TPTP Problem File: DAT228^1.p
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%------------------------------------------------------------------------------
% File : DAT228^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Tllist 950
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : tllist__950.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 413 ( 199 unt; 101 typ; 0 def)
% Number of atoms : 584 ( 376 equ; 0 cnn)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 3094 ( 71 ~; 9 |; 30 &;2780 @)
% ( 0 <=>; 204 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 5 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 375 ( 375 >; 0 *; 0 +; 0 <<)
% Number of symbols : 94 ( 91 usr; 3 con; 0-10 aty)
% Number of variables : 981 ( 46 ^; 826 !; 14 ?; 981 :)
% ( 95 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:54:40.359
%------------------------------------------------------------------------------
%----Could-be-implicit typings (18)
thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist,type,
tLList446370796tllist: $tType > $tType > $tType ).
thf(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: $tType > $tType > $tType > $tType ).
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_d,type,
d: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (83)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__Nat_Oco_Oenat_Oepred,type,
coinductive_epred: extended_enat > extended_enat ).
thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
coindu1491768222unfold:
!>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
extended_size_enat: extended_enat > nat ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Obool_Osize__bool,type,
size_bool: $o > nat ).
thf(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( A > product_unit > A ) ).
thf(sy_c_Product__Type_Ounit_Osize__unit,type,
product_size_unit: product_unit > nat ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
tuple_742722141rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( ( product_prod @ B @ C ) > A ) > ( tuple_isomorphism @ A @ B @ C ) ) ).
thf(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
tuple_1907371454rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > nat ) > ( B > nat ) > ( C > nat ) > ( tuple_isomorphism @ A @ B @ C ) > nat ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_String_OCode_Oabort,type,
abort:
!>[A: $tType] : ( literal > ( product_unit > A ) > A ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ogen__tlength,type,
tLList767988081length:
!>[A: $tType,B: $tType] : ( nat > ( tLList446370796tllist @ A @ B ) > extended_enat ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Olappendt,type,
tLList98099029ppendt:
!>[A: $tType,B: $tType] : ( ( coinductive_llist @ A ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otconcat,type,
tLList365522113concat:
!>[B: $tType,A: $tType] : ( B > ( tLList446370796tllist @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otconcat_H,type,
tLList1329345514concat:
!>[B: $tType,A: $tType] : ( ( product_unit > B ) > ( tLList446370796tllist @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otdropn,type,
tLList1881248882tdropn:
!>[A: $tType,B: $tType] : ( nat > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otfilter,type,
tLList1813626245filter:
!>[B: $tType,A: $tType] : ( B > ( A > $o ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otfilter_H,type,
tLList1122525870filter:
!>[B: $tType,A: $tType] : ( ( product_unit > B ) > ( A > $o ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otlength,type,
tLList942171283length:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > extended_enat ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_OTCons,type,
tLList1992840728_TCons:
!>[A: $tType,B: $tType] : ( A > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_OTNil,type,
tLList57907754e_TNil:
!>[B: $tType,A: $tType] : ( B > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocorec__tllist,type,
tLList1614408749tllist:
!>[E: $tType,B: $tType,A: $tType] : ( ( E > $o ) > ( E > B ) > ( E > A ) > ( E > $o ) > ( E > ( tLList446370796tllist @ A @ B ) ) > ( E > E ) > E > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oset2__tllist,type,
tLList457864275tllist:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Otllist__all2,type,
tLList1380991092t_all2:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ C @ D ) > $o ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otnth,type,
tLList2115636725e_tnth:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > nat > A ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ounfold__tllist,type,
tLList636831159tllist:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > $o ) > ( A > B ) > ( A > C ) > ( A > A ) > A > ( tLList446370796tllist @ C @ B ) ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a2: c > d > $o ).
thf(sy_v_B,type,
b2: a > b > $o ).
%----Relevant facts (252)
thf(fact_0_tllist__all2__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,P: A > B > $o,Q: C > D > $o,Xs: tLList446370796tllist @ A @ C,Ys: tLList446370796tllist @ B @ D,P2: A > B > $o,Q2: C > D > $o] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Xs @ Ys )
=> ( ! [X: A,Y: B] :
( ( P @ X @ Y )
=> ( P2 @ X @ Y ) )
=> ( ! [X: C,Y: D] :
( ( Q @ X @ Y )
=> ( Q2 @ X @ Y ) )
=> ( tLList1380991092t_all2 @ A @ B @ C @ D @ P2 @ Q2 @ Xs @ Ys ) ) ) ) ).
% tllist_all2_mono
thf(fact_1_tllist_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( tLList1380991092t_all2 @ A @ A @ B @ B
@ ^ [Y2: A,Z: A] : Y2 = Z
@ ^ [Y2: B,Z: B] : Y2 = Z )
= ( ^ [Y2: tLList446370796tllist @ A @ B,Z: tLList446370796tllist @ A @ B] : Y2 = Z ) ) ).
% tllist.rel_eq
thf(fact_2_tllist_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X2: tLList446370796tllist @ C @ D] :
( ! [X: C] : ( R1a @ X @ X )
=> ( ! [X: D] : ( R2a @ X @ X )
=> ( tLList1380991092t_all2 @ C @ C @ D @ D @ R1a @ R2a @ X2 @ X2 ) ) ) ).
% tllist.rel_refl
thf(fact_3_tfilter__TNil,axiom,
! [A: $tType,B: $tType,B2: B,P: A > $o,B3: B] :
( ( tLList1813626245filter @ B @ A @ B2 @ P @ ( tLList57907754e_TNil @ B @ A @ B3 ) )
= ( tLList57907754e_TNil @ B @ A @ B3 ) ) ).
% tfilter_TNil
thf(fact_4_tfilter__TCons,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,B3: B,Tr: tLList446370796tllist @ A @ B] :
( ( ( P @ A2 )
=> ( ( tLList1813626245filter @ B @ A @ B3 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1992840728_TCons @ A @ B @ A2 @ ( tLList1813626245filter @ B @ A @ B3 @ P @ Tr ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( tLList1813626245filter @ B @ A @ B3 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1813626245filter @ B @ A @ B3 @ P @ Tr ) ) ) ) ).
% tfilter_TCons
thf(fact_5_tllist__all2__tlengthD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,P: A > B > $o,Q: C > D > $o,Xs: tLList446370796tllist @ A @ C,Ys: tLList446370796tllist @ B @ D] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Xs @ Ys )
=> ( ( tLList942171283length @ A @ C @ Xs )
= ( tLList942171283length @ B @ D @ Ys ) ) ) ).
% tllist_all2_tlengthD
thf(fact_6_tllist__all2__TNil,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,X1: B,Y1: D] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) @ ( tLList57907754e_TNil @ D @ C @ Y1 ) )
= ( R2 @ X1 @ Y1 ) ) ).
% tllist_all2_TNil
thf(fact_7_tllist__all2__TCons,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X21: A,X22: tLList446370796tllist @ A @ B,Y21: C,Y22: tLList446370796tllist @ C @ D] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) @ ( tLList1992840728_TCons @ C @ D @ Y21 @ Y22 ) )
= ( ( R1 @ X21 @ Y21 )
& ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X22 @ Y22 ) ) ) ).
% tllist_all2_TCons
thf(fact_8_tllist__all2__trans,axiom,
! [B: $tType,A: $tType,P: A > A > $o,Q: B > B > $o,Xs: tLList446370796tllist @ A @ B,Ys: tLList446370796tllist @ A @ B,Zs: tLList446370796tllist @ A @ B] :
( ( tLList1380991092t_all2 @ A @ A @ B @ B @ P @ Q @ Xs @ Ys )
=> ( ( tLList1380991092t_all2 @ A @ A @ B @ B @ P @ Q @ Ys @ Zs )
=> ( ( transp @ A @ P )
=> ( ( transp @ B @ Q )
=> ( tLList1380991092t_all2 @ A @ A @ B @ B @ P @ Q @ Xs @ Zs ) ) ) ) ) ).
% tllist_all2_trans
thf(fact_9_tllist_Orel__intros_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R2: B > D > $o,X1: B,Y1: D,R1: A > C > $o] :
( ( R2 @ X1 @ Y1 )
=> ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) @ ( tLList57907754e_TNil @ D @ C @ Y1 ) ) ) ).
% tllist.rel_intros(1)
thf(fact_10_tllist__all2__TNil1,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,P: A > B > $o,Q: C > D > $o,B3: C,Ts: tLList446370796tllist @ B @ D] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ ( tLList57907754e_TNil @ C @ A @ B3 ) @ Ts )
= ( ? [B4: D] :
( ( Ts
= ( tLList57907754e_TNil @ D @ B @ B4 ) )
& ( Q @ B3 @ B4 ) ) ) ) ).
% tllist_all2_TNil1
thf(fact_11_tllist__all2__TNil2,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P: A > B > $o,Q: C > D > $o,Ts: tLList446370796tllist @ A @ C,B2: D] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Ts @ ( tLList57907754e_TNil @ D @ B @ B2 ) )
= ( ? [B5: C] :
( ( Ts
= ( tLList57907754e_TNil @ C @ A @ B5 ) )
& ( Q @ B5 @ B2 ) ) ) ) ).
% tllist_all2_TNil2
thf(fact_12_tllist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,X21: A,Y21: C,R2: B > D > $o,X22: tLList446370796tllist @ A @ B,Y22: tLList446370796tllist @ C @ D] :
( ( R1 @ X21 @ Y21 )
=> ( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X22 @ Y22 )
=> ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) @ ( tLList1992840728_TCons @ C @ D @ Y21 @ Y22 ) ) ) ) ).
% tllist.rel_intros(2)
thf(fact_13_tllist__all2__TCons1,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,P: A > B > $o,Q: C > D > $o,X2: A,Ts: tLList446370796tllist @ A @ C,Ts2: tLList446370796tllist @ B @ D] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ ( tLList1992840728_TCons @ A @ C @ X2 @ Ts ) @ Ts2 )
= ( ? [X3: B,Ts3: tLList446370796tllist @ B @ D] :
( ( Ts2
= ( tLList1992840728_TCons @ B @ D @ X3 @ Ts3 ) )
& ( P @ X2 @ X3 )
& ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Ts @ Ts3 ) ) ) ) ).
% tllist_all2_TCons1
thf(fact_14_tllist__all2__TCons2,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,P: A > B > $o,Q: C > D > $o,Ts2: tLList446370796tllist @ A @ C,X2: B,Ts: tLList446370796tllist @ B @ D] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Ts2 @ ( tLList1992840728_TCons @ B @ D @ X2 @ Ts ) )
= ( ? [X3: A,Ts3: tLList446370796tllist @ A @ C] :
( ( Ts2
= ( tLList1992840728_TCons @ A @ C @ X3 @ Ts3 ) )
& ( P @ X3 @ X2 )
& ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Ts3 @ Ts ) ) ) ) ).
% tllist_all2_TCons2
thf(fact_15_tllist_Oinject_I2_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B,Y21: A,Y22: tLList446370796tllist @ A @ B] :
( ( ( tLList1992840728_TCons @ A @ B @ X21 @ X22 )
= ( tLList1992840728_TCons @ A @ B @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% tllist.inject(2)
thf(fact_16_tllist_Oinject_I1_J,axiom,
! [A: $tType,B: $tType,X1: B,Y1: B] :
( ( ( tLList57907754e_TNil @ B @ A @ X1 )
= ( tLList57907754e_TNil @ B @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tllist.inject(1)
thf(fact_17_tllist_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X1: B,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList57907754e_TNil @ B @ A @ X1 )
!= ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) ) ).
% tllist.distinct(1)
thf(fact_18_tllist_Oexhaust,axiom,
! [B: $tType,A: $tType,Y3: tLList446370796tllist @ A @ B] :
( ! [X12: B] :
( Y3
!= ( tLList57907754e_TNil @ B @ A @ X12 ) )
=> ~ ! [X212: A,X222: tLList446370796tllist @ A @ B] :
( Y3
!= ( tLList1992840728_TCons @ A @ B @ X212 @ X222 ) ) ) ).
% tllist.exhaust
thf(fact_19_tllist__all2__cases,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,P: A > B > $o,Q: C > D > $o,Xs: tLList446370796tllist @ A @ C,Ys: tLList446370796tllist @ B @ D] :
( ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Xs @ Ys )
=> ( ! [B6: C] :
( ( Xs
= ( tLList57907754e_TNil @ C @ A @ B6 ) )
=> ! [B7: D] :
( ( Ys
= ( tLList57907754e_TNil @ D @ B @ B7 ) )
=> ~ ( Q @ B6 @ B7 ) ) )
=> ~ ! [X: A,Xs2: tLList446370796tllist @ A @ C] :
( ( Xs
= ( tLList1992840728_TCons @ A @ C @ X @ Xs2 ) )
=> ! [Y: B,Ys2: tLList446370796tllist @ B @ D] :
( ( Ys
= ( tLList1992840728_TCons @ B @ D @ Y @ Ys2 ) )
=> ( ( P @ X @ Y )
=> ~ ( tLList1380991092t_all2 @ A @ B @ C @ D @ P @ Q @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% tllist_all2_cases
thf(fact_20_tllist_Orel__cases,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,A2: tLList446370796tllist @ A @ B,B3: tLList446370796tllist @ C @ D] :
( ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ A2 @ B3 )
=> ( ! [X: B] :
( ( A2
= ( tLList57907754e_TNil @ B @ A @ X ) )
=> ! [Y: D] :
( ( B3
= ( tLList57907754e_TNil @ D @ C @ Y ) )
=> ~ ( R2 @ X @ Y ) ) )
=> ~ ! [X1a: A,X23: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ X1a @ X23 ) )
=> ! [Y1a: C,Y23: tLList446370796tllist @ C @ D] :
( ( B3
= ( tLList1992840728_TCons @ C @ D @ Y1a @ Y23 ) )
=> ( ( R1 @ X1a @ Y1a )
=> ~ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ X23 @ Y23 ) ) ) ) ) ) ).
% tllist.rel_cases
thf(fact_21_tllist_Orel__distinct_I1_J,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X1: B,Y21: C,Y22: tLList446370796tllist @ C @ D] :
~ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) @ ( tLList1992840728_TCons @ C @ D @ Y21 @ Y22 ) ) ).
% tllist.rel_distinct(1)
thf(fact_22_tllist_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,Y21: A,Y22: tLList446370796tllist @ A @ B,X1: D] :
~ ( tLList1380991092t_all2 @ A @ C @ B @ D @ R1 @ R2 @ ( tLList1992840728_TCons @ A @ B @ Y21 @ Y22 ) @ ( tLList57907754e_TNil @ D @ C @ X1 ) ) ).
% tllist.rel_distinct(2)
thf(fact_23_tllist_Orel__transp,axiom,
! [A: $tType,B: $tType,R1: A > A > $o,R2: B > B > $o] :
( ( transp @ A @ R1 )
=> ( ( transp @ B @ R2 )
=> ( transp @ ( tLList446370796tllist @ A @ B ) @ ( tLList1380991092t_all2 @ A @ A @ B @ B @ R1 @ R2 ) ) ) ) ).
% tllist.rel_transp
thf(fact_24_transpD,axiom,
! [A: $tType,R: A > A > $o,X2: A,Y3: A,Z2: A] :
( ( transp @ A @ R )
=> ( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X2 @ Z2 ) ) ) ) ).
% transpD
thf(fact_25_transpE,axiom,
! [A: $tType,R: A > A > $o,X2: A,Y3: A,Z2: A] :
( ( transp @ A @ R )
=> ( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X2 @ Z2 ) ) ) ) ).
% transpE
thf(fact_26_transpI,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X: A,Y: A,Z3: A] :
( ( R @ X @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( transp @ A @ R ) ) ).
% transpI
thf(fact_27_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R3: A > A > $o] :
! [X4: A,Y4: A,Z4: A] :
( ( R3 @ X4 @ Y4 )
=> ( ( R3 @ Y4 @ Z4 )
=> ( R3 @ X4 @ Z4 ) ) ) ) ) ).
% transp_def
thf(fact_28_DEADID_Orel__transp,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y2: A,Z: A] : Y2 = Z ) ).
% DEADID.rel_transp
thf(fact_29_tconcat__TNil,axiom,
! [A: $tType,B: $tType,B3: B,B2: B] :
( ( tLList365522113concat @ B @ A @ B3 @ ( tLList57907754e_TNil @ B @ ( coinductive_llist @ A ) @ B2 ) )
= ( tLList57907754e_TNil @ B @ A @ B2 ) ) ).
% tconcat_TNil
thf(fact_30_tlength__TNil,axiom,
! [B: $tType,A: $tType,B3: B] :
( ( tLList942171283length @ A @ B @ ( tLList57907754e_TNil @ B @ A @ B3 ) )
= ( zero_zero @ extended_enat ) ) ).
% tlength_TNil
thf(fact_31_tllist_Ocorec__code,axiom,
! [B: $tType,A: $tType,E: $tType] :
( ( tLList1614408749tllist @ E @ B @ A )
= ( ^ [P3: E > $o,G1: E > B,G21: E > A,Q22: E > $o,G221: E > ( tLList446370796tllist @ A @ B ),G222: E > E,A3: E] : ( if @ ( tLList446370796tllist @ A @ B ) @ ( P3 @ A3 ) @ ( tLList57907754e_TNil @ B @ A @ ( G1 @ A3 ) ) @ ( tLList1992840728_TCons @ A @ B @ ( G21 @ A3 ) @ ( if @ ( tLList446370796tllist @ A @ B ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( tLList1614408749tllist @ E @ B @ A @ P3 @ G1 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% tllist.corec_code
thf(fact_32_unfold__tllist_Ocode,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( tLList636831159tllist @ A @ B @ C )
= ( ^ [P3: A > $o,G1: A > B,G21: A > C,G22: A > A,A3: A] : ( if @ ( tLList446370796tllist @ C @ B ) @ ( P3 @ A3 ) @ ( tLList57907754e_TNil @ B @ C @ ( G1 @ A3 ) ) @ ( tLList1992840728_TCons @ C @ B @ ( G21 @ A3 ) @ ( tLList636831159tllist @ A @ B @ C @ P3 @ G1 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% unfold_tllist.code
thf(fact_33_tconcat_H__code_I1_J,axiom,
! [A: $tType,B: $tType,B3: product_unit > B,B2: B] :
( ( tLList1329345514concat @ B @ A @ B3 @ ( tLList57907754e_TNil @ B @ ( coinductive_llist @ A ) @ B2 ) )
= ( tLList57907754e_TNil @ B @ A @ B2 ) ) ).
% tconcat'_code(1)
thf(fact_34_unfold__tllist__eq__TNil,axiom,
! [A: $tType,B: $tType,C: $tType,IS_TNIL: C > $o,TNIL: C > B,THD: C > A,TTL: C > C,A2: C,B3: B] :
( ( ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ A2 )
= ( tLList57907754e_TNil @ B @ A @ B3 ) )
= ( ( IS_TNIL @ A2 )
& ( B3
= ( TNIL @ A2 ) ) ) ) ).
% unfold_tllist_eq_TNil
thf(fact_35_TNil__eq__unfold__tllist,axiom,
! [A: $tType,B: $tType,C: $tType,B3: B,IS_TNIL: C > $o,TNIL: C > B,THD: C > A,TTL: C > C,A2: C] :
( ( ( tLList57907754e_TNil @ B @ A @ B3 )
= ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ A2 ) )
= ( ( IS_TNIL @ A2 )
& ( B3
= ( TNIL @ A2 ) ) ) ) ).
% TNil_eq_unfold_tllist
thf(fact_36_unfold__tllist_Octr_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,P4: A > $o,A2: A,G12: A > B,G212: A > C,G223: A > A] :
( ~ ( P4 @ A2 )
=> ( ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ A2 )
= ( tLList1992840728_TCons @ C @ B @ ( G212 @ A2 ) @ ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ ( G223 @ A2 ) ) ) ) ) ).
% unfold_tllist.ctr(2)
thf(fact_37_tllist_Ocorec_I2_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ( ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( tLList1992840728_TCons @ A @ B @ ( G212 @ A2 ) @ ( if @ ( tLList446370796tllist @ A @ B ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% tllist.corec(2)
thf(fact_38_unfold__tllist_Octr_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,P4: A > $o,A2: A,G12: A > B,G212: A > C,G223: A > A] :
( ( P4 @ A2 )
=> ( ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ A2 )
= ( tLList57907754e_TNil @ B @ C @ ( G12 @ A2 ) ) ) ) ).
% unfold_tllist.ctr(1)
thf(fact_39_tllist_Ocorec_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ( P4 @ A2 )
=> ( ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( tLList57907754e_TNil @ B @ A @ ( G12 @ A2 ) ) ) ) ).
% tllist.corec(1)
thf(fact_40_tconcat_H__def,axiom,
! [A: $tType,B: $tType] :
( ( tLList1329345514concat @ B @ A )
= ( ^ [B5: product_unit > B] : ( tLList365522113concat @ B @ A @ ( B5 @ product_Unity ) ) ) ) ).
% tconcat'_def
thf(fact_41_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F: B > A,X4: B,Y4: B] : ( R3 @ ( F @ X4 ) @ ( F @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_42_tconcat__TCons,axiom,
! [B: $tType,A: $tType,B3: B,A2: coinductive_llist @ A,Tr: tLList446370796tllist @ ( coinductive_llist @ A ) @ B] :
( ( tLList365522113concat @ B @ A @ B3 @ ( tLList1992840728_TCons @ ( coinductive_llist @ A ) @ B @ A2 @ Tr ) )
= ( tLList98099029ppendt @ A @ B @ A2 @ ( tLList365522113concat @ B @ A @ B3 @ Tr ) ) ) ).
% tconcat_TCons
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X: A] :
( ( F2 @ X )
= ( G @ X ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_47_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_48_tllist_Oset__induct_I2_J,axiom,
! [A: $tType,Aa: $tType,X2: A,A2: tLList446370796tllist @ Aa @ A,P: A > ( tLList446370796tllist @ Aa @ A ) > $o] :
( ( member @ A @ X2 @ ( tLList457864275tllist @ Aa @ A @ A2 ) )
=> ( ! [Z3: A] : ( P @ Z3 @ ( tLList57907754e_TNil @ A @ Aa @ Z3 ) )
=> ( ! [Z1: Aa,Z22: tLList446370796tllist @ Aa @ A,Xb: A] :
( ( member @ A @ Xb @ ( tLList457864275tllist @ Aa @ A @ Z22 ) )
=> ( ( P @ Xb @ Z22 )
=> ( P @ Xb @ ( tLList1992840728_TCons @ Aa @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% tllist.set_induct(2)
thf(fact_49_tllist_Oset__cases_I2_J,axiom,
! [B: $tType,A: $tType,E2: B,A2: tLList446370796tllist @ A @ B] :
( ( member @ B @ E2 @ ( tLList457864275tllist @ A @ B @ A2 ) )
=> ( ( A2
!= ( tLList57907754e_TNil @ B @ A @ E2 ) )
=> ~ ! [Z1: A,Z22: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ Z1 @ Z22 ) )
=> ~ ( member @ B @ E2 @ ( tLList457864275tllist @ A @ B @ Z22 ) ) ) ) ) ).
% tllist.set_cases(2)
thf(fact_50_tfilter_H__code_I1_J,axiom,
! [A: $tType,B: $tType,B2: product_unit > B,P: A > $o,B3: B] :
( ( tLList1122525870filter @ B @ A @ B2 @ P @ ( tLList57907754e_TNil @ B @ A @ B3 ) )
= ( tLList57907754e_TNil @ B @ A @ B3 ) ) ).
% tfilter'_code(1)
thf(fact_51_tfilter_H__def,axiom,
! [A: $tType,B: $tType] :
( ( tLList1122525870filter @ B @ A )
= ( ^ [B5: product_unit > B] : ( tLList1813626245filter @ B @ A @ ( B5 @ product_Unity ) ) ) ) ).
% tfilter'_def
thf(fact_52_tllist_Osimps_I23_J,axiom,
! [B: $tType,A: $tType,X21: A,X22: tLList446370796tllist @ A @ B] :
( ( tLList457864275tllist @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X21 @ X22 ) )
= ( tLList457864275tllist @ A @ B @ X22 ) ) ).
% tllist.simps(23)
thf(fact_53_tllist_Oset__intros_I4_J,axiom,
! [B: $tType,A: $tType,Xc: B,A2a: tLList446370796tllist @ A @ B,A1a: A] :
( ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ A2a ) )
=> ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1a @ A2a ) ) ) ) ).
% tllist.set_intros(4)
thf(fact_54_tllist_Oset__intros_I3_J,axiom,
! [A: $tType,B: $tType,Aa2: B] : ( member @ B @ Aa2 @ ( tLList457864275tllist @ A @ B @ ( tLList57907754e_TNil @ B @ A @ Aa2 ) ) ) ).
% tllist.set_intros(3)
thf(fact_55_tconcat_H__code_I2_J,axiom,
! [B: $tType,C: $tType,B3: product_unit > B,A2: coinductive_llist @ C,Tr: tLList446370796tllist @ ( coinductive_llist @ C ) @ B] :
( ( tLList1329345514concat @ B @ C @ B3 @ ( tLList1992840728_TCons @ ( coinductive_llist @ C ) @ B @ A2 @ Tr ) )
= ( tLList98099029ppendt @ C @ B @ A2 @ ( tLList1329345514concat @ B @ C @ B3 @ Tr ) ) ) ).
% tconcat'_code(2)
thf(fact_56_tfilter_H__code_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,B2: product_unit > B,Tr: tLList446370796tllist @ A @ B] :
( ( ( P @ A2 )
=> ( ( tLList1122525870filter @ B @ A @ B2 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1992840728_TCons @ A @ B @ A2 @ ( tLList1122525870filter @ B @ A @ B2 @ P @ Tr ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( tLList1122525870filter @ B @ A @ B2 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1122525870filter @ B @ A @ B2 @ P @ Tr ) ) ) ) ).
% tfilter'_code(2)
thf(fact_57_case__enat__0,axiom,
! [A: $tType,F2: nat > A,I: A] :
( ( extended_case_enat @ A @ F2 @ I @ ( zero_zero @ extended_enat ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_58_old_Ounit_Oexhaust,axiom,
! [Y3: product_unit] : Y3 = product_Unity ).
% old.unit.exhaust
thf(fact_59_old_Ounit_Oinducts,axiom,
! [P: product_unit > $o,Unit: product_unit] :
( ( P @ product_Unity )
=> ( P @ Unit ) ) ).
% old.unit.inducts
thf(fact_60_unit__all__impI,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( ( P @ product_Unity )
=> ( Q @ product_Unity ) )
=> ! [X5: product_unit] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ).
% unit_all_impI
thf(fact_61_eq__sym__Unity__conv,axiom,
! [X2: $o] :
( ( X2
= ( product_Unity = product_Unity ) )
= X2 ) ).
% eq_sym_Unity_conv
thf(fact_62_tlength__code,axiom,
! [A: $tType,B: $tType] :
( ( tLList942171283length @ A @ B )
= ( tLList767988081length @ A @ B @ ( zero_zero @ nat ) ) ) ).
% tlength_code
thf(fact_63_enat__unfold__eq__0,axiom,
! [A: $tType,Stop: A > $o,Next: A > A,A2: A] :
( ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
= ( zero_zero @ extended_enat ) )
= ( Stop @ A2 ) ) ).
% enat_unfold_eq_0
thf(fact_64_enat__unfold__stop,axiom,
! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
( ( Stop @ A2 )
=> ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
= ( zero_zero @ extended_enat ) ) ) ).
% enat_unfold_stop
thf(fact_65_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_66_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_67_unit_Osize_I1_J,axiom,
( ( product_size_unit @ product_Unity )
= ( zero_zero @ nat ) ) ).
% unit.size(1)
thf(fact_68_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_69_tnth__0,axiom,
! [B: $tType,A: $tType,X2: A,Xs: tLList446370796tllist @ A @ B] :
( ( tLList2115636725e_tnth @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X2 @ Xs ) @ ( zero_zero @ nat ) )
= X2 ) ).
% tnth_0
thf(fact_70_unit_Osize_I2_J,axiom,
( ( size_size @ product_unit @ product_Unity )
= ( zero_zero @ nat ) ) ).
% unit.size(2)
thf(fact_71_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_72_size__unit__overloaded__def,axiom,
( ( size_size @ product_unit )
= ( product_rec_unit @ nat @ ( zero_zero @ nat ) ) ) ).
% size_unit_overloaded_def
thf(fact_73_size__unit__def,axiom,
( product_size_unit
= ( product_rec_unit @ nat @ ( zero_zero @ nat ) ) ) ).
% size_unit_def
thf(fact_74_bool_Osize_I2_J,axiom,
( ( size_bool @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(2)
thf(fact_75_bool_Osize_I1_J,axiom,
( ( size_bool @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(1)
thf(fact_76_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_77_bool_Osize_I4_J,axiom,
( ( size_size @ $o @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(4)
thf(fact_78_bool_Osize_I3_J,axiom,
( ( size_size @ $o @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(3)
thf(fact_79_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_80_size__bool,axiom,
( ( size_size @ $o )
= ( ^ [B5: $o] : ( zero_zero @ nat ) ) ) ).
% size_bool
thf(fact_81_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_82_size__bool__def,axiom,
( size_bool
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_def
thf(fact_83_size__bool__overloaded__def,axiom,
( ( size_size @ $o )
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_overloaded_def
thf(fact_84_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_85_size__ne__size__imp__ne,axiom,
! [A: $tType] :
( ( size @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y3 ) )
=> ( X2 != Y3 ) ) ) ).
% size_ne_size_imp_ne
thf(fact_86_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_87_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_88_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_89_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_90_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural ) ) ).
% natural.simps(5)
thf(fact_91_option_Osize__neq,axiom,
! [A: $tType,X2: option @ A] :
( ( size_size @ ( option @ A ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_92_typerep_Osize__neq,axiom,
! [X2: typerep] :
( ( size_size @ typerep @ X2 )
!= ( zero_zero @ nat ) ) ).
% typerep.size_neq
thf(fact_93_tuple__isomorphism_Osize__neq,axiom,
! [A: $tType,B: $tType,C: $tType,X2: tuple_isomorphism @ A @ B @ C] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% tuple_isomorphism.size_neq
thf(fact_94_size__literal__def,axiom,
( ( size_size @ literal )
= ( ^ [S: literal] : ( zero_zero @ nat ) ) ) ).
% size_literal_def
thf(fact_95_old_Ounit_Ocase,axiom,
! [A: $tType,F2: A] :
( ( product_case_unit @ A @ F2 @ product_Unity )
= F2 ) ).
% old.unit.case
thf(fact_96_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_97_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_98_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F2: A,Unit: product_unit] :
( ( H @ ( product_case_unit @ A @ F2 @ Unit ) )
= ( product_case_unit @ B @ ( H @ F2 ) @ Unit ) ) ).
% unit.case_distrib
thf(fact_99_case__unit__Unity,axiom,
! [A: $tType] :
( ( product_case_unit @ A )
= ( ^ [F: A,U: product_unit] : F ) ) ).
% case_unit_Unity
thf(fact_100_natural_Oinducts,axiom,
! [P: code_natural > $o,Natural: code_natural] :
( ( P @ ( zero_zero @ code_natural ) )
=> ( ! [Natural3: code_natural] :
( ( P @ Natural3 )
=> ( P @ ( code_Suc @ Natural3 ) ) )
=> ( P @ Natural ) ) ) ).
% natural.inducts
thf(fact_101_natural_Oexhaust,axiom,
! [Y3: code_natural] :
( ( Y3
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural3: code_natural] :
( Y3
!= ( code_Suc @ Natural3 ) ) ) ).
% natural.exhaust
thf(fact_102_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_103_natural_Odistinct_I2_J,axiom,
! [Natural4: code_natural] :
( ( code_Suc @ Natural4 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_104_Code_Oabort__def,axiom,
! [A: $tType] :
( ( abort @ A )
= ( ^ [Uu: literal,F: product_unit > A] : ( F @ product_Unity ) ) ) ).
% Code.abort_def
thf(fact_105_epred__enat__unfold,axiom,
! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
( ( ( Stop @ A2 )
=> ( ( coinductive_epred @ ( coindu1491768222unfold @ A @ Stop @ Next @ A2 ) )
= ( zero_zero @ extended_enat ) ) )
& ( ~ ( Stop @ A2 )
=> ( ( coinductive_epred @ ( coindu1491768222unfold @ A @ Stop @ Next @ A2 ) )
= ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A2 ) ) ) ) ) ).
% epred_enat_unfold
thf(fact_106_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [Inc: A > A,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_107_tnth__Suc__TCons,axiom,
! [A: $tType,B: $tType,X2: A,Xs: tLList446370796tllist @ A @ B,N: nat] :
( ( tLList2115636725e_tnth @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X2 @ Xs ) @ ( suc @ N ) )
= ( tLList2115636725e_tnth @ A @ B @ Xs @ N ) ) ).
% tnth_Suc_TCons
thf(fact_108_lappendt__LCons,axiom,
! [B: $tType,A: $tType,X2: A,Xs: coinductive_llist @ A,Tr: tLList446370796tllist @ A @ B] :
( ( tLList98099029ppendt @ A @ B @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Tr )
= ( tLList1992840728_TCons @ A @ B @ X2 @ ( tLList98099029ppendt @ A @ B @ Xs @ Tr ) ) ) ).
% lappendt_LCons
thf(fact_109_nat_Odistinct_I1_J,axiom,
! [X24: nat] :
( ( zero_zero @ nat )
!= ( suc @ X24 ) ) ).
% nat.distinct(1)
thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_111_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_112_nat_OdiscI,axiom,
! [Nat3: nat,X24: nat] :
( ( Nat3
= ( suc @ X24 ) )
=> ( Nat3
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_113_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_114_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ ( zero_zero @ nat ) )
=> ( ! [Y: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( P @ X @ Y )
=> ( P @ ( suc @ X ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_115_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_116_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_117_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_118_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_119_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y3
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_120_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat3: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat4: nat] :
( ( P @ Nat4 )
=> ( P @ ( suc @ Nat4 ) ) )
=> ( P @ Nat3 ) ) ) ).
% old.nat.inducts
thf(fact_121_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_122_epred__0,axiom,
( ( coinductive_epred @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% epred_0
thf(fact_123_epred__inject,axiom,
! [X2: extended_enat,Y3: extended_enat] :
( ( X2
!= ( zero_zero @ extended_enat ) )
=> ( ( Y3
!= ( zero_zero @ extended_enat ) )
=> ( ( ( coinductive_epred @ X2 )
= ( coinductive_epred @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% epred_inject
thf(fact_124_enat__coinduct,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N2: extended_enat] :
( ( P @ M2 @ N2 )
=> ( ( ( M2
= ( zero_zero @ extended_enat ) )
= ( N2
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N2
!= ( zero_zero @ extended_enat ) )
=> ( ( P @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N2 ) )
| ( ( coinductive_epred @ M2 )
= ( coinductive_epred @ N2 ) ) ) ) ) ) )
=> ( M = N ) ) ) ).
% enat_coinduct
thf(fact_125_co_Oenat_Oexpand,axiom,
! [Enat: extended_enat,Enat2: extended_enat] :
( ( ( Enat
= ( zero_zero @ extended_enat ) )
= ( Enat2
= ( zero_zero @ extended_enat ) ) )
=> ( ( ( Enat
!= ( zero_zero @ extended_enat ) )
=> ( ( Enat2
!= ( zero_zero @ extended_enat ) )
=> ( ( coinductive_epred @ Enat )
= ( coinductive_epred @ Enat2 ) ) ) )
=> ( Enat = Enat2 ) ) ) ).
% co.enat.expand
thf(fact_126_enat__coinduct2,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N2: extended_enat] :
( ( P @ M2 @ N2 )
=> ( ( M2
= ( zero_zero @ extended_enat ) )
= ( N2
= ( zero_zero @ extended_enat ) ) ) )
=> ( ! [M2: extended_enat,N2: extended_enat] :
( ( P @ M2 @ N2 )
=> ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N2
!= ( zero_zero @ extended_enat ) )
=> ( ( P @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N2 ) )
| ( ( coinductive_epred @ M2 )
= ( coinductive_epred @ N2 ) ) ) ) ) )
=> ( M = N ) ) ) ) ).
% enat_coinduct2
thf(fact_127_tuple__isomorphism_Osize_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,X1: A > ( product_prod @ B @ C ),X24: ( product_prod @ B @ C ) > A] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ ( tuple_742722141rphism @ A @ B @ C @ X1 @ X24 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size(2)
thf(fact_128_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_129_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X13: A] : ( P @ ( zero_zero @ nat ) @ X13 )
=> ( ! [X: A,N2: nat] :
( ( P @ N2 @ X )
=> ? [Y5: A] :
( ( P @ ( suc @ N2 ) @ Y5 )
& ( Q @ N2 @ X @ Y5 ) ) )
=> ? [F3: nat > A] :
! [N3: nat] :
( ( P @ N3 @ ( F3 @ N3 ) )
& ( Q @ N3 @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_130_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X13: nat] : ( P @ X13 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_131_natural_Osize_I2_J,axiom,
! [Natural: code_natural] :
( ( code_size_natural @ ( code_Suc @ Natural ) )
= ( plus_plus @ nat @ ( code_size_natural @ Natural ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% natural.size(2)
thf(fact_132_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add_left_cancel
thf(fact_133_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B3 = C2 ) ) ) ).
% add_right_cancel
thf(fact_134_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_135_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( plus_plus @ A @ B3 @ A2 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_136_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= A2 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_137_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= A2 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_138_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_139_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_140_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_141_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_142_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_143_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_144_natural_Osize_I4_J,axiom,
! [Natural: code_natural] :
( ( size_size @ code_natural @ ( code_Suc @ Natural ) )
= ( plus_plus @ nat @ ( size_size @ code_natural @ Natural ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% natural.size(4)
thf(fact_145_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_146_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_147_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_148_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_149_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_150_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add.left_cancel
thf(fact_151_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B3 = C2 ) ) ) ).
% add.right_cancel
thf(fact_152_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B5: A] : ( plus_plus @ A @ B5 @ A3 ) ) ) ) ).
% add.commute
thf(fact_153_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_154_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_155_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_156_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( plus_plus @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_157_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_158_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_159_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_160_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_161_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_162_tuple__isomorphism_Osize__gen,axiom,
! [B: $tType,C: $tType,A: $tType,Xb2: A > nat,Xa: B > nat,X2: C > nat,X1: A > ( product_prod @ B @ C ),X24: ( product_prod @ B @ C ) > A] :
( ( tuple_1907371454rphism @ A @ B @ C @ Xb2 @ Xa @ X2 @ ( tuple_742722141rphism @ A @ B @ C @ X1 @ X24 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size_gen
thf(fact_163_ind__euclid,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B6: nat] :
( ( P @ A5 @ B6 )
= ( P @ B6 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ A5 @ B6 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B6 ) ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% ind_euclid
thf(fact_164_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( B3
= ( plus_plus @ A @ B3 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_165_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(5)
thf(fact_166_plus__enat__eq__0__conv,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% plus_enat_eq_0_conv
thf(fact_167_epred__iadd1,axiom,
! [A2: extended_enat,B3: extended_enat] :
( ( A2
!= ( zero_zero @ extended_enat ) )
=> ( ( coinductive_epred @ ( plus_plus @ extended_enat @ A2 @ B3 ) )
= ( plus_plus @ extended_enat @ ( coinductive_epred @ A2 ) @ B3 ) ) ) ).
% epred_iadd1
thf(fact_168_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(6)
thf(fact_169_tdropn__Suc__TCons,axiom,
! [B: $tType,A: $tType,N: nat,X2: A,Xs: tLList446370796tllist @ A @ B] :
( ( tLList1881248882tdropn @ A @ B @ ( suc @ N ) @ ( tLList1992840728_TCons @ A @ B @ X2 @ Xs ) )
= ( tLList1881248882tdropn @ A @ B @ N @ Xs ) ) ).
% tdropn_Suc_TCons
thf(fact_170_gen__tlength__code_I2_J,axiom,
! [D: $tType,C: $tType,N: nat,X2: C,Xs: tLList446370796tllist @ C @ D] :
( ( tLList767988081length @ C @ D @ N @ ( tLList1992840728_TCons @ C @ D @ X2 @ Xs ) )
= ( tLList767988081length @ C @ D @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ).
% gen_tlength_code(2)
thf(fact_171_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_172_tdropn__0,axiom,
! [B: $tType,A: $tType,Xs: tLList446370796tllist @ A @ B] :
( ( tLList1881248882tdropn @ A @ B @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% tdropn_0
thf(fact_173_tdropn__TNil,axiom,
! [A: $tType,B: $tType,N: nat,B3: B] :
( ( tLList1881248882tdropn @ A @ B @ N @ ( tLList57907754e_TNil @ B @ A @ B3 ) )
= ( tLList57907754e_TNil @ B @ A @ B3 ) ) ).
% tdropn_TNil
thf(fact_174_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_175_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_176_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_177_option_Osize__gen_I1_J,axiom,
! [A: $tType,X2: A > nat] :
( ( size_option @ A @ X2 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_178_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_179_ln__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_180_epred__1,axiom,
( ( coinductive_epred @ ( one_one @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% epred_1
thf(fact_181_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( ( X2
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X2
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_182_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [W: A,Z2: A] :
( ( ( powr @ A @ W @ Z2 )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_183_powr__0,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [Z2: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z2 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_184_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_185_option_Osize__gen_I2_J,axiom,
! [A: $tType,X2: A > nat,X24: A] :
( ( size_option @ A @ X2 @ ( some @ A @ X24 ) )
= ( plus_plus @ nat @ ( X2 @ X24 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_186_option_Osize_I4_J,axiom,
! [A: $tType,X24: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X24 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_187_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P3: $o] : ( if @ A @ P3 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_188_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_189_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_190_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_191_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M3: nat,N4: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N4
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_192_gen__tlength__def,axiom,
! [A: $tType,B: $tType] :
( ( tLList767988081length @ A @ B )
= ( ^ [N4: nat,Xs3: tLList446370796tllist @ A @ B] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N4 ) @ ( tLList942171283length @ A @ B @ Xs3 ) ) ) ) ).
% gen_tlength_def
thf(fact_193_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_194_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_195_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_196_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_197_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_198_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_199_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_200_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( minus_minus @ A @ A2 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_201_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ A2 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_202_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ A2 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_203_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_204_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_205_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_206_enat_Osize_I3_J,axiom,
! [Nat3: nat] :
( ( size_size @ extended_enat @ ( extended_enat2 @ Nat3 ) )
= ( zero_zero @ nat ) ) ).
% enat.size(3)
thf(fact_207_enat__add1__eq,axiom,
! [X2: nat,Y3: extended_enat,Z2: extended_enat] :
( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X2 ) @ Y3 )
= ( plus_plus @ extended_enat @ ( extended_enat2 @ X2 ) @ Z2 ) )
= ( Y3 = Z2 ) ) ).
% enat_add1_eq
thf(fact_208_enat__add2__eq,axiom,
! [Y3: extended_enat,X2: nat,Z2: extended_enat] :
( ( ( plus_plus @ extended_enat @ Y3 @ ( extended_enat2 @ X2 ) )
= ( plus_plus @ extended_enat @ Z2 @ ( extended_enat2 @ X2 ) ) )
= ( Y3 = Z2 ) ) ).
% enat_add2_eq
thf(fact_209_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F22: T,Nat3: nat] :
( ( extended_case_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat3 ) )
= ( F1 @ Nat3 ) ) ).
% enat.simps(4)
thf(fact_210_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B3 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_211_epred__enat,axiom,
! [M: nat] :
( ( coinductive_epred @ ( extended_enat2 @ M ) )
= ( extended_enat2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ).
% epred_enat
thf(fact_212_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_213_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B3 ) )
= ( ( plus_plus @ A @ A2 @ B3 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_214_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_215_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_216_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_217_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_218_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% diff_diff_add
thf(fact_219_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [C2: A,B3: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B3 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_220_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A3: A,B5: A] :
( ( minus_minus @ A @ A3 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_221_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_222_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B3 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_223_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_224_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M )
= ( zero_zero @ nat ) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_225_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_226_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_227_enat__0__iff_I2_J,axiom,
! [X2: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X2 ) )
= ( X2
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_228_enat__0__iff_I1_J,axiom,
! [X2: nat] :
( ( ( extended_enat2 @ X2 )
= ( zero_zero @ extended_enat ) )
= ( X2
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_229_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_230_gen__tlength__code_I1_J,axiom,
! [B: $tType,A: $tType,N: nat,B3: B] :
( ( tLList767988081length @ A @ B @ N @ ( tLList57907754e_TNil @ B @ A @ B3 ) )
= ( extended_enat2 @ N ) ) ).
% gen_tlength_code(1)
thf(fact_231_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_232_enat_Osize_I1_J,axiom,
! [Nat3: nat] :
( ( extended_size_enat @ ( extended_enat2 @ Nat3 ) )
= ( zero_zero @ nat ) ) ).
% enat.size(1)
thf(fact_233_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_234_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_235_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_236_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N ) ).
% idiff_enat_0_right
thf(fact_237_epred__minus__epred,axiom,
! [M: extended_enat,N: extended_enat] :
( ( M
!= ( zero_zero @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ ( coinductive_epred @ N ) @ ( coinductive_epred @ M ) )
= ( minus_minus @ extended_enat @ N @ M ) ) ) ).
% epred_minus_epred
thf(fact_238_tlength__tdropn,axiom,
! [A: $tType,B: $tType,N: nat,Xs: tLList446370796tllist @ A @ B] :
( ( tLList942171283length @ A @ B @ ( tLList1881248882tdropn @ A @ B @ N @ Xs ) )
= ( minus_minus @ extended_enat @ ( tLList942171283length @ A @ B @ Xs ) @ ( extended_enat2 @ N ) ) ) ).
% tlength_tdropn
thf(fact_239_natural__zero__minus__one,axiom,
( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
= ( zero_zero @ code_natural ) ) ).
% natural_zero_minus_one
thf(fact_240_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
= N ) ).
% Suc_natural_minus_one
thf(fact_241_epred__conv__minus,axiom,
( coinductive_epred
= ( ^ [N4: extended_enat] : ( minus_minus @ extended_enat @ N4 @ ( one_one @ extended_enat ) ) ) ) ).
% epred_conv_minus
thf(fact_242_tnth__tdropn,axiom,
! [A: $tType,B: $tType,N: nat,M: nat,Xs: tLList446370796tllist @ A @ B] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N @ M ) ) @ ( tLList942171283length @ A @ B @ Xs ) )
=> ( ( tLList2115636725e_tnth @ A @ B @ ( tLList1881248882tdropn @ A @ B @ N @ Xs ) @ M )
= ( tLList2115636725e_tnth @ A @ B @ Xs @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% tnth_tdropn
thf(fact_243_tdropn__Suc__conv__tdropn,axiom,
! [B: $tType,A: $tType,N: nat,Xs: tLList446370796tllist @ A @ B] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( tLList942171283length @ A @ B @ Xs ) )
=> ( ( tLList1992840728_TCons @ A @ B @ ( tLList2115636725e_tnth @ A @ B @ Xs @ N ) @ ( tLList1881248882tdropn @ A @ B @ ( suc @ N ) @ Xs ) )
= ( tLList1881248882tdropn @ A @ B @ N @ Xs ) ) ) ).
% tdropn_Suc_conv_tdropn
thf(fact_244_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_245_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_246_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_247_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_248_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_249_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_250_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B3 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_251_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
%----Type constructors (56)
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Nat_Osize_1,axiom,
size @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_2,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_3,axiom,
! [A6: $tType] : ( size @ ( option @ A6 ) @ ( type2 @ ( option @ A6 ) ) ) ).
thf(tcon_String_Oliteral___Nat_Osize_4,axiom,
size @ literal @ ( type2 @ literal ) ).
thf(tcon_Typerep_Otyperep___Nat_Osize_5,axiom,
size @ typerep @ ( type2 @ typerep ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_6,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_7,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_8,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_9,axiom,
comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_10,axiom,
comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_11,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_12,axiom,
zero_neq_one @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_13,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_14,axiom,
semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_15,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_16,axiom,
one @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Nat_Osize_17,axiom,
size @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_18,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ).
thf(tcon_Product__Type_Ounit___Nat_Osize_19,axiom,
size @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_20,axiom,
ordere516151231imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_21,axiom,
ordere236663937imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_22,axiom,
ordere779506340up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_23,axiom,
cancel146912293up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_24,axiom,
cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_25,axiom,
cancel_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_26,axiom,
comm_monoid_diff @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_27,axiom,
ab_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_28,axiom,
comm_monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_29,axiom,
comm_semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_30,axiom,
semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_31,axiom,
zero_neq_one @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_32,axiom,
monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_33,axiom,
semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_34,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_35,axiom,
one @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_36,axiom,
size @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Record_Otuple__isomorphism___Nat_Osize_37,axiom,
! [A6: $tType,A7: $tType,A8: $tType] : ( size @ ( tuple_isomorphism @ A6 @ A7 @ A8 ) @ ( type2 @ ( tuple_isomorphism @ A6 @ A7 @ A8 ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $true @ X2 @ Y3 )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
! [X: a,Y: b] :
( ~ ( b2 @ X @ Y )
| ! [Xa2: c > $o,Ya: d > $o] :
( ? [Xb3: c,Yb: d] :
( ( a2 @ Xb3 @ Yb )
& ( ( Xa2 @ Xb3 )
!= ( Ya @ Yb ) ) )
| ! [Xb: tLList446370796tllist @ c @ a,Yb2: tLList446370796tllist @ d @ b] :
( ~ ( tLList1380991092t_all2 @ c @ d @ a @ b @ a2 @ b2 @ Xb @ Yb2 )
| ( tLList1380991092t_all2 @ c @ d @ a @ b @ a2 @ b2 @ ( tLList1813626245filter @ a @ c @ X @ Xa2 @ Xb ) @ ( tLList1813626245filter @ b @ d @ Y @ Ya @ Yb2 ) ) ) ) ) ).
%------------------------------------------------------------------------------