso_tu2011167877assist:
% 0.16/0.43 !>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Relation_Oinv__image,type,
% 0.16/0.43 inv_image:
% 0.16/0.43 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Relation_Orelcompp,type,
% 0.16/0.43 relcompp:
% 0.16/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Set_OCollect,type,
% 0.16/0.43 collect:
% 0.16/0.43 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_Wfrec_Osame__fst,type,
% 0.16/0.43 same_fst:
% 0.16/0.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_c_member,type,
% 0.16/0.43 member:
% 0.16/0.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_v_xs,type,
% 0.16/0.43 xs: coinductive_llist @ a ).
% 0.16/0.43
% 0.16/0.43 %----Relevant facts (256)
% 0.16/0.43 thf(fact_0_local_Oinf__llist__def,axiom,
% 0.16/0.43 ( ( inf_inf @ ( coinductive_llist @ a ) )
% 0.16/0.43 = ( ^ [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.16/0.43 ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ a
% 0.16/0.43 @ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
% 0.16/0.43 @ ^ [Xt: coinductive_llist @ a,Yt: coinductive_llist @ a] :
% 0.16/0.43 ( ( Xt
% 0.16/0.43 != ( coinductive_LNil @ a ) )
% 0.16/0.43 => ( ( Yt
% 0.16/0.43 != ( coinductive_LNil @ a ) )
% 0.16/0.43 => ( ( coinductive_lhd @ a @ Xt )
% 0.16/0.43 != ( coinductive_lhd @ a @ Yt ) ) ) ) )
% 0.16/0.43 @ ( comp @ ( coinductive_llist @ a ) @ a @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( coinductive_lhd @ a ) @ ( product_snd @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) )
% 0.16/0.43 @ ( product_map_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_ltl @ a ) @ ( coinductive_ltl @ a ) )
% 0.16/0.43 @ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ Xs @ Ys ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % local.inf_llist_def
% 0.16/0.43 thf(fact_1_snd__comp__map__prod,axiom,
% 0.16/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
% 0.16/0.43 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_comp_map_prod
% 0.16/0.43 thf(fact_2_case__prodI,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.16/0.43 ( ( F @ A2 @ B2 )
% 0.16/0.43 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodI
% 0.16/0.43 thf(fact_3_case__prodI2,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
% 0.16/0.43 ( ! [A3: A,B3: B] :
% 0.16/0.43 ( ( P
% 0.16/0.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.16/0.43 => ( C2 @ A3 @ B3 ) )
% 0.16/0.43 => ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodI2
% 0.16/0.43 thf(fact_4_case__prod__conv,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 0.16/0.43 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.16/0.43 = ( F @ A2 @ B2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_conv
% 0.16/0.43 thf(fact_5_snd__map__prod,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 0.16/0.43 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 0.16/0.43 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_map_prod
% 0.16/0.43 thf(fact_6_map__prod__simp,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 0.16/0.43 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod_simp
% 0.16/0.43 thf(fact_7_Coinductive__List__Prefix__Mirabelle__yfrixeyiok_Oinf__llist__def,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( inf_inf @ ( coinductive_llist @ A ) )
% 0.16/0.43 = ( ^ [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.16/0.43 ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ A
% 0.16/0.43 @ ( product_case_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ $o
% 0.16/0.43 @ ^ [Xt: coinductive_llist @ A,Yt: coinductive_llist @ A] :
% 0.16/0.43 ( ( Xt
% 0.16/0.43 != ( coinductive_LNil @ A ) )
% 0.16/0.43 => ( ( Yt
% 0.16/0.43 != ( coinductive_LNil @ A ) )
% 0.16/0.43 => ( ( coinductive_lhd @ A @ Xt )
% 0.16/0.43 != ( coinductive_lhd @ A @ Yt ) ) ) ) )
% 0.16/0.43 @ ( comp @ ( coinductive_llist @ A ) @ A @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( coinductive_lhd @ A ) @ ( product_snd @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) )
% 0.16/0.43 @ ( product_map_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_ltl @ A ) @ ( coinductive_ltl @ A ) )
% 0.16/0.43 @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs @ Ys ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Coinductive_List_Prefix_Mirabelle_yfrixeyiok.inf_llist_def
% 0.16/0.43 thf(fact_8_map__prod__ident,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_map_prod @ A @ A @ B @ B
% 0.16/0.43 @ ^ [X2: A] : X2
% 0.16/0.43 @ ^ [Y: B] : Y )
% 0.16/0.43 = ( ^ [Z: product_prod @ A @ B] : Z ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod_ident
% 0.16/0.43 thf(fact_9_split__part,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ $o
% 0.16/0.43 @ ^ [A4: A,B4: B] :
% 0.16/0.43 ( P2
% 0.16/0.43 & ( Q @ A4 @ B4 ) ) )
% 0.16/0.43 = ( ^ [Ab: product_prod @ A @ B] :
% 0.16/0.43 ( P2
% 0.16/0.43 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % split_part
% 0.16/0.43 thf(fact_10_ltl__unfold__llist,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
% 0.16/0.43 ( ( ( IS_LNIL @ A2 )
% 0.16/0.43 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 0.16/0.43 = ( coinductive_LNil @ A ) ) )
% 0.16/0.43 & ( ~ ( IS_LNIL @ A2 )
% 0.16/0.43 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 0.16/0.43 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % ltl_unfold_llist
% 0.16/0.43 thf(fact_11_comp__apply,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType] :
% 0.16/0.43 ( ( comp @ B @ A @ C )
% 0.16/0.43 = ( ^ [F2: B > A,G2: C > B,X2: C] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_apply
% 0.16/0.43 thf(fact_12_old_Oprod_Oinject,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 0.16/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.16/0.43 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.16/0.43 = ( ( A2 = A5 )
% 0.16/0.43 & ( B2 = B5 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.prod.inject
% 0.16/0.43 thf(fact_13_prod_Oinject,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.16/0.43 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.16/0.43 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.16/0.43 = ( ( X1 = Y1 )
% 0.16/0.43 & ( X22 = Y2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.inject
% 0.16/0.43 thf(fact_14_mem__case__prodI2,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
% 0.16/0.43 ( ! [A3: A,B3: B] :
% 0.16/0.43 ( ( P
% 0.16/0.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.16/0.43 => ( member @ C @ Z2 @ ( C2 @ A3 @ B3 ) ) )
% 0.16/0.43 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mem_case_prodI2
% 0.16/0.43 thf(fact_15_mem__case__prodI,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
% 0.16/0.43 ( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) )
% 0.16/0.43 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mem_case_prodI
% 0.16/0.43 thf(fact_16_case__prodI2_H,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.16/0.43 ( ! [A3: A,B3: B] :
% 0.16/0.43 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.16/0.43 = P )
% 0.16/0.43 => ( C2 @ A3 @ B3 @ X ) )
% 0.16/0.43 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodI2'
% 0.16/0.43 thf(fact_17_mem__case__prodE,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P: product_prod @ B @ C] :
% 0.16/0.43 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P ) )
% 0.16/0.43 => ~ ! [X3: B,Y3: C] :
% 0.16/0.43 ( ( P
% 0.16/0.43 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.16/0.43 => ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % mem_case_prodE
% 0.16/0.43 thf(fact_18_case__prodE_H,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P: product_prod @ A @ B,Z2: C] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ Z2 )
% 0.16/0.43 => ~ ! [X3: A,Y3: B] :
% 0.16/0.43 ( ( P
% 0.16/0.43 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.16/0.43 => ~ ( C2 @ X3 @ Y3 @ Z2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodE'
% 0.16/0.43 thf(fact_19_case__prodD_H,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
% 0.16/0.43 => ( R @ A2 @ B2 @ C2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodD'
% 0.16/0.43 thf(fact_20_map__prod__def,axiom,
% 0.16/0.43 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.16/0.43 ( ( product_map_prod @ A @ C @ B @ D )
% 0.16/0.43 = ( ^ [F2: A > C,G2: B > D] :
% 0.16/0.43 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.16/0.43 @ ^ [X2: A,Y: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G2 @ Y ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod_def
% 0.16/0.43 thf(fact_21_old_Oprod_Oinducts,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( ! [A3: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.16/0.43 => ( P2 @ Prod ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.prod.inducts
% 0.16/0.43 thf(fact_22_old_Oprod_Oexhaust,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
% 0.16/0.43 ~ ! [A3: A,B3: B] :
% 0.16/0.43 ( Y4
% 0.16/0.43 != ( product_Pair @ A @ B @ A3 @ B3 ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.prod.exhaust
% 0.16/0.43 thf(fact_23_prod__induct7,axiom,
% 0.16/0.43 ! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.16/0.43 ( ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.16/0.43 => ( P2 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_induct7
% 0.16/0.43 thf(fact_24_prod__induct6,axiom,
% 0.16/0.43 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.16/0.43 ( ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 0.16/0.43 => ( P2 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_induct6
% 0.16/0.43 thf(fact_25_prod__induct5,axiom,
% 0.16/0.43 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.16/0.43 ( ! [A3: A,B3: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.16/0.43 => ( P2 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_induct5
% 0.16/0.43 thf(fact_26_prod__induct4,axiom,
% 0.16/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.16/0.43 ( ! [A3: A,B3: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.16/0.43 => ( P2 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_induct4
% 0.16/0.43 thf(fact_27_prod__induct3,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.16/0.43 ( ! [A3: A,B3: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) )
% 0.16/0.43 => ( P2 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_induct3
% 0.16/0.43 thf(fact_28_prod__cases7,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.16/0.43 ~ ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3,G4: G3] :
% 0.16/0.43 ( Y4
% 0.16/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_cases7
% 0.16/0.43 thf(fact_29_prod__cases6,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.16/0.43 ~ ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3] :
% 0.16/0.43 ( Y4
% 0.16/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_cases6
% 0.16/0.43 thf(fact_30_prod__cases5,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.16/0.43 ~ ! [A3: A,B3: B,C3: C,D2: D,E2: E] :
% 0.16/0.43 ( Y4
% 0.16/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_cases5
% 0.16/0.43 thf(fact_31_prod__cases4,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.16/0.43 ~ ! [A3: A,B3: B,C3: C,D2: D] :
% 0.16/0.43 ( Y4
% 0.16/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_cases4
% 0.16/0.43 thf(fact_32_prod__cases3,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.16/0.43 ~ ! [A3: A,B3: B,C3: C] :
% 0.16/0.43 ( Y4
% 0.16/0.43 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_cases3
% 0.16/0.43 thf(fact_33_Pair__inject,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 0.16/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.16/0.43 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.16/0.43 => ~ ( ( A2 = A5 )
% 0.16/0.43 => ( B2 != B5 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Pair_inject
% 0.16/0.43 thf(fact_34_prod__cases,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.16/0.43 ( ! [A3: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.16/0.43 => ( P2 @ P ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_cases
% 0.16/0.43 thf(fact_35_surj__pair,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.16/0.43 ? [X3: A,Y3: B] :
% 0.16/0.43 ( P
% 0.16/0.43 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.16/0.43
% 0.16/0.43 % surj_pair
% 0.16/0.43 thf(fact_36_comp__eq__dest__lhs,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.16/0.43 = C2 )
% 0.16/0.43 => ( ( A2 @ ( B2 @ V ) )
% 0.16/0.43 = ( C2 @ V ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_eq_dest_lhs
% 0.16/0.43 thf(fact_37_comp__eq__elim,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D3: A > D] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.16/0.43 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.16/0.43 => ! [V2: A] :
% 0.16/0.43 ( ( A2 @ ( B2 @ V2 ) )
% 0.16/0.43 = ( C2 @ ( D3 @ V2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_eq_elim
% 0.16/0.43 thf(fact_38_comp__eq__dest,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.16/0.43 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.16/0.43 => ( ( A2 @ ( B2 @ V ) )
% 0.16/0.43 = ( C2 @ ( D3 @ V ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_eq_dest
% 0.16/0.43 thf(fact_39_comp__assoc,axiom,
% 0.16/0.43 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 0.16/0.43 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 0.16/0.43 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_assoc
% 0.16/0.43 thf(fact_40_comp__def,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType] :
% 0.16/0.43 ( ( comp @ B @ C @ A )
% 0.16/0.43 = ( ^ [F2: B > C,G2: A > B,X2: A] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_def
% 0.16/0.43 thf(fact_41_prod_Ocase__distrib,axiom,
% 0.16/0.43 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.16/0.43 = ( product_case_prod @ A @ B @ D
% 0.16/0.43 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 0.16/0.43 @ Prod ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.case_distrib
% 0.16/0.43 thf(fact_42_prod_Odisc__eq__case,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( product_case_prod @ A @ B @ $o
% 0.16/0.43 @ ^ [Uu: A,Uv: B] : $true
% 0.16/0.43 @ Prod ) ).
% 0.16/0.43
% 0.16/0.43 % prod.disc_eq_case
% 0.16/0.43 thf(fact_43_old_Oprod_Ocase,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.16/0.43 = ( F @ X1 @ X22 ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.prod.case
% 0.16/0.43 thf(fact_44_snd__conv,axiom,
% 0.16/0.43 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.16/0.43 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.16/0.43 = X22 ) ).
% 0.16/0.43
% 0.16/0.43 % snd_conv
% 0.16/0.43 thf(fact_45_mem__Collect__eq,axiom,
% 0.16/0.43 ! [A: $tType,A2: A,P2: A > $o] :
% 0.16/0.43 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.16/0.43 = ( P2 @ A2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % mem_Collect_eq
% 0.16/0.43 thf(fact_46_Collect__mem__eq,axiom,
% 0.16/0.43 ! [A: $tType,A6: set @ A] :
% 0.16/0.43 ( ( collect @ A
% 0.16/0.43 @ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
% 0.16/0.43 = A6 ) ).
% 0.16/0.43
% 0.16/0.43 % Collect_mem_eq
% 0.16/0.43 thf(fact_47_Collect__cong,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.16/0.43 ( ! [X3: A] :
% 0.16/0.43 ( ( P2 @ X3 )
% 0.16/0.43 = ( Q @ X3 ) )
% 0.16/0.43 => ( ( collect @ A @ P2 )
% 0.16/0.43 = ( collect @ A @ Q ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Collect_cong
% 0.16/0.43 thf(fact_48_ext,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.16/0.43 ( ! [X3: A] :
% 0.16/0.43 ( ( F @ X3 )
% 0.16/0.43 = ( G @ X3 ) )
% 0.16/0.43 => ( F = G ) ) ).
% 0.16/0.43
% 0.16/0.43 % ext
% 0.16/0.43 thf(fact_49_snd__eqD,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X: B,Y4: A,A2: A] :
% 0.16/0.43 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) )
% 0.16/0.43 = A2 )
% 0.16/0.43 => ( Y4 = A2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_eqD
% 0.16/0.43 thf(fact_50_ltl__simps_I1_J,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
% 0.16/0.43 = ( coinductive_LNil @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % ltl_simps(1)
% 0.16/0.43 thf(fact_51_map__prod_Ocompositionality,axiom,
% 0.16/0.43 ! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F3,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_map_prod @ C @ E @ D @ F3 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
% 0.16/0.43 = ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Prod ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod.compositionality
% 0.16/0.43 thf(fact_52_map__prod__compose,axiom,
% 0.16/0.43 ! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: E > C,F22: A > E,G1: F3 > D,G22: B > F3] :
% 0.16/0.43 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F3 @ D @ B @ G1 @ G22 ) )
% 0.16/0.43 = ( comp @ ( product_prod @ E @ F3 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F3 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F3 @ F22 @ G22 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod_compose
% 0.16/0.43 thf(fact_53_map__prod_Ocomp,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F: C > E,G: D > F3,H: A > C,I: B > D] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F3 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F3 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
% 0.16/0.43 = ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod.comp
% 0.16/0.43 thf(fact_54_unfold__llist__ltl__unroll,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B] :
% 0.16/0.43 ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) )
% 0.16/0.43 = ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % unfold_llist_ltl_unroll
% 0.16/0.43 thf(fact_55_unfold__llist_Octr_I1_J,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
% 0.16/0.43 ( ( P @ A2 )
% 0.16/0.43 => ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 )
% 0.16/0.43 = ( coinductive_LNil @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % unfold_llist.ctr(1)
% 0.16/0.43 thf(fact_56_unfold__llist_Osimps_I4_J,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
% 0.16/0.43 ( ~ ( P @ A2 )
% 0.16/0.43 => ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 ) )
% 0.16/0.43 = ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ ( G222 @ A2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % unfold_llist.simps(4)
% 0.16/0.43 thf(fact_57_unfold__llist_Osimps_I3_J,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
% 0.16/0.43 ( ~ ( P @ A2 )
% 0.16/0.43 => ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 ) )
% 0.16/0.43 = ( G21 @ A2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % unfold_llist.simps(3)
% 0.16/0.43 thf(fact_58_cond__case__prod__eta,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.16/0.43 ( ! [X3: A,Y3: B] :
% 0.16/0.43 ( ( F @ X3 @ Y3 )
% 0.16/0.43 = ( G @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.16/0.43 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.16/0.43 = G ) ) ).
% 0.16/0.43
% 0.16/0.43 % cond_case_prod_eta
% 0.16/0.43 thf(fact_59_case__prod__eta,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ C
% 0.16/0.43 @ ^ [X2: A,Y: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
% 0.16/0.43 = F ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_eta
% 0.16/0.43 thf(fact_60_case__prodE2,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z2: product_prod @ B @ C] :
% 0.16/0.43 ( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z2 ) )
% 0.16/0.43 => ~ ! [X3: B,Y3: C] :
% 0.16/0.43 ( ( Z2
% 0.16/0.43 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.16/0.43 => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodE2
% 0.16/0.43 thf(fact_61_case__prodE,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ $o @ C2 @ P )
% 0.16/0.43 => ~ ! [X3: A,Y3: B] :
% 0.16/0.43 ( ( P
% 0.16/0.43 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.16/0.43 => ~ ( C2 @ X3 @ Y3 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodE
% 0.16/0.43 thf(fact_62_case__prodD,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.16/0.43 => ( F @ A2 @ B2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prodD
% 0.16/0.43 thf(fact_63_snd__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_snd @ A @ B )
% 0.16/0.43 = ( product_case_prod @ A @ B @ B
% 0.16/0.43 @ ^ [X12: A,X23: B] : X23 ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_def
% 0.16/0.43 thf(fact_64_case__prod__o__map__prod,axiom,
% 0.16/0.43 ! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F: D > E > C,G1: A > D,G22: B > E] :
% 0.16/0.43 ( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
% 0.16/0.43 = ( product_case_prod @ A @ B @ C
% 0.16/0.43 @ ^ [L: A,R2: B] : ( F @ ( G1 @ L ) @ ( G22 @ R2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_o_map_prod
% 0.16/0.43 thf(fact_65_snd__diag__snd,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
% 0.16/0.43 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 0.16/0.43 @ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
% 0.16/0.43 @ ( product_snd @ A @ B ) ) )
% 0.16/0.43 = ( product_snd @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_diag_snd
% 0.16/0.43 thf(fact_66_old_Oprod_Orec,axiom,
% 0.16/0.43 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.16/0.43 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.16/0.43 = ( F1 @ A2 @ B2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.prod.rec
% 0.16/0.43 thf(fact_67_case__prod__map__prod,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X: product_prod @ D @ E] :
% 0.16/0.43 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X ) )
% 0.16/0.43 = ( product_case_prod @ D @ E @ A
% 0.16/0.43 @ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G @ R2 ) )
% 0.16/0.43 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_map_prod
% 0.16/0.43 thf(fact_68_inf__apply,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.16/0.43 => ( ( inf_inf @ ( A > B ) )
% 0.16/0.43 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_apply
% 0.16/0.43 thf(fact_69_inf_Oidem,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [A2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ A2 @ A2 )
% 0.16/0.43 = A2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf.idem
% 0.16/0.43 thf(fact_70_inf__idem,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A] :
% 0.16/0.43 ( ( inf_inf @ A @ X @ X )
% 0.16/0.43 = X ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_idem
% 0.16/0.43 thf(fact_71_inf_Oleft__idem,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [A2: A,B2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.16/0.43 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf.left_idem
% 0.16/0.43 thf(fact_72_inf__left__idem,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A] :
% 0.16/0.43 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y4 ) )
% 0.16/0.43 = ( inf_inf @ A @ X @ Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_left_idem
% 0.16/0.43 thf(fact_73_inf_Oright__idem,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [A2: A,B2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 0.16/0.43 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf.right_idem
% 0.16/0.43 thf(fact_74_inf__right__idem,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A] :
% 0.16/0.43 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Y4 )
% 0.16/0.43 = ( inf_inf @ A @ X @ Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_right_idem
% 0.16/0.43 thf(fact_75_inf__left__commute,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A,Z2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) )
% 0.16/0.43 = ( inf_inf @ A @ Y4 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_left_commute
% 0.16/0.43 thf(fact_76_inf_Oleft__commute,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [B2: A,A2: A,C2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C2 ) )
% 0.16/0.43 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf.left_commute
% 0.16/0.43 thf(fact_77_inf__commute,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ( ( inf_inf @ A )
% 0.16/0.43 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_commute
% 0.16/0.43 thf(fact_78_inf_Ocommute,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ( ( inf_inf @ A )
% 0.16/0.43 = ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf.commute
% 0.16/0.43 thf(fact_79_inf__assoc,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A,Z2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Z2 )
% 0.16/0.43 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_assoc
% 0.16/0.43 thf(fact_80_inf_Oassoc,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [A2: A,B2: A,C2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 )
% 0.16/0.43 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf.assoc
% 0.16/0.43 thf(fact_81_inf__fun__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.16/0.43 => ( ( inf_inf @ ( A > B ) )
% 0.16/0.43 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_fun_def
% 0.16/0.43 thf(fact_82_inf__sup__aci_I1_J,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( lattice @ A @ ( type2 @ A ) )
% 0.16/0.43 => ( ( inf_inf @ A )
% 0.16/0.43 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_sup_aci(1)
% 0.16/0.43 thf(fact_83_inf__sup__aci_I2_J,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( lattice @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A,Z2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Z2 )
% 0.16/0.43 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_sup_aci(2)
% 0.16/0.43 thf(fact_84_inf__sup__aci_I3_J,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( lattice @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A,Z2: A] :
% 0.16/0.43 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) )
% 0.16/0.43 = ( inf_inf @ A @ Y4 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_sup_aci(3)
% 0.16/0.43 thf(fact_85_inf__sup__aci_I4_J,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( lattice @ A @ ( type2 @ A ) )
% 0.16/0.43 => ! [X: A,Y4: A] :
% 0.16/0.43 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y4 ) )
% 0.16/0.43 = ( inf_inf @ A @ X @ Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_sup_aci(4)
% 0.16/0.43 thf(fact_86_rewriteL__comp__comp,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L2: A > B,H: D > A] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.16/0.43 = L2 )
% 0.16/0.43 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 0.16/0.43 = ( comp @ A @ B @ D @ L2 @ H ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % rewriteL_comp_comp
% 0.16/0.43 thf(fact_87_rewriteR__comp__comp,axiom,
% 0.16/0.43 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.16/0.43 = R3 )
% 0.16/0.43 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 0.16/0.43 = ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % rewriteR_comp_comp
% 0.16/0.43 thf(fact_88_rewriteL__comp__comp2,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L22: A > D,H: E > A,R3: E > D] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.16/0.43 = ( comp @ D @ B @ A @ L1 @ L22 ) )
% 0.16/0.43 => ( ( ( comp @ A @ D @ E @ L22 @ H )
% 0.16/0.43 = R3 )
% 0.16/0.43 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 0.16/0.43 = ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % rewriteL_comp_comp2
% 0.16/0.43 thf(fact_89_rewriteR__comp__comp2,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L2: D > E] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.16/0.43 = ( comp @ D @ B @ A @ R1 @ R22 ) )
% 0.16/0.43 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 0.16/0.43 = L2 )
% 0.16/0.43 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 0.16/0.43 = ( comp @ D @ E @ A @ L2 @ R22 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % rewriteR_comp_comp2
% 0.16/0.43 thf(fact_90_case__prod__Pair__iden,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
% 0.16/0.43 = P ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_Pair_iden
% 0.16/0.43 thf(fact_91_case__prod__app,axiom,
% 0.16/0.43 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.16/0.43 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.16/0.43 = ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y: D] :
% 0.16/0.43 ( product_case_prod @ B @ C @ A
% 0.16/0.43 @ ^ [L: B,R2: C] : ( F2 @ L @ R2 @ Y )
% 0.16/0.43 @ X2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_app
% 0.16/0.43 thf(fact_92_prod_Omap__comp,axiom,
% 0.16/0.43 ! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F3,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_map_prod @ C @ E @ D @ F3 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 0.16/0.43 = ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G22 @ F22 ) @ V ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.map_comp
% 0.16/0.43 thf(fact_93_sndI,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y4: A,Z2: B] :
% 0.16/0.43 ( ( X
% 0.16/0.43 = ( product_Pair @ A @ B @ Y4 @ Z2 ) )
% 0.16/0.43 => ( ( product_snd @ A @ B @ X )
% 0.16/0.43 = Z2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % sndI
% 0.16/0.43 thf(fact_94_eq__snd__iff,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
% 0.16/0.43 ( ( B2
% 0.16/0.43 = ( product_snd @ B @ A @ P ) )
% 0.16/0.43 = ( ? [A4: B] :
% 0.16/0.43 ( P
% 0.16/0.43 = ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % eq_snd_iff
% 0.16/0.43 thf(fact_95_split__cong,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G: A > B > C,P: product_prod @ A @ B] :
% 0.16/0.43 ( ! [X3: A,Y3: B] :
% 0.16/0.43 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 0.16/0.43 = Q2 )
% 0.16/0.43 => ( ( F @ X3 @ Y3 )
% 0.16/0.43 = ( G @ X3 @ Y3 ) ) )
% 0.16/0.43 => ( ( P = Q2 )
% 0.16/0.43 => ( ( product_case_prod @ A @ B @ C @ F @ P )
% 0.16/0.43 = ( product_case_prod @ A @ B @ C @ G @ Q2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % split_cong
% 0.16/0.43 thf(fact_96_snd__comp__apsnd,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.16/0.43 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_comp_apsnd
% 0.16/0.43 thf(fact_97_map__prod__o__convol,axiom,
% 0.16/0.43 ! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H2: E > C,F: A > D,G: A > E] :
% 0.16/0.43 ( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H2 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
% 0.16/0.43 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H2 @ G ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod_o_convol
% 0.16/0.43 thf(fact_98_apsnd__conv,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y4: C] :
% 0.16/0.43 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y4 ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ X @ ( F @ Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_conv
% 0.16/0.43 thf(fact_99_apsnd__eq__conv,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
% 0.16/0.43 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 0.16/0.43 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 0.16/0.43 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 0.16/0.43 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_eq_conv
% 0.16/0.43 thf(fact_100_snd__apsnd,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 0.16/0.43 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 0.16/0.43 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_apsnd
% 0.16/0.43 thf(fact_101_convol__def,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( bNF_convol @ A @ B @ C )
% 0.16/0.43 = ( ^ [F2: A > B,G2: A > C,A4: A] : ( product_Pair @ B @ C @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % convol_def
% 0.16/0.43 thf(fact_102_apsnd__compose,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
% 0.16/0.43 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
% 0.16/0.43 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_compose
% 0.16/0.43 thf(fact_103_snd__convol_H,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X: C] :
% 0.16/0.43 ( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X ) )
% 0.16/0.43 = ( G @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_convol'
% 0.16/0.43 thf(fact_104_snd__convol,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
% 0.16/0.43 = G ) ).
% 0.16/0.43
% 0.16/0.43 % snd_convol
% 0.16/0.43 thf(fact_105_fun_Omap__comp,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 0.16/0.43 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 0.16/0.43 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.16/0.43
% 0.16/0.43 % fun.map_comp
% 0.16/0.43 thf(fact_106_comp__apply__eq,axiom,
% 0.16/0.43 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
% 0.16/0.43 ( ( ( F @ ( G @ X ) )
% 0.16/0.43 = ( H @ ( K @ X ) ) )
% 0.16/0.43 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.16/0.43 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_apply_eq
% 0.16/0.43 thf(fact_107_comp__cong,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F5: D > A,G5: E > D,X4: E] :
% 0.16/0.43 ( ( ( F @ ( G @ X ) )
% 0.16/0.43 = ( F5 @ ( G5 @ X4 ) ) )
% 0.16/0.43 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.16/0.43 = ( comp @ D @ A @ E @ F5 @ G5 @ X4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_cong
% 0.16/0.43 thf(fact_108_convol__o,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H: A > D] :
% 0.16/0.43 ( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H )
% 0.16/0.43 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H ) @ ( comp @ D @ C @ A @ G @ H ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % convol_o
% 0.16/0.43 thf(fact_109_fun_Omap__ident,axiom,
% 0.16/0.43 ! [A: $tType,D: $tType,T2: D > A] :
% 0.16/0.43 ( ( comp @ A @ A @ D
% 0.16/0.43 @ ^ [X2: A] : X2
% 0.16/0.43 @ T2 )
% 0.16/0.43 = T2 ) ).
% 0.16/0.43
% 0.16/0.43 % fun.map_ident
% 0.16/0.43 thf(fact_110_prod_Omap__ident,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_map_prod @ A @ A @ B @ B
% 0.16/0.43 @ ^ [X2: A] : X2
% 0.16/0.43 @ ^ [X2: B] : X2
% 0.16/0.43 @ T2 )
% 0.16/0.43 = T2 ) ).
% 0.16/0.43
% 0.16/0.43 % prod.map_ident
% 0.16/0.43 thf(fact_111_snd__sndOp,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,P2: B > C > $o,Q: C > A > $o] :
% 0.16/0.43 ( ( product_snd @ B @ A )
% 0.16/0.43 = ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P2 @ Q ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_sndOp
% 0.16/0.43 thf(fact_112_conj__comp__iff,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: B > $o,Q: B > $o,G: A > B] :
% 0.16/0.43 ( ( comp @ B @ $o @ A
% 0.16/0.43 @ ^ [X2: B] :
% 0.16/0.43 ( ( P2 @ X2 )
% 0.16/0.43 & ( Q @ X2 ) )
% 0.16/0.43 @ G )
% 0.16/0.43 = ( ^ [X2: A] :
% 0.16/0.43 ( ( comp @ B @ $o @ A @ P2 @ G @ X2 )
% 0.16/0.43 & ( comp @ B @ $o @ A @ Q @ G @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % conj_comp_iff
% 0.16/0.43 thf(fact_113_snd__diag__fst,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
% 0.16/0.43 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 0.16/0.43 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.16/0.43 @ ( product_fst @ A @ B ) ) )
% 0.16/0.43 = ( product_fst @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_diag_fst
% 0.16/0.43 thf(fact_114_fst__diag__snd,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
% 0.16/0.43 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 0.16/0.43 @ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
% 0.16/0.43 @ ( product_snd @ A @ B ) ) )
% 0.16/0.43 = ( product_snd @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_diag_snd
% 0.16/0.43 thf(fact_115_K__record__comp,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
% 0.16/0.43 ( ( comp @ C @ B @ A
% 0.16/0.43 @ ^ [X2: C] : C2
% 0.16/0.43 @ F )
% 0.16/0.43 = ( ^ [X2: A] : C2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % K_record_comp
% 0.16/0.43 thf(fact_116_fst__map__prod,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.16/0.43 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 0.16/0.43 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_map_prod
% 0.16/0.43 thf(fact_117_fst__apsnd,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 0.16/0.43 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 0.16/0.43 = ( product_fst @ A @ C @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_apsnd
% 0.16/0.43 thf(fact_118_prod_Ocollapse,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.16/0.43 = Prod ) ).
% 0.16/0.43
% 0.16/0.43 % prod.collapse
% 0.16/0.43 thf(fact_119_fst__comp__apsnd,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.16/0.43 = ( product_fst @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_comp_apsnd
% 0.16/0.43 thf(fact_120_fst__comp__map__prod,axiom,
% 0.16/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
% 0.16/0.43 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_comp_map_prod
% 0.16/0.43 thf(fact_121_fstI,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y4: A,Z2: B] :
% 0.16/0.43 ( ( X
% 0.16/0.43 = ( product_Pair @ A @ B @ Y4 @ Z2 ) )
% 0.16/0.43 => ( ( product_fst @ A @ B @ X )
% 0.16/0.43 = Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % fstI
% 0.16/0.43 thf(fact_122_eq__fst__iff,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
% 0.16/0.43 ( ( A2
% 0.16/0.43 = ( product_fst @ A @ B @ P ) )
% 0.16/0.43 = ( ? [B4: B] :
% 0.16/0.43 ( P
% 0.16/0.43 = ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % eq_fst_iff
% 0.16/0.43 thf(fact_123_fst__conv,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.16/0.43 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.16/0.43 = X1 ) ).
% 0.16/0.43
% 0.16/0.43 % fst_conv
% 0.16/0.43 thf(fact_124_fst__eqD,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X: A,Y4: B,A2: A] :
% 0.16/0.43 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.16/0.43 = A2 )
% 0.16/0.43 => ( X = A2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_eqD
% 0.16/0.43 thf(fact_125_prod__eq__iff,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( ^ [Y5: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y5 = Z3 ) )
% 0.16/0.43 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.16/0.43 ( ( ( product_fst @ A @ B @ S )
% 0.16/0.43 = ( product_fst @ A @ B @ T3 ) )
% 0.16/0.43 & ( ( product_snd @ A @ B @ S )
% 0.16/0.43 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_eq_iff
% 0.16/0.43 thf(fact_126_prod_Oexpand,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.16/0.43 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.16/0.43 = ( product_fst @ A @ B @ Prod2 ) )
% 0.16/0.43 & ( ( product_snd @ A @ B @ Prod )
% 0.16/0.43 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.16/0.43 => ( Prod = Prod2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.expand
% 0.16/0.43 thf(fact_127_prod__eqI,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.16/0.43 ( ( ( product_fst @ A @ B @ P )
% 0.16/0.43 = ( product_fst @ A @ B @ Q2 ) )
% 0.16/0.43 => ( ( ( product_snd @ A @ B @ P )
% 0.16/0.43 = ( product_snd @ A @ B @ Q2 ) )
% 0.16/0.43 => ( P = Q2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod_eqI
% 0.16/0.43 thf(fact_128_fst__convol_H,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
% 0.16/0.43 ( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X ) )
% 0.16/0.43 = ( F @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_convol'
% 0.16/0.43 thf(fact_129_fst__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_fst @ A @ B )
% 0.16/0.43 = ( product_case_prod @ A @ B @ A
% 0.16/0.43 @ ^ [X12: A,X23: B] : X12 ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_def
% 0.16/0.43 thf(fact_130_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y4: B,A2: product_prod @ A @ B] :
% 0.16/0.43 ( ( P2 @ X @ Y4 )
% 0.16/0.43 => ( ( A2
% 0.16/0.43 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.16/0.43 => ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % BNF_Greatest_Fixpoint.subst_Pair
% 0.16/0.43 thf(fact_131_prod_Oexhaust__sel,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( Prod
% 0.16/0.43 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.exhaust_sel
% 0.16/0.43 thf(fact_132_surjective__pairing,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.16/0.43 ( T2
% 0.16/0.43 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % surjective_pairing
% 0.16/0.43 thf(fact_133_case__prod__beta,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType] :
% 0.16/0.43 ( ( product_case_prod @ B @ C @ A )
% 0.16/0.43 = ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_beta
% 0.16/0.43 thf(fact_134_prod_Ocase__eq__if,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ C )
% 0.16/0.43 = ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.case_eq_if
% 0.16/0.43 thf(fact_135_Product__Type_OCollect__case__prodD,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
% 0.16/0.43 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
% 0.16/0.43 => ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Product_Type.Collect_case_prodD
% 0.16/0.43 thf(fact_136_split__comp__eq,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
% 0.16/0.43 ( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
% 0.16/0.43 = ( product_case_prod @ D @ B @ C
% 0.16/0.43 @ ^ [X2: D] : ( F @ ( G @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % split_comp_eq
% 0.16/0.43 thf(fact_137_case__prod__beta_H,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ C )
% 0.16/0.43 = ( ^ [F2: A > B > C,X2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_beta'
% 0.16/0.43 thf(fact_138_case__prod__unfold,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ C )
% 0.16/0.43 = ( ^ [C4: A > B > C,P3: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P3 ) @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_unfold
% 0.16/0.43 thf(fact_139_case__prod__comp,axiom,
% 0.16/0.43 ! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
% 0.16/0.43 ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
% 0.16/0.43 = ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_comp
% 0.16/0.43 thf(fact_140_fst__convol,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
% 0.16/0.43 = F ) ).
% 0.16/0.43
% 0.16/0.43 % fst_convol
% 0.16/0.43 thf(fact_141_prod_Osplit__sel,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.16/0.43 = ( ( Prod
% 0.16/0.43 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.16/0.43 => ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.split_sel
% 0.16/0.43 thf(fact_142_prod_Osplit__sel__asm,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.16/0.43 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.16/0.43 = ( ~ ( ( Prod
% 0.16/0.43 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.16/0.43 & ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.split_sel_asm
% 0.16/0.43 thf(fact_143_type__copy__map__cong0,axiom,
% 0.16/0.43 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
% 0.16/0.43 ( ( ( M @ ( G @ X ) )
% 0.16/0.43 = ( N @ ( H @ X ) ) )
% 0.16/0.43 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
% 0.16/0.43 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % type_copy_map_cong0
% 0.16/0.43 thf(fact_144_convol__expand__snd_H,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H: A > C] :
% 0.16/0.43 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 0.16/0.43 = G )
% 0.16/0.43 => ( ( H
% 0.16/0.43 = ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 0.16/0.43 = ( ( bNF_convol @ A @ B @ C @ G @ H )
% 0.16/0.43 = F ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % convol_expand_snd'
% 0.16/0.43 thf(fact_145_convol__expand__snd,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
% 0.16/0.43 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 0.16/0.43 = G )
% 0.16/0.43 => ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 0.16/0.43 = F ) ) ).
% 0.16/0.43
% 0.16/0.43 % convol_expand_snd
% 0.16/0.43 thf(fact_146_fst__diag__fst,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
% 0.16/0.43 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 0.16/0.43 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.16/0.43 @ ( product_fst @ A @ B ) ) )
% 0.16/0.43 = ( product_fst @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_diag_fst
% 0.16/0.43 thf(fact_147_fst__snd__flip,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_fst @ A @ B )
% 0.16/0.43 = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
% 0.16/0.43 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 0.16/0.43 @ ^ [X2: A,Y: B] : ( product_Pair @ B @ A @ Y @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_snd_flip
% 0.16/0.43 thf(fact_148_snd__fst__flip,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType] :
% 0.16/0.43 ( ( product_snd @ B @ A )
% 0.16/0.43 = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
% 0.16/0.43 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.16/0.43 @ ^ [X2: B,Y: A] : ( product_Pair @ A @ B @ Y @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_fst_flip
% 0.16/0.43 thf(fact_149_exE__realizer,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
% 0.16/0.43 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 0.16/0.43 => ( ! [X3: B,Y3: A] :
% 0.16/0.43 ( ( P2 @ Y3 @ X3 )
% 0.16/0.43 => ( Q @ ( F @ X3 @ Y3 ) ) )
% 0.16/0.43 => ( Q @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % exE_realizer
% 0.16/0.43 thf(fact_150_sndOp__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType] :
% 0.16/0.43 ( ( bNF_sndOp @ C @ A @ B )
% 0.16/0.43 = ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % sndOp_def
% 0.16/0.43 thf(fact_151_exI__realizer,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: A > B > $o,Y4: A,X: B] :
% 0.16/0.43 ( ( P2 @ Y4 @ X )
% 0.16/0.43 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % exI_realizer
% 0.16/0.43 thf(fact_152_conjI__realizer,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 0.16/0.43 ( ( P2 @ P )
% 0.16/0.43 => ( ( Q @ Q2 )
% 0.16/0.43 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 0.16/0.43 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % conjI_realizer
% 0.16/0.43 thf(fact_153_exE__realizer_H,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
% 0.16/0.43 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 0.16/0.43 => ~ ! [X3: B,Y3: A] :
% 0.16/0.43 ~ ( P2 @ Y3 @ X3 ) ) ).
% 0.16/0.43
% 0.16/0.43 % exE_realizer'
% 0.16/0.43 thf(fact_154_fstOp__def,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( bNF_fstOp @ A @ B @ C )
% 0.16/0.43 = ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fstOp_def
% 0.16/0.43 thf(fact_155_apfst__apsnd,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.16/0.43 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_apsnd
% 0.16/0.43 thf(fact_156_apsnd__apfst,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
% 0.16/0.43 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_apfst
% 0.16/0.43 thf(fact_157_apfst__conv,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y4: B] :
% 0.16/0.43 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y4 ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ ( F @ X ) @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_conv
% 0.16/0.43 thf(fact_158_snd__apfst,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 0.16/0.43 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 0.16/0.43 = ( product_snd @ C @ A @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_apfst
% 0.16/0.43 thf(fact_159_fst__apfst,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 0.16/0.43 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 0.16/0.43 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_apfst
% 0.16/0.43 thf(fact_160_apfst__eq__conv,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
% 0.16/0.43 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 0.16/0.43 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 0.16/0.43 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 0.16/0.43 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_eq_conv
% 0.16/0.43 thf(fact_161_snd__comp__apfst,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.16/0.43 = ( product_snd @ A @ B ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_comp_apfst
% 0.16/0.43 thf(fact_162_fst__comp__apfst,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.16/0.43 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_comp_apfst
% 0.16/0.43 thf(fact_163_apfst__compose,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
% 0.16/0.43 ( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
% 0.16/0.43 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_compose
% 0.16/0.43 thf(fact_164_apsnd__apfst__commute,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
% 0.16/0.43 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
% 0.16/0.43 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_apfst_commute
% 0.16/0.43 thf(fact_165_fst__fstOp,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,P2: A > C > $o,Q: C > B > $o] :
% 0.16/0.43 ( ( product_fst @ A @ B )
% 0.16/0.43 = ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_fstOp
% 0.16/0.43 thf(fact_166_apfst__convE,axiom,
% 0.16/0.43 ! [C: $tType,A: $tType,B: $tType,Q2: product_prod @ A @ B,F: C > A,P: product_prod @ C @ B] :
% 0.16/0.43 ( ( Q2
% 0.16/0.43 = ( product_apfst @ C @ A @ B @ F @ P ) )
% 0.16/0.43 => ~ ! [X3: C,Y3: B] :
% 0.16/0.43 ( ( P
% 0.16/0.43 = ( product_Pair @ C @ B @ X3 @ Y3 ) )
% 0.16/0.43 => ( Q2
% 0.16/0.43 != ( product_Pair @ A @ B @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_convE
% 0.16/0.43 thf(fact_167_scomp__unfold,axiom,
% 0.16/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_scomp @ A @ B @ C @ D )
% 0.16/0.43 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X2: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X2 ) ) @ ( product_snd @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % scomp_unfold
% 0.16/0.43 thf(fact_168_Collect__case__prod__Grp__eqD,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A6: set @ A,F: A > B] :
% 0.16/0.43 ( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
% 0.16/0.43 => ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ Z2 )
% 0.16/0.43 = ( product_snd @ A @ B @ Z2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Collect_case_prod_Grp_eqD
% 0.16/0.43 thf(fact_169_scomp__apply,axiom,
% 0.16/0.43 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.16/0.43 ( ( product_scomp @ B @ C @ D @ A )
% 0.16/0.43 = ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X2: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % scomp_apply
% 0.16/0.43 thf(fact_170_scomp__scomp,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F: A > ( product_prod @ E @ F3 ),G: E > F3 > ( product_prod @ C @ D ),H: C > D > B] :
% 0.16/0.43 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
% 0.16/0.43 = ( product_scomp @ A @ E @ F3 @ B @ F
% 0.16/0.43 @ ^ [X2: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G @ X2 ) @ H ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % scomp_scomp
% 0.16/0.43 thf(fact_171_Grp__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( bNF_Grp @ A @ B )
% 0.16/0.43 = ( ^ [A7: set @ A,F2: A > B,A4: A,B4: B] :
% 0.16/0.43 ( ( B4
% 0.16/0.43 = ( F2 @ A4 ) )
% 0.16/0.43 & ( member @ A @ A4 @ A7 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % Grp_def
% 0.16/0.43 thf(fact_172_GrpI,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,F: B > A,X: B,Y4: A,A6: set @ B] :
% 0.16/0.43 ( ( ( F @ X )
% 0.16/0.43 = Y4 )
% 0.16/0.43 => ( ( member @ B @ X @ A6 )
% 0.16/0.43 => ( bNF_Grp @ B @ A @ A6 @ F @ X @ Y4 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % GrpI
% 0.16/0.43 thf(fact_173_GrpE,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,A6: set @ A,F: A > B,X: A,Y4: B] :
% 0.16/0.43 ( ( bNF_Grp @ A @ B @ A6 @ F @ X @ Y4 )
% 0.16/0.43 => ~ ( ( ( F @ X )
% 0.16/0.43 = Y4 )
% 0.16/0.43 => ~ ( member @ A @ X @ A6 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % GrpE
% 0.16/0.43 thf(fact_174_scomp__Pair,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.16/0.43 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.16/0.43 = X ) ).
% 0.16/0.43
% 0.16/0.43 % scomp_Pair
% 0.16/0.43 thf(fact_175_Pair__scomp,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
% 0.16/0.43 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
% 0.16/0.43 = ( F @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % Pair_scomp
% 0.16/0.43 thf(fact_176_scomp__def,axiom,
% 0.16/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_scomp @ A @ B @ C @ D )
% 0.16/0.43 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X2: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % scomp_def
% 0.16/0.43 thf(fact_177_Collect__case__prod__Grp__in,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A6: set @ A,F: A > B] :
% 0.16/0.43 ( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
% 0.16/0.43 => ( member @ A @ ( product_fst @ A @ B @ Z2 ) @ A6 ) ) ).
% 0.16/0.43
% 0.16/0.43 % Collect_case_prod_Grp_in
% 0.16/0.43 thf(fact_178_prod_Oswap__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_swap @ A @ B )
% 0.16/0.43 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.swap_def
% 0.16/0.43 thf(fact_179_map__prod__o__convol__id,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X )
% 0.16/0.43 = ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod_o_convol_id
% 0.16/0.43 thf(fact_180_internal__case__prod__def,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.16/0.43 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.16/0.43
% 0.16/0.43 % internal_case_prod_def
% 0.16/0.43 thf(fact_181_id__apply,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( id @ A )
% 0.16/0.43 = ( ^ [X2: A] : X2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % id_apply
% 0.16/0.43 thf(fact_182_swap__swap,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.16/0.43 = P ) ).
% 0.16/0.43
% 0.16/0.43 % swap_swap
% 0.16/0.43 thf(fact_183_comp__id,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,F: A > B] :
% 0.16/0.43 ( ( comp @ A @ B @ A @ F @ ( id @ A ) )
% 0.16/0.43 = F ) ).
% 0.16/0.43
% 0.16/0.43 % comp_id
% 0.16/0.43 thf(fact_184_id__comp,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,G: A > B] :
% 0.16/0.43 ( ( comp @ B @ B @ A @ ( id @ B ) @ G )
% 0.16/0.43 = G ) ).
% 0.16/0.43
% 0.16/0.43 % id_comp
% 0.16/0.43 thf(fact_185_fun_Omap__id,axiom,
% 0.16/0.43 ! [A: $tType,D: $tType,T2: D > A] :
% 0.16/0.43 ( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
% 0.16/0.43 = T2 ) ).
% 0.16/0.43
% 0.16/0.43 % fun.map_id
% 0.16/0.43 thf(fact_186_case__prod__Pair,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
% 0.16/0.43 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_prod_Pair
% 0.16/0.43 thf(fact_187_swap__simp,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,X: B,Y4: A] :
% 0.16/0.43 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ Y4 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % swap_simp
% 0.16/0.43 thf(fact_188_apfst__id,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_apfst @ A @ A @ B @ ( id @ A ) )
% 0.16/0.43 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_id
% 0.16/0.43 thf(fact_189_apsnd__id,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
% 0.16/0.43 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_id
% 0.16/0.43 thf(fact_190_case__swap,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
% 0.16/0.43 ( ( product_case_prod @ B @ C @ A
% 0.16/0.43 @ ^ [Y: B,X2: C] : ( F @ X2 @ Y )
% 0.16/0.43 @ ( product_swap @ C @ B @ P ) )
% 0.16/0.43 = ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% 0.16/0.43
% 0.16/0.43 % case_swap
% 0.16/0.43 thf(fact_191_swap__comp__swap,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
% 0.16/0.43 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % swap_comp_swap
% 0.16/0.43 thf(fact_192_fst__swap,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.16/0.43 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.16/0.43 = ( product_snd @ B @ A @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_swap
% 0.16/0.43 thf(fact_193_snd__swap,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.16/0.43 = ( product_fst @ A @ B @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_swap
% 0.16/0.43 thf(fact_194_map__prod_Oidentity,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_map_prod @ A @ A @ B @ B
% 0.16/0.43 @ ^ [X2: A] : X2
% 0.16/0.43 @ ^ [X2: B] : X2 )
% 0.16/0.43 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % map_prod.identity
% 0.16/0.43 thf(fact_195_fun_Omap__id0,axiom,
% 0.16/0.43 ! [A: $tType,D: $tType] :
% 0.16/0.43 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 0.16/0.43 = ( id @ ( D > A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fun.map_id0
% 0.16/0.43 thf(fact_196_comp__eq__id__dest,axiom,
% 0.16/0.43 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.16/0.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.16/0.43 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 0.16/0.43 => ( ( A2 @ ( B2 @ V ) )
% 0.16/0.43 = ( C2 @ V ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % comp_eq_id_dest
% 0.16/0.43 thf(fact_197_pointfree__idE,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
% 0.16/0.43 ( ( ( comp @ B @ A @ A @ F @ G )
% 0.16/0.43 = ( id @ A ) )
% 0.16/0.43 => ( ( F @ ( G @ X ) )
% 0.16/0.43 = X ) ) ).
% 0.16/0.43
% 0.16/0.43 % pointfree_idE
% 0.16/0.43 thf(fact_198_prod_Omap__id0,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
% 0.16/0.43 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.map_id0
% 0.16/0.43 thf(fact_199_id__def,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( id @ A )
% 0.16/0.43 = ( ^ [X2: A] : X2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % id_def
% 0.16/0.43 thf(fact_200_eq__id__iff,axiom,
% 0.16/0.43 ! [A: $tType,F: A > A] :
% 0.16/0.43 ( ( ! [X2: A] :
% 0.16/0.43 ( ( F @ X2 )
% 0.16/0.43 = X2 ) )
% 0.16/0.43 = ( F
% 0.16/0.43 = ( id @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % eq_id_iff
% 0.16/0.43 thf(fact_201_prod_Omap__id,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.16/0.43 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
% 0.16/0.43 = T2 ) ).
% 0.16/0.43
% 0.16/0.43 % prod.map_id
% 0.16/0.43 thf(fact_202_apfst__def,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,A: $tType] :
% 0.16/0.43 ( ( product_apfst @ A @ C @ B )
% 0.16/0.43 = ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apfst_def
% 0.16/0.43 thf(fact_203_apsnd__def,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,A: $tType] :
% 0.16/0.43 ( ( product_apsnd @ B @ C @ A )
% 0.16/0.43 = ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % apsnd_def
% 0.16/0.43 thf(fact_204_convol__mem__GrpI,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,X: A,A6: set @ A,G: A > B] :
% 0.16/0.43 ( ( member @ A @ X @ A6 )
% 0.16/0.43 => ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ G ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % convol_mem_GrpI
% 0.16/0.43 thf(fact_205_snd__diag__id,axiom,
% 0.16/0.43 ! [A: $tType,Z2: A] :
% 0.16/0.43 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
% 0.16/0.43 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.16/0.43 @ Z2 )
% 0.16/0.43 = ( id @ A @ Z2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % snd_diag_id
% 0.16/0.43 thf(fact_206_fst__diag__id,axiom,
% 0.16/0.43 ! [A: $tType,Z2: A] :
% 0.16/0.43 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
% 0.16/0.43 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.16/0.43 @ Z2 )
% 0.16/0.43 = ( id @ A @ Z2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % fst_diag_id
% 0.16/0.43 thf(fact_207_internal__case__prod__conv,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.16/0.43 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.16/0.43 = ( C2 @ A2 @ B2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % internal_case_prod_conv
% 0.16/0.43 thf(fact_208_inf__Int__eq2,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.16/0.43 ( ( inf_inf @ ( A > B > $o )
% 0.16/0.43 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
% 0.16/0.43 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) )
% 0.16/0.43 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inf_Int_eq2
% 0.16/0.43 thf(fact_209_pred__equals__eq2,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.16/0.43 ( ( ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
% 0.16/0.43 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) ) )
% 0.16/0.43 = ( R = S2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % pred_equals_eq2
% 0.16/0.43 thf(fact_210_prod_Osize__gen__o__map,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
% 0.16/0.43 ( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
% 0.16/0.43 = ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % prod.size_gen_o_map
% 0.16/0.43 thf(fact_211_inv__image__def,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType] :
% 0.16/0.43 ( ( inv_image @ B @ A )
% 0.16/0.43 = ( ^ [R2: set @ ( product_prod @ B @ B ),F2: A > B] :
% 0.16/0.43 ( collect @ ( product_prod @ A @ A )
% 0.16/0.43 @ ( product_case_prod @ A @ A @ $o
% 0.16/0.43 @ ^ [X2: A,Y: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % inv_image_def
% 0.16/0.43 thf(fact_212_iso__tuple__update__accessor__eq__assist__idI,axiom,
% 0.16/0.43 ! [A: $tType,V3: A,F: A > A,V: A] :
% 0.16/0.43 ( ( V3
% 0.16/0.43 = ( F @ V ) )
% 0.16/0.43 => ( iso_tu2011167877assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) @ V @ F @ V3 @ V ) ) ).
% 0.16/0.43
% 0.16/0.43 % iso_tuple_update_accessor_eq_assist_idI
% 0.16/0.43 thf(fact_213_in__inv__image,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,X: A,Y4: A,R3: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.16/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( inv_image @ B @ A @ R3 @ F ) )
% 0.16/0.43 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y4 ) ) @ R3 ) ) ).
% 0.16/0.43
% 0.16/0.43 % in_inv_image
% 0.16/0.43 thf(fact_214_update__accessor__updator__eqE,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.16/0.43 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.16/0.43 => ( ( Upd @ F @ V )
% 0.16/0.43 = V3 ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_updator_eqE
% 0.16/0.43 thf(fact_215_update__accessor__accessor__eqE,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.16/0.43 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.16/0.43 => ( ( Ac2 @ V )
% 0.16/0.43 = X ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_accessor_eqE
% 0.16/0.43 thf(fact_216_iso__tuple__update__accessor__eq__assist__triv,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.16/0.43 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.16/0.43 => ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % iso_tuple_update_accessor_eq_assist_triv
% 0.16/0.43 thf(fact_217_rp__inv__image__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( fun_rp_inv_image @ A @ B )
% 0.16/0.43 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.16/0.43 @ ^ [R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S3 @ F2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % rp_inv_image_def
% 0.16/0.43 thf(fact_218_fun_Opred__map,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
% 0.16/0.43 ( ( basic_pred_fun @ D @ B
% 0.16/0.43 @ ^ [Uu: D] : $true
% 0.16/0.43 @ Q
% 0.16/0.43 @ ( comp @ A @ B @ D @ F @ X ) )
% 0.16/0.43 = ( basic_pred_fun @ D @ A
% 0.16/0.43 @ ^ [Uu: D] : $true
% 0.16/0.43 @ ( comp @ B @ $o @ A @ Q @ F )
% 0.16/0.43 @ X ) ) ).
% 0.16/0.43
% 0.16/0.43 % fun.pred_map
% 0.16/0.43 thf(fact_219_pred__funI,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,A6: A > $o,B6: B > $o,F: A > B] :
% 0.16/0.43 ( ! [X3: A] :
% 0.16/0.43 ( ( A6 @ X3 )
% 0.16/0.43 => ( B6 @ ( F @ X3 ) ) )
% 0.16/0.43 => ( basic_pred_fun @ A @ B @ A6 @ B6 @ F ) ) ).
% 0.16/0.43
% 0.16/0.43 % pred_funI
% 0.16/0.43 thf(fact_220_fun_Opred__True,axiom,
% 0.16/0.43 ! [A: $tType,D: $tType] :
% 0.16/0.43 ( ( basic_pred_fun @ D @ A
% 0.16/0.43 @ ^ [Uu: D] : $true
% 0.16/0.43 @ ^ [Uu: A] : $true )
% 0.16/0.43 = ( ^ [Uu: D > A] : $true ) ) ).
% 0.16/0.43
% 0.16/0.43 % fun.pred_True
% 0.16/0.43 thf(fact_221_fun_Omap__cong__pred,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
% 0.16/0.43 ( ( X = Ya )
% 0.16/0.43 => ( ( basic_pred_fun @ D @ A
% 0.16/0.43 @ ^ [Uu: D] : $true
% 0.16/0.43 @ ^ [Z: A] :
% 0.16/0.43 ( ( F @ Z )
% 0.16/0.43 = ( G @ Z ) )
% 0.16/0.43 @ Ya )
% 0.16/0.43 => ( ( comp @ A @ B @ D @ F @ X )
% 0.16/0.43 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fun.map_cong_pred
% 0.16/0.43 thf(fact_222_rp__inv__image__rp,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 0.16/0.43 ( ( fun_reduction_pair @ A @ P2 )
% 0.16/0.43 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % rp_inv_image_rp
% 0.16/0.43 thf(fact_223_csquare__fstOp__sndOp,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q ) ) ).
% 0.16/0.43
% 0.16/0.43 % csquare_fstOp_sndOp
% 0.16/0.43 thf(fact_224_fstOp__in,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 0.16/0.43 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 0.16/0.43 => ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % fstOp_in
% 0.16/0.43 thf(fact_225_sndOp__in,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 0.16/0.43 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 0.16/0.43 => ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % sndOp_in
% 0.16/0.43 thf(fact_226_csquare__def,axiom,
% 0.16/0.43 ! [B: $tType,C: $tType,D: $tType,A: $tType] :
% 0.16/0.43 ( ( bNF_csquare @ A @ B @ C @ D )
% 0.16/0.43 = ( ^ [A7: set @ A,F12: B > C,F23: D > C,P1: A > B,P22: A > D] :
% 0.16/0.43 ! [X2: A] :
% 0.16/0.43 ( ( member @ A @ X2 @ A7 )
% 0.16/0.43 => ( ( F12 @ ( P1 @ X2 ) )
% 0.16/0.43 = ( F23 @ ( P22 @ X2 ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % csquare_def
% 0.16/0.43 thf(fact_227_pick__middlep,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q: B > C > $o,A2: A,C2: C] :
% 0.16/0.43 ( ( relcompp @ A @ B @ C @ P2 @ Q @ A2 @ C2 )
% 0.16/0.43 => ( ( P2 @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A2 @ C2 ) )
% 0.16/0.43 & ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A2 @ C2 ) @ C2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % pick_middlep
% 0.16/0.43 thf(fact_228_update__accessor__cong__assist__idI,axiom,
% 0.16/0.43 ! [A: $tType] : ( iso_tu2017585022assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_cong_assist_idI
% 0.16/0.43 thf(fact_229_The__case__prod,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType,P2: A > B > $o] :
% 0.16/0.43 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
% 0.16/0.43 = ( the @ ( product_prod @ A @ B )
% 0.16/0.43 @ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % The_case_prod
% 0.16/0.43 thf(fact_230_The__split__eq,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,X: A,Y4: B] :
% 0.16/0.43 ( ( the @ ( product_prod @ A @ B )
% 0.16/0.43 @ ( product_case_prod @ A @ B @ $o
% 0.16/0.43 @ ^ [X5: A,Y6: B] :
% 0.16/0.43 ( ( X = X5 )
% 0.16/0.43 & ( Y4 = Y6 ) ) ) )
% 0.16/0.43 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 0.16/0.43
% 0.16/0.43 % The_split_eq
% 0.16/0.43 thf(fact_231_iso__tuple__update__accessor__cong__from__eq,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.16/0.43 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.16/0.43 => ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % iso_tuple_update_accessor_cong_from_eq
% 0.16/0.43 thf(fact_232_iso__tuple__update__accessor__eq__assist__def,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType] :
% 0.16/0.43 ( ( iso_tu2011167877assist @ B @ A )
% 0.16/0.43 = ( ^ [Upd2: ( B > B ) > A > A,Ac: A > B,V4: A,F2: B > B,V5: A,X2: B] :
% 0.16/0.43 ( ( ( Upd2 @ F2 @ V4 )
% 0.16/0.43 = V5 )
% 0.16/0.43 & ( ( Ac @ V4 )
% 0.16/0.43 = X2 )
% 0.16/0.43 & ( iso_tu2017585022assist @ B @ A @ Upd2 @ Ac ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % iso_tuple_update_accessor_eq_assist_def
% 0.16/0.43 thf(fact_233_update__accessor__congruence__unfoldE,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,R3: B,R5: B,V3: A,F: A > A,F5: A > A] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.16/0.43 => ( ( R3 = R5 )
% 0.16/0.43 => ( ( ( Ac2 @ R5 )
% 0.16/0.43 = V3 )
% 0.16/0.43 => ( ! [V6: A] :
% 0.16/0.43 ( ( V6 = V3 )
% 0.16/0.43 => ( ( F @ V6 )
% 0.16/0.43 = ( F5 @ V6 ) ) )
% 0.16/0.43 => ( ( Upd @ F @ R3 )
% 0.16/0.43 = ( Upd @ F5 @ R5 ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_congruence_unfoldE
% 0.16/0.43 thf(fact_234_update__accessor__congruence__foldE,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,R3: B,R5: B,V3: A,F: A > A,F5: A > A] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.16/0.43 => ( ( R3 = R5 )
% 0.16/0.43 => ( ( ( Ac2 @ R5 )
% 0.16/0.43 = V3 )
% 0.16/0.43 => ( ! [V6: A] :
% 0.16/0.43 ( ( V3 = V6 )
% 0.16/0.43 => ( ( F @ V6 )
% 0.16/0.43 = ( F5 @ V6 ) ) )
% 0.16/0.43 => ( ( Upd @ F @ R3 )
% 0.16/0.43 = ( Upd @ F5 @ R5 ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_congruence_foldE
% 0.16/0.43 thf(fact_235_update__accessor__cong__assist__triv,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.16/0.43 => ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_cong_assist_triv
% 0.16/0.43 thf(fact_236_update__accessor__noopE,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X: B] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.16/0.43 => ( ( ( F @ ( Ac2 @ X ) )
% 0.16/0.43 = ( Ac2 @ X ) )
% 0.16/0.43 => ( ( Upd @ F @ X )
% 0.16/0.43 = X ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_noopE
% 0.16/0.43 thf(fact_237_iso__tuple__update__accessor__cong__assist__def,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ B @ A )
% 0.16/0.43 = ( ^ [Upd2: ( B > B ) > A > A,Ac: A > B] :
% 0.16/0.43 ( ! [F2: B > B,V4: A] :
% 0.16/0.43 ( ( Upd2
% 0.16/0.43 @ ^ [X2: B] : ( F2 @ ( Ac @ V4 ) )
% 0.16/0.43 @ V4 )
% 0.16/0.43 = ( Upd2 @ F2 @ V4 ) )
% 0.16/0.43 & ! [V4: A] :
% 0.16/0.43 ( ( Upd2 @ ( id @ B ) @ V4 )
% 0.16/0.43 = V4 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % iso_tuple_update_accessor_cong_assist_def
% 0.16/0.43 thf(fact_238_update__accessor__noop__compE,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X: B,G: A > A] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.16/0.43 => ( ( ( F @ ( Ac2 @ X ) )
% 0.16/0.43 = ( Ac2 @ X ) )
% 0.16/0.43 => ( ( Upd @ ( comp @ A @ A @ A @ G @ F ) @ X )
% 0.16/0.43 = ( Upd @ G @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % update_accessor_noop_compE
% 0.16/0.43 thf(fact_239_iso__tuple__update__accessor__cong__assist__id,axiom,
% 0.16/0.43 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A] :
% 0.16/0.43 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.16/0.43 => ( ( Upd @ ( id @ A ) )
% 0.16/0.43 = ( id @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % iso_tuple_update_accessor_cong_assist_id
% 0.16/0.43 thf(fact_240_old_Orec__prod__def,axiom,
% 0.16/0.43 ! [T: $tType,B: $tType,A: $tType] :
% 0.16/0.43 ( ( product_rec_prod @ A @ B @ T )
% 0.16/0.43 = ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.rec_prod_def
% 0.16/0.43 thf(fact_241_the__sym__eq__trivial,axiom,
% 0.16/0.43 ! [A: $tType,X: A] :
% 0.16/0.43 ( ( the @ A
% 0.16/0.43 @ ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 )
% 0.16/0.43 @ X ) )
% 0.16/0.43 = X ) ).
% 0.16/0.43
% 0.16/0.43 % the_sym_eq_trivial
% 0.16/0.43 thf(fact_242_the__equality,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o,A2: A] :
% 0.16/0.43 ( ( P2 @ A2 )
% 0.16/0.43 => ( ! [X3: A] :
% 0.16/0.43 ( ( P2 @ X3 )
% 0.16/0.43 => ( X3 = A2 ) )
% 0.16/0.43 => ( ( the @ A @ P2 )
% 0.16/0.43 = A2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % the_equality
% 0.16/0.43 thf(fact_243_the__eq__trivial,axiom,
% 0.16/0.43 ! [A: $tType,A2: A] :
% 0.16/0.43 ( ( the @ A
% 0.16/0.43 @ ^ [X2: A] : ( X2 = A2 ) )
% 0.16/0.43 = A2 ) ).
% 0.16/0.43
% 0.16/0.43 % the_eq_trivial
% 0.16/0.43 thf(fact_244_theI,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o,A2: A] :
% 0.16/0.43 ( ( P2 @ A2 )
% 0.16/0.43 => ( ! [X3: A] :
% 0.16/0.43 ( ( P2 @ X3 )
% 0.16/0.43 => ( X3 = A2 ) )
% 0.16/0.43 => ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % theI
% 0.16/0.43 thf(fact_245_theI_H,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o] :
% 0.16/0.43 ( ? [X6: A] :
% 0.16/0.43 ( ( P2 @ X6 )
% 0.16/0.43 & ! [Y3: A] :
% 0.16/0.43 ( ( P2 @ Y3 )
% 0.16/0.43 => ( Y3 = X6 ) ) )
% 0.16/0.43 => ( P2 @ ( the @ A @ P2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % theI'
% 0.16/0.43 thf(fact_246_theI2,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o,A2: A,Q: A > $o] :
% 0.16/0.43 ( ( P2 @ A2 )
% 0.16/0.43 => ( ! [X3: A] :
% 0.16/0.43 ( ( P2 @ X3 )
% 0.16/0.43 => ( X3 = A2 ) )
% 0.16/0.43 => ( ! [X3: A] :
% 0.16/0.43 ( ( P2 @ X3 )
% 0.16/0.43 => ( Q @ X3 ) )
% 0.16/0.43 => ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % theI2
% 0.16/0.43 thf(fact_247_If__def,axiom,
% 0.16/0.43 ! [A: $tType] :
% 0.16/0.43 ( ( if @ A )
% 0.16/0.43 = ( ^ [P4: $o,X2: A,Y: A] :
% 0.16/0.43 ( the @ A
% 0.16/0.43 @ ^ [Z: A] :
% 0.16/0.43 ( ( P4
% 0.16/0.43 => ( Z = X2 ) )
% 0.16/0.43 & ( ~ P4
% 0.16/0.43 => ( Z = Y ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % If_def
% 0.16/0.43 thf(fact_248_the1I2,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.16/0.43 ( ? [X6: A] :
% 0.16/0.43 ( ( P2 @ X6 )
% 0.16/0.43 & ! [Y3: A] :
% 0.16/0.43 ( ( P2 @ Y3 )
% 0.16/0.43 => ( Y3 = X6 ) ) )
% 0.16/0.43 => ( ! [X3: A] :
% 0.16/0.43 ( ( P2 @ X3 )
% 0.16/0.43 => ( Q @ X3 ) )
% 0.16/0.43 => ( Q @ ( the @ A @ P2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % the1I2
% 0.16/0.43 thf(fact_249_the1__equality,axiom,
% 0.16/0.43 ! [A: $tType,P2: A > $o,A2: A] :
% 0.16/0.43 ( ? [X6: A] :
% 0.16/0.43 ( ( P2 @ X6 )
% 0.16/0.43 & ! [Y3: A] :
% 0.16/0.43 ( ( P2 @ Y3 )
% 0.16/0.43 => ( Y3 = X6 ) ) )
% 0.16/0.43 => ( ( P2 @ A2 )
% 0.16/0.43 => ( ( the @ A @ P2 )
% 0.16/0.43 = A2 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % the1_equality
% 0.16/0.43 thf(fact_250_old_Orec__unit__def,axiom,
% 0.16/0.43 ! [T: $tType] :
% 0.16/0.43 ( ( product_rec_unit @ T )
% 0.16/0.43 = ( ^ [F12: T,X2: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.rec_unit_def
% 0.16/0.43 thf(fact_251_image2__eqI,axiom,
% 0.16/0.43 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A6: set @ B] :
% 0.16/0.43 ( ( B2
% 0.16/0.43 = ( F @ X ) )
% 0.16/0.43 => ( ( C2
% 0.16/0.43 = ( G @ X ) )
% 0.16/0.43 => ( ( member @ B @ X @ A6 )
% 0.16/0.43 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % image2_eqI
% 0.16/0.43 thf(fact_252_same__fst__def,axiom,
% 0.16/0.43 ! [B: $tType,A: $tType] :
% 0.16/0.43 ( ( same_fst @ A @ B )
% 0.16/0.43 = ( ^ [P4: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.16/0.43 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.16/0.43 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.16/0.43 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.16/0.43 @ ^ [X5: A,Y6: B] :
% 0.16/0.43 ( product_case_prod @ A @ B @ $o
% 0.16/0.43 @ ^ [X2: A,Y: B] :
% 0.16/0.43 ( ( X5 = X2 )
% 0.16/0.43 & ( P4 @ X2 )
% 0.16/0.43 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % same_fst_def
% 0.16/0.43 thf(fact_253_old_Orec__bool__def,axiom,
% 0.16/0.43 ! [T: $tType] :
% 0.16/0.43 ( ( product_rec_bool @ T )
% 0.16/0.43 = ( ^ [F12: T,F23: T,X2: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 % old.rec_bool_def
% 0.16/0.43 thf(fact_254_old_Obool_Osimps_I6_J,axiom,
% 0.16/0.43 ! [T: $tType,F1: T,F22: T] :
% 0.16/0.43 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.16/0.43 = F22 ) ).
% 0.16/0.43
% 0.16/0.43 % old.bool.simps(6)
% 0.16/0.43 thf(fact_255_old_Obool_Osimps_I5_J,axiom,
% 0.16/0.43 ! [T: $tType,F1: T,F22: T] :
% 0.16/0.43 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.16/0.43 = F1 ) ).
% 0.16/0.43
% 0.16/0.43 % old.bool.simps(5)
% 0.16/0.43
% 0.16/0.43 %----Type constructors (10)
% 0.16/0.43 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.16/0.43 ! [A8: $tType,A9: $tType] :
% 0.16/0.43 ( ( semilattice_inf @ A9 @ ( type2 @ A9 ) )
% 0.16/0.43 => ( semilattice_inf @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.16/0.43 ! [A8: $tType,A9: $tType] :
% 0.16/0.43 ( ( lattice @ A9 @ ( type2 @ A9 ) )
% 0.16/0.43 => ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1,axiom,
% 0.16/0.43 semilattice_inf @ nat @ ( type2 @ nat ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
% 0.16/0.43 lattice @ nat @ ( type2 @ nat ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3,axiom,
% 0.16/0.43 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
% 0.16/0.43 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_5,axiom,
% 0.16/0.43 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_HOL_Obool___Lattices_Olattice_6,axiom,
% 0.16/0.43 lattice @ $o @ ( type2 @ $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_7,axiom,
% 0.16/0.43 semilattice_inf @ product_unit @ ( type2 @ product_unit ) ).
% 0.16/0.43
% 0.16/0.43 thf(tcon_Product__Type_Ounit___Lattices_Olattice_8,axiom,
% 0.16/0.43 lattice @ product_unit @ ( type2 @ product_unit ) ).
% 0.16/0.43
% 0.16/0.43 %----Helper facts (3)
% 0.16/0.43 thf(help_If_3_1_T,axiom,
% 0.16/0.43 ! [P2: $o] :
% 0.16/0.43 ( ( P2 = $true )
% 0.16/0.43 | ( P2 = $false ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(help_If_2_1_T,axiom,
% 0.16/0.43 ! [A: $tType,X: A,Y4: A] :
% 0.16/0.43 ( ( if @ A @ $false @ X @ Y4 )
% 0.16/0.43 = Y4 ) ).
% 0.16/0.43
% 0.16/0.43 thf(help_If_1_1_T,axiom,
% 0.16/0.43 ! [A: $tType,X: A,Y4: A] :
% 0.16/0.43 ( ( if @ A @ $true @ X @ Y4 )
% 0.16/0.43 = X ) ).
% 0.16/0.43
% 0.16/0.43 %----Conjectures (1)
% 0.16/0.43 thf(conj_0,conjecture,
% 0.16/0.43 ( ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ a
% 0.16/0.43 @ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
% 0.16/0.43 @ ^ [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.16/0.43 ( ( Xs
% 0.16/0.43 != ( coinductive_LNil @ a ) )
% 0.16/0.43 => ( ( Ys
% 0.16/0.43 != ( coinductive_LNil @ a ) )
% 0.16/0.43 => ( ( coinductive_lhd @ a @ Xs )
% 0.16/0.43 != ( coinductive_lhd @ a @ Ys ) ) ) ) )
% 0.16/0.43 @ ( comp @ ( coinductive_llist @ a ) @ a @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( coinductive_lhd @ a ) @ ( product_snd @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) )
% 0.16/0.43 @ ( product_map_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_ltl @ a ) @ ( coinductive_ltl @ a ) )
% 0.16/0.43 @ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_LNil @ a ) @ xs ) )
% 0.16/0.43 = ( coinductive_LNil @ a ) ) ).
% 0.16/0.43
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 ------------------------
% 0.24/1.40 ---- Embedded in THF ---
% 0.24/1.40 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.24/1.40 %%% Generated on Tue Feb 24 06:11:15 EST 2026
% 0.24/1.40 %%% using '$$dhol' embedding, version 1.3.0.
% 0.24/1.40 %%% Logic specification used:
% 0.24/1.40 %%% thf(spec, logic, $$dhol).
% 0.24/1.40
% 0.24/1.40 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.pFALYqNRSO/DTF2DTF23795.p
% 0.24/1.40 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 0.24/1.40 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 0.24/1.40 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 0.24/1.40 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.24/1.40 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.24/1.40 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 0.24/1.40 thf(ty_tf_a, type, a: $tType).
% 0.24/1.40 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.24/1.40 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.24/1.40 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.24/1.40 thf(sy_c_BNF__Def_OGrp, type, bNF_Grp: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (A > (B > $o))))))).
% 0.24/1.40 thf(sy_c_BNF__Def_Oconvol, type, bNF_convol: (!> [A:$tType,B:$tType,C:$tType]: (((A > B) > ((A > C) > (A > ((product_prod @ B) @ C))))))).
% 0.24/1.40 thf(sy_c_BNF__Def_Ocsquare, type, bNF_csquare: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((set @ A) > ((B > C) > ((D > C) > ((A > B) > ((A > D) > $o)))))))).
% 0.24/1.40 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 0.24/1.40 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 0.24/1.40 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 0.24/1.40 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 0.24/1.40 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod, type, basic_BNF_size_prod: (!> [A:$tType,B:$tType]: (((A > nat) > ((B > nat) > (((product_prod @ A) @ B) > nat)))))).
% 0.24/1.40 thf(sy_c_Basic__BNFs_Opred__fun, type, basic_pred_fun: (!> [A:$tType,B:$tType]: (((A > $o) > ((B > $o) > ((A > B) > $o)))))).
% 0.24/1.40 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 0.24/1.40 thf(sy_c_Coinductive__List_Ollist_Olhd, type, coinductive_lhd: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 0.24/1.40 thf(sy_c_Coinductive__List_Ollist_Oltl, type, coinductive_ltl: (!> [A:$tType]: (((coinductive_llist @ A) > (coinductive_llist @ A))))).
% 0.24/1.40 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 0.24/1.40 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 0.24/1.40 thf(sy_c_Fun_Oid, type, id: (!> [A:$tType]: ((A > A)))).
% 0.24/1.40 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))).
% 0.24/1.40 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 0.24/1.40 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 0.24/1.40 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 0.24/1.40 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 0.24/1.40 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 0.24/1.40 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.24/1.40 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool, type, product_rec_set_bool: (!> [T:$tType]: ((T > (T > ($o > (T > $o))))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 0.24/1.40 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 0.24/1.40 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 0.24/1.40 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 0.24/1.40 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 0.24/1.40 thf(sy_c_Record_Oiso__tuple__update__accessor__cong__assist, type, iso_tu2017585022assist: (!> [B:$tType,A:$tType]: ((((B > B) > (A > A)) > ((A > B) > $o))))).
% 0.24/1.40 thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist, type, iso_tu2011167877assist: (!> [B:$tType,A:$tType]: ((((B > B) > (A > A)) > ((A > B) > (A > ((B > B) > (A > (B > $o))))))))).
% 0.24/1.40 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 0.24/1.40 thf(sy_c_Relation_Orelcompp, type, relcompp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (A > (C > $o))))))).
% 0.24/1.40 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.24/1.40 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 0.24/1.40 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.24/1.40 thf(sy_v_xs, type, xs: (coinductive_llist @ a)).
% 0.24/1.40 thf(fact_0_local_Oinf__llist__def, axiom, ((inf_inf @ (coinductive_llist @ a)) = (^ [Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: (((((((coindu1441602521_llist @ ((product_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a))) @ a) @ ((((product_case_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ $o) @ (^ [Xt:(coinductive_llist @ a),Yt:(coinductive_llist @ a)]: (((Xt != (coinductive_LNil @ a)) => ((Yt != (coinductive_LNil @ a)) => (((coinductive_lhd @ a) @ Xt) != ((coinductive_lhd @ a) @ Yt)))))))) @ (((((comp @ (coinductive_llist @ a)) @ a) @ ((product_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a))) @ (coinductive_lhd @ a)) @ ((product_snd @ (coinductive_llist @ a)) @ (coinductive_llist @ a)))) @ ((((((product_map_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_ltl @ a)) @ (coinductive_ltl @ a))) @ ((((product_Pair @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ Xs) @ Ys)))))).
% 0.24/1.40 thf(fact_1_snd__comp__map__prod, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F:(A > D),G:(B > C)]: (((((((comp @ ((product_prod @ D) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ D) @ C)) @ ((((((product_map_prod @ A) @ D) @ B) @ C) @ F) @ G)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ G) @ ((product_snd @ A) @ B)))))).
% 0.24/1.40 thf(fact_2_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B2:B]: ((((F @ A2) @ B2) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 0.24/1.40 thf(fact_3_case__prodI2, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),C2:(A > (B > $o))]: (((! [A3:A,B3:B]: (((P = ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((C2 @ A3) @ B3)))) => (((((product_case_prod @ A) @ B) @ $o) @ C2) @ P))))).
% 0.24/1.40 thf(fact_4_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A2:B,B2:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((F @ A2) @ B2))))).
% 0.24/1.40 thf(fact_5_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G) @ X)) = (G @ (((product_snd @ C) @ D) @ X)))))).
% 0.24/1.40 thf(fact_6_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F @ A2)) @ (G @ B2)))))).
% 0.24/1.40 thf(fact_7_Coinductive__List__Prefix__Mirabelle__yfrixeyiok_Oinf__llist__def, axiom, (! [A:$tType]: (((inf_inf @ (coinductive_llist @ A)) = (^ [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((((coindu1441602521_llist @ ((product_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A))) @ A) @ ((((product_case_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ $o) @ (^ [Xt:(coinductive_llist @ A),Yt:(coinductive_llist @ A)]: (((Xt != (coinductive_LNil @ A)) => ((Yt != (coinductive_LNil @ A)) => (((coinductive_lhd @ A) @ Xt) != ((coinductive_lhd @ A) @ Yt)))))))) @ (((((comp @ (coinductive_llist @ A)) @ A) @ ((product_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A))) @ (coinductive_lhd @ A)) @ ((product_snd @ (coinductive_llist @ A)) @ (coinductive_llist @ A)))) @ ((((((product_map_prod @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ (coinductive_ltl @ A)) @ (coinductive_ltl @ A))) @ ((((product_Pair @ (coinductive_llist @ A)) @ (coinductive_llist @ A)) @ Xs) @ Ys)))))))).
% 0.24/1.40 thf(fact_8_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X2:A]: (X2))) @ (^ [Y:B]: (Y))) = (^ [Z:((product_prod @ A) @ B)]: (Z)))))).
% 0.24/1.40 thf(fact_9_split__part, axiom, (! [B:$tType,A:$tType,P2:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: ((P2 & ((Q @ A4) @ B4))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P2 & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 0.24/1.40 thf(fact_10_ltl__unfold__llist, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),A2:B,LHD:(B > A),LTL:(B > B)]: ((((IS_LNIL @ A2) => (((coinductive_ltl @ A) @ ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ A2)) = (coinductive_LNil @ A))) & ((~ ((IS_LNIL @ A2))) => (((coinductive_ltl @ A) @ ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ A2)) = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ A2)))))))).
% 0.24/1.40 thf(fact_11_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F2:(B > A),G2:(C > B),X2:C]: ((F2 @ (G2 @ X2)))))))).
% 0.24/1.40 thf(fact_12_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B5)) = ((A2 = A5) & (B2 = B5)))))).
% 0.24/1.40 thf(fact_13_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 0.24/1.40 thf(fact_14_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P:((product_prod @ A) @ B),Z2:C,C2:(A > (B > (set @ C)))]: (((! [A3:A,B3:B]: (((P = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (((member @ C) @ Z2) @ ((C2 @ A3) @ B3))))) => (((member @ C) @ Z2) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C2) @ P)))))).
% 0.24/1.40 thf(fact_15_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),A2:B,B2:C]: (((((member @ A) @ Z2) @ ((C2 @ A2) @ B2)) => (((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2))))))).
% 0.24/1.40 thf(fact_16_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P:((product_prod @ A) @ B),C2:(A > (B > (C > $o))),X:C]: (((! [A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = P) => (((C2 @ A3) @ B3) @ X)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P) @ X))))).
% 0.24/1.40 thf(fact_17_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),P:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P)) => (~ ((! [X3:B,Y3:C]: (((P = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((((member @ A) @ Z2) @ ((C2 @ X3) @ Y3))))))))))))).
% 0.24/1.40 thf(fact_18_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(A > (B > (C > $o))),P:((product_prod @ A) @ B),Z2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P) @ Z2) => (~ ((! [X3:A,Y3:B]: (((P = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (~ ((((C2 @ X3) @ Y3) @ Z2)))))))))))).
% 0.24/1.40 thf(fact_19_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R:(A > (B > (C > $o))),A2:A,B2:B,C2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ C2) => (((R @ A2) @ B2) @ C2))))).
% 0.24/1.40 thf(fact_20_map__prod__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((product_map_prod @ A) @ C) @ B) @ D) = (^ [F2:(A > C),G2:(B > D)]: (((((product_case_prod @ A) @ B) @ ((product_prod @ C) @ D)) @ (^ [X2:A,Y:B]: (((((product_Pair @ C) @ D) @ (F2 @ X2)) @ (G2 @ Y))))))))))).
% 0.24/1.40 thf(fact_21_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A3:A,B3:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A3) @ B3)))) => (P2 @ Prod))))).
% 0.24/1.40 thf(fact_22_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y4:((product_prod @ A) @ B)]: ((~ ((! [A3:A,B3:B]: ((Y4 != ((((product_Pair @ A) @ B) @ A3) @ B3))))))))).
% 0.24/1.40 thf(fact_23_prod__induct7, axiom, (! [G3:$tType,F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: (((! [A3:A,B3:B,C3:C,D2:D,E2:E,F4:F3,G4:G3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4))))))))) => (P2 @ X))))).
% 0.24/1.40 thf(fact_24_prod__induct6, axiom, (! [F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: (((! [A3:A,B3:B,C3:C,D2:D,E2:E,F4:F3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4)))))))) => (P2 @ X))))).
% 0.24/1.40 thf(fact_25_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A3:A,B3:B,C3:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X))))).
% 0.24/1.40 thf(fact_26_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A3:A,B3:B,C3:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P2 @ X))))).
% 0.24/1.40 thf(fact_27_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A3:A,B3:B,C3:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A3) @ ((((product_Pair @ B) @ C) @ B3) @ C3))))) => (P2 @ X))))).
% 0.24/1.40 thf(fact_28_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,G3:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: ((~ ((! [A3:A,B3:B,C3:C,D2:D,E2:E,F4:F3,G4:G3]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4)))))))))))))).
% 0.24/1.40 thf(fact_29_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A3:A,B3:B,C3:C,D2:D,E2:E,F4:F3]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4))))))))))))).
% 0.24/1.40 thf(fact_30_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A3:A,B3:B,C3:C,D2:D,E2:E]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 0.24/1.40 thf(fact_31_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A3:A,B3:B,C3:C,D2:D]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A3) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 0.24/1.40 thf(fact_32_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y4:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A3:A,B3:B,C3:C]: ((Y4 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A3) @ ((((product_Pair @ B) @ C) @ B3) @ C3)))))))))).
% 0.24/1.40 thf(fact_33_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (~ (((A2 = A5) => (B2 != B5)))))))).
% 0.24/1.40 thf(fact_34_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A3:A,B3:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A3) @ B3)))) => (P2 @ P))))).
% 0.24/1.40 thf(fact_35_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 0.24/1.40 thf(fact_36_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = C2) => ((A2 @ (B2 @ V)) = (C2 @ V)))))).
% 0.24/1.40 thf(fact_37_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(D > B),D3:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D3)) => (! [V2:A]: (((A2 @ (B2 @ V2)) = (C2 @ (D3 @ V2))))))))).
% 0.24/1.40 thf(fact_38_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(D > B),D3:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D3)) => ((A2 @ (B2 @ V)) = (C2 @ (D3 @ V))))))).
% 0.24/1.40 thf(fact_39_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G) @ H)))))).
% 0.24/1.40 thf(fact_40_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F2:(B > C),G2:(A > B),X2:A]: ((F2 @ (G2 @ X2)))))))).
% 0.24/1.40 thf(fact_41_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X12:A,X23:B]: ((H @ ((F @ X12) @ X23))))) @ Prod))))).
% 0.24/1.40 thf(fact_42_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 0.24/1.40 thf(fact_43_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 0.24/1.40 thf(fact_44_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 0.24/1.40 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 0.24/1.40 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A6)))) = A6)))).
% 0.24/1.40 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P2 @ X3) = (Q @ X3)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 0.24/1.40 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 0.24/1.40 thf(fact_49_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y4:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y4)) = A2) => (Y4 = A2))))).
% 0.24/1.40 thf(fact_50_ltl__simps_I1_J, axiom, (! [A:$tType]: ((((coinductive_ltl @ A) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 0.24/1.40 thf(fact_51_map__prod_Ocompositionality, axiom, (! [D:$tType,F3:$tType,E:$tType,C:$tType,B:$tType,A:$tType,F:(C > E),G:(D > F3),H:(A > C),I:(B > D),Prod:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F3) @ F) @ G) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ H) @ I) @ Prod)) = (((((((product_map_prod @ A) @ E) @ B) @ F3) @ (((((comp @ C) @ E) @ A) @ F) @ H)) @ (((((comp @ D) @ F3) @ B) @ G) @ I)) @ Prod))))).
% 0.24/1.40 thf(fact_52_map__prod__compose, axiom, (! [D:$tType,C:$tType,A:$tType,E:$tType,F3:$tType,B:$tType,F1:(E > C),F22:(A > E),G1:(F3 > D),G22:(B > F3)]: ((((((((product_map_prod @ A) @ C) @ B) @ D) @ (((((comp @ E) @ C) @ A) @ F1) @ F22)) @ (((((comp @ F3) @ D) @ B) @ G1) @ G22)) = (((((comp @ ((product_prod @ E) @ F3)) @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ E) @ C) @ F3) @ D) @ F1) @ G1)) @ ((((((product_map_prod @ A) @ E) @ B) @ F3) @ F22) @ G22)))))).
% 0.24/1.40 thf(fact_53_map__prod_Ocomp, axiom, (! [A:$tType,C:$tType,E:$tType,F3:$tType,D:$tType,B:$tType,F:(C > E),G:(D > F3),H:(A > C),I:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ ((product_prod @ E) @ F3)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ E) @ D) @ F3) @ F) @ G)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ H) @ I)) = ((((((product_map_prod @ A) @ E) @ B) @ F3) @ (((((comp @ C) @ E) @ A) @ F) @ H)) @ (((((comp @ D) @ F3) @ B) @ G) @ I)))))).
% 0.24/1.40 thf(fact_54_unfold__llist__ltl__unroll, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B2:B]: ((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B2)) = ((((((coindu1441602521_llist @ B) @ A) @ (((((comp @ B) @ $o) @ B) @ IS_LNIL) @ LTL)) @ (((((comp @ B) @ A) @ B) @ LHD) @ LTL)) @ LTL) @ B2))))).
% 0.24/1.40 thf(fact_55_unfold__llist_Octr_I1_J, axiom, (! [A:$tType,B:$tType,P:(A > $o),A2:A,G21:(A > B),G222:(A > A)]: (((P @ A2) => (((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G222) @ A2) = (coinductive_LNil @ B)))))).
% 0.24/1.40 thf(fact_56_unfold__llist_Osimps_I4_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G222:(A > A)]: (((~ ((P @ A2))) => (((coinductive_ltl @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G222) @ A2)) = ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G222) @ (G222 @ A2))))))).
% 0.24/1.40 thf(fact_57_unfold__llist_Osimps_I3_J, axiom, (! [B:$tType,A:$tType,P:(A > $o),A2:A,G21:(A > B),G222:(A > A)]: (((~ ((P @ A2))) => (((coinductive_lhd @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P) @ G21) @ G222) @ A2)) = (G21 @ A2)))))).
% 0.24/1.40 thf(fact_58_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G:(((product_prod @ A) @ B) > C)]: (((! [X3:A,Y3:B]: ((((F @ X3) @ Y3) = (G @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G))))).
% 0.24/1.40 thf(fact_59_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X2:A,Y:B]: ((F @ ((((product_Pair @ A) @ B) @ X2) @ Y))))) = F)))).
% 0.24/1.40 thf(fact_60_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P2:(B > (C > A)),Z2:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P2) @ Z2)) => (~ ((! [X3:B,Y3:C]: (((Z2 = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((Q @ ((P2 @ X3) @ Y3))))))))))))).
% 0.24/1.40 thf(fact_61_case__prodE, axiom, (! [A:$tType,B:$tType,C2:(A > (B > $o)),P:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C2) @ P) => (~ ((! [X3:A,Y3:B]: (((P = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (~ (((C2 @ X3) @ Y3)))))))))))).
% 0.24/1.40 thf(fact_62_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B2:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((F @ A2) @ B2))))).
% 0.24/1.40 thf(fact_63_snd__def, axiom, (! [B:$tType,A:$tType]: ((((product_snd @ A) @ B) = ((((product_case_prod @ A) @ B) @ B) @ (^ [X12:A,X23:B]: (X23))))))).
% 0.24/1.40 thf(fact_64_case__prod__o__map__prod, axiom, (! [B:$tType,D:$tType,C:$tType,E:$tType,A:$tType,F:(D > (E > C)),G1:(A > D),G22:(B > E)]: (((((((comp @ ((product_prod @ D) @ E)) @ C) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ D) @ E) @ C) @ F)) @ ((((((product_map_prod @ A) @ D) @ B) @ E) @ G1) @ G22)) = ((((product_case_prod @ A) @ B) @ C) @ (^ [L:A,R2:B]: (((F @ (G1 @ L)) @ (G22 @ R2))))))))).
% 0.24/1.40 thf(fact_65_snd__diag__snd, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ B) @ B)) @ (((((comp @ B) @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ B)) @ (^ [X2:B]: (((((product_Pair @ B) @ B) @ X2) @ X2)))) @ ((product_snd @ A) @ B))) = ((product_snd @ A) @ B))))).
% 0.24/1.40 thf(fact_66_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 0.24/1.40 thf(fact_67_case__prod__map__prod, axiom, (! [C:$tType,A:$tType,B:$tType,E:$tType,D:$tType,H:(B > (C > A)),F:(D > B),G:(E > C),X:((product_prod @ D) @ E)]: (((((((product_case_prod @ B) @ C) @ A) @ H) @ (((((((product_map_prod @ D) @ B) @ E) @ C) @ F) @ G) @ X)) = (((((product_case_prod @ D) @ E) @ A) @ (^ [L:D,R2:E]: (((H @ (F @ L)) @ (G @ R2))))) @ X))))).
% 0.24/1.40 thf(fact_68_inf__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.24/1.40 thf(fact_69_inf_Oidem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 0.24/1.40 thf(fact_70_inf__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 0.24/1.40 thf(fact_71_inf_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B2)) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 0.24/1.40 thf(fact_72_inf__left__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y4)) = (((inf_inf @ A) @ X) @ Y4)))))))).
% 0.24/1.40 thf(fact_73_inf_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ B2) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 0.24/1.40 thf(fact_74_inf__right__idem, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y4)) @ Y4) = (((inf_inf @ A) @ X) @ Y4)))))))).
% 0.24/1.40 thf(fact_75_inf__left__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y4) @ Z2)) = (((inf_inf @ A) @ Y4) @ (((inf_inf @ A) @ X) @ Z2))))))))).
% 0.24/1.40 thf(fact_76_inf_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((inf_inf @ A) @ B2) @ (((inf_inf @ A) @ A2) @ C2)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C2))))))))).
% 0.24/1.40 thf(fact_77_inf__commute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X2:A,Y:A]: ((((inf_inf @ A) @ Y) @ X2)))))))).
% 0.24/1.40 thf(fact_78_inf_Ocommute, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [A4:A,B4:A]: ((((inf_inf @ A) @ B4) @ A4)))))))).
% 0.24/1.40 thf(fact_79_inf__assoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y4)) @ Z2) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y4) @ Z2))))))))).
% 0.24/1.40 thf(fact_80_inf_Oassoc, axiom, (! [A:$tType]: ((((semilattice_inf @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C2) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C2))))))))).
% 0.24/1.40 thf(fact_81_inf__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ B) @ (type2 @ B)) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.24/1.40 thf(fact_82_inf__sup__aci_I1_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((inf_inf @ A) = (^ [X2:A,Y:A]: ((((inf_inf @ A) @ Y) @ X2)))))))).
% 0.24/1.40 thf(fact_83_inf__sup__aci_I2_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y4)) @ Z2) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y4) @ Z2))))))))).
% 0.24/1.40 thf(fact_84_inf__sup__aci_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y4) @ Z2)) = (((inf_inf @ A) @ Y4) @ (((inf_inf @ A) @ X) @ Z2))))))))).
% 0.24/1.40 thf(fact_85_inf__sup__aci_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y4)) = (((inf_inf @ A) @ X) @ Y4)))))))).
% 0.24/1.40 thf(fact_86_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(C > B),G:(A > C),L2:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = L2) => ((((((comp @ C) @ B) @ D) @ F) @ (((((comp @ A) @ C) @ D) @ G) @ H)) = (((((comp @ A) @ B) @ D) @ L2) @ H)))))).
% 0.24/1.40 thf(fact_87_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G:(C > B),H:(A > C),R3:(A > B),F:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = R3) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F) @ G)) @ H) = (((((comp @ B) @ D) @ A) @ F) @ R3)))))).
% 0.24/1.40 thf(fact_88_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F:(C > B),G:(A > C),L1:(D > B),L22:(A > D),H:(E > A),R3:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ D) @ B) @ A) @ L1) @ L22)) => (((((((comp @ A) @ D) @ E) @ L22) @ H) = R3) => ((((((comp @ C) @ B) @ E) @ F) @ (((((comp @ A) @ C) @ E) @ G) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R3))))))).
% 0.24/1.40 thf(fact_89_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G:(C > B),H:(A > C),R1:(D > B),R22:(A > D),F:(B > E),L2:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = (((((comp @ D) @ B) @ A) @ R1) @ R22)) => (((((((comp @ B) @ E) @ D) @ F) @ R1) = L2) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ E) @ A) @ L2) @ R22))))))).
% 0.24/1.40 thf(fact_90_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P) = P)))).
% 0.24/1.40 thf(fact_91_case__prod__app, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F2:(B > (C > (D > A))),X2:((product_prod @ B) @ C),Y:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R2:C]: ((((F2 @ L) @ R2) @ Y)))) @ X2))))))).
% 0.24/1.40 thf(fact_92_prod_Omap__comp, axiom, (! [D:$tType,F3:$tType,E:$tType,C:$tType,B:$tType,A:$tType,G1:(C > E),G22:(D > F3),F1:(A > C),F22:(B > D),V:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F3) @ G1) @ G22) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ F1) @ F22) @ V)) = (((((((product_map_prod @ A) @ E) @ B) @ F3) @ (((((comp @ C) @ E) @ A) @ G1) @ F1)) @ (((((comp @ D) @ F3) @ B) @ G22) @ F22)) @ V))))).
% 0.24/1.40 thf(fact_93_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y4:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y4) @ Z2)) => ((((product_snd @ A) @ B) @ X) = Z2))))).
% 0.24/1.40 thf(fact_94_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P)) = (? [A4:B]: ((P = ((((product_Pair @ B) @ A) @ A4) @ B2)))))))).
% 0.24/1.40 thf(fact_95_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F:(A > (B > C)),G:(A > (B > C)),P:((product_prod @ A) @ B)]: (((! [X3:A,Y3:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y3) = Q2) => (((F @ X3) @ Y3) = ((G @ X3) @ Y3))))) => ((P = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P) = (((((product_case_prod @ A) @ B) @ C) @ G) @ Q2))))))).
% 0.24/1.40 thf(fact_96_snd__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_snd @ A) @ B)))))).
% 0.24/1.40 thf(fact_97_map__prod__o__convol, axiom, (! [D:$tType,B:$tType,C:$tType,E:$tType,A:$tType,H1:(D > B),H2:(E > C),F:(A > D),G:(A > E)]: (((((((comp @ ((product_prod @ D) @ E)) @ ((product_prod @ B) @ C)) @ A) @ ((((((product_map_prod @ D) @ B) @ E) @ C) @ H1) @ H2)) @ (((((bNF_convol @ A) @ D) @ E) @ F) @ G)) = (((((bNF_convol @ A) @ B) @ C) @ (((((comp @ D) @ B) @ A) @ H1) @ F)) @ (((((comp @ E) @ C) @ A) @ H2) @ G)))))).
% 0.24/1.40 thf(fact_98_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),X:A,Y4:C]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ ((((product_Pair @ A) @ C) @ X) @ Y4)) = ((((product_Pair @ A) @ B) @ X) @ (F @ Y4)))))).
% 0.24/1.40 thf(fact_99_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C),G:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F) @ X) = (((((product_apsnd @ C) @ B) @ A) @ G) @ X)) = ((F @ (((product_snd @ A) @ C) @ X)) = (G @ (((product_snd @ A) @ C) @ X))))))).
% 0.24/1.40 thf(fact_100_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),X:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_snd @ B) @ C) @ X)))))).
% 0.24/1.40 thf(fact_101_convol__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_convol @ A) @ B) @ C) = (^ [F2:(A > B),G2:(A > C),A4:A]: (((((product_Pair @ B) @ C) @ (F2 @ A4)) @ (G2 @ A4)))))))).
% 0.24/1.40 thf(fact_102_apsnd__compose, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,F:(C > B),G:(D > C),X:((product_prod @ A) @ D)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apsnd @ D) @ C) @ A) @ G) @ X)) = (((((product_apsnd @ D) @ B) @ A) @ (((((comp @ C) @ B) @ D) @ F) @ G)) @ X))))).
% 0.24/1.40 thf(fact_103_snd__convol_H, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > B),G:(C > A),X:C]: (((((product_snd @ B) @ A) @ ((((((bNF_convol @ C) @ B) @ A) @ F) @ G) @ X)) = (G @ X))))).
% 0.24/1.40 thf(fact_104_snd__convol, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C),G:(A > B)]: (((((((comp @ ((product_prod @ C) @ B)) @ B) @ A) @ ((product_snd @ C) @ B)) @ (((((bNF_convol @ A) @ C) @ B) @ F) @ G)) = G)))).
% 0.24/1.40 thf(fact_105_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G:(B > C),F:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G) @ (((((comp @ A) @ B) @ D) @ F) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 0.24/1.40 thf(fact_106_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),X:C,H:(D > A),K:(C > D)]: ((((F @ (G @ X)) = (H @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ C) @ H) @ K) @ X)))))).
% 0.24/1.40 thf(fact_107_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F:(B > A),G:(C > B),X:C,F5:(D > A),G5:(E > D),X4:E]: ((((F @ (G @ X)) = (F5 @ (G5 @ X4))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ E) @ F5) @ G5) @ X4)))))).
% 0.24/1.40 thf(fact_108_convol__o, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,F:(D > B),G:(D > C),H:(A > D)]: (((((((comp @ D) @ ((product_prod @ B) @ C)) @ A) @ (((((bNF_convol @ D) @ B) @ C) @ F) @ G)) @ H) = (((((bNF_convol @ A) @ B) @ C) @ (((((comp @ D) @ B) @ A) @ F) @ H)) @ (((((comp @ D) @ C) @ A) @ G) @ H)))))).
% 0.24/1.40 thf(fact_109_fun_Omap__ident, axiom, (! [A:$tType,D:$tType,T2:(D > A)]: (((((((comp @ A) @ A) @ D) @ (^ [X2:A]: (X2))) @ T2) = T2)))).
% 0.24/1.40 thf(fact_110_prod_Omap__ident, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X2:A]: (X2))) @ (^ [X2:B]: (X2))) @ T2) = T2)))).
% 0.24/1.40 thf(fact_111_snd__sndOp, axiom, (! [B:$tType,A:$tType,C:$tType,P2:(B > (C > $o)),Q:(C > (A > $o))]: ((((product_snd @ B) @ A) = (((((comp @ ((product_prod @ C) @ A)) @ A) @ ((product_prod @ B) @ A)) @ ((product_snd @ C) @ A)) @ (((((bNF_sndOp @ B) @ C) @ A) @ P2) @ Q)))))).
% 0.24/1.40 thf(fact_112_conj__comp__iff, axiom, (! [B:$tType,A:$tType,P2:(B > $o),Q:(B > $o),G:(A > B)]: (((((((comp @ B) @ $o) @ A) @ (^ [X2:B]: (((P2 @ X2) & (Q @ X2))))) @ G) = (^ [X2:A]: ((((((((comp @ B) @ $o) @ A) @ P2) @ G) @ X2) & ((((((comp @ B) @ $o) @ A) @ Q) @ G) @ X2)))))))).
% 0.24/1.40 thf(fact_113_snd__diag__fst, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ A) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ A)) @ (((((comp @ A) @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ B)) @ (^ [X2:A]: (((((product_Pair @ A) @ A) @ X2) @ X2)))) @ ((product_fst @ A) @ B))) = ((product_fst @ A) @ B))))).
% 0.24/1.40 thf(fact_114_fst__diag__snd, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_fst @ B) @ B)) @ (((((comp @ B) @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ B)) @ (^ [X2:B]: (((((product_Pair @ B) @ B) @ X2) @ X2)))) @ ((product_snd @ A) @ B))) = ((product_snd @ A) @ B))))).
% 0.24/1.40 thf(fact_115_K__record__comp, axiom, (! [C:$tType,B:$tType,A:$tType,C2:B,F:(A > C)]: (((((((comp @ C) @ B) @ A) @ (^ [X2:C]: (C2))) @ F) = (^ [X2:A]: (C2)))))).
% 0.24/1.40 thf(fact_116_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ X)) = (F @ (((product_fst @ C) @ D) @ X)))))).
% 0.24/1.40 thf(fact_117_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 0.24/1.40 thf(fact_118_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 0.24/1.40 thf(fact_119_fst__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) = ((product_fst @ A) @ B))))).
% 0.24/1.40 thf(fact_120_fst__comp__map__prod, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F:(A > C),G:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)))))).
% 0.24/1.40 thf(fact_121_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y4:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y4) @ Z2)) => ((((product_fst @ A) @ B) @ X) = Y4))))).
% 0.24/1.40 thf(fact_122_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P)) = (? [B4:B]: ((P = ((((product_Pair @ A) @ B) @ A2) @ B4)))))))).
% 0.24/1.40 thf(fact_123_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 0.24/1.40 thf(fact_124_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y4:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) = A2) => (X = A2))))).
% 0.24/1.40 thf(fact_125_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z3:((product_prod @ A) @ B)]: ((Y5 = Z3))) = (^ [S:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S) = (((product_snd @ A) @ B) @ T3))))))))).
% 0.24/1.40 thf(fact_126_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 0.24/1.40 thf(fact_127_prod__eqI, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P) = (((product_snd @ A) @ B) @ Q2)) => (P = Q2)))))).
% 0.24/1.40 thf(fact_128_fst__convol_H, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > A),G:(C > B),X:C]: (((((product_fst @ A) @ B) @ ((((((bNF_convol @ C) @ A) @ B) @ F) @ G) @ X)) = (F @ X))))).
% 0.24/1.41 thf(fact_129_fst__def, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = ((((product_case_prod @ A) @ B) @ A) @ (^ [X12:A,X23:B]: (X12))))))).
% 0.24/1.41 thf(fact_130_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),X:A,Y4:B,A2:((product_prod @ A) @ B)]: ((((P2 @ X) @ Y4) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y4)) => ((P2 @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 0.24/1.41 thf(fact_131_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 0.24/1.41 thf(fact_132_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 0.24/1.41 thf(fact_133_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F2:(B > (C > A)),P3:((product_prod @ B) @ C)]: (((F2 @ (((product_fst @ B) @ C) @ P3)) @ (((product_snd @ B) @ C) @ P3)))))))).
% 0.24/1.41 thf(fact_134_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F2:(A > (B > C)),Prod3:((product_prod @ A) @ B)]: (((F2 @ (((product_fst @ A) @ B) @ Prod3)) @ (((product_snd @ A) @ B) @ Prod3)))))))).
% 0.24/1.41 thf(fact_135_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A6:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A6))) => ((A6 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))).
% 0.24/1.41 thf(fact_136_split__comp__eq, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,F:(A > (B > C)),G:(D > A)]: (((^ [U:((product_prod @ D) @ B)]: (((F @ (G @ (((product_fst @ D) @ B) @ U))) @ (((product_snd @ D) @ B) @ U)))) = ((((product_case_prod @ D) @ B) @ C) @ (^ [X2:D]: ((F @ (G @ X2))))))))).
% 0.24/1.41 thf(fact_137_case__prod__beta_H, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F2:(A > (B > C)),X2:((product_prod @ A) @ B)]: (((F2 @ (((product_fst @ A) @ B) @ X2)) @ (((product_snd @ A) @ B) @ X2)))))))).
% 0.24/1.41 thf(fact_138_case__prod__unfold, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [C4:(A > (B > C)),P3:((product_prod @ A) @ B)]: (((C4 @ (((product_fst @ A) @ B) @ P3)) @ (((product_snd @ A) @ B) @ P3)))))))).
% 0.24/1.41 thf(fact_139_case__prod__comp, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType,F:(D > (C > A)),G:(B > D),X:((product_prod @ B) @ C)]: (((((((product_case_prod @ B) @ C) @ A) @ (((((comp @ D) @ (C > A)) @ B) @ F) @ G)) @ X) = ((F @ (G @ (((product_fst @ B) @ C) @ X))) @ (((product_snd @ B) @ C) @ X)))))).
% 0.24/1.41 thf(fact_140_fst__convol, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > B),G:(A > C)]: (((((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ (((((bNF_convol @ A) @ B) @ C) @ F) @ G)) = F)))).
% 0.24/1.41 thf(fact_141_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P2 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P2 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 0.24/1.41 thf(fact_142_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P2 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P2 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 0.24/1.41 thf(fact_143_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M:(B > A),G:(C > B),X:C,N:(D > A),H:(C > D),F:(A > E)]: ((((M @ (G @ X)) = (N @ (H @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F) @ M)) @ G) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F) @ N)) @ H) @ X)))))).
% 0.24/1.41 thf(fact_144_convol__expand__snd_H, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > ((product_prod @ B) @ C)),G:(A > B),H:(A > C)]: ((((((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ F) = G) => ((H = (((((comp @ ((product_prod @ B) @ C)) @ C) @ A) @ ((product_snd @ B) @ C)) @ F)) = ((((((bNF_convol @ A) @ B) @ C) @ G) @ H) = F)))))).
% 0.24/1.41 thf(fact_145_convol__expand__snd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > ((product_prod @ B) @ C)),G:(A > B)]: ((((((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ F) = G) => ((((((bNF_convol @ A) @ B) @ C) @ G) @ (((((comp @ ((product_prod @ B) @ C)) @ C) @ A) @ ((product_snd @ B) @ C)) @ F)) = F))))).
% 0.24/1.41 thf(fact_146_fst__diag__fst, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ A) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ A)) @ (((((comp @ A) @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ B)) @ (^ [X2:A]: (((((product_Pair @ A) @ A) @ X2) @ X2)))) @ ((product_fst @ A) @ B))) = ((product_fst @ A) @ B))))).
% 0.24/1.41 thf(fact_147_fst__snd__flip, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = (((((comp @ ((product_prod @ B) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_snd @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [X2:A,Y:B]: (((((product_Pair @ B) @ A) @ Y) @ X2))))))))).
% 0.24/1.41 thf(fact_148_snd__fst__flip, axiom, (! [A:$tType,B:$tType]: ((((product_snd @ B) @ A) = (((((comp @ ((product_prod @ A) @ B)) @ A) @ ((product_prod @ B) @ A)) @ ((product_fst @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [X2:B,Y:A]: (((((product_Pair @ A) @ B) @ Y) @ X2))))))))).
% 0.24/1.41 thf(fact_149_exE__realizer, axiom, (! [C:$tType,A:$tType,B:$tType,P2:(A > (B > $o)),P:((product_prod @ B) @ A),Q:(C > $o),F:(B > (A > C))]: ((((P2 @ (((product_snd @ B) @ A) @ P)) @ (((product_fst @ B) @ A) @ P)) => ((! [X3:B,Y3:A]: ((((P2 @ Y3) @ X3) => (Q @ ((F @ X3) @ Y3))))) => (Q @ (((((product_case_prod @ B) @ A) @ C) @ F) @ P))))))).
% 0.24/1.41 thf(fact_150_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P4:(C > (A > $o)),Q3:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P4) @ Q3) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 0.24/1.41 thf(fact_151_exI__realizer, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),Y4:A,X:B]: ((((P2 @ Y4) @ X) => ((P2 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y4))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y4))))))).
% 0.24/1.41 thf(fact_152_conjI__realizer, axiom, (! [A:$tType,B:$tType,P2:(A > $o),P:A,Q:(B > $o),Q2:B]: (((P2 @ P) => ((Q @ Q2) => ((P2 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))))))))).
% 0.24/1.41 thf(fact_153_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),P:((product_prod @ B) @ A)]: ((((P2 @ (((product_snd @ B) @ A) @ P)) @ (((product_fst @ B) @ A) @ P)) => (~ ((! [X3:B,Y3:A]: ((~ (((P2 @ Y3) @ X3))))))))))).
% 0.24/1.41 thf(fact_154_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P4:(A > (B > $o)),Q3:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P4) @ Q3) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 0.24/1.41 thf(fact_155_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apsnd @ D) @ B) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (F @ (((product_fst @ C) @ D) @ X))) @ (G @ (((product_snd @ C) @ D) @ X))))))).
% 0.24/1.41 thf(fact_156_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),X:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (G @ (((product_fst @ D) @ C) @ X))) @ (F @ (((product_snd @ D) @ C) @ X))))))).
% 0.24/1.41 thf(fact_157_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F:(C > A),X:C,Y4:B]: (((((((product_apfst @ C) @ A) @ B) @ F) @ ((((product_Pair @ C) @ B) @ X) @ Y4)) = ((((product_Pair @ A) @ B) @ (F @ X)) @ Y4))))).
% 0.24/1.41 thf(fact_158_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > B),X:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F) @ X)) = (((product_snd @ C) @ A) @ X))))).
% 0.24/1.41 thf(fact_159_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_fst @ C) @ B) @ X)))))).
% 0.24/1.41 thf(fact_160_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B),G:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F) @ X) = (((((product_apfst @ C) @ A) @ B) @ G) @ X)) = ((F @ (((product_fst @ C) @ B) @ X)) = (G @ (((product_fst @ C) @ B) @ X))))))).
% 0.24/1.41 thf(fact_161_snd__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) = ((product_snd @ A) @ B))))).
% 0.24/1.41 thf(fact_162_fst__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)))))).
% 0.24/1.41 thf(fact_163_apfst__compose, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > C),X:((product_prod @ D) @ B)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apfst @ D) @ C) @ B) @ G) @ X)) = (((((product_apfst @ D) @ A) @ B) @ (((((comp @ C) @ A) @ D) @ F) @ G)) @ X))))).
% 0.24/1.41 thf(fact_164_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),P:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ P)) = (((((product_apfst @ D) @ A) @ B) @ G) @ (((((product_apsnd @ C) @ B) @ D) @ F) @ P)))))).
% 0.24/1.41 thf(fact_165_fst__fstOp, axiom, (! [A:$tType,B:$tType,C:$tType,P2:(A > (C > $o)),Q:(C > (B > $o))]: ((((product_fst @ A) @ B) = (((((comp @ ((product_prod @ A) @ C)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ C)) @ (((((bNF_fstOp @ A) @ C) @ B) @ P2) @ Q)))))).
% 0.24/1.41 thf(fact_166_apfst__convE, axiom, (! [C:$tType,A:$tType,B:$tType,Q2:((product_prod @ A) @ B),F:(C > A),P:((product_prod @ C) @ B)]: (((Q2 = (((((product_apfst @ C) @ A) @ B) @ F) @ P)) => (~ ((! [X3:C,Y3:B]: (((P = ((((product_Pair @ C) @ B) @ X3) @ Y3)) => (Q2 != ((((product_Pair @ A) @ B) @ (F @ X3)) @ Y3))))))))))).
% 0.24/1.41 thf(fact_167_scomp__unfold, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F2:(A > ((product_prod @ B) @ C)),G2:(B > (C > D)),X2:A]: (((G2 @ (((product_fst @ B) @ C) @ (F2 @ X2))) @ (((product_snd @ B) @ C) @ (F2 @ X2))))))))).
% 0.24/1.41 thf(fact_168_Collect__case__prod__Grp__eqD, axiom, (! [B:$tType,A:$tType,Z2:((product_prod @ A) @ B),A6:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ Z2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ ((((bNF_Grp @ A) @ B) @ A6) @ F)))) => (((((((comp @ A) @ B) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)) @ Z2) = (((product_snd @ A) @ B) @ Z2)))))).
% 0.24/1.41 thf(fact_169_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F2:(B > ((product_prod @ C) @ D)),G2:(C > (D > A)),X2:B]: ((((((product_case_prod @ C) @ D) @ A) @ G2) @ (F2 @ X2)))))))).
% 0.24/1.41 thf(fact_170_scomp__scomp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F3:$tType,E:$tType,F:(A > ((product_prod @ E) @ F3)),G:(E > (F3 > ((product_prod @ C) @ D))),H:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ ((((((product_scomp @ A) @ E) @ F3) @ ((product_prod @ C) @ D)) @ F) @ G)) @ H) = ((((((product_scomp @ A) @ E) @ F3) @ B) @ F) @ (^ [X2:E]: (((((((product_scomp @ F3) @ C) @ D) @ B) @ (G @ X2)) @ H)))))))).
% 0.24/1.41 thf(fact_171_Grp__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Grp @ A) @ B) = (^ [A7:(set @ A),F2:(A > B),A4:A,B4:B]: (((B4 = (F2 @ A4)) & (((member @ A) @ A4) @ A7)))))))).
% 0.24/1.41 thf(fact_172_GrpI, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y4:A,A6:(set @ B)]: ((((F @ X) = Y4) => ((((member @ B) @ X) @ A6) => ((((((bNF_Grp @ B) @ A) @ A6) @ F) @ X) @ Y4)))))).
% 0.24/1.41 thf(fact_173_GrpE, axiom, (! [B:$tType,A:$tType,A6:(set @ A),F:(A > B),X:A,Y4:B]: ((((((((bNF_Grp @ A) @ B) @ A6) @ F) @ X) @ Y4) => (~ ((((F @ X) = Y4) => (~ ((((member @ A) @ X) @ A6)))))))))).
% 0.24/1.41 thf(fact_174_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 0.24/1.41 thf(fact_175_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F) = (F @ X))))).
% 0.24/1.41 thf(fact_176_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F2:(A > ((product_prod @ B) @ C)),G2:(B > (C > D)),X2:A]: ((((((product_case_prod @ B) @ C) @ D) @ G2) @ (F2 @ X2)))))))).
% 0.24/1.41 thf(fact_177_Collect__case__prod__Grp__in, axiom, (! [B:$tType,A:$tType,Z2:((product_prod @ A) @ B),A6:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ Z2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ ((((bNF_Grp @ A) @ B) @ A6) @ F)))) => (((member @ A) @ (((product_fst @ A) @ B) @ Z2)) @ A6))))).
% 0.24/1.41 thf(fact_178_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 0.24/1.41 thf(fact_179_map__prod__o__convol__id, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > A),G:(C > B),X:C]: ((((((((comp @ ((product_prod @ C) @ B)) @ ((product_prod @ A) @ B)) @ C) @ ((((((product_map_prod @ C) @ A) @ B) @ B) @ F) @ (id @ B))) @ (((((bNF_convol @ C) @ C) @ B) @ (id @ C)) @ G)) @ X) = ((((((bNF_convol @ C) @ A) @ B) @ (((((comp @ A) @ A) @ C) @ (id @ A)) @ F)) @ G) @ X))))).
% 0.24/1.41 thf(fact_180_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 0.24/1.41 thf(fact_181_id__apply, axiom, (! [A:$tType]: (((id @ A) = (^ [X2:A]: (X2)))))).
% 0.24/1.41 thf(fact_182_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 0.24/1.41 thf(fact_183_comp__id, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((comp @ A) @ B) @ A) @ F) @ (id @ A)) = F)))).
% 0.24/1.41 thf(fact_184_id__comp, axiom, (! [B:$tType,A:$tType,G:(A > B)]: (((((((comp @ B) @ B) @ A) @ (id @ B)) @ G) = G)))).
% 0.24/1.41 thf(fact_185_fun_Omap__id, axiom, (! [A:$tType,D:$tType,T2:(D > A)]: (((((((comp @ A) @ A) @ D) @ (id @ A)) @ T2) = T2)))).
% 0.24/1.41 thf(fact_186_case__prod__Pair, axiom, (! [B:$tType,A:$tType]: ((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 0.24/1.41 thf(fact_187_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y4:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y4)) = ((((product_Pair @ A) @ B) @ Y4) @ X))))).
% 0.24/1.41 thf(fact_188_apfst__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apfst @ A) @ A) @ B) @ (id @ A)) = (id @ ((product_prod @ A) @ B)))))).
% 0.24/1.41 thf(fact_189_apsnd__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apsnd @ B) @ B) @ A) @ (id @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 0.24/1.41 thf(fact_190_case__swap, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > (B > A)),P:((product_prod @ C) @ B)]: (((((((product_case_prod @ B) @ C) @ A) @ (^ [Y:B,X2:C]: (((F @ X2) @ Y)))) @ (((product_swap @ C) @ B) @ P)) = (((((product_case_prod @ C) @ B) @ A) @ F) @ P))))).
% 0.24/1.41 thf(fact_191_swap__comp__swap, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((product_swap @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 0.24/1.41 thf(fact_192_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 0.24/1.41 thf(fact_193_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 0.24/1.41 thf(fact_194_map__prod_Oidentity, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X2:A]: (X2))) @ (^ [X2:B]: (X2))) = (id @ ((product_prod @ A) @ B)))))).
% 0.24/1.41 thf(fact_195_fun_Omap__id0, axiom, (! [A:$tType,D:$tType]: ((((((comp @ A) @ A) @ D) @ (id @ A)) = (id @ (D > A)))))).
% 0.24/1.41 thf(fact_196_comp__eq__id__dest, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ B) @ B) @ A) @ (id @ B)) @ C2)) => ((A2 @ (B2 @ V)) = (C2 @ V)))))).
% 0.24/1.41 thf(fact_197_pointfree__idE, axiom, (! [B:$tType,A:$tType,F:(B > A),G:(A > B),X:A]: ((((((((comp @ B) @ A) @ A) @ F) @ G) = (id @ A)) => ((F @ (G @ X)) = X))))).
% 0.24/1.41 thf(fact_198_prod_Omap__id0, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (id @ A)) @ (id @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 0.24/1.41 thf(fact_199_id__def, axiom, (! [A:$tType]: (((id @ A) = (^ [X2:A]: (X2)))))).
% 0.24/1.41 thf(fact_200_eq__id__iff, axiom, (! [A:$tType,F:(A > A)]: (((! [X2:A]: (((F @ X2) = X2))) = (F = (id @ A)))))).
% 0.24/1.41 thf(fact_201_prod_Omap__id, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (id @ A)) @ (id @ B)) @ T2) = T2)))).
% 0.24/1.41 thf(fact_202_apfst__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((product_apfst @ A) @ C) @ B) = (^ [F2:(A > C)]: (((((((product_map_prod @ A) @ C) @ B) @ B) @ F2) @ (id @ B)))))))).
% 0.24/1.41 thf(fact_203_apsnd__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((product_apsnd @ B) @ C) @ A) = (((((product_map_prod @ A) @ A) @ B) @ C) @ (id @ A)))))).
% 0.24/1.41 thf(fact_204_convol__mem__GrpI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),G:(A > B)]: (((((member @ A) @ X) @ A6) => (((member @ ((product_prod @ A) @ B)) @ ((((((bNF_convol @ A) @ A) @ B) @ (id @ A)) @ G) @ X)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ ((((bNF_Grp @ A) @ B) @ A6) @ G)))))))).
% 0.24/1.41 thf(fact_205_snd__diag__id, axiom, (! [A:$tType,Z2:A]: ((((((((comp @ ((product_prod @ A) @ A)) @ A) @ A) @ ((product_snd @ A) @ A)) @ (^ [X2:A]: (((((product_Pair @ A) @ A) @ X2) @ X2)))) @ Z2) = ((id @ A) @ Z2))))).
% 0.24/1.41 thf(fact_206_fst__diag__id, axiom, (! [A:$tType,Z2:A]: ((((((((comp @ ((product_prod @ A) @ A)) @ A) @ A) @ ((product_fst @ A) @ A)) @ (^ [X2:A]: (((((product_Pair @ A) @ A) @ X2) @ X2)))) @ Z2) = ((id @ A) @ Z2))))).
% 0.24/1.41 thf(fact_207_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C2 @ A2) @ B2))))).
% 0.24/1.41 thf(fact_208_inf__Int__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((((inf_inf @ (A > (B > $o))) @ (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R)))) @ (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ S2)))) = (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ R) @ S2)))))))).
% 0.24/1.41 thf(fact_209_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: ((((^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R))) = (^ [X2:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ S2)))) = (R = S2))))).
% 0.24/1.41 thf(fact_210_prod_Osize__gen__o__map, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(C > nat),Fa:(D > nat),G:(A > C),Ga:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ nat) @ ((product_prod @ A) @ B)) @ ((((basic_BNF_size_prod @ C) @ D) @ F) @ Fa)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ G) @ Ga)) = ((((basic_BNF_size_prod @ A) @ B) @ (((((comp @ C) @ nat) @ A) @ F) @ G)) @ (((((comp @ D) @ nat) @ B) @ Fa) @ Ga)))))).
% 0.24/1.41 thf(fact_211_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R2:(set @ ((product_prod @ B) @ B)),F2:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X2:A,Y:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F2 @ X2)) @ (F2 @ Y))) @ R2))))))))))).
% 0.24/1.41 thf(fact_212_iso__tuple__update__accessor__eq__assist__idI, axiom, (! [A:$tType,V3:A,F:(A > A),V:A]: (((V3 = (F @ V)) => ((((((((iso_tu2011167877assist @ A) @ A) @ (id @ (A > A))) @ (id @ A)) @ V) @ F) @ V3) @ V))))).
% 0.24/1.41 thf(fact_213_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y4:A,R3:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ ((((inv_image @ B) @ A) @ R3) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y4))) @ R3))))).
% 0.24/1.41 thf(fact_214_update__accessor__updator__eqE, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),V:B,F:(A > A),V3:B,X:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac2) @ V) @ F) @ V3) @ X) => (((Upd @ F) @ V) = V3))))).
% 0.24/1.41 thf(fact_215_update__accessor__accessor__eqE, axiom, (! [B:$tType,A:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),V:B,F:(A > A),V3:B,X:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac2) @ V) @ F) @ V3) @ X) => ((Ac2 @ V) = X))))).
% 0.24/1.41 thf(fact_216_iso__tuple__update__accessor__eq__assist__triv, axiom, (! [B:$tType,A:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),V:B,F:(A > A),V3:B,X:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac2) @ V) @ F) @ V3) @ X) => ((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac2) @ V) @ F) @ V3) @ X))))).
% 0.24/1.41 thf(fact_217_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R4:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A)),F2:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R4) @ F2)) @ ((((inv_image @ A) @ B) @ S3) @ F2))))))))).
% 0.24/1.41 thf(fact_218_fun_Opred__map, axiom, (! [B:$tType,A:$tType,D:$tType,Q:(B > $o),F:(A > B),X:(D > A)]: (((((((basic_pred_fun @ D) @ B) @ (^ [Uu:D]: ($true))) @ Q) @ (((((comp @ A) @ B) @ D) @ F) @ X)) = (((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (((((comp @ B) @ $o) @ A) @ Q) @ F)) @ X))))).
% 0.24/1.41 thf(fact_219_pred__funI, axiom, (! [B:$tType,A:$tType,A6:(A > $o),B6:(B > $o),F:(A > B)]: (((! [X3:A]: (((A6 @ X3) => (B6 @ (F @ X3))))) => (((((basic_pred_fun @ A) @ B) @ A6) @ B6) @ F))))).
% 0.24/1.41 thf(fact_220_fun_Opred__True, axiom, (! [A:$tType,D:$tType]: ((((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (^ [Uu:A]: ($true))) = (^ [Uu:(D > A)]: ($true)))))).
% 0.24/1.41 thf(fact_221_fun_Omap__cong__pred, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Ya:(D > A),F:(A > B),G:(A > B)]: (((X = Ya) => ((((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (^ [Z:A]: (((F @ Z) = (G @ Z))))) @ Ya) => ((((((comp @ A) @ B) @ D) @ F) @ X) = (((((comp @ A) @ B) @ D) @ G) @ Ya))))))).
% 0.24/1.41 thf(fact_222_rp__inv__image__rp, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))),F:(B > A)]: ((((fun_reduction_pair @ A) @ P2) => ((fun_reduction_pair @ B) @ ((((fun_rp_inv_image @ A) @ B) @ P2) @ F)))))).
% 0.24/1.41 thf(fact_223_csquare__fstOp__sndOp, axiom, (! [A:$tType,B:$tType,C:$tType,F:((A > (B > $o)) > (((product_prod @ A) @ B) > $o)),P2:(A > (C > $o)),Q:(C > (B > $o))]: ((((((((((bNF_csquare @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ C) @ B)) @ ((collect @ ((product_prod @ A) @ B)) @ (F @ (((((relcompp @ A) @ C) @ B) @ P2) @ Q)))) @ ((product_snd @ A) @ C)) @ ((product_fst @ C) @ B)) @ (((((bNF_fstOp @ A) @ C) @ B) @ P2) @ Q)) @ (((((bNF_sndOp @ A) @ C) @ B) @ P2) @ Q))))).
% 0.24/1.41 thf(fact_224_fstOp__in, axiom, (! [B:$tType,C:$tType,A:$tType,Ac2:((product_prod @ A) @ B),P2:(A > (C > $o)),Q:(C > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ Ac2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((((relcompp @ A) @ C) @ B) @ P2) @ Q)))) => (((member @ ((product_prod @ A) @ C)) @ ((((((bNF_fstOp @ A) @ C) @ B) @ P2) @ Q) @ Ac2)) @ ((collect @ ((product_prod @ A) @ C)) @ ((((product_case_prod @ A) @ C) @ $o) @ P2))))))).
% 0.24/1.41 thf(fact_225_sndOp__in, axiom, (! [A:$tType,B:$tType,C:$tType,Ac2:((product_prod @ A) @ B),P2:(A > (C > $o)),Q:(C > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ Ac2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((((relcompp @ A) @ C) @ B) @ P2) @ Q)))) => (((member @ ((product_prod @ C) @ B)) @ ((((((bNF_sndOp @ A) @ C) @ B) @ P2) @ Q) @ Ac2)) @ ((collect @ ((product_prod @ C) @ B)) @ ((((product_case_prod @ C) @ B) @ $o) @ Q))))))).
% 0.24/1.41 thf(fact_226_csquare__def, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType]: ((((((bNF_csquare @ A) @ B) @ C) @ D) = (^ [A7:(set @ A),F12:(B > C),F23:(D > C),P1:(A > B),P22:(A > D)]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => ((F12 @ (P1 @ X2)) = (F23 @ (P22 @ X2)))))))))))).
% 0.24/1.41 thf(fact_227_pick__middlep, axiom, (! [B:$tType,A:$tType,C:$tType,P2:(A > (B > $o)),Q:(B > (C > $o)),A2:A,C2:C]: (((((((((relcompp @ A) @ B) @ C) @ P2) @ Q) @ A2) @ C2) => (((P2 @ A2) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P2) @ Q) @ A2) @ C2)) & ((Q @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P2) @ Q) @ A2) @ C2)) @ C2)))))).
% 0.24/1.41 thf(fact_228_update__accessor__cong__assist__idI, axiom, (! [A:$tType]: (((((iso_tu2017585022assist @ A) @ A) @ (id @ (A > A))) @ (id @ A))))).
% 0.24/1.41 thf(fact_229_The__case__prod, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o))]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P2)) = ((the @ ((product_prod @ A) @ B)) @ (^ [Xy:((product_prod @ A) @ B)]: (((P2 @ (((product_fst @ A) @ B) @ Xy)) @ (((product_snd @ A) @ B) @ Xy))))))))).
% 0.24/1.41 thf(fact_230_The__split__eq, axiom, (! [A:$tType,B:$tType,X:A,Y4:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X5:A,Y6:B]: (((X = X5) & (Y4 = Y6)))))) = ((((product_Pair @ A) @ B) @ X) @ Y4))))).
% 0.24/1.41 thf(fact_231_iso__tuple__update__accessor__cong__from__eq, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),V:B,F:(A > A),V3:B,X:A]: ((((((((((iso_tu2011167877assist @ A) @ B) @ Upd) @ Ac2) @ V) @ F) @ V3) @ X) => ((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2))))).
% 0.24/1.41 thf(fact_232_iso__tuple__update__accessor__eq__assist__def, axiom, (! [A:$tType,B:$tType]: ((((iso_tu2011167877assist @ B) @ A) = (^ [Upd2:((B > B) > (A > A)),Ac:(A > B),V4:A,F2:(B > B),V5:A,X2:B]: (((((Upd2 @ F2) @ V4) = V5) & (((Ac @ V4) = X2) & ((((iso_tu2017585022assist @ B) @ A) @ Upd2) @ Ac))))))))).
% 0.24/1.41 thf(fact_233_update__accessor__congruence__unfoldE, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),R3:B,R5:B,V3:A,F:(A > A),F5:(A > A)]: ((((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2) => ((R3 = R5) => (((Ac2 @ R5) = V3) => ((! [V6:A]: (((V6 = V3) => ((F @ V6) = (F5 @ V6))))) => (((Upd @ F) @ R3) = ((Upd @ F5) @ R5))))))))).
% 0.24/1.41 thf(fact_234_update__accessor__congruence__foldE, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),R3:B,R5:B,V3:A,F:(A > A),F5:(A > A)]: ((((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2) => ((R3 = R5) => (((Ac2 @ R5) = V3) => ((! [V6:A]: (((V3 = V6) => ((F @ V6) = (F5 @ V6))))) => (((Upd @ F) @ R3) = ((Upd @ F5) @ R5))))))))).
% 0.24/1.41 thf(fact_235_update__accessor__cong__assist__triv, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A)]: ((((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2) => ((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2))))).
% 0.24/1.41 thf(fact_236_update__accessor__noopE, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),F:(A > A),X:B]: ((((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2) => (((F @ (Ac2 @ X)) = (Ac2 @ X)) => (((Upd @ F) @ X) = X)))))).
% 0.24/1.41 thf(fact_237_iso__tuple__update__accessor__cong__assist__def, axiom, (! [A:$tType,B:$tType]: ((((iso_tu2017585022assist @ B) @ A) = (^ [Upd2:((B > B) > (A > A)),Ac:(A > B)]: (((! [F2:(B > B),V4:A]: ((((Upd2 @ (^ [X2:B]: ((F2 @ (Ac @ V4))))) @ V4) = ((Upd2 @ F2) @ V4)))) & (! [V4:A]: ((((Upd2 @ (id @ B)) @ V4) = V4)))))))))).
% 0.24/1.41 thf(fact_238_update__accessor__noop__compE, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A),F:(A > A),X:B,G:(A > A)]: ((((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2) => (((F @ (Ac2 @ X)) = (Ac2 @ X)) => (((Upd @ (((((comp @ A) @ A) @ A) @ G) @ F)) @ X) = ((Upd @ G) @ X))))))).
% 0.24/1.41 thf(fact_239_iso__tuple__update__accessor__cong__assist__id, axiom, (! [A:$tType,B:$tType,Upd:((A > A) > (B > B)),Ac2:(B > A)]: ((((((iso_tu2017585022assist @ A) @ B) @ Upd) @ Ac2) => ((Upd @ (id @ A)) = (id @ B)))))).
% 0.24/1.41 thf(fact_240_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X2:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X2)))))))).
% 0.24/1.41 thf(fact_241_the__sym__eq__trivial, axiom, (! [A:$tType,X:A]: ((((the @ A) @ ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) @ X)) = X)))).
% 0.24/1.41 thf(fact_242_the__equality, axiom, (! [A:$tType,P2:(A > $o),A2:A]: (((P2 @ A2) => ((! [X3:A]: (((P2 @ X3) => (X3 = A2)))) => (((the @ A) @ P2) = A2)))))).
% 0.24/1.41 thf(fact_243_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X2:A]: ((X2 = A2)))) = A2)))).
% 0.24/1.41 thf(fact_244_theI, axiom, (! [A:$tType,P2:(A > $o),A2:A]: (((P2 @ A2) => ((! [X3:A]: (((P2 @ X3) => (X3 = A2)))) => (P2 @ ((the @ A) @ P2))))))).
% 0.24/1.41 thf(fact_245_theI_H, axiom, (! [A:$tType,P2:(A > $o)]: (((? [X6:A]: (((P2 @ X6) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X6))))))) => (P2 @ ((the @ A) @ P2)))))).
% 0.24/1.41 thf(fact_246_theI2, axiom, (! [A:$tType,P2:(A > $o),A2:A,Q:(A > $o)]: (((P2 @ A2) => ((! [X3:A]: (((P2 @ X3) => (X3 = A2)))) => ((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P2)))))))).
% 0.24/1.41 thf(fact_247_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P4:$o,X2:A,Y:A]: (((the @ A) @ (^ [Z:A]: (((P4 => (Z = X2)) & ((~ (P4)) => (Z = Y)))))))))))).
% 0.24/1.41 thf(fact_248_the1I2, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((? [X6:A]: (((P2 @ X6) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X6))))))) => ((! [X3:A]: (((P2 @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P2))))))).
% 0.24/1.41 thf(fact_249_the1__equality, axiom, (! [A:$tType,P2:(A > $o),A2:A]: (((? [X6:A]: (((P2 @ X6) & (! [Y3:A]: (((P2 @ Y3) => (Y3 = X6))))))) => ((P2 @ A2) => (((the @ A) @ P2) = A2)))))).
% 0.24/1.41 thf(fact_250_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X2:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X2)))))))).
% 0.24/1.41 thf(fact_251_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F:(B > A),X:B,C2:C,G:(B > C),A6:(set @ B)]: (((B2 = (F @ X)) => ((C2 = (G @ X)) => ((((member @ B) @ X) @ A6) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C2)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A6) @ F) @ G)))))))).
% 0.24/1.41 thf(fact_252_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P4:(A > $o),R4:(A > (set @ ((product_prod @ B) @ B)))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [X5:A,Y6:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X2:A,Y:B]: (((X5 = X2) & ((P4 @ X2) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y6) @ Y)) @ (R4 @ X2)))))))))))))))))).
% 0.24/1.41 thf(fact_253_old_Orec__bool__def, axiom, (! [T:$tType]: (((product_rec_bool @ T) = (^ [F12:T,F23:T,X2:$o]: (((the @ T) @ ((((product_rec_set_bool @ T) @ F12) @ F23) @ X2)))))))).
% 0.24/1.41 thf(fact_254_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $false) = F22)))).
% 0.24/1.41 thf(fact_255_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $true) = F1)))).
% 0.24/1.41 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A8:$tType,A9:$tType]: ((((semilattice_inf @ A9) @ (type2 @ A9)) => ((semilattice_inf @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 0.24/1.41 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: ((((lattice @ A9) @ (type2 @ A9)) => ((lattice @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 0.24/1.41 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1, axiom, ((semilattice_inf @ nat) @ (type2 @ nat))).
% 0.24/1.41 thf(tcon_Nat_Onat___Lattices_Olattice_2, axiom, ((lattice @ nat) @ (type2 @ nat))).
% 0.24/1.41 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3, axiom, (! [A8:$tType]: (((semilattice_inf @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 0.24/1.41 thf(tcon_Set_Oset___Lattices_Olattice_4, axiom, (! [A8:$tType]: (((lattice @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 0.24/1.41 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_5, axiom, ((semilattice_inf @ $o) @ (type2 @ $o))).
% 0.24/1.41 thf(tcon_HOL_Obool___Lattices_Olattice_6, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 0.24/1.41 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_7, axiom, ((semilattice_inf @ product_unit) @ (type2 @ product_unit))).
% 0.24/1.41 thf(tcon_Product__Type_Ounit___Lattices_Olattice_8, axiom, ((lattice @ product_unit) @ (type2 @ product_unit))).
% 0.24/1.41 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 0.24/1.41 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y4:A]: ((((((if @ A) @ $false) @ X) @ Y4) = Y4)))).
% 0.24/1.41 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y4:A]: ((((((if @ A) @ $true) @ X) @ Y4) = X)))).
% 0.24/1.41 thf(conj_0, conjecture, (((((((coindu1441602521_llist @ ((product_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a))) @ a) @ ((((product_case_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ $o) @ (^ [Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: (((Xs != (coinductive_LNil @ a)) => ((Ys != (coinductive_LNil @ a)) => (((coinductive_lhd @ a) @ Xs) != ((coinductive_lhd @ a) @ Ys)))))))) @ (((((comp @ (coinductive_llist @ a)) @ a) @ ((product_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a))) @ (coinductive_lhd @ a)) @ ((product_snd @ (coinductive_llist @ a)) @ (coinductive_llist @ a)))) @ ((((((product_map_prod @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_ltl @ a)) @ (coinductive_ltl @ a))) @ ((((product_Pair @ (coinductive_llist @ a)) @ (coinductive_llist @ a)) @ (coinductive_LNil @ a)) @ xs)) = (coinductive_LNil @ a))).
% 0.24/1.41 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.pFALYqNRSO/DTF2DTF23795.p
% 0.24/1.41 ------------------------
% 0.24/1.43 ---- Cleaned THF ---
% 0.24/1.43 thf(ty_t_Coinductive__List_Ollist,type,
% 0.24/1.43 coinductive_llist: $tType > $tType ).
% 0.24/1.43
% 0.24/1.43 thf(ty_t_Product__Type_Ounit,type,
% 0.24/1.43 product_unit: $tType ).
% 0.24/1.43
% 0.24/1.43 thf(ty_t_Product__Type_Oprod,type,
% 0.24/1.43 product_prod: $tType > $tType > $tType ).
% 0.24/1.43
% 0.24/1.43 thf(ty_t_Set_Oset,type,
% 0.24/1.43 set: $tType > $tType ).
% 0.24/1.43
% 0.24/1.43 thf(ty_t_Nat_Onat,type,
% 0.24/1.43 nat: $tType ).
% 0.24/1.43
% 0.24/1.43 thf(ty_t_itself,type,
% 0.24/1.43 itself: $tType > $tType ).
% 0.24/1.43
% 0.24/1.43 thf(ty_tf_a,type,
% 0.24/1.43 a: $tType ).
% 0.24/1.43
% 0.24/1.43 thf(sy_cl_HOL_Otype,type,
% 0.24/1.43 type:
% 0.24/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_cl_Lattices_Olattice,type,
% 0.24/1.43 lattice:
% 0.24/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.24/1.43 semilattice_inf:
% 0.24/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Def_OGrp,type,
% 0.24/1.43 bNF_Grp:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Def_Oconvol,type,
% 0.24/1.43 bNF_convol:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Def_Ocsquare,type,
% 0.24/1.43 bNF_csquare:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Def_OfstOp,type,
% 0.24/1.43 bNF_fstOp:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.24/1.43 bNF_pick_middlep:
% 0.24/1.43 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Def_OsndOp,type,
% 0.24/1.43 bNF_sndOp:
% 0.24/1.43 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.24/1.43 bNF_Greatest_image2:
% 0.24/1.43 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
% 0.24/1.43 basic_BNF_size_prod:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Basic__BNFs_Opred__fun,type,
% 0.24/1.43 basic_pred_fun:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.24/1.43 coinductive_LNil:
% 0.24/1.43 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 0.24/1.43 coinductive_lhd:
% 0.24/1.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Coinductive__List_Ollist_Oltl,type,
% 0.24/1.43 coinductive_ltl:
% 0.24/1.43 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.24/1.43 coindu1441602521_llist:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Fun_Ocomp,type,
% 0.24/1.43 comp:
% 0.24/1.43 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Fun_Oid,type,
% 0.24/1.43 id:
% 0.24/1.43 !>[A: $tType] : ( A > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.24/1.43 fun_reduction_pair:
% 0.24/1.43 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.24/1.43 fun_rp_inv_image:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_HOL_OThe,type,
% 0.24/1.43 the:
% 0.24/1.43 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_If,type,
% 0.24/1.43 if:
% 0.24/1.43 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.24/1.43 inf_inf:
% 0.24/1.43 !>[A: $tType] : ( A > A > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_OPair,type,
% 0.24/1.43 product_Pair:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oapfst,type,
% 0.24/1.43 product_apfst:
% 0.24/1.43 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oapsnd,type,
% 0.24/1.43 product_apsnd:
% 0.24/1.43 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.24/1.43 produc2004651681e_prod:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Omap__prod,type,
% 0.24/1.43 product_map_prod:
% 0.24/1.43 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.24/1.43 product_rec_bool:
% 0.24/1.43 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 0.24/1.43 product_rec_set_bool:
% 0.24/1.43 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.24/1.43 product_rec_prod:
% 0.24/1.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.24/1.43 product_rec_set_prod:
% 0.24/1.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.24/1.43 product_rec_set_unit:
% 0.24/1.43 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.24/1.43 product_rec_unit:
% 0.24/1.43 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.24/1.43 product_case_prod:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.24/1.43 product_fst:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.24/1.43 product_snd:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.24/1.43 product_swap:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Product__Type_Oscomp,type,
% 0.24/1.43 product_scomp:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Pure_Otype,type,
% 0.24/1.43 type2:
% 0.24/1.43 !>[A: $tType] : ( itself @ A ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Record_Oiso__tuple__update__accessor__cong__assist,type,
% 0.24/1.43 iso_tu2017585022assist:
% 0.24/1.43 !>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
% 0.24/1.43 iso_tu2011167877assist:
% 0.24/1.43 !>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Relation_Oinv__image,type,
% 0.24/1.43 inv_image:
% 0.24/1.43 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Relation_Orelcompp,type,
% 0.24/1.43 relcompp:
% 0.24/1.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Set_OCollect,type,
% 0.24/1.43 collect:
% 0.24/1.43 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_Wfrec_Osame__fst,type,
% 0.24/1.43 same_fst:
% 0.24/1.43 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_c_member,type,
% 0.24/1.43 member:
% 0.24/1.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.24/1.43
% 0.24/1.43 thf(sy_v_xs,type,
% 0.24/1.43 xs: coinductive_llist @ a ).
% 0.24/1.43
% 0.24/1.43 thf(fact_0_local_Oinf__llist__def,axiom,
% 0.24/1.43 ( ( inf_inf @ ( coinductive_llist @ a ) )
% 0.24/1.43 = ( ^ [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.24/1.43 ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ a
% 0.24/1.43 @ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
% 0.24/1.43 @ ^ [Xt: coinductive_llist @ a,Yt: coinductive_llist @ a] :
% 0.24/1.43 ( ( Xt
% 0.24/1.43 != ( coinductive_LNil @ a ) )
% 0.24/1.43 => ( ( Yt
% 0.24/1.43 != ( coinductive_LNil @ a ) )
% 0.24/1.43 => ( ( coinductive_lhd @ a @ Xt )
% 0.24/1.43 != ( coinductive_lhd @ a @ Yt ) ) ) ) )
% 0.24/1.43 @ ( comp @ ( coinductive_llist @ a ) @ a @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( coinductive_lhd @ a ) @ ( product_snd @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) )
% 0.24/1.43 @ ( product_map_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_ltl @ a ) @ ( coinductive_ltl @ a ) )
% 0.24/1.43 @ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ Xs @ Ys ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_1_snd__comp__map__prod,axiom,
% 0.24/1.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
% 0.24/1.43 ( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
% 0.24/1.43 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_2_case__prodI,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.24/1.43 ( ( F @ A2 @ B2 )
% 0.24/1.43 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_3_case__prodI2,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
% 0.24/1.43 ( ! [A3: A,B3: B] :
% 0.24/1.43 ( ( P
% 0.24/1.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.24/1.43 => ( C2 @ A3 @ B3 ) )
% 0.24/1.43 => ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_4_case__prod__conv,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 0.24/1.43 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.24/1.43 = ( F @ A2 @ B2 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_5_snd__map__prod,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 0.24/1.43 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 0.24/1.43 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_6_map__prod__simp,axiom,
% 0.24/1.43 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
% 0.24/1.43 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.24/1.43 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_7_Coinductive__List__Prefix__Mirabelle__yfrixeyiok_Oinf__llist__def,axiom,
% 0.24/1.43 ! [A: $tType] :
% 0.24/1.43 ( ( inf_inf @ ( coinductive_llist @ A ) )
% 0.24/1.43 = ( ^ [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.24/1.43 ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ A
% 0.24/1.43 @ ( product_case_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ $o
% 0.24/1.43 @ ^ [Xt: coinductive_llist @ A,Yt: coinductive_llist @ A] :
% 0.24/1.43 ( ( Xt
% 0.24/1.43 != ( coinductive_LNil @ A ) )
% 0.24/1.43 => ( ( Yt
% 0.24/1.43 != ( coinductive_LNil @ A ) )
% 0.24/1.43 => ( ( coinductive_lhd @ A @ Xt )
% 0.24/1.43 != ( coinductive_lhd @ A @ Yt ) ) ) ) )
% 0.24/1.43 @ ( comp @ ( coinductive_llist @ A ) @ A @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( coinductive_lhd @ A ) @ ( product_snd @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) )
% 0.24/1.43 @ ( product_map_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_ltl @ A ) @ ( coinductive_ltl @ A ) )
% 0.24/1.43 @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs @ Ys ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_8_map__prod__ident,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType] :
% 0.24/1.43 ( ( product_map_prod @ A @ A @ B @ B
% 0.24/1.43 @ ^ [X2: A] : X2
% 0.24/1.43 @ ^ [Y: B] : Y )
% 0.24/1.43 = ( ^ [Z: product_prod @ A @ B] : Z ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_9_split__part,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
% 0.24/1.43 ( ( product_case_prod @ A @ B @ $o
% 0.24/1.43 @ ^ [A4: A,B4: B] :
% 0.24/1.43 ( P2
% 0.24/1.43 & ( Q @ A4 @ B4 ) ) )
% 0.24/1.43 = ( ^ [Ab: product_prod @ A @ B] :
% 0.24/1.43 ( P2
% 0.24/1.43 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_10_ltl__unfold__llist,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
% 0.24/1.43 ( ( ( IS_LNIL @ A2 )
% 0.24/1.43 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 0.24/1.43 = ( coinductive_LNil @ A ) ) )
% 0.24/1.43 & ( ~ ( IS_LNIL @ A2 )
% 0.24/1.43 => ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
% 0.24/1.43 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_11_comp__apply,axiom,
% 0.24/1.43 ! [C: $tType,A: $tType,B: $tType] :
% 0.24/1.43 ( ( comp @ B @ A @ C )
% 0.24/1.43 = ( ^ [F2: B > A,G2: C > B,X2: C] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_12_old_Oprod_Oinject,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 0.24/1.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.24/1.43 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.24/1.43 = ( ( A2 = A5 )
% 0.24/1.43 & ( B2 = B5 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_13_prod_Oinject,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.24/1.43 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.24/1.43 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.24/1.43 = ( ( X1 = Y1 )
% 0.24/1.43 & ( X22 = Y2 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_14_mem__case__prodI2,axiom,
% 0.24/1.43 ! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
% 0.24/1.43 ( ! [A3: A,B3: B] :
% 0.24/1.43 ( ( P
% 0.24/1.43 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.24/1.43 => ( member @ C @ Z2 @ ( C2 @ A3 @ B3 ) ) )
% 0.24/1.43 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_15_mem__case__prodI,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
% 0.24/1.43 ( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) )
% 0.24/1.43 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_16_case__prodI2_H,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.24/1.43 ( ! [A3: A,B3: B] :
% 0.24/1.43 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.24/1.43 = P )
% 0.24/1.43 => ( C2 @ A3 @ B3 @ X ) )
% 0.24/1.43 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_17_mem__case__prodE,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P: product_prod @ B @ C] :
% 0.24/1.43 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P ) )
% 0.24/1.43 => ~ ! [X3: B,Y3: C] :
% 0.24/1.43 ( ( P
% 0.24/1.43 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.24/1.43 => ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_18_case__prodE_H,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P: product_prod @ A @ B,Z2: C] :
% 0.24/1.43 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ Z2 )
% 0.24/1.43 => ~ ! [X3: A,Y3: B] :
% 0.24/1.43 ( ( P
% 0.24/1.43 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.24/1.43 => ~ ( C2 @ X3 @ Y3 @ Z2 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_19_case__prodD_H,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
% 0.24/1.43 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
% 0.24/1.43 => ( R @ A2 @ B2 @ C2 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_20_map__prod__def,axiom,
% 0.24/1.43 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.24/1.43 ( ( product_map_prod @ A @ C @ B @ D )
% 0.24/1.43 = ( ^ [F2: A > C,G2: B > D] :
% 0.24/1.43 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.24/1.43 @ ^ [X2: A,Y: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G2 @ Y ) ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_21_old_Oprod_Oinducts,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.24/1.43 ( ! [A3: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.24/1.43 => ( P2 @ Prod ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_22_old_Oprod_Oexhaust,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
% 0.24/1.43 ~ ! [A3: A,B3: B] :
% 0.24/1.43 ( Y4
% 0.24/1.43 != ( product_Pair @ A @ B @ A3 @ B3 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_23_prod__induct7,axiom,
% 0.24/1.43 ! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.24/1.43 ( ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.24/1.43 => ( P2 @ X ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_24_prod__induct6,axiom,
% 0.24/1.43 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.24/1.43 ( ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 0.24/1.43 => ( P2 @ X ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_25_prod__induct5,axiom,
% 0.24/1.43 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.24/1.43 ( ! [A3: A,B3: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.24/1.43 => ( P2 @ X ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_26_prod__induct4,axiom,
% 0.24/1.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.24/1.43 ( ! [A3: A,B3: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.24/1.43 => ( P2 @ X ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_27_prod__induct3,axiom,
% 0.24/1.43 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.24/1.43 ( ! [A3: A,B3: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) )
% 0.24/1.43 => ( P2 @ X ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_28_prod__cases7,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.24/1.43 ~ ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3,G4: G3] :
% 0.24/1.43 ( Y4
% 0.24/1.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_29_prod__cases6,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.24/1.43 ~ ! [A3: A,B3: B,C3: C,D2: D,E2: E,F4: F3] :
% 0.24/1.43 ( Y4
% 0.24/1.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_30_prod__cases5,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.24/1.43 ~ ! [A3: A,B3: B,C3: C,D2: D,E2: E] :
% 0.24/1.43 ( Y4
% 0.24/1.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_31_prod__cases4,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.24/1.43 ~ ! [A3: A,B3: B,C3: C,D2: D] :
% 0.24/1.43 ( Y4
% 0.24/1.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_32_prod__cases3,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,C: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.24/1.43 ~ ! [A3: A,B3: B,C3: C] :
% 0.24/1.43 ( Y4
% 0.24/1.43 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_33_Pair__inject,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 0.24/1.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.24/1.43 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.24/1.43 => ~ ( ( A2 = A5 )
% 0.24/1.43 => ( B2 != B5 ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_34_prod__cases,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.24/1.43 ( ! [A3: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.24/1.43 => ( P2 @ P ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_35_surj__pair,axiom,
% 0.24/1.43 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.24/1.43 ? [X3: A,Y3: B] :
% 0.24/1.43 ( P
% 0.24/1.43 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_36_comp__eq__dest__lhs,axiom,
% 0.24/1.43 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.24/1.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.24/1.43 = C2 )
% 0.24/1.43 => ( ( A2 @ ( B2 @ V ) )
% 0.24/1.43 = ( C2 @ V ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_37_comp__eq__elim,axiom,
% 0.24/1.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D3: A > D] :
% 0.24/1.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.24/1.43 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.24/1.43 => ! [V2: A] :
% 0.24/1.43 ( ( A2 @ ( B2 @ V2 ) )
% 0.24/1.43 = ( C2 @ ( D3 @ V2 ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_38_comp__eq__dest,axiom,
% 0.24/1.43 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
% 0.24/1.43 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.24/1.43 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.24/1.43 => ( ( A2 @ ( B2 @ V ) )
% 0.24/1.43 = ( C2 @ ( D3 @ V ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_39_comp__assoc,axiom,
% 0.24/1.43 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 0.24/1.43 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 0.24/1.43 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_40_comp__def,axiom,
% 0.24/1.43 ! [A: $tType,C: $tType,B: $tType] :
% 0.24/1.43 ( ( comp @ B @ C @ A )
% 0.24/1.43 = ( ^ [F2: B > C,G2: A > B,X2: A] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_41_prod_Ocase__distrib,axiom,
% 0.24/1.43 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.24/1.43 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.24/1.43 = ( product_case_prod @ A @ B @ D
% 0.24/1.43 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 0.24/1.43 @ Prod ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_42_prod_Odisc__eq__case,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.24/1.43 ( product_case_prod @ A @ B @ $o
% 0.24/1.43 @ ^ [Uu: A,Uv: B] : $true
% 0.24/1.43 @ Prod ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_43_old_Oprod_Ocase,axiom,
% 0.24/1.43 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.24/1.43 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.24/1.43 = ( F @ X1 @ X22 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_44_snd__conv,axiom,
% 0.24/1.43 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.24/1.43 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.24/1.43 = X22 ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_45_mem__Collect__eq,axiom,
% 0.24/1.43 ! [A: $tType,A2: A,P2: A > $o] :
% 0.24/1.43 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.24/1.43 = ( P2 @ A2 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_46_Collect__mem__eq,axiom,
% 0.24/1.43 ! [A: $tType,A6: set @ A] :
% 0.24/1.43 ( ( collect @ A
% 0.24/1.43 @ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
% 0.24/1.43 = A6 ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_47_Collect__cong,axiom,
% 0.24/1.43 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.24/1.43 ( ! [X3: A] :
% 0.24/1.43 ( ( P2 @ X3 )
% 0.24/1.43 = ( Q @ X3 ) )
% 0.24/1.43 => ( ( collect @ A @ P2 )
% 0.24/1.43 = ( collect @ A @ Q ) ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_48_ext,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.24/1.43 ( ! [X3: A] :
% 0.24/1.43 ( ( F @ X3 )
% 0.24/1.43 = ( G @ X3 ) )
% 0.24/1.43 => ( F = G ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_49_snd__eqD,axiom,
% 0.24/1.43 ! [B: $tType,A: $tType,X: B,Y4: A,A2: A] :
% 0.24/1.43 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) )
% 0.24/1.43 = A2 )
% 0.24/1.43 => ( Y4 = A2 ) ) ).
% 0.24/1.43
% 0.24/1.43 thf(fact_50_ltl__simps_I1_J,axiom,
% 0.24/1.43 ! [A: $tType] :
% 0.24/1.43 ( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
% 0.24/1.44 = ( coinductive_LNil @ A ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_51_map__prod_Ocompositionality,axiom,
% 0.24/1.44 ! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F3,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_map_prod @ C @ E @ D @ F3 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
% 0.24/1.44 = ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Prod ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_52_map__prod__compose,axiom,
% 0.24/1.44 ! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: E > C,F22: A > E,G1: F3 > D,G22: B > F3] :
% 0.24/1.44 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F3 @ D @ B @ G1 @ G22 ) )
% 0.24/1.44 = ( comp @ ( product_prod @ E @ F3 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F3 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F3 @ F22 @ G22 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_53_map__prod_Ocomp,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F: C > E,G: D > F3,H: A > C,I: B > D] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F3 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F3 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
% 0.24/1.44 = ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_54_unfold__llist__ltl__unroll,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B] :
% 0.24/1.44 ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) )
% 0.24/1.44 = ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_55_unfold__llist_Octr_I1_J,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
% 0.24/1.44 ( ( P @ A2 )
% 0.24/1.44 => ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 )
% 0.24/1.44 = ( coinductive_LNil @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_56_unfold__llist_Osimps_I4_J,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
% 0.24/1.44 ( ~ ( P @ A2 )
% 0.24/1.44 => ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 ) )
% 0.24/1.44 = ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ ( G222 @ A2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_57_unfold__llist_Osimps_I3_J,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G222: A > A] :
% 0.24/1.44 ( ~ ( P @ A2 )
% 0.24/1.44 => ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G222 @ A2 ) )
% 0.24/1.44 = ( G21 @ A2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_58_cond__case__prod__eta,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.24/1.44 ( ! [X3: A,Y3: B] :
% 0.24/1.44 ( ( F @ X3 @ Y3 )
% 0.24/1.44 = ( G @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.24/1.44 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.24/1.44 = G ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_59_case__prod__eta,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ C
% 0.24/1.44 @ ^ [X2: A,Y: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
% 0.24/1.44 = F ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_60_case__prodE2,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z2: product_prod @ B @ C] :
% 0.24/1.44 ( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z2 ) )
% 0.24/1.44 => ~ ! [X3: B,Y3: C] :
% 0.24/1.44 ( ( Z2
% 0.24/1.44 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.24/1.44 => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_61_case__prodE,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ $o @ C2 @ P )
% 0.24/1.44 => ~ ! [X3: A,Y3: B] :
% 0.24/1.44 ( ( P
% 0.24/1.44 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.24/1.44 => ~ ( C2 @ X3 @ Y3 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_62_case__prodD,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.24/1.44 => ( F @ A2 @ B2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_63_snd__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_snd @ A @ B )
% 0.24/1.44 = ( product_case_prod @ A @ B @ B
% 0.24/1.44 @ ^ [X12: A,X23: B] : X23 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_64_case__prod__o__map__prod,axiom,
% 0.24/1.44 ! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F: D > E > C,G1: A > D,G22: B > E] :
% 0.24/1.44 ( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
% 0.24/1.44 = ( product_case_prod @ A @ B @ C
% 0.24/1.44 @ ^ [L: A,R2: B] : ( F @ ( G1 @ L ) @ ( G22 @ R2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_65_snd__diag__snd,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
% 0.24/1.44 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 0.24/1.44 @ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
% 0.24/1.44 @ ( product_snd @ A @ B ) ) )
% 0.24/1.44 = ( product_snd @ A @ B ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_66_old_Oprod_Orec,axiom,
% 0.24/1.44 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.24/1.44 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.24/1.44 = ( F1 @ A2 @ B2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_67_case__prod__map__prod,axiom,
% 0.24/1.44 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X: product_prod @ D @ E] :
% 0.24/1.44 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X ) )
% 0.24/1.44 = ( product_case_prod @ D @ E @ A
% 0.24/1.44 @ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G @ R2 ) )
% 0.24/1.44 @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_68_inf__apply,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.24/1.44 => ( ( inf_inf @ ( A > B ) )
% 0.24/1.44 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_69_inf_Oidem,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [A2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ A2 @ A2 )
% 0.24/1.44 = A2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_70_inf__idem,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A] :
% 0.24/1.44 ( ( inf_inf @ A @ X @ X )
% 0.24/1.44 = X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_71_inf_Oleft__idem,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [A2: A,B2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.24/1.44 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_72_inf__left__idem,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A] :
% 0.24/1.44 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y4 ) )
% 0.24/1.44 = ( inf_inf @ A @ X @ Y4 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_73_inf_Oright__idem,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [A2: A,B2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 0.24/1.44 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_74_inf__right__idem,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A] :
% 0.24/1.44 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Y4 )
% 0.24/1.44 = ( inf_inf @ A @ X @ Y4 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_75_inf__left__commute,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A,Z2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) )
% 0.24/1.44 = ( inf_inf @ A @ Y4 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_76_inf_Oleft__commute,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [B2: A,A2: A,C2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C2 ) )
% 0.24/1.44 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_77_inf__commute,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ( ( inf_inf @ A )
% 0.24/1.44 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_78_inf_Ocommute,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ( ( inf_inf @ A )
% 0.24/1.44 = ( ^ [A4: A,B4: A] : ( inf_inf @ A @ B4 @ A4 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_79_inf__assoc,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A,Z2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Z2 )
% 0.24/1.44 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_80_inf_Oassoc,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [A2: A,B2: A,C2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 )
% 0.24/1.44 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_81_inf__fun__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ B @ ( type2 @ B ) )
% 0.24/1.44 => ( ( inf_inf @ ( A > B ) )
% 0.24/1.44 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_82_inf__sup__aci_I1_J,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( lattice @ A @ ( type2 @ A ) )
% 0.24/1.44 => ( ( inf_inf @ A )
% 0.24/1.44 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_83_inf__sup__aci_I2_J,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( lattice @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A,Z2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y4 ) @ Z2 )
% 0.24/1.44 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_84_inf__sup__aci_I3_J,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( lattice @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A,Z2: A] :
% 0.24/1.44 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y4 @ Z2 ) )
% 0.24/1.44 = ( inf_inf @ A @ Y4 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_85_inf__sup__aci_I4_J,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( lattice @ A @ ( type2 @ A ) )
% 0.24/1.44 => ! [X: A,Y4: A] :
% 0.24/1.44 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y4 ) )
% 0.24/1.44 = ( inf_inf @ A @ X @ Y4 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_86_rewriteL__comp__comp,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L2: A > B,H: D > A] :
% 0.24/1.44 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.24/1.44 = L2 )
% 0.24/1.44 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 0.24/1.44 = ( comp @ A @ B @ D @ L2 @ H ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_87_rewriteR__comp__comp,axiom,
% 0.24/1.44 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D] :
% 0.24/1.44 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.24/1.44 = R3 )
% 0.24/1.44 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 0.24/1.44 = ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_88_rewriteL__comp__comp2,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L22: A > D,H: E > A,R3: E > D] :
% 0.24/1.44 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.24/1.44 = ( comp @ D @ B @ A @ L1 @ L22 ) )
% 0.24/1.44 => ( ( ( comp @ A @ D @ E @ L22 @ H )
% 0.24/1.44 = R3 )
% 0.24/1.44 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 0.24/1.44 = ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_89_rewriteR__comp__comp2,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L2: D > E] :
% 0.24/1.44 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.24/1.44 = ( comp @ D @ B @ A @ R1 @ R22 ) )
% 0.24/1.44 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 0.24/1.44 = L2 )
% 0.24/1.44 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 0.24/1.44 = ( comp @ D @ E @ A @ L2 @ R22 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_90_case__prod__Pair__iden,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
% 0.24/1.44 = P ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_91_case__prod__app,axiom,
% 0.24/1.44 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.24/1.44 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.24/1.44 = ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y: D] :
% 0.24/1.44 ( product_case_prod @ B @ C @ A
% 0.24/1.44 @ ^ [L: B,R2: C] : ( F2 @ L @ R2 @ Y )
% 0.24/1.44 @ X2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_92_prod_Omap__comp,axiom,
% 0.24/1.44 ! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F3,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_map_prod @ C @ E @ D @ F3 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 0.24/1.44 = ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G22 @ F22 ) @ V ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_93_sndI,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y4: A,Z2: B] :
% 0.24/1.44 ( ( X
% 0.24/1.44 = ( product_Pair @ A @ B @ Y4 @ Z2 ) )
% 0.24/1.44 => ( ( product_snd @ A @ B @ X )
% 0.24/1.44 = Z2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_94_eq__snd__iff,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
% 0.24/1.44 ( ( B2
% 0.24/1.44 = ( product_snd @ B @ A @ P ) )
% 0.24/1.44 = ( ? [A4: B] :
% 0.24/1.44 ( P
% 0.24/1.44 = ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_95_split__cong,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G: A > B > C,P: product_prod @ A @ B] :
% 0.24/1.44 ( ! [X3: A,Y3: B] :
% 0.24/1.44 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 0.24/1.44 = Q2 )
% 0.24/1.44 => ( ( F @ X3 @ Y3 )
% 0.24/1.44 = ( G @ X3 @ Y3 ) ) )
% 0.24/1.44 => ( ( P = Q2 )
% 0.24/1.44 => ( ( product_case_prod @ A @ B @ C @ F @ P )
% 0.24/1.44 = ( product_case_prod @ A @ B @ C @ G @ Q2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_96_snd__comp__apsnd,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.24/1.44 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.24/1.44 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_97_map__prod__o__convol,axiom,
% 0.24/1.44 ! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H2: E > C,F: A > D,G: A > E] :
% 0.24/1.44 ( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H2 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
% 0.24/1.44 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H2 @ G ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_98_apsnd__conv,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y4: C] :
% 0.24/1.44 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y4 ) )
% 0.24/1.44 = ( product_Pair @ A @ B @ X @ ( F @ Y4 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_99_apsnd__eq__conv,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
% 0.24/1.44 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 0.24/1.44 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 0.24/1.44 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 0.24/1.44 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_100_snd__apsnd,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 0.24/1.44 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 0.24/1.44 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_101_convol__def,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( bNF_convol @ A @ B @ C )
% 0.24/1.44 = ( ^ [F2: A > B,G2: A > C,A4: A] : ( product_Pair @ B @ C @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_102_apsnd__compose,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
% 0.24/1.44 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
% 0.24/1.44 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_103_snd__convol_H,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X: C] :
% 0.24/1.44 ( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X ) )
% 0.24/1.44 = ( G @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_104_snd__convol,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
% 0.24/1.44 = G ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_105_fun_Omap__comp,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 0.24/1.44 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 0.24/1.44 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_106_comp__apply__eq,axiom,
% 0.24/1.44 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
% 0.24/1.44 ( ( ( F @ ( G @ X ) )
% 0.24/1.44 = ( H @ ( K @ X ) ) )
% 0.24/1.44 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.24/1.44 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_107_comp__cong,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F5: D > A,G5: E > D,X4: E] :
% 0.24/1.44 ( ( ( F @ ( G @ X ) )
% 0.24/1.44 = ( F5 @ ( G5 @ X4 ) ) )
% 0.24/1.44 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.24/1.44 = ( comp @ D @ A @ E @ F5 @ G5 @ X4 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_108_convol__o,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H: A > D] :
% 0.24/1.44 ( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H )
% 0.24/1.44 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H ) @ ( comp @ D @ C @ A @ G @ H ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_109_fun_Omap__ident,axiom,
% 0.24/1.44 ! [A: $tType,D: $tType,T2: D > A] :
% 0.24/1.44 ( ( comp @ A @ A @ D
% 0.24/1.44 @ ^ [X2: A] : X2
% 0.24/1.44 @ T2 )
% 0.24/1.44 = T2 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_110_prod_Omap__ident,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_map_prod @ A @ A @ B @ B
% 0.24/1.44 @ ^ [X2: A] : X2
% 0.24/1.44 @ ^ [X2: B] : X2
% 0.24/1.44 @ T2 )
% 0.24/1.44 = T2 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_111_snd__sndOp,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,P2: B > C > $o,Q: C > A > $o] :
% 0.24/1.44 ( ( product_snd @ B @ A )
% 0.24/1.44 = ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P2 @ Q ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_112_conj__comp__iff,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P2: B > $o,Q: B > $o,G: A > B] :
% 0.24/1.44 ( ( comp @ B @ $o @ A
% 0.24/1.44 @ ^ [X2: B] :
% 0.24/1.44 ( ( P2 @ X2 )
% 0.24/1.44 & ( Q @ X2 ) )
% 0.24/1.44 @ G )
% 0.24/1.44 = ( ^ [X2: A] :
% 0.24/1.44 ( ( comp @ B @ $o @ A @ P2 @ G @ X2 )
% 0.24/1.44 & ( comp @ B @ $o @ A @ Q @ G @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_113_snd__diag__fst,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
% 0.24/1.44 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 0.24/1.44 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.24/1.44 @ ( product_fst @ A @ B ) ) )
% 0.24/1.44 = ( product_fst @ A @ B ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_114_fst__diag__snd,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
% 0.24/1.44 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 0.24/1.44 @ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
% 0.24/1.44 @ ( product_snd @ A @ B ) ) )
% 0.24/1.44 = ( product_snd @ A @ B ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_115_K__record__comp,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
% 0.24/1.44 ( ( comp @ C @ B @ A
% 0.24/1.44 @ ^ [X2: C] : C2
% 0.24/1.44 @ F )
% 0.24/1.44 = ( ^ [X2: A] : C2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_116_fst__map__prod,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.24/1.44 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 0.24/1.44 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_117_fst__apsnd,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 0.24/1.44 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 0.24/1.44 = ( product_fst @ A @ C @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_118_prod_Ocollapse,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.24/1.44 = Prod ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_119_fst__comp__apsnd,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.24/1.44 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.24/1.44 = ( product_fst @ A @ B ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_120_fst__comp__map__prod,axiom,
% 0.24/1.44 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
% 0.24/1.44 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_121_fstI,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y4: A,Z2: B] :
% 0.24/1.44 ( ( X
% 0.24/1.44 = ( product_Pair @ A @ B @ Y4 @ Z2 ) )
% 0.24/1.44 => ( ( product_fst @ A @ B @ X )
% 0.24/1.44 = Y4 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_122_eq__fst__iff,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
% 0.24/1.44 ( ( A2
% 0.24/1.44 = ( product_fst @ A @ B @ P ) )
% 0.24/1.44 = ( ? [B4: B] :
% 0.24/1.44 ( P
% 0.24/1.44 = ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_123_fst__conv,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.24/1.44 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.24/1.44 = X1 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_124_fst__eqD,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,X: A,Y4: B,A2: A] :
% 0.24/1.44 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.24/1.44 = A2 )
% 0.24/1.44 => ( X = A2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_125_prod__eq__iff,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( ^ [Y5: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y5 = Z3 ) )
% 0.24/1.44 = ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.24/1.44 ( ( ( product_fst @ A @ B @ S )
% 0.24/1.44 = ( product_fst @ A @ B @ T3 ) )
% 0.24/1.44 & ( ( product_snd @ A @ B @ S )
% 0.24/1.44 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_126_prod_Oexpand,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.24/1.44 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.24/1.44 = ( product_fst @ A @ B @ Prod2 ) )
% 0.24/1.44 & ( ( product_snd @ A @ B @ Prod )
% 0.24/1.44 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.24/1.44 => ( Prod = Prod2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_127_prod__eqI,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.24/1.44 ( ( ( product_fst @ A @ B @ P )
% 0.24/1.44 = ( product_fst @ A @ B @ Q2 ) )
% 0.24/1.44 => ( ( ( product_snd @ A @ B @ P )
% 0.24/1.44 = ( product_snd @ A @ B @ Q2 ) )
% 0.24/1.44 => ( P = Q2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_128_fst__convol_H,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
% 0.24/1.44 ( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X ) )
% 0.24/1.44 = ( F @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_129_fst__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_fst @ A @ B )
% 0.24/1.44 = ( product_case_prod @ A @ B @ A
% 0.24/1.44 @ ^ [X12: A,X23: B] : X12 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_130_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y4: B,A2: product_prod @ A @ B] :
% 0.24/1.44 ( ( P2 @ X @ Y4 )
% 0.24/1.44 => ( ( A2
% 0.24/1.44 = ( product_Pair @ A @ B @ X @ Y4 ) )
% 0.24/1.44 => ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_131_prod_Oexhaust__sel,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.24/1.44 ( Prod
% 0.24/1.44 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_132_surjective__pairing,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.24/1.44 ( T2
% 0.24/1.44 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_133_case__prod__beta,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,B: $tType] :
% 0.24/1.44 ( ( product_case_prod @ B @ C @ A )
% 0.24/1.44 = ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_134_prod_Ocase__eq__if,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ C )
% 0.24/1.44 = ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_135_Product__Type_OCollect__case__prodD,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
% 0.24/1.44 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
% 0.24/1.44 => ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_136_split__comp__eq,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
% 0.24/1.44 ( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
% 0.24/1.44 = ( product_case_prod @ D @ B @ C
% 0.24/1.44 @ ^ [X2: D] : ( F @ ( G @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_137_case__prod__beta_H,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ C )
% 0.24/1.44 = ( ^ [F2: A > B > C,X2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_138_case__prod__unfold,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ C )
% 0.24/1.44 = ( ^ [C4: A > B > C,P3: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P3 ) @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_139_case__prod__comp,axiom,
% 0.24/1.44 ! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
% 0.24/1.44 ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
% 0.24/1.44 = ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_140_fst__convol,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
% 0.24/1.44 ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
% 0.24/1.44 = F ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_141_prod_Osplit__sel,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.24/1.44 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.24/1.44 = ( ( Prod
% 0.24/1.44 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.24/1.44 => ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_142_prod_Osplit__sel__asm,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.24/1.44 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.24/1.44 = ( ~ ( ( Prod
% 0.24/1.44 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.24/1.44 & ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_143_type__copy__map__cong0,axiom,
% 0.24/1.44 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
% 0.24/1.44 ( ( ( M @ ( G @ X ) )
% 0.24/1.44 = ( N @ ( H @ X ) ) )
% 0.24/1.44 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
% 0.24/1.44 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_144_convol__expand__snd_H,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H: A > C] :
% 0.24/1.44 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 0.24/1.44 = G )
% 0.24/1.44 => ( ( H
% 0.24/1.44 = ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 0.24/1.44 = ( ( bNF_convol @ A @ B @ C @ G @ H )
% 0.24/1.44 = F ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_145_convol__expand__snd,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
% 0.24/1.44 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 0.24/1.44 = G )
% 0.24/1.44 => ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 0.24/1.44 = F ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_146_fst__diag__fst,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
% 0.24/1.44 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 0.24/1.44 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.24/1.44 @ ( product_fst @ A @ B ) ) )
% 0.24/1.44 = ( product_fst @ A @ B ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_147_fst__snd__flip,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_fst @ A @ B )
% 0.24/1.44 = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
% 0.24/1.44 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 0.24/1.44 @ ^ [X2: A,Y: B] : ( product_Pair @ B @ A @ Y @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_148_snd__fst__flip,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType] :
% 0.24/1.44 ( ( product_snd @ B @ A )
% 0.24/1.44 = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
% 0.24/1.44 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.24/1.44 @ ^ [X2: B,Y: A] : ( product_Pair @ A @ B @ Y @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_149_exE__realizer,axiom,
% 0.24/1.44 ! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
% 0.24/1.44 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 0.24/1.44 => ( ! [X3: B,Y3: A] :
% 0.24/1.44 ( ( P2 @ Y3 @ X3 )
% 0.24/1.44 => ( Q @ ( F @ X3 @ Y3 ) ) )
% 0.24/1.44 => ( Q @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_150_sndOp__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType] :
% 0.24/1.44 ( ( bNF_sndOp @ C @ A @ B )
% 0.24/1.44 = ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_151_exI__realizer,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P2: A > B > $o,Y4: A,X: B] :
% 0.24/1.44 ( ( P2 @ Y4 @ X )
% 0.24/1.44 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_152_conjI__realizer,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 0.24/1.44 ( ( P2 @ P )
% 0.24/1.44 => ( ( Q @ Q2 )
% 0.24/1.44 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 0.24/1.44 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_153_exE__realizer_H,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
% 0.24/1.44 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 0.24/1.44 => ~ ! [X3: B,Y3: A] :
% 0.24/1.44 ~ ( P2 @ Y3 @ X3 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_154_fstOp__def,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( bNF_fstOp @ A @ B @ C )
% 0.24/1.44 = ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_155_apfst__apsnd,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.24/1.44 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 0.24/1.44 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_156_apsnd__apfst,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
% 0.24/1.44 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 0.24/1.44 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_157_apfst__conv,axiom,
% 0.24/1.44 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y4: B] :
% 0.24/1.44 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y4 ) )
% 0.24/1.44 = ( product_Pair @ A @ B @ ( F @ X ) @ Y4 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_158_snd__apfst,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 0.24/1.44 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 0.24/1.44 = ( product_snd @ C @ A @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_159_fst__apfst,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 0.24/1.44 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 0.24/1.44 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_160_apfst__eq__conv,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
% 0.24/1.44 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 0.24/1.44 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 0.24/1.44 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 0.24/1.44 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_161_snd__comp__apfst,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.24/1.44 = ( product_snd @ A @ B ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_162_fst__comp__apfst,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.24/1.44 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_163_apfst__compose,axiom,
% 0.24/1.44 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
% 0.24/1.44 ( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
% 0.24/1.44 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_164_apsnd__apfst__commute,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
% 0.24/1.44 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
% 0.24/1.44 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_165_fst__fstOp,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,P2: A > C > $o,Q: C > B > $o] :
% 0.24/1.44 ( ( product_fst @ A @ B )
% 0.24/1.44 = ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_166_apfst__convE,axiom,
% 0.24/1.44 ! [C: $tType,A: $tType,B: $tType,Q2: product_prod @ A @ B,F: C > A,P: product_prod @ C @ B] :
% 0.24/1.44 ( ( Q2
% 0.24/1.44 = ( product_apfst @ C @ A @ B @ F @ P ) )
% 0.24/1.44 => ~ ! [X3: C,Y3: B] :
% 0.24/1.44 ( ( P
% 0.24/1.44 = ( product_Pair @ C @ B @ X3 @ Y3 ) )
% 0.24/1.44 => ( Q2
% 0.24/1.44 != ( product_Pair @ A @ B @ ( F @ X3 ) @ Y3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_167_scomp__unfold,axiom,
% 0.24/1.44 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_scomp @ A @ B @ C @ D )
% 0.24/1.44 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X2: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X2 ) ) @ ( product_snd @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_168_Collect__case__prod__Grp__eqD,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A6: set @ A,F: A > B] :
% 0.24/1.44 ( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
% 0.24/1.44 => ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ Z2 )
% 0.24/1.44 = ( product_snd @ A @ B @ Z2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_169_scomp__apply,axiom,
% 0.24/1.44 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.24/1.44 ( ( product_scomp @ B @ C @ D @ A )
% 0.24/1.44 = ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X2: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_170_scomp__scomp,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F: A > ( product_prod @ E @ F3 ),G: E > F3 > ( product_prod @ C @ D ),H: C > D > B] :
% 0.24/1.44 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
% 0.24/1.44 = ( product_scomp @ A @ E @ F3 @ B @ F
% 0.24/1.44 @ ^ [X2: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G @ X2 ) @ H ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_171_Grp__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( bNF_Grp @ A @ B )
% 0.24/1.44 = ( ^ [A7: set @ A,F2: A > B,A4: A,B4: B] :
% 0.24/1.44 ( ( B4
% 0.24/1.44 = ( F2 @ A4 ) )
% 0.24/1.44 & ( member @ A @ A4 @ A7 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_172_GrpI,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,F: B > A,X: B,Y4: A,A6: set @ B] :
% 0.24/1.44 ( ( ( F @ X )
% 0.24/1.44 = Y4 )
% 0.24/1.44 => ( ( member @ B @ X @ A6 )
% 0.24/1.44 => ( bNF_Grp @ B @ A @ A6 @ F @ X @ Y4 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_173_GrpE,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,A6: set @ A,F: A > B,X: A,Y4: B] :
% 0.24/1.44 ( ( bNF_Grp @ A @ B @ A6 @ F @ X @ Y4 )
% 0.24/1.44 => ~ ( ( ( F @ X )
% 0.24/1.44 = Y4 )
% 0.24/1.44 => ~ ( member @ A @ X @ A6 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_174_scomp__Pair,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.24/1.44 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.24/1.44 = X ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_175_Pair__scomp,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
% 0.24/1.44 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
% 0.24/1.44 = ( F @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_176_scomp__def,axiom,
% 0.24/1.44 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_scomp @ A @ B @ C @ D )
% 0.24/1.44 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X2: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_177_Collect__case__prod__Grp__in,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Z2: product_prod @ A @ B,A6: set @ A,F: A > B] :
% 0.24/1.44 ( ( member @ ( product_prod @ A @ B ) @ Z2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
% 0.24/1.44 => ( member @ A @ ( product_fst @ A @ B @ Z2 ) @ A6 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_178_prod_Oswap__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_swap @ A @ B )
% 0.24/1.44 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_179_map__prod__o__convol__id,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X )
% 0.24/1.44 = ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_180_internal__case__prod__def,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.24/1.44 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_181_id__apply,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( id @ A )
% 0.24/1.44 = ( ^ [X2: A] : X2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_182_swap__swap,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.24/1.44 = P ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_183_comp__id,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,F: A > B] :
% 0.24/1.44 ( ( comp @ A @ B @ A @ F @ ( id @ A ) )
% 0.24/1.44 = F ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_184_id__comp,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,G: A > B] :
% 0.24/1.44 ( ( comp @ B @ B @ A @ ( id @ B ) @ G )
% 0.24/1.44 = G ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_185_fun_Omap__id,axiom,
% 0.24/1.44 ! [A: $tType,D: $tType,T2: D > A] :
% 0.24/1.44 ( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
% 0.24/1.44 = T2 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_186_case__prod__Pair,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
% 0.24/1.44 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_187_swap__simp,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,X: B,Y4: A] :
% 0.24/1.44 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y4 ) )
% 0.24/1.44 = ( product_Pair @ A @ B @ Y4 @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_188_apfst__id,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_apfst @ A @ A @ B @ ( id @ A ) )
% 0.24/1.44 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_189_apsnd__id,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
% 0.24/1.44 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_190_case__swap,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
% 0.24/1.44 ( ( product_case_prod @ B @ C @ A
% 0.24/1.44 @ ^ [Y: B,X2: C] : ( F @ X2 @ Y )
% 0.24/1.44 @ ( product_swap @ C @ B @ P ) )
% 0.24/1.44 = ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_191_swap__comp__swap,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
% 0.24/1.44 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_192_fst__swap,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.24/1.44 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.24/1.44 = ( product_snd @ B @ A @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_193_snd__swap,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.24/1.44 = ( product_fst @ A @ B @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_194_map__prod_Oidentity,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_map_prod @ A @ A @ B @ B
% 0.24/1.44 @ ^ [X2: A] : X2
% 0.24/1.44 @ ^ [X2: B] : X2 )
% 0.24/1.44 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_195_fun_Omap__id0,axiom,
% 0.24/1.44 ! [A: $tType,D: $tType] :
% 0.24/1.44 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 0.24/1.44 = ( id @ ( D > A ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_196_comp__eq__id__dest,axiom,
% 0.24/1.44 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
% 0.24/1.44 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.24/1.44 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 0.24/1.44 => ( ( A2 @ ( B2 @ V ) )
% 0.24/1.44 = ( C2 @ V ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_197_pointfree__idE,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
% 0.24/1.44 ( ( ( comp @ B @ A @ A @ F @ G )
% 0.24/1.44 = ( id @ A ) )
% 0.24/1.44 => ( ( F @ ( G @ X ) )
% 0.24/1.44 = X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_198_prod_Omap__id0,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
% 0.24/1.44 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_199_id__def,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( id @ A )
% 0.24/1.44 = ( ^ [X2: A] : X2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_200_eq__id__iff,axiom,
% 0.24/1.44 ! [A: $tType,F: A > A] :
% 0.24/1.44 ( ( ! [X2: A] :
% 0.24/1.44 ( ( F @ X2 )
% 0.24/1.44 = X2 ) )
% 0.24/1.44 = ( F
% 0.24/1.44 = ( id @ A ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_201_prod_Omap__id,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.24/1.44 ( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
% 0.24/1.44 = T2 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_202_apfst__def,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,A: $tType] :
% 0.24/1.44 ( ( product_apfst @ A @ C @ B )
% 0.24/1.44 = ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_203_apsnd__def,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,A: $tType] :
% 0.24/1.44 ( ( product_apsnd @ B @ C @ A )
% 0.24/1.44 = ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_204_convol__mem__GrpI,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,X: A,A6: set @ A,G: A > B] :
% 0.24/1.44 ( ( member @ A @ X @ A6 )
% 0.24/1.44 => ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ G ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_205_snd__diag__id,axiom,
% 0.24/1.44 ! [A: $tType,Z2: A] :
% 0.24/1.44 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
% 0.24/1.44 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.24/1.44 @ Z2 )
% 0.24/1.44 = ( id @ A @ Z2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_206_fst__diag__id,axiom,
% 0.24/1.44 ! [A: $tType,Z2: A] :
% 0.24/1.44 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
% 0.24/1.44 @ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
% 0.24/1.44 @ Z2 )
% 0.24/1.44 = ( id @ A @ Z2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_207_internal__case__prod__conv,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.24/1.44 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.24/1.44 = ( C2 @ A2 @ B2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_208_inf__Int__eq2,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.24/1.44 ( ( inf_inf @ ( A > B > $o )
% 0.24/1.44 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
% 0.24/1.44 @ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) )
% 0.24/1.44 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_209_pred__equals__eq2,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.24/1.44 ( ( ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
% 0.24/1.44 = ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) ) )
% 0.24/1.44 = ( R = S2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_210_prod_Osize__gen__o__map,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
% 0.24/1.44 ( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
% 0.24/1.44 = ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_211_inv__image__def,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType] :
% 0.24/1.44 ( ( inv_image @ B @ A )
% 0.24/1.44 = ( ^ [R2: set @ ( product_prod @ B @ B ),F2: A > B] :
% 0.24/1.44 ( collect @ ( product_prod @ A @ A )
% 0.24/1.44 @ ( product_case_prod @ A @ A @ $o
% 0.24/1.44 @ ^ [X2: A,Y: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ R2 ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_212_iso__tuple__update__accessor__eq__assist__idI,axiom,
% 0.24/1.44 ! [A: $tType,V3: A,F: A > A,V: A] :
% 0.24/1.44 ( ( V3
% 0.24/1.44 = ( F @ V ) )
% 0.24/1.44 => ( iso_tu2011167877assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) @ V @ F @ V3 @ V ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_213_in__inv__image,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,X: A,Y4: A,R3: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.24/1.44 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( inv_image @ B @ A @ R3 @ F ) )
% 0.24/1.44 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y4 ) ) @ R3 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_214_update__accessor__updator__eqE,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.24/1.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.24/1.44 => ( ( Upd @ F @ V )
% 0.24/1.44 = V3 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_215_update__accessor__accessor__eqE,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.24/1.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.24/1.44 => ( ( Ac2 @ V )
% 0.24/1.44 = X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_216_iso__tuple__update__accessor__eq__assist__triv,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.24/1.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.24/1.44 => ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_217_rp__inv__image__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( fun_rp_inv_image @ A @ B )
% 0.24/1.44 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.24/1.44 @ ^ [R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S3 @ F2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_218_fun_Opred__map,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
% 0.24/1.44 ( ( basic_pred_fun @ D @ B
% 0.24/1.44 @ ^ [Uu: D] : $true
% 0.24/1.44 @ Q
% 0.24/1.44 @ ( comp @ A @ B @ D @ F @ X ) )
% 0.24/1.44 = ( basic_pred_fun @ D @ A
% 0.24/1.44 @ ^ [Uu: D] : $true
% 0.24/1.44 @ ( comp @ B @ $o @ A @ Q @ F )
% 0.24/1.44 @ X ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_219_pred__funI,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,A6: A > $o,B6: B > $o,F: A > B] :
% 0.24/1.44 ( ! [X3: A] :
% 0.24/1.44 ( ( A6 @ X3 )
% 0.24/1.44 => ( B6 @ ( F @ X3 ) ) )
% 0.24/1.44 => ( basic_pred_fun @ A @ B @ A6 @ B6 @ F ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_220_fun_Opred__True,axiom,
% 0.24/1.44 ! [A: $tType,D: $tType] :
% 0.24/1.44 ( ( basic_pred_fun @ D @ A
% 0.24/1.44 @ ^ [Uu: D] : $true
% 0.24/1.44 @ ^ [Uu: A] : $true )
% 0.24/1.44 = ( ^ [Uu: D > A] : $true ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_221_fun_Omap__cong__pred,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
% 0.24/1.44 ( ( X = Ya )
% 0.24/1.44 => ( ( basic_pred_fun @ D @ A
% 0.24/1.44 @ ^ [Uu: D] : $true
% 0.24/1.44 @ ^ [Z: A] :
% 0.24/1.44 ( ( F @ Z )
% 0.24/1.44 = ( G @ Z ) )
% 0.24/1.44 @ Ya )
% 0.24/1.44 => ( ( comp @ A @ B @ D @ F @ X )
% 0.24/1.44 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_222_rp__inv__image__rp,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 0.24/1.44 ( ( fun_reduction_pair @ A @ P2 )
% 0.24/1.44 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_223_csquare__fstOp__sndOp,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_224_fstOp__in,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 0.24/1.44 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 0.24/1.44 => ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_225_sndOp__in,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 0.24/1.44 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 0.24/1.44 => ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_226_csquare__def,axiom,
% 0.24/1.44 ! [B: $tType,C: $tType,D: $tType,A: $tType] :
% 0.24/1.44 ( ( bNF_csquare @ A @ B @ C @ D )
% 0.24/1.44 = ( ^ [A7: set @ A,F12: B > C,F23: D > C,P1: A > B,P22: A > D] :
% 0.24/1.44 ! [X2: A] :
% 0.24/1.44 ( ( member @ A @ X2 @ A7 )
% 0.24/1.44 => ( ( F12 @ ( P1 @ X2 ) )
% 0.24/1.44 = ( F23 @ ( P22 @ X2 ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_227_pick__middlep,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q: B > C > $o,A2: A,C2: C] :
% 0.24/1.44 ( ( relcompp @ A @ B @ C @ P2 @ Q @ A2 @ C2 )
% 0.24/1.44 => ( ( P2 @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A2 @ C2 ) )
% 0.24/1.44 & ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A2 @ C2 ) @ C2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_228_update__accessor__cong__assist__idI,axiom,
% 0.24/1.44 ! [A: $tType] : ( iso_tu2017585022assist @ A @ A @ ( id @ ( A > A ) ) @ ( id @ A ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_229_The__case__prod,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType,P2: A > B > $o] :
% 0.24/1.44 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
% 0.24/1.44 = ( the @ ( product_prod @ A @ B )
% 0.24/1.44 @ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_230_The__split__eq,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,X: A,Y4: B] :
% 0.24/1.44 ( ( the @ ( product_prod @ A @ B )
% 0.24/1.44 @ ( product_case_prod @ A @ B @ $o
% 0.24/1.44 @ ^ [X5: A,Y6: B] :
% 0.24/1.44 ( ( X = X5 )
% 0.24/1.44 & ( Y4 = Y6 ) ) ) )
% 0.24/1.44 = ( product_Pair @ A @ B @ X @ Y4 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_231_iso__tuple__update__accessor__cong__from__eq,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,V: B,F: A > A,V3: B,X: A] :
% 0.24/1.44 ( ( iso_tu2011167877assist @ A @ B @ Upd @ Ac2 @ V @ F @ V3 @ X )
% 0.24/1.44 => ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_232_iso__tuple__update__accessor__eq__assist__def,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType] :
% 0.24/1.44 ( ( iso_tu2011167877assist @ B @ A )
% 0.24/1.44 = ( ^ [Upd2: ( B > B ) > A > A,Ac: A > B,V4: A,F2: B > B,V5: A,X2: B] :
% 0.24/1.44 ( ( ( Upd2 @ F2 @ V4 )
% 0.24/1.44 = V5 )
% 0.24/1.44 & ( ( Ac @ V4 )
% 0.24/1.44 = X2 )
% 0.24/1.44 & ( iso_tu2017585022assist @ B @ A @ Upd2 @ Ac ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_233_update__accessor__congruence__unfoldE,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,R3: B,R5: B,V3: A,F: A > A,F5: A > A] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.24/1.44 => ( ( R3 = R5 )
% 0.24/1.44 => ( ( ( Ac2 @ R5 )
% 0.24/1.44 = V3 )
% 0.24/1.44 => ( ! [V6: A] :
% 0.24/1.44 ( ( V6 = V3 )
% 0.24/1.44 => ( ( F @ V6 )
% 0.24/1.44 = ( F5 @ V6 ) ) )
% 0.24/1.44 => ( ( Upd @ F @ R3 )
% 0.24/1.44 = ( Upd @ F5 @ R5 ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_234_update__accessor__congruence__foldE,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,R3: B,R5: B,V3: A,F: A > A,F5: A > A] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.24/1.44 => ( ( R3 = R5 )
% 0.24/1.44 => ( ( ( Ac2 @ R5 )
% 0.24/1.44 = V3 )
% 0.24/1.44 => ( ! [V6: A] :
% 0.24/1.44 ( ( V3 = V6 )
% 0.24/1.44 => ( ( F @ V6 )
% 0.24/1.44 = ( F5 @ V6 ) ) )
% 0.24/1.44 => ( ( Upd @ F @ R3 )
% 0.24/1.44 = ( Upd @ F5 @ R5 ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_235_update__accessor__cong__assist__triv,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.24/1.44 => ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_236_update__accessor__noopE,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X: B] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.24/1.44 => ( ( ( F @ ( Ac2 @ X ) )
% 0.24/1.44 = ( Ac2 @ X ) )
% 0.24/1.44 => ( ( Upd @ F @ X )
% 0.24/1.44 = X ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_237_iso__tuple__update__accessor__cong__assist__def,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ B @ A )
% 0.24/1.44 = ( ^ [Upd2: ( B > B ) > A > A,Ac: A > B] :
% 0.24/1.44 ( ! [F2: B > B,V4: A] :
% 0.24/1.44 ( ( Upd2
% 0.24/1.44 @ ^ [X2: B] : ( F2 @ ( Ac @ V4 ) )
% 0.24/1.44 @ V4 )
% 0.24/1.44 = ( Upd2 @ F2 @ V4 ) )
% 0.24/1.44 & ! [V4: A] :
% 0.24/1.44 ( ( Upd2 @ ( id @ B ) @ V4 )
% 0.24/1.44 = V4 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_238_update__accessor__noop__compE,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X: B,G: A > A] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.24/1.44 => ( ( ( F @ ( Ac2 @ X ) )
% 0.24/1.44 = ( Ac2 @ X ) )
% 0.24/1.44 => ( ( Upd @ ( comp @ A @ A @ A @ G @ F ) @ X )
% 0.24/1.44 = ( Upd @ G @ X ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_239_iso__tuple__update__accessor__cong__assist__id,axiom,
% 0.24/1.44 ! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A] :
% 0.24/1.44 ( ( iso_tu2017585022assist @ A @ B @ Upd @ Ac2 )
% 0.24/1.44 => ( ( Upd @ ( id @ A ) )
% 0.24/1.44 = ( id @ B ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_240_old_Orec__prod__def,axiom,
% 0.24/1.44 ! [T: $tType,B: $tType,A: $tType] :
% 0.24/1.44 ( ( product_rec_prod @ A @ B @ T )
% 0.24/1.44 = ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_241_the__sym__eq__trivial,axiom,
% 0.24/1.44 ! [A: $tType,X: A] :
% 0.24/1.44 ( ( the @ A
% 0.24/1.44 @ ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 )
% 0.24/1.44 @ X ) )
% 0.24/1.44 = X ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_242_the__equality,axiom,
% 0.24/1.44 ! [A: $tType,P2: A > $o,A2: A] :
% 0.24/1.44 ( ( P2 @ A2 )
% 0.24/1.44 => ( ! [X3: A] :
% 0.24/1.44 ( ( P2 @ X3 )
% 0.24/1.44 => ( X3 = A2 ) )
% 0.24/1.44 => ( ( the @ A @ P2 )
% 0.24/1.44 = A2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_243_the__eq__trivial,axiom,
% 0.24/1.44 ! [A: $tType,A2: A] :
% 0.24/1.44 ( ( the @ A
% 0.24/1.44 @ ^ [X2: A] : ( X2 = A2 ) )
% 0.24/1.44 = A2 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_244_theI,axiom,
% 0.24/1.44 ! [A: $tType,P2: A > $o,A2: A] :
% 0.24/1.44 ( ( P2 @ A2 )
% 0.24/1.44 => ( ! [X3: A] :
% 0.24/1.44 ( ( P2 @ X3 )
% 0.24/1.44 => ( X3 = A2 ) )
% 0.24/1.44 => ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_245_theI_H,axiom,
% 0.24/1.44 ! [A: $tType,P2: A > $o] :
% 0.24/1.44 ( ? [X6: A] :
% 0.24/1.44 ( ( P2 @ X6 )
% 0.24/1.44 & ! [Y3: A] :
% 0.24/1.44 ( ( P2 @ Y3 )
% 0.24/1.44 => ( Y3 = X6 ) ) )
% 0.24/1.44 => ( P2 @ ( the @ A @ P2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_246_theI2,axiom,
% 0.24/1.44 ! [A: $tType,P2: A > $o,A2: A,Q: A > $o] :
% 0.24/1.44 ( ( P2 @ A2 )
% 0.24/1.44 => ( ! [X3: A] :
% 0.24/1.44 ( ( P2 @ X3 )
% 0.24/1.44 => ( X3 = A2 ) )
% 0.24/1.44 => ( ! [X3: A] :
% 0.24/1.44 ( ( P2 @ X3 )
% 0.24/1.44 => ( Q @ X3 ) )
% 0.24/1.44 => ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_247_If__def,axiom,
% 0.24/1.44 ! [A: $tType] :
% 0.24/1.44 ( ( if @ A )
% 0.24/1.44 = ( ^ [P4: $o,X2: A,Y: A] :
% 0.24/1.44 ( the @ A
% 0.24/1.44 @ ^ [Z: A] :
% 0.24/1.44 ( ( P4
% 0.24/1.44 => ( Z = X2 ) )
% 0.24/1.44 & ( ~ P4
% 0.24/1.44 => ( Z = Y ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_248_the1I2,axiom,
% 0.24/1.44 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.24/1.44 ( ? [X6: A] :
% 0.24/1.44 ( ( P2 @ X6 )
% 0.24/1.44 & ! [Y3: A] :
% 0.24/1.44 ( ( P2 @ Y3 )
% 0.24/1.44 => ( Y3 = X6 ) ) )
% 0.24/1.44 => ( ! [X3: A] :
% 0.24/1.44 ( ( P2 @ X3 )
% 0.24/1.44 => ( Q @ X3 ) )
% 0.24/1.44 => ( Q @ ( the @ A @ P2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_249_the1__equality,axiom,
% 0.24/1.44 ! [A: $tType,P2: A > $o,A2: A] :
% 0.24/1.44 ( ? [X6: A] :
% 0.24/1.44 ( ( P2 @ X6 )
% 0.24/1.44 & ! [Y3: A] :
% 0.24/1.44 ( ( P2 @ Y3 )
% 0.24/1.44 => ( Y3 = X6 ) ) )
% 0.24/1.44 => ( ( P2 @ A2 )
% 0.24/1.44 => ( ( the @ A @ P2 )
% 0.24/1.44 = A2 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_250_old_Orec__unit__def,axiom,
% 0.24/1.44 ! [T: $tType] :
% 0.24/1.44 ( ( product_rec_unit @ T )
% 0.24/1.44 = ( ^ [F12: T,X2: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_251_image2__eqI,axiom,
% 0.24/1.44 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A6: set @ B] :
% 0.24/1.44 ( ( B2
% 0.24/1.44 = ( F @ X ) )
% 0.24/1.44 => ( ( C2
% 0.24/1.44 = ( G @ X ) )
% 0.24/1.44 => ( ( member @ B @ X @ A6 )
% 0.24/1.44 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_252_same__fst__def,axiom,
% 0.24/1.44 ! [B: $tType,A: $tType] :
% 0.24/1.44 ( ( same_fst @ A @ B )
% 0.24/1.44 = ( ^ [P4: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.24/1.44 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.24/1.44 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.24/1.44 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.24/1.44 @ ^ [X5: A,Y6: B] :
% 0.24/1.44 ( product_case_prod @ A @ B @ $o
% 0.24/1.44 @ ^ [X2: A,Y: B] :
% 0.24/1.44 ( ( X5 = X2 )
% 0.24/1.44 & ( P4 @ X2 )
% 0.24/1.44 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_253_old_Orec__bool__def,axiom,
% 0.24/1.44 ! [T: $tType] :
% 0.24/1.44 ( ( product_rec_bool @ T )
% 0.24/1.44 = ( ^ [F12: T,F23: T,X2: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_254_old_Obool_Osimps_I6_J,axiom,
% 0.24/1.44 ! [T: $tType,F1: T,F22: T] :
% 0.24/1.44 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.24/1.44 = F22 ) ).
% 0.24/1.44
% 0.24/1.44 thf(fact_255_old_Obool_Osimps_I5_J,axiom,
% 0.24/1.44 ! [T: $tType,F1: T,F22: T] :
% 0.24/1.44 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.24/1.44 = F1 ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.24/1.44 ! [A8: $tType,A9: $tType] :
% 0.24/1.44 ( ( semilattice_inf @ A9 @ ( type2 @ A9 ) )
% 0.24/1.44 => ( semilattice_inf @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.24/1.44 ! [A8: $tType,A9: $tType] :
% 0.24/1.44 ( ( lattice @ A9 @ ( type2 @ A9 ) )
% 0.24/1.44 => ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_1,axiom,
% 0.24/1.44 semilattice_inf @ nat @ ( type2 @ nat ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
% 0.24/1.44 lattice @ nat @ ( type2 @ nat ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3,axiom,
% 0.24/1.44 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
% 0.24/1.44 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_5,axiom,
% 0.24/1.44 semilattice_inf @ $o @ ( type2 @ $o ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_HOL_Obool___Lattices_Olattice_6,axiom,
% 0.24/1.44 lattice @ $o @ ( type2 @ $o ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_7,axiom,
% 0.24/1.44 semilattice_inf @ product_unit @ ( type2 @ product_unit ) ).
% 0.24/1.44
% 0.24/1.44 thf(tcon_Product__Type_Ounit___Lattices_Olattice_8,axiom,
% 0.24/1.44 lattice @ product_unit @ ( type2 @ product_unit ) ).
% 0.24/1.44
% 0.24/1.44 thf(help_If_3_1_T,axiom,
% 0.24/1.44 ! [P2: $o] :
% 0.24/1.44 ( ( P2 = $true )
% 0.24/1.44 | ( P2 = $false ) ) ).
% 0.24/1.44
% 0.24/1.44 thf(help_If_2_1_T,axiom,
% 0.24/1.44 ! [A: $tType,X: A,Y4: A] :
% 0.24/1.44 ( ( if @ A @ $false @ X @ Y4 )
% 0.24/1.44 = Y4 ) ).
% 0.24/1.44
% 0.24/1.44 thf(help_If_1_1_T,axiom,
% 0.24/1.44 ! [A: $tType,X: A,Y4: A] :
% 0.24/1.44 ( ( if @ A @ $true @ X @ Y4 )
% 0.24/1.44 = X ) ).
% 0.24/1.44
% 0.24/1.44 thf(conj_0,conjecture,
% 0.24/1.44 ( ( coindu1441602521_llist @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ a
% 0.24/1.44 @ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
% 0.24/1.44 @ ^ [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.24/1.44 ( ( Xs
% 0.24/1.44 != ( coinductive_LNil @ a ) )
% 0.24/1.44 => ( ( Ys
% 0.24/1.44 != ( coinductive_LNil @ a ) )
% 0.24/1.44 => ( ( coinductive_lhd @ a @ Xs )
% 0.24/1.44 != ( coinductive_lhd @ a @ Ys ) ) ) ) )
% 0.24/1.44 @ ( comp @ ( coinductive_llist @ a ) @ a @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( coinductive_lhd @ a ) @ ( product_snd @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) )
% 0.24/1.44 @ ( product_map_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_ltl @ a ) @ ( coinductive_ltl @ a ) )
% 0.24/1.44 @ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_LNil @ a ) @ xs ) )
% 0.24/1.44 = ( coinductive_LNil @ a ) ) ).
% 0.24/1.44 ------------------------
% 0.24/1.47 % (24001)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_23795 for (2999ds/4Mi)
% 0.24/1.47 % (24000)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/183Mi)
% 0.24/1.47 % (24004)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/2Mi)
% 0.24/1.47 % (24005)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_23795 for (2999ds/2Mi)
% 0.24/1.47 % (24007)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/18Mi)
% 0.24/1.47 % (24004)Instruction limit reached!
% 0.24/1.47 % (24004)------------------------------
% 0.24/1.47 % (24004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.47 % (24004)Termination reason: Unknown
% 0.24/1.47 % (24004)Termination phase: shuffling
% 0.24/1.47
% 0.24/1.47 % (24004)Memory used [KB]: 1407
% 0.24/1.47 % (24004)Time elapsed: 0.002 s
% 0.24/1.47 % (24004)Instructions burned: 2 (million)
% 0.24/1.47 % (24004)------------------------------
% 0.24/1.47 % (24004)------------------------------
% 0.24/1.47 % (24005)Instruction limit reached!
% 0.24/1.47 % (24005)------------------------------
% 0.24/1.47 % (24005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.47 % (24005)Termination reason: Unknown
% 0.24/1.47 % (24005)Termination phase: shuffling
% 0.24/1.47
% 0.24/1.47 % (24005)Memory used [KB]: 1407
% 0.24/1.47 % (24005)Time elapsed: 0.002 s
% 0.24/1.47 % (24005)Instructions burned: 2 (million)
% 0.24/1.47 % (24005)------------------------------
% 0.24/1.47 % (24005)------------------------------
% 0.24/1.47 % (24001)Instruction limit reached!
% 0.24/1.47 % (24001)------------------------------
% 0.24/1.47 % (24001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.47 % (24001)Termination reason: Unknown
% 0.24/1.47 % (24001)Termination phase: shuffling
% 0.24/1.47
% 0.24/1.47 % (24001)Memory used [KB]: 1535
% 0.24/1.47 % (24001)Time elapsed: 0.003 s
% 0.24/1.47 % (24001)Instructions burned: 5 (million)
% 0.24/1.47 % (24001)------------------------------
% 0.24/1.47 % (24001)------------------------------
% 0.24/1.48 % (24003)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_23795 for (2999ds/27Mi)
% 0.24/1.48 % (24007)Instruction limit reached!
% 0.24/1.48 % (24007)------------------------------
% 0.24/1.48 % (24007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.48 % (24007)Termination reason: Unknown
% 0.24/1.48 % (24007)Termination phase: shuffling
% 0.24/1.48
% 0.24/1.48 % (24007)Memory used [KB]: 1791
% 0.24/1.48 % (24007)Time elapsed: 0.008 s
% 0.24/1.48 % (24007)Instructions burned: 20 (million)
% 0.24/1.48 % (24007)------------------------------
% 0.24/1.48 % (24007)------------------------------
% 0.24/1.48 % (24006)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_23795 for (2999ds/275Mi)
% 0.24/1.48 % (24014)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/3Mi)
% 0.24/1.48 % (24015)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_23795 for (2999ds/37Mi)
% 0.24/1.48 % (24016)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_23795 for (2999ds/15Mi)
% 0.24/1.48 % (24014)Instruction limit reached!
% 0.24/1.48 % (24014)------------------------------
% 0.24/1.48 % (24014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.48 % (24014)Termination reason: Unknown
% 0.24/1.48 % (24014)Termination phase: shuffling
% 0.24/1.48
% 0.24/1.48 % (24014)Memory used [KB]: 1407
% 0.24/1.48 % (24014)Time elapsed: 0.002 s
% 0.24/1.48 % (24014)Instructions burned: 3 (million)
% 0.24/1.48 % (24014)------------------------------
% 0.24/1.48 % (24014)------------------------------
% 0.24/1.49 % (24016)Instruction limit reached!
% 0.24/1.49 % (24016)------------------------------
% 0.24/1.49 % (24016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.49 % (24016)Termination reason: Unknown
% 0.24/1.49 % (24016)Termination phase: shuffling
% 0.24/1.49
% 0.24/1.49 % (24016)Memory used [KB]: 1791
% 0.24/1.49 % (24016)Time elapsed: 0.006 s
% 0.24/1.49 % (24016)Instructions burned: 15 (million)
% 0.24/1.49 % (24016)------------------------------
% 0.24/1.49 % (24016)------------------------------
% 0.24/1.49 % (24021)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_23795 for (2999ds/3Mi)
% 0.24/1.49 % (24021)Instruction limit reached!
% 0.24/1.49 % (24021)------------------------------
% 0.24/1.49 % (24021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.49 % (24021)Termination reason: Unknown
% 0.24/1.49 % (24021)Termination phase: shuffling
% 0.24/1.49
% 0.24/1.49 % (24021)Memory used [KB]: 1407
% 0.24/1.49 % (24021)Time elapsed: 0.002 s
% 0.24/1.49 % (24021)Instructions burned: 3 (million)
% 0.24/1.49 % (24021)------------------------------
% 0.24/1.49 % (24021)------------------------------
% 0.24/1.49 % (24024)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_23795 for (2999ds/1041Mi)
% 0.24/1.49 % (24015)Instruction limit reached!
% 0.24/1.49 % (24015)------------------------------
% 0.24/1.49 % (24015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.49 % (24015)Termination reason: Unknown
% 0.24/1.49 % (24015)Termination phase: shuffling
% 0.24/1.49
% 0.24/1.49 % (24015)Memory used [KB]: 2302
% 0.24/1.49 % (24015)Time elapsed: 0.012 s
% 0.24/1.49 % (24015)Instructions burned: 38 (million)
% 0.24/1.49 % (24015)------------------------------
% 0.24/1.49 % (24015)------------------------------
% 0.24/1.49 % (24003)Instruction limit reached!
% 0.24/1.49 % (24003)------------------------------
% 0.24/1.49 % (24003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.24/1.49 % (24003)Termination reason: Unknown
% 0.24/1.49 % (24003)Termination phase: Property scanning
% 0.24/1.49
% 0.24/1.49 % (24003)Memory used [KB]: 1918
% 0.24/1.49 % (24003)Time elapsed: 0.017 s
% 0.24/1.49 % (24003)Instructions burned: 27 (million)
% 0.24/1.49 % (24003)------------------------------
% 0.24/1.49 % (24003)------------------------------
% 2.56/1.50 % (24025)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/7Mi)
% 2.56/1.50 % (24026)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/16Mi)
% 2.56/1.50 % (24025)Instruction limit reached!
% 2.56/1.50 % (24025)------------------------------
% 2.56/1.50 % (24025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.50 % (24025)Termination reason: Unknown
% 2.56/1.50 % (24025)Termination phase: shuffling
% 2.56/1.50
% 2.56/1.50 % (24025)Memory used [KB]: 1663
% 2.56/1.50 % (24025)Time elapsed: 0.004 s
% 2.56/1.50 % (24025)Instructions burned: 8 (million)
% 2.56/1.50 % (24025)------------------------------
% 2.56/1.50 % (24025)------------------------------
% 2.56/1.50 % (24027)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_23795 for (2999ds/3Mi)
% 2.56/1.50 % (24027)Instruction limit reached!
% 2.56/1.50 % (24027)------------------------------
% 2.56/1.50 % (24027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.50 % (24027)Termination reason: Unknown
% 2.56/1.50 % (24027)Termination phase: shuffling
% 2.56/1.50
% 2.56/1.50 % (24027)Memory used [KB]: 1407
% 2.56/1.50 % (24027)Time elapsed: 0.002 s
% 2.56/1.50 % (24027)Instructions burned: 3 (million)
% 2.56/1.50 % (24027)------------------------------
% 2.56/1.50 % (24027)------------------------------
% 2.56/1.50 % (24026)Instruction limit reached!
% 2.56/1.50 % (24026)------------------------------
% 2.56/1.50 % (24026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.50 % (24026)Termination reason: Unknown
% 2.56/1.50 % (24026)Termination phase: shuffling
% 2.56/1.50
% 2.56/1.50 % (24026)Memory used [KB]: 1791
% 2.56/1.50 % (24026)Time elapsed: 0.007 s
% 2.56/1.50 % (24026)Instructions burned: 18 (million)
% 2.56/1.50 % (24026)------------------------------
% 2.56/1.50 % (24026)------------------------------
% 2.56/1.50 % (24028)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_23795 for (2999ds/3Mi)
% 2.56/1.50 % (24028)Instruction limit reached!
% 2.56/1.50 % (24028)------------------------------
% 2.56/1.50 % (24028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.50 % (24028)Termination reason: Unknown
% 2.56/1.50 % (24028)Termination phase: shuffling
% 2.56/1.50
% 2.56/1.50 % (24028)Memory used [KB]: 1535
% 2.56/1.50 % (24028)Time elapsed: 0.002 s
% 2.56/1.50 % (24028)Instructions burned: 4 (million)
% 2.56/1.50 % (24028)------------------------------
% 2.56/1.50 % (24028)------------------------------
% 2.56/1.51 % (24033)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/7Mi)
% 2.56/1.51 % (24033)Instruction limit reached!
% 2.56/1.51 % (24033)------------------------------
% 2.56/1.51 % (24033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (24033)Termination reason: Unknown
% 2.56/1.51 % (24033)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (24033)Memory used [KB]: 1663
% 2.56/1.51 % (24033)Time elapsed: 0.004 s
% 2.56/1.51 % (24033)Instructions burned: 9 (million)
% 2.56/1.51 % (24033)------------------------------
% 2.56/1.51 % (24033)------------------------------
% 2.56/1.51 % (24038)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/3Mi)
% 2.56/1.51 % (24038)Instruction limit reached!
% 2.56/1.51 % (24038)------------------------------
% 2.56/1.51 % (24038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (24038)Termination reason: Unknown
% 2.56/1.51 % (24038)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (24038)Memory used [KB]: 1535
% 2.56/1.51 % (24038)Time elapsed: 0.002 s
% 2.56/1.51 % (24038)Instructions burned: 3 (million)
% 2.56/1.51 % (24038)------------------------------
% 2.56/1.51 % (24038)------------------------------
% 2.56/1.51 % (24039)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/4Mi)
% 2.56/1.51 % (24042)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/18Mi)
% 2.56/1.51 % (24039)Instruction limit reached!
% 2.56/1.51 % (24039)------------------------------
% 2.56/1.51 % (24039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (24039)Termination reason: Unknown
% 2.56/1.51 % (24039)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (24039)Memory used [KB]: 1535
% 2.56/1.51 % (24039)Time elapsed: 0.003 s
% 2.56/1.51 % (24039)Instructions burned: 6 (million)
% 2.56/1.51 % (24039)------------------------------
% 2.56/1.51 % (24039)------------------------------
% 2.56/1.52 % (24042)Instruction limit reached!
% 2.56/1.52 % (24042)------------------------------
% 2.56/1.52 % (24042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (24042)Termination reason: Unknown
% 2.56/1.52 % (24042)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (24042)Memory used [KB]: 1791
% 2.56/1.52 % (24042)Time elapsed: 0.007 s
% 2.56/1.52 % (24042)Instructions burned: 18 (million)
% 2.56/1.52 % (24042)------------------------------
% 2.56/1.52 % (24042)------------------------------
% 2.56/1.52 % (24048)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/710Mi)
% 2.56/1.52 % (24049)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/6Mi)
% 2.56/1.52 % (24049)Instruction limit reached!
% 2.56/1.52 % (24049)------------------------------
% 2.56/1.52 % (24049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (24049)Termination reason: Unknown
% 2.56/1.52 % (24049)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (24049)Memory used [KB]: 1535
% 2.56/1.52 % (24049)Time elapsed: 0.003 s
% 2.56/1.52 % (24049)Instructions burned: 6 (million)
% 2.56/1.52 % (24049)------------------------------
% 2.56/1.52 % (24049)------------------------------
% 2.56/1.52 % (24050)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/902Mi)
% 2.56/1.53 % (24000)Instruction limit reached!
% 2.56/1.53 % (24000)------------------------------
% 2.56/1.53 % (24000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (24000)Termination reason: Unknown
% 2.56/1.53 % (24000)Termination phase: Saturation
% 2.56/1.53
% 2.56/1.53 % (24000)Memory used [KB]: 7547
% 2.56/1.53 % (24000)Time elapsed: 0.057 s
% 2.56/1.53 % (24000)Instructions burned: 186 (million)
% 2.56/1.53 % (24000)------------------------------
% 2.56/1.53 % (24000)------------------------------
% 2.56/1.53 % (24051)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_23795 for (2999ds/21Mi)
% 2.56/1.53 % (24006)First to succeed.
% 2.56/1.53 % (24006)Refutation found. Thanks to Tanya!
% 2.56/1.53 % SZS status Theorem for DTF2THF_23795
% 2.56/1.53 % SZS output start Proof for DTF2THF_23795
% 2.56/1.53 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.56/1.53 thf(type_def_7, type, set: ($tType) > $tType).
% 2.56/1.53 thf(type_def_8, type, product_prod: ($tType * $tType) > $tType).
% 2.56/1.53 thf(type_def_9, type, nat: $tType).
% 2.56/1.53 thf(type_def_10, type, coinductive_llist: ($tType) > $tType).
% 2.56/1.53 thf(type_def_11, type, product_unit: $tType).
% 2.56/1.53 thf(type_def_12, type, a: $tType).
% 2.56/1.53 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.56/1.53 thf(func_def_1, type, product_unit: $tType).
% 2.56/1.53 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 2.56/1.53 thf(func_def_3, type, set: ($tType) > $tType).
% 2.56/1.53 thf(func_def_4, type, nat: $tType).
% 2.56/1.53 thf(func_def_5, type, itself: ($tType) > $tType).
% 2.56/1.53 thf(func_def_6, type, a: $tType).
% 2.56/1.53 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.56/1.53 thf(func_def_8, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.56/1.53 thf(func_def_9, type, semilattice_inf: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.56/1.53 thf(func_def_10, type, bNF_Grp: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X0 > X1 > $o)).
% 2.56/1.53 thf(func_def_11, type, bNF_convol: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X0 > product_prod @ X1 @ X2)).
% 2.56/1.53 thf(func_def_12, type, bNF_csquare: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ X0 > (X1 > X2) > (X3 > X2) > (X0 > X1) > (X0 > X3) > $o)).
% 2.56/1.53 thf(func_def_13, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 2.56/1.53 thf(func_def_14, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 2.56/1.53 thf(func_def_15, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.56/1.53 thf(func_def_16, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 2.56/1.53 thf(func_def_17, type, basic_BNF_size_prod: !>[X0: $tType, X1: $tType]:((X0 > nat) > (X1 > nat) > product_prod @ X0 @ X1 > nat)).
% 2.56/1.53 thf(func_def_18, type, basic_pred_fun: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > $o) > (X0 > X1) > $o)).
% 2.56/1.53 thf(func_def_19, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 2.56/1.53 thf(func_def_20, type, coinductive_lhd: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.56/1.53 thf(func_def_21, type, coinductive_ltl: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.56/1.53 thf(func_def_22, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 2.56/1.53 thf(func_def_23, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 2.56/1.53 thf(func_def_24, type, id: !>[X0: $tType]:(X0 > X0)).
% 2.56/1.53 thf(func_def_25, type, fun_reduction_pair: !>[X0: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > $o)).
% 2.56/1.53 thf(func_def_26, type, fun_rp_inv_image: !>[X0: $tType, X1: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > (X1 > X0) > product_prod @ set @ product_prod @ X1 @ X1 @ set @ product_prod @ X1 @ X1)).
% 2.56/1.53 thf(func_def_27, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_28, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.56/1.53 thf(func_def_29, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.56/1.53 thf(func_def_30, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.56/1.53 thf(func_def_31, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.56/1.53 thf(func_def_32, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 2.56/1.53 thf(func_def_33, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.56/1.53 thf(func_def_34, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.56/1.53 thf(func_def_35, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 2.56/1.53 thf(func_def_36, type, product_rec_set_bool: !>[X0: $tType]:(X0 > X0 > $o > X0 > $o)).
% 2.56/1.53 thf(func_def_37, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.56/1.53 thf(func_def_38, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 2.56/1.53 thf(func_def_39, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 2.56/1.53 thf(func_def_40, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 2.56/1.53 thf(func_def_41, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.56/1.53 thf(func_def_42, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.56/1.53 thf(func_def_43, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.56/1.53 thf(func_def_44, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.56/1.53 thf(func_def_45, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 2.56/1.53 thf(func_def_46, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.56/1.53 thf(func_def_47, type, iso_tu2017585022assist: !>[X0: $tType, X1: $tType]:(((X0 > X0) > X1 > X1) > (X1 > X0) > $o)).
% 2.56/1.53 thf(func_def_48, type, iso_tu2011167877assist: !>[X0: $tType, X1: $tType]:(((X0 > X0) > X1 > X1) > (X1 > X0) > X1 > (X0 > X0) > X1 > X0 > $o)).
% 2.56/1.53 thf(func_def_49, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.56/1.53 thf(func_def_50, type, relcompp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > $o)).
% 2.56/1.53 thf(func_def_51, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.56/1.53 thf(func_def_52, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 2.56/1.53 thf(func_def_53, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.56/1.53 thf(func_def_54, type, xs: coinductive_llist @ a).
% 2.56/1.53 thf(func_def_70, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.56/1.53 thf(func_def_71, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_72, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_73, type, sK3: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0 > X1)).
% 2.56/1.53 thf(func_def_74, type, sK4: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > (X0 > X1 > $o) > X0)).
% 2.56/1.53 thf(func_def_75, type, sK5: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > (X0 > X1 > $o) > X1)).
% 2.56/1.53 thf(func_def_76, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.56/1.53 thf(func_def_77, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.56/1.53 thf(func_def_78, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.56/1.53 thf(func_def_79, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.56/1.53 thf(func_def_80, type, sK10: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_81, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.56/1.53 thf(func_def_82, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.56/1.53 thf(func_def_83, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.56/1.53 thf(func_def_84, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.56/1.53 thf(func_def_85, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.56/1.53 thf(func_def_86, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.56/1.53 thf(func_def_87, type, sK17: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > X0 > X1)).
% 2.56/1.53 thf(func_def_88, type, sK18: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.56/1.53 thf(func_def_89, type, sK19: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.56/1.53 thf(func_def_90, type, sK20: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_91, type, sK21: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.56/1.53 thf(func_def_92, type, sK22: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.56/1.53 thf(func_def_93, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X2 > X1 > X0) > product_prod @ X2 @ X1 > X2)).
% 2.56/1.53 thf(func_def_94, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X2 > X1 > X0) > product_prod @ X2 @ X1 > X1)).
% 2.56/1.53 thf(func_def_95, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X0 > $o) > (X1 > X2 > $o) > X2)).
% 2.56/1.53 thf(func_def_96, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1 > X0) > (X0 > $o) > (X1 > X2 > $o) > X1)).
% 2.56/1.53 thf(func_def_97, type, sK27: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_98, type, sK28: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_99, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > (X0 > X1 > X2 > $o) > product_prod @ X0 @ X1 > X0)).
% 2.56/1.53 thf(func_def_100, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > (X0 > X1 > X2 > $o) > product_prod @ X0 @ X1 > X1)).
% 2.56/1.53 thf(func_def_101, type, sK31: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_102, type, sK32: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_103, type, sK33: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1)).
% 2.56/1.53 thf(func_def_104, type, sK34: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0)).
% 2.56/1.53 thf(func_def_105, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.56/1.53 thf(func_def_106, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.56/1.53 thf(func_def_107, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.56/1.53 thf(func_def_108, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_109, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_110, type, sK40: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_111, type, sK41: !>[X0: $tType]:((X0 > X0) > X0)).
% 2.56/1.53 thf(func_def_112, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ X1 > X0) > (X2 > X1 > X0) > X2)).
% 2.56/1.53 thf(func_def_113, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ X1 > X0) > (X2 > X1 > X0) > X1)).
% 2.56/1.53 thf(func_def_114, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > product_prod @ X1 @ X0 > (X1 > X0 > X2 > $o) > X1)).
% 2.56/1.53 thf(func_def_115, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > product_prod @ X1 @ X0 > (X1 > X0 > X2 > $o) > X0)).
% 2.56/1.53 thf(func_def_116, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_117, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.56/1.53 thf(func_def_118, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.56/1.53 thf(func_def_119, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_120, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_121, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.56/1.53 thf(func_def_122, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.56/1.53 thf(func_def_123, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.56/1.53 thf(func_def_124, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.56/1.53 thf(func_def_125, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.56/1.53 thf(func_def_126, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.56/1.53 thf(func_def_127, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.56/1.53 thf(func_def_128, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.56/1.53 thf(func_def_129, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.56/1.53 thf(func_def_130, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_131, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_132, type, sK62: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > product_prod @ X1 @ X0 > X1)).
% 2.56/1.53 thf(func_def_133, type, sK63: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > product_prod @ X1 @ X0 > X0)).
% 2.56/1.53 thf(func_def_134, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ X2 > (X0 > X2 > X1) > (X1 > $o) > X0)).
% 2.56/1.53 thf(func_def_135, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ X2 > (X0 > X2 > X1) > (X1 > $o) > X2)).
% 2.56/1.53 thf(func_def_136, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.56/1.53 thf(func_def_137, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.56/1.53 thf(func_def_138, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.56/1.53 thf(func_def_139, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.56/1.53 thf(func_def_140, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.56/1.53 thf(func_def_141, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.56/1.53 thf(func_def_142, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.56/1.53 thf(func_def_143, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.56/1.53 thf(func_def_144, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.56/1.53 thf(func_def_145, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.56/1.53 thf(func_def_146, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.56/1.53 thf(func_def_147, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.56/1.53 thf(func_def_148, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.56/1.53 thf(func_def_149, type, sK79: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_150, type, sK80: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_151, type, sK81: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_152, type, sK82: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.56/1.53 thf(func_def_153, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.56/1.53 thf(func_def_154, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.56/1.53 thf(func_def_155, type, sK85: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_156, type, sK86: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 2.56/1.53 thf(func_def_157, type, sK87: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.56/1.53 thf(func_def_159, type, ph89: !>[X0: $tType]:(X0)).
% 2.56/1.53 thf(f1143,plain,(
% 2.56/1.53 $false),
% 2.56/1.53 inference(subsumption_resolution,[status(thm)],[f1133,f1072])).
% 2.56/1.53 thf(f1072,plain,(
% 2.56/1.53 ( ! [X1 : $tType,X0 : $tType,X2 : X1,X3 : X0] : (((product_fst @ X1 @ X0 @ (product_Pair @ X1 @ X0 @ X2 @ X3)) = X2)) )),
% 2.56/1.53 inference(equality_resolution,[status(thm)],[f926])).
% 2.56/1.53 thf(f926,plain,(
% 2.56/1.53 ( ! [X1 : $tType,X0 : $tType,X2 : X1,X3 : X0,X4 : X1] : ((X2 = X4) | ((product_fst @ X1 @ X0 @ (product_Pair @ X1 @ X0 @ X2 @ X3)) != X4)) )),
% 2.56/1.53 inference(cnf_transformation,[status(thm)],[f715])).
% 2.56/1.53 thf(f715,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : X1,X3 : X0,X4 : X1] : ((X2 = X4) | ((product_fst @ X1 @ X0 @ (product_Pair @ X1 @ X0 @ X2 @ X3)) != X4))),
% 2.56/1.53 inference(ennf_transformation,[status(thm)],[f678])).
% 2.56/1.53 thf(f678,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : X1,X3 : X0,X4 : X1] : (((product_fst @ X1 @ X0 @ (product_Pair @ X1 @ X0 @ X2 @ X3)) = X4) => (X2 = X4))),
% 2.56/1.53 inference(rectify,[status(thm)],[f125])).
% 2.56/1.53 thf(f125,axiom,(
% 2.56/1.53 ! [X1 : $tType,X0 : $tType,X17 : X0,X42 : X1,X11 : X0] : (((product_fst @ X0 @ X1 @ (product_Pair @ X0 @ X1 @ X17 @ X42)) = X11) => (X11 = X17))),
% 2.56/1.53 file('/export/starexec/sandbox/tmp/tmp.pFALYqNRSO/DTF2THF_23795.p',fact_124_fst__eqD)).
% 2.56/1.53 thf(f1133,plain,(
% 2.56/1.53 (coinductive_LNil @ a != (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(equality_proxy_clausification,[status(thm)],[f1132])).
% 2.56/1.53 thf(f1132,plain,(
% 2.56/1.53 ($false = (coinductive_LNil @ a = (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs))))),
% 2.56/1.53 inference(not_proxy_clausification,[status(thm)],[f1131])).
% 2.56/1.53 thf(f1131,plain,(
% 2.56/1.53 ($true = (~ (coinductive_LNil @ a = (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))))),
% 2.56/1.53 inference(binary_proxy_clausification,[status(thm)],[f1129])).
% 2.56/1.53 thf(f1129,plain,(
% 2.56/1.53 ($true != ((~ (coinductive_LNil @ a = (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))) => ((~ (coinductive_LNil @ a = (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))) => (~ ((coinductive_lhd @ a @ (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs))) = (coinductive_lhd @ a @ (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs))))))))),
% 2.56/1.53 inference(beta_eta_normalization,[status(thm)],[f1128])).
% 2.56/1.53 thf(f1128,plain,(
% 2.56/1.53 ($true != ((^[Y0 : product_prod @ coinductive_llist @ a @ coinductive_llist @ a]: ((~ (coinductive_LNil @ a = (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))) => ((~ (coinductive_LNil @ a = (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))) => (~ ((coinductive_lhd @ a @ (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y0)) = (coinductive_lhd @ a @ (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))))))) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(trivial_inequality_removal,[status(thm)],[f1127])).
% 2.56/1.53 thf(f1127,plain,(
% 2.56/1.53 ($true != ((^[Y0 : product_prod @ coinductive_llist @ a @ coinductive_llist @ a]: ((~ (coinductive_LNil @ a = (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))) => ((~ (coinductive_LNil @ a = (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))) => (~ ((coinductive_lhd @ a @ (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y0)) = (coinductive_lhd @ a @ (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))))))) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs))) | (coinductive_LNil @ a != coinductive_LNil @ a)),
% 2.56/1.53 inference(superposition,[status(thm)],[f1126,f877])).
% 2.56/1.53 thf(f877,plain,(
% 2.56/1.53 ( ! [X1 : $tType,X0 : $tType,X2 : X0 > $o,X3 : X0,X4 : X0 > X1,X5 : X0 > X0] : ((coinductive_LNil @ X1 = (coindu1441602521_llist @ X0 @ X1 @ X2 @ X4 @ X5 @ X3)) | ($true != (X2 @ X3))) )),
% 2.56/1.53 inference(cnf_transformation,[status(thm)],[f729])).
% 2.56/1.53 thf(f729,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : X0 > $o,X3 : X0,X4 : X0 > X1,X5 : X0 > X0] : (($true != (X2 @ X3)) | (coinductive_LNil @ X1 = (coindu1441602521_llist @ X0 @ X1 @ X2 @ X4 @ X5 @ X3)))),
% 2.56/1.53 inference(ennf_transformation,[status(thm)],[f312])).
% 2.56/1.53 thf(f312,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : X0 > $o,X3 : X0,X4 : X0 > X1,X5 : X0 > X0] : (($true = (X2 @ X3)) => (coinductive_LNil @ X1 = (coindu1441602521_llist @ X0 @ X1 @ X2 @ X4 @ X5 @ X3)))),
% 2.56/1.53 inference(fool_elimination,[status(thm)],[f311])).
% 2.56/1.53 thf(f311,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : X0 > $o,X3 : X0,X4 : X0 > X1,X5 : X0 > X0] : ((X2 @ X3) => (coinductive_LNil @ X1 = (coindu1441602521_llist @ X0 @ X1 @ X2 @ X4 @ X5 @ X3)))),
% 2.56/1.53 inference(rectify,[status(thm)],[f56])).
% 2.56/1.53 thf(f56,axiom,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X13 : X0 > $o,X11 : X0,X66 : X0 > X1,X67 : X0 > X0] : ((X13 @ X11) => ((coindu1441602521_llist @ X0 @ X1 @ X13 @ X66 @ X67 @ X11) = coinductive_LNil @ X1))),
% 2.56/1.53 file('/export/starexec/sandbox/tmp/tmp.pFALYqNRSO/DTF2THF_23795.p',fact_55_unfold__llist_Octr_I1_J)).
% 2.56/1.53 thf(f1126,plain,(
% 2.56/1.53 (coinductive_LNil @ a != (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (^[Y0 : product_prod @ coinductive_llist @ a @ coinductive_llist @ a]: ((~ (coinductive_LNil @ a = (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))) => ((~ (coinductive_LNil @ a = (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))) => (~ ((coinductive_lhd @ a @ (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y0)) = (coinductive_lhd @ a @ (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y0))))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(beta_eta_normalization,[status(thm)],[f1044])).
% 2.56/1.53 thf(f1044,plain,(
% 2.56/1.53 (coinductive_LNil @ a != (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ ((^[Y0 : coinductive_llist @ a > coinductive_llist @ a > $o]: ((^[Y1 : product_prod @ coinductive_llist @ a @ coinductive_llist @ a]: (Y0 @ (product_fst @ coinductive_llist @ a @ coinductive_llist @ a @ Y1) @ (product_snd @ coinductive_llist @ a @ coinductive_llist @ a @ Y1))))) @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : coinductive_llist @ a]: ((~ (coinductive_LNil @ a = Y0)) => ((~ (coinductive_LNil @ a = Y1)) => (~ ((coinductive_lhd @ a @ Y1) = (coinductive_lhd @ a @ Y0))))))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(definition_unfolding,[status(thm)],[f956,f860])).
% 2.56/1.53 thf(f860,plain,(
% 2.56/1.53 ( ! [X1 : $tType,X0 : $tType,X2 : $tType] : ((product_case_prod @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1 > X0]: ((^[Y1 : product_prod @ X2 @ X1]: (Y0 @ (product_fst @ X2 @ X1 @ Y1) @ (product_snd @ X2 @ X1 @ Y1))))))) )),
% 2.56/1.53 inference(cnf_transformation,[status(thm)],[f546])).
% 2.56/1.53 thf(f546,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (product_case_prod @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1 > X0]: ((^[Y1 : product_prod @ X2 @ X1]: (Y0 @ (product_fst @ X2 @ X1 @ Y1) @ (product_snd @ X2 @ X1 @ Y1))))))),
% 2.56/1.53 inference(fool_elimination,[status(thm)],[f545])).
% 2.56/1.53 thf(f545,plain,(
% 2.56/1.53 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (product_case_prod @ X2 @ X1 @ X0 = (^[X3 : X2 > X1 > X0, X4 : product_prod @ X2 @ X1] : (X3 @ (product_fst @ X2 @ X1 @ X4) @ (product_snd @ X2 @ X1 @ X4))))),
% 2.56/1.53 inference(rectify,[status(thm)],[f135])).
% 2.56/1.53 thf(f135,axiom,(
% 2.56/1.53 ! [X2 : $tType,X1 : $tType,X0 : $tType] : (product_case_prod @ X0 @ X1 @ X2 = (^[X29 : X0 > X1 > X2, X90 : product_prod @ X0 @ X1] : (X29 @ (product_fst @ X0 @ X1 @ X90) @ (product_snd @ X0 @ X1 @ X90))))),
% 2.56/1.53 file('/export/starexec/sandbox/tmp/tmp.pFALYqNRSO/DTF2THF_23795.p',fact_134_prod_Ocase__eq__if)).
% 2.56/1.53 thf(f956,plain,(
% 2.56/1.53 (coinductive_LNil @ a != (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (product_case_prod @ coinductive_llist @ a @ coinductive_llist @ a @ $o @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : coinductive_llist @ a]: ((~ (coinductive_LNil @ a = Y0)) => ((~ (coinductive_LNil @ a = Y1)) => (~ ((coinductive_lhd @ a @ Y1) = (coinductive_lhd @ a @ Y0))))))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(cnf_transformation,[status(thm)],[f611])).
% 2.56/1.53 thf(f611,plain,(
% 2.56/1.53 (coinductive_LNil @ a != (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (product_case_prod @ coinductive_llist @ a @ coinductive_llist @ a @ $o @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : coinductive_llist @ a]: ((~ (coinductive_LNil @ a = Y0)) => ((~ (coinductive_LNil @ a = Y1)) => (~ ((coinductive_lhd @ a @ Y1) = (coinductive_lhd @ a @ Y0))))))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(flattening,[status(thm)],[f584])).
% 2.56/1.53 thf(f584,plain,(
% 2.56/1.53 ~(coinductive_LNil @ a = (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (product_case_prod @ coinductive_llist @ a @ coinductive_llist @ a @ $o @ (^[Y0 : coinductive_llist @ a]: ((^[Y1 : coinductive_llist @ a]: ((~ (coinductive_LNil @ a = Y0)) => ((~ (coinductive_LNil @ a = Y1)) => (~ ((coinductive_lhd @ a @ Y1) = (coinductive_lhd @ a @ Y0))))))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(fool_elimination,[status(thm)],[f583])).
% 2.56/1.53 thf(f583,plain,(
% 2.56/1.53 ~(coinductive_LNil @ a = (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (product_case_prod @ coinductive_llist @ a @ coinductive_llist @ a @ $o @ (^[X0 : coinductive_llist @ a, X1 : coinductive_llist @ a] : ((coinductive_LNil @ a != X0) => ((coinductive_LNil @ a != X1) => ((coinductive_lhd @ a @ X1) != (coinductive_lhd @ a @ X0)))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(rectify,[status(thm)],[f271])).
% 2.56/1.53 thf(f271,negated_conjecture,(
% 2.56/1.53 ~(coinductive_LNil @ a = (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (product_case_prod @ coinductive_llist @ a @ coinductive_llist @ a @ $o @ (^[X5 : coinductive_llist @ a, X6 : coinductive_llist @ a] : ((coinductive_LNil @ a != X5) => ((coinductive_LNil @ a != X6) => ((coinductive_lhd @ a @ X5) != (coinductive_lhd @ a @ X6)))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 inference(negated_conjecture,[status(cth)],[f270])).
% 2.56/1.53 thf(f270,conjecture,(
% 2.56/1.53 (coinductive_LNil @ a = (coindu1441602521_llist @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ a @ (product_case_prod @ coinductive_llist @ a @ coinductive_llist @ a @ $o @ (^[X5 : coinductive_llist @ a, X6 : coinductive_llist @ a] : ((coinductive_LNil @ a != X5) => ((coinductive_LNil @ a != X6) => ((coinductive_lhd @ a @ X5) != (coinductive_lhd @ a @ X6)))))) @ (comp @ coinductive_llist @ a @ a @ product_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_lhd @ a @ product_snd @ coinductive_llist @ a @ coinductive_llist @ a) @ (product_map_prod @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_ltl @ a @ coinductive_ltl @ a) @ (product_Pair @ coinductive_llist @ a @ coinductive_llist @ a @ coinductive_LNil @ a @ xs)))),
% 2.56/1.53 file('/export/starexec/sandbox/tmp/tmp.pFALYqNRSO/DTF2THF_23795.p',conj_0)).
% 2.56/1.53 % SZS output end Proof for DTF2THF_23795
% 2.56/1.53 % (24006)------------------------------
% 2.56/1.53 % (24006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (24006)Termination reason: Refutation
% 2.56/1.53
% 2.56/1.53 % (24006)Memory used [KB]: 7291
% 2.56/1.53 % (24006)Time elapsed: 0.054 s
% 2.56/1.53 % (24006)Instructions burned: 111 (million)
% 2.56/1.53 % (24006)------------------------------
% 2.56/1.53 % (24006)------------------------------
% 2.56/1.53 % (23988)Success in time 0.084 s
%------------------------------------------------------------------------------