TPTP Problem File: DAT122^1.p
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%------------------------------------------------------------------------------
% File : DAT122^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 1876
% 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 : coinductive_list__1876.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.1.0
% Syntax : Number of formulae : 402 ( 206 unt; 73 typ; 0 def)
% Number of atoms : 717 ( 340 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4016 ( 156 ~; 40 |; 81 &;3506 @)
% ( 0 <=>; 233 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 303 ( 303 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 65 usr; 4 con; 0-6 aty)
% Number of variables : 1090 ( 36 ^; 955 !; 39 ?;1090 :)
% ( 60 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:53:07.332
%------------------------------------------------------------------------------
%----Could-be-implicit typings (15)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: $tType > $tType > $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $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_a,type,
a: $tType ).
%----Explicit typings (58)
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Divides_Osemiring__numeral__div,type,
semiring_numeral_div:
!>[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__Mirabelle__kmikjhschf_Oiterates,type,
coindu1529956631erates:
!>[A: $tType] : ( ( A > A ) > A > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord,type,
coindu300403952lexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocase__llist,type,
coindu882539134_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coindu1593790203_llist @ A ) > B ) > ( coindu1593790203_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ollist__all2,type,
coindu987187967t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olmap,type,
coindu1062782156e_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
coindu1802687541_ltake:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
coindu721411036eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olzip,type,
coindu847746867e_lzip:
!>[A: $tType,B: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > ( coindu1593790203_llist @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
coindu1599971794_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_llist @ B ) ) ).
thf(sy_c_Coinductive__Nat_Oco_Oenat_Ocase__enat,type,
coindu440805660e_enat:
!>[A: $tType] : ( A > ( extended_enat > A ) > extended_enat > 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_Divides_Osemiring__numeral__div__class_Odivides__aux,type,
semiri577515795es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
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_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
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_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
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 ) ).
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_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > 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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
%----Relevant facts (241)
thf(fact_0_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_1_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_2_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_3_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
& ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_4_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_5_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_6_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_7_ltakeWhile_Osimps_I3_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.simps(3)
thf(fact_8_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_9_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_10_lzip_Odisc__iff_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ~ ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
& ~ ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.disc_iff(2)
thf(fact_11_lzip_Odisc__iff_I1_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.disc_iff(1)
thf(fact_12_Coinductive__List__Mirabelle__kmikjhschf_Oiterates_Osimps_I2_J,axiom,
! [A: $tType,F: A > A,X: A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1529956631erates @ A @ F @ X ) )
= X ) ).
% Coinductive_List_Mirabelle_kmikjhschf.iterates.simps(2)
thf(fact_13_ltake_Oexhaust,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ~ ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ~ ( ( N
!= ( zero_zero @ extended_enat ) )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.exhaust
thf(fact_14_lzip_Odisc_I1_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% lzip.disc(1)
thf(fact_15_lzip_Odisc_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ~ ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) ) ) ) ).
% lzip.disc(2)
thf(fact_16_iterates_Odisc__iff,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coindu335574135_lnull @ A @ ( coindu1529956631erates @ A @ F @ X ) ) ).
% iterates.disc_iff
thf(fact_17_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_18_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_19_lzip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( ( coindu1046438764le_lhd @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ B @ Ys ) ) ) ) ) ).
% lzip.simps(3)
thf(fact_20_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_21_ltakeWhile_Osimps_I4_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% ltakeWhile.simps(4)
thf(fact_22_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_23_lnth__0,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% lnth_0
thf(fact_24_llist_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( F @ ( coindu1046438764le_lhd @ A @ A2 ) ) ) ) ).
% llist.map_sel(1)
thf(fact_25_lhd__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_26_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F2: nat > T > T] :
( ( rec_nat @ T @ F1 @ F2 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_27_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_28_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_29_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_30_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_31_llist_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( coindu335574135_lnull @ A @ A2 ) ) ).
% llist.map_disc_iff
thf(fact_32_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_33_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_34_ltl__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1047225960le_ltl @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu1047225960le_ltl @ B @ Xs ) ) ) ).
% ltl_lmap
thf(fact_35_ltl__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1047225960le_ltl @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_36_lhd__LCons__ltl,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) )
= Llist ) ) ).
% lhd_LCons_ltl
thf(fact_37_eq__LConsD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
& ( ( coindu1046438764le_lhd @ A @ Xs )
= Y )
& ( ( coindu1047225960le_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_38_llist_Oexhaust__sel,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
!= ( coindu1598213697e_LNil @ A ) )
=> ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ).
% llist.exhaust_sel
thf(fact_39_lappend__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys )
= ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_40_llist_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1047225960le_ltl @ A @ A2 ) ) ) ) ).
% llist.map_sel(2)
thf(fact_41_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_43_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_44_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_46_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Ys: coindu1593790203_llist @ B] :
( ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu268472904append @ B @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) @ ( coindu1062782156e_lmap @ B @ A @ F @ Ys ) ) ) ).
% lmap_lappend_distrib
thf(fact_47_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_48_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_49_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ? [X2: B,Xs2: coindu1593790203_llist @ B] :
( ( Xs
= ( coindu1121789889_LCons @ B @ X2 @ Xs2 ) )
& ( Y
= ( F @ X2 ) )
& ( Ys
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_50_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_51_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X2: A,Xs2: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_52_llist_Oexhaust,axiom,
! [A: $tType,Y: coindu1593790203_llist @ A] :
( ( Y
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
( Y
!= ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_53_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_54_lmap__eq__LNil,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% lmap_eq_LNil
thf(fact_55_LNil__eq__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% LNil_eq_lmap
thf(fact_56_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_57_llist_Osimps_I12_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ B ) ) ).
% llist.simps(12)
thf(fact_58_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( coindu1121789889_LCons @ B @ ( F @ X21 ) @ ( coindu1062782156e_lmap @ A @ B @ F @ X22 ) ) ) ).
% llist.simps(13)
thf(fact_59_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_60_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_61_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_62_lzip_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu847746867e_lzip @ A @ B )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ B] :
( if @ ( coindu1593790203_llist @ ( product_prod @ A @ B ) )
@ ( ( coindu335574135_lnull @ A @ Xs3 )
| ( coindu335574135_lnull @ B @ Ys2 ) )
@ ( coindu1598213697e_LNil @ ( product_prod @ A @ B ) )
@ ( coindu1121789889_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ ( coindu1046438764le_lhd @ B @ Ys2 ) ) @ ( coindu847746867e_lzip @ A @ B @ ( coindu1047225960le_ltl @ A @ Xs3 ) @ ( coindu1047225960le_ltl @ B @ Ys2 ) ) ) ) ) ) ).
% lzip.code
thf(fact_63_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu721411036eWhile @ A )
= ( ^ [P2: A > $o,Xs3: coindu1593790203_llist @ A] :
( if @ ( coindu1593790203_llist @ A )
@ ( ( coindu335574135_lnull @ A @ Xs3 )
| ~ ( P2 @ ( coindu1046438764le_lhd @ A @ Xs3 ) ) )
@ ( coindu1598213697e_LNil @ A )
@ ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ ( coindu721411036eWhile @ A @ P2 @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_64_lzip_Octr_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( ( coindu847746867e_lzip @ A @ B @ Xs @ Ys )
= ( coindu1121789889_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ B @ Ys ) ) @ ( coindu847746867e_lzip @ A @ B @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ B @ Ys ) ) ) ) ) ) ).
% lzip.ctr(2)
thf(fact_65_lzip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( ( coindu1047225960le_ltl @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( coindu847746867e_lzip @ A @ B @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ B @ Ys ) ) ) ) ) ).
% lzip.simps(4)
thf(fact_66_lnull__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_67_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_68_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_69_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs2: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% not_lnull_conv
thf(fact_70_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_71_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_72_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_73_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_74_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_75_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_76_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_77_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_78_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_79_Coinductive__List__Mirabelle__kmikjhschf_Oiterates_Osimps_I3_J,axiom,
! [A: $tType,F: A > A,X: A] :
( ( coindu1047225960le_ltl @ A @ ( coindu1529956631erates @ A @ F @ X ) )
= ( coindu1529956631erates @ A @ F @ ( F @ X ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.iterates.simps(3)
thf(fact_80_iterates_Ocode,axiom,
! [A: $tType] :
( ( coindu1529956631erates @ A )
= ( ^ [F3: A > A,X2: A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1529956631erates @ A @ F3 @ ( F3 @ X2 ) ) ) ) ) ).
% iterates.code
thf(fact_81_ltakeWhile_Octr_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltakeWhile.ctr(2)
thf(fact_82_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist3 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_83_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_84_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist4 )
= ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_85_lzip_Octr_I1_J,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ( ( coindu847746867e_lzip @ A @ B @ Xs @ Ys )
= ( coindu1598213697e_LNil @ ( product_prod @ A @ B ) ) ) ) ).
% lzip.ctr(1)
thf(fact_86_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_87_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_88_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F2 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_89_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
=> ( P @ ( F2 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_90_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F2 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_91_unfold__llist__id,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1599971794_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu335574135_lnull @ A ) @ ( coindu1046438764le_lhd @ A ) @ ( coindu1047225960le_ltl @ A ) @ Xs )
= Xs ) ).
% unfold_llist_id
thf(fact_92_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F2 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_93_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu882539134_llist @ B @ A )
= ( ^ [F12: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist2: coindu1593790203_llist @ A] : ( if @ B @ ( coindu335574135_lnull @ A @ Llist2 ) @ F12 @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist2 ) @ ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_94_divides__aux__eq,axiom,
! [A: $tType] :
( ( semiring_numeral_div @ A @ ( type2 @ A ) )
=> ! [Q2: A,R2: A] :
( ( semiri577515795es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_95_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) )
= ( P3 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_96_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P3 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_97_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_98_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P3 @ A2 )
=> ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_99_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_100_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 )
= ( coindu1121789889_LCons @ B @ ( G21 @ A2 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_101_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P3 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_102_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) )
= ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_103_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_104_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_105_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_106_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P4: A > $o,G212: A > B,G222: A > A,A5: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P4 @ A5 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A5 ) @ ( coindu1599971794_llist @ A @ B @ P4 @ G212 @ G222 @ ( G222 @ A5 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_107_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1598213697e_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_108_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_109_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_110_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_111_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_112_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_113_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_114_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D,E2: E,F5: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_115_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D,E2: E,F5: F4,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F5 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_116_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_117_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_118_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( 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 ) ) )] :
( ! [A6: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_119_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A6: A,B4: B,C2: C,D2: D,E2: E,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_120_prod__induct7,axiom,
! [G2: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
( ! [A6: A,B4: B,C2: C,D2: D,E2: E,F5: F4,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F5 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_121_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_122_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_123_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 ) ) ).
% old.prod.rec
thf(fact_124_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_125_llexord_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ A1 @ A22 )
=> ( ! [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X3: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys3 ) )
=> ~ ( coindu300403952lexord @ A @ R2 @ Xs4 @ Ys3 ) ) )
=> ( ! [X3: A] :
( ? [Xs4: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
=> ! [Y3: A] :
( ? [Ys3: coindu1593790203_llist @ A] :
( A22
= ( coindu1121789889_LCons @ A @ Y3 @ Ys3 ) )
=> ~ ( R2 @ X3 @ Y3 ) ) )
=> ~ ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys3: coindu1593790203_llist @ A] : A22 != Ys3 ) ) ) ) ).
% llexord.cases
thf(fact_126_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R3: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys2 ) )
& ( coindu300403952lexord @ A @ R3 @ Xs3 @ Ys2 ) )
| ? [X2: A,Y4: A,Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ Y4 @ Ys2 ) )
& ( R3 @ X2 @ Y4 ) )
| ? [Ys2: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Ys2 ) ) ) ) ) ).
% llexord.simps
thf(fact_127_llexord_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R2: A > A > $o] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X4 @ X3 @ Xa2 )
=> ( ? [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X3
= ( coindu1121789889_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X4 @ Xs5 @ Ys4 )
| ( coindu300403952lexord @ A @ R2 @ Xs5 @ Ys4 ) ) )
| ? [Xb: A,Y5: A,Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( X3
= ( coindu1121789889_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Y5 @ Ys4 ) )
& ( R2 @ Xb @ Y5 ) )
| ? [Ys4: coindu1593790203_llist @ A] :
( ( X3
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Ys4 ) ) ) )
=> ( coindu300403952lexord @ A @ R2 @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_128_llexord__LCons__less,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R2 @ X @ Y )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_129_llexord__LCons__eq,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_130_llexord__LNil,axiom,
! [A: $tType,R2: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_131_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F2 @ ( code_Suc @ Natural ) )
= ( F2 @ Natural ) ) ).
% natural.simps(5)
thf(fact_132_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_133_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_134_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_135_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F2 @ ( code_Suc @ Natural ) )
= ( F2 @ Natural @ ( code_rec_natural @ T @ F1 @ F2 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_136_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_137_natural_Oexhaust,axiom,
! [Y: code_natural] :
( ( Y
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural3: code_natural] :
( Y
!= ( code_Suc @ Natural3 ) ) ) ).
% natural.exhaust
thf(fact_138_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_139_natural_Odistinct_I2_J,axiom,
! [Natural4: code_natural] :
( ( code_Suc @ Natural4 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_140_random__aux__set_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
( ! [J: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( zero_zero @ code_natural ) @ J ) )
=> ~ ! [I2: code_natural,J: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( code_Suc @ I2 ) @ J ) ) ) ).
% random_aux_set.cases
thf(fact_141_random__aux__rec,axiom,
! [A: $tType,Random_aux: code_natural > A,Rhs: code_natural > A,K: code_natural] :
( ( ( Random_aux @ ( zero_zero @ code_natural ) )
= ( Rhs @ ( zero_zero @ code_natural ) ) )
=> ( ! [K2: code_natural] :
( ( Random_aux @ ( code_Suc @ K2 ) )
= ( Rhs @ ( code_Suc @ K2 ) ) )
=> ( ( Random_aux @ K )
= ( Rhs @ K ) ) ) ) ).
% random_aux_rec
thf(fact_142_random__aux__set_Oinduct,axiom,
! [B: $tType] :
( ( quickcheck_random @ B @ ( type2 @ B ) )
=> ! [P: code_natural > code_natural > $o,A0: code_natural,A1: code_natural] :
( ! [X12: code_natural] : ( P @ ( zero_zero @ code_natural ) @ X12 )
=> ( ! [I2: code_natural,J: code_natural] :
( ! [X5: product_prod @ B @ ( product_unit > code_term )] : ( P @ I2 @ J )
=> ( P @ ( code_Suc @ I2 ) @ J ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% random_aux_set.induct
thf(fact_143_case__enat__0,axiom,
! [A: $tType,F: nat > A,I: A] :
( ( extended_case_enat @ A @ F @ I @ ( zero_zero @ extended_enat ) )
= ( F @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_144_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_145_size__ne__size__imp__ne,axiom,
! [A: $tType] :
( ( size @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_ne_size_imp_ne
thf(fact_146_lprefix__expand,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ Ys )
& ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) )
& ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ A @ Ys ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lprefix_expand
thf(fact_147_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_148_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_149_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_150_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_151_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_152_lnull__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_153_lprefix__lnull,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_154_lprefix__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_155_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_156_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys5: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys5 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys5 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_157_Le__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_158_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_159_lprefix__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ A @ Ys ) ) ) ).
% lprefix_ltlI
thf(fact_160_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_161_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_162_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_163_llist_Oleq__trans,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ Z )
=> ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_164_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_165_llist_Oleq__antisym,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_166_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_167_lprefixI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X4: set @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) )] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs @ Ys ) @ X4 )
=> ( ! [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs4 @ Ys3 ) @ X4 )
=> ( ( coindu335574135_lnull @ A @ Xs4 )
| ? [X5: A,Xs6: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A] :
( ( Xs4
= ( coindu1121789889_LCons @ A @ X5 @ Xs6 ) )
& ( Ys3
= ( coindu1121789889_LCons @ A @ X5 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs6 @ Ys6 ) @ X4 )
| ( coindu1696667936prefix @ A @ Xs6 @ Ys6 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_168_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_169_lprefix_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Xs4: coindu1593790203_llist @ A] : A22 != Xs4 )
=> ~ ! [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X3: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys3 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs4 @ Ys3 ) ) ) ) ) ).
% lprefix.cases
thf(fact_170_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Xs3 ) )
| ? [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys2 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys2 ) ) ) ) ) ).
% lprefix.simps
thf(fact_171_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X4 @ X3 @ Xa2 )
=> ( ? [Xs5: coindu1593790203_llist @ A] :
( ( X3
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs5 ) )
| ? [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X3
= ( coindu1121789889_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X4 @ Xs5 @ Ys4 )
| ( coindu1696667936prefix @ A @ Xs5 @ Ys4 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_172_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs2 ) )
& ( coindu1696667936prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_173_lprefix__lhdD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) ) ) ) ).
% lprefix_lhdD
thf(fact_174_lprefix__coinduct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( P @ Xs @ Ys )
=> ( ! [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( P @ Xs4 @ Ys3 )
=> ( ( ( coindu335574135_lnull @ A @ Ys3 )
=> ( coindu335574135_lnull @ A @ Xs4 ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs4 )
=> ( ~ ( coindu335574135_lnull @ A @ Ys3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Xs4 )
= ( coindu1046438764le_lhd @ A @ Ys3 ) )
& ( ( P @ ( coindu1047225960le_ltl @ A @ Xs4 ) @ ( coindu1047225960le_ltl @ A @ Ys3 ) )
| ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs4 ) @ ( coindu1047225960le_ltl @ A @ Ys3 ) ) ) ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_175_bool_Osize_I4_J,axiom,
( ( size_size @ $o @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(4)
thf(fact_176_bool_Osize_I3_J,axiom,
( ( size_size @ $o @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(3)
thf(fact_177_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs3 @ Ys2 )
& ( Xs3 != Ys2 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_178_size__bool,axiom,
( ( size_size @ $o )
= ( ^ [B5: $o] : ( zero_zero @ nat ) ) ) ).
% size_bool
thf(fact_179_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_180_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_181_bool_Osize_I2_J,axiom,
( ( size_bool @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(2)
thf(fact_182_bool_Osize_I1_J,axiom,
( ( size_bool @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(1)
thf(fact_183_size__bool__overloaded__def,axiom,
( ( size_size @ $o )
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_overloaded_def
thf(fact_184_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F2: T] :
( ( product_rec_bool @ T @ F1 @ F2 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_185_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F2: T] :
( ( product_rec_bool @ T @ F1 @ F2 @ $false )
= F2 ) ).
% old.bool.simps(6)
thf(fact_186_size__bool__def,axiom,
( size_bool
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_def
thf(fact_187_tuple__isomorphism_Osize__neq,axiom,
! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism @ A @ B @ C] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ X )
!= ( zero_zero @ nat ) ) ).
% tuple_isomorphism.size_neq
thf(fact_188_size__literal__def,axiom,
( ( size_size @ literal )
= ( ^ [S: literal] : ( zero_zero @ nat ) ) ) ).
% size_literal_def
thf(fact_189_typerep_Osize__neq,axiom,
! [X: typerep] :
( ( size_size @ typerep @ X )
!= ( zero_zero @ nat ) ) ).
% typerep.size_neq
thf(fact_190_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_191_size__unit__overloaded__def,axiom,
( ( size_size @ product_unit )
= ( product_rec_unit @ nat @ ( zero_zero @ nat ) ) ) ).
% size_unit_overloaded_def
thf(fact_192_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_193_size__unit__def,axiom,
( product_size_unit
= ( product_rec_unit @ nat @ ( zero_zero @ nat ) ) ) ).
% size_unit_def
thf(fact_194_unit_Osize_I2_J,axiom,
( ( size_size @ product_unit @ product_Unity )
= ( zero_zero @ nat ) ) ).
% unit.size(2)
thf(fact_195_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_196_unit_Osize_I1_J,axiom,
( ( product_size_unit @ product_Unity )
= ( zero_zero @ nat ) ) ).
% unit.size(1)
thf(fact_197_old_Ounit_Oinducts,axiom,
! [P: product_unit > $o,Unit: product_unit] :
( ( P @ product_Unity )
=> ( P @ Unit ) ) ).
% old.unit.inducts
thf(fact_198_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
thf(fact_199_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_200_old_Ounit_Ocase,axiom,
! [A: $tType,F: A] :
( ( product_case_unit @ A @ F @ product_Unity )
= F ) ).
% old.unit.case
thf(fact_201_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F: A,Unit: product_unit] :
( ( H @ ( product_case_unit @ A @ F @ Unit ) )
= ( product_case_unit @ B @ ( H @ F ) @ Unit ) ) ).
% unit.case_distrib
thf(fact_202_ltake_Osimps_I3_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltake.simps(3)
thf(fact_203_llist_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( coindu987187967t_all2 @ A @ B )
= ( ^ [R4: A > B > $o,A5: coindu1593790203_llist @ A,B5: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ A5 )
= ( coindu335574135_lnull @ B @ B5 ) )
& ( ~ ( coindu335574135_lnull @ A @ A5 )
=> ( ~ ( coindu335574135_lnull @ B @ B5 )
=> ( ( R4 @ ( coindu1046438764le_lhd @ A @ A5 ) @ ( coindu1046438764le_lhd @ B @ B5 ) )
& ( coindu987187967t_all2 @ A @ B @ R4 @ ( coindu1047225960le_ltl @ A @ A5 ) @ ( coindu1047225960le_ltl @ B @ B5 ) ) ) ) ) ) ) ) ).
% llist.rel_sel
thf(fact_204_ltake__is__lprefix,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ Xs ) ).
% ltake_is_lprefix
thf(fact_205_ltake__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_LNil
thf(fact_206_llist_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: coindu1593790203_llist @ A,Y21: B,Y22: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% llist.rel_inject(2)
thf(fact_207_ltake__lmap,axiom,
! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu1802687541_ltake @ B @ N @ Xs ) ) ) ).
% ltake_lmap
thf(fact_208_lnull__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% lnull_ltake
thf(fact_209_ltake_Odisc__iff_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.disc_iff(1)
thf(fact_210_ltake_Odisc__iff_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) )
= ( ( N
!= ( zero_zero @ extended_enat ) )
& ~ ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.disc_iff(2)
thf(fact_211_LNil__eq__ltake__iff,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% LNil_eq_ltake_iff
thf(fact_212_ltake__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( zero_zero @ extended_enat ) @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_0
thf(fact_213_lhd__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ).
% lhd_ltake
thf(fact_214_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: coindu1593790203_llist @ A,B2: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ A2 @ B2 )
=> ( ( ( A2
= ( coindu1598213697e_LNil @ A ) )
=> ( B2
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X12: A,X24: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X12 @ X24 ) )
=> ! [Y12: B,Y23: coindu1593790203_llist @ B] :
( ( B2
= ( coindu1121789889_LCons @ B @ Y12 @ Y23 ) )
=> ( ( R @ X12 @ Y12 )
=> ~ ( coindu987187967t_all2 @ A @ B @ R @ X24 @ Y23 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_215_llist_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: coindu1593790203_llist @ B] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ).
% llist.rel_distinct(1)
thf(fact_216_llist_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: coindu1593790203_llist @ A] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.rel_distinct(2)
thf(fact_217_ltake_Odisc_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ) ).
% ltake.disc(2)
thf(fact_218_ltake_Odisc_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake.disc(1)
thf(fact_219_ltake__eq__LNil__iff,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% ltake_eq_LNil_iff
thf(fact_220_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: coindu1593790203_llist @ A,Y22: coindu1593790203_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_221_llist_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.ctr_transfer(1)
thf(fact_222_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu987187967t_all2 @ A @ A
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [Y6: coindu1593790203_llist @ A,Z2: coindu1593790203_llist @ A] : Y6 = Z2 ) ) ).
% llist.rel_eq
thf(fact_223_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coindu1593790203_llist @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( coindu987187967t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_224_ltake_Octr_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltake.ctr(1)
thf(fact_225_llist_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,R: A > B > $o] :
( ( P @ X @ Y )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ B] :
( ( P @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ B @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ B @ Llist5 )
=> ( ( R @ ( coindu1046438764le_lhd @ A @ Llist4 ) @ ( coindu1046438764le_lhd @ B @ Llist5 ) )
& ( P @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ B @ Llist5 ) ) ) ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% llist.rel_coinduct
thf(fact_226_ltake_Ocode,axiom,
! [A: $tType] :
( ( coindu1802687541_ltake @ A )
= ( ^ [N2: extended_enat,Xs3: coindu1593790203_llist @ A] :
( if @ ( coindu1593790203_llist @ A )
@ ( ( N2
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs3 ) )
@ ( coindu1598213697e_LNil @ A )
@ ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ ( coindu1802687541_ltake @ A @ ( coinductive_epred @ N2 ) @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) ) ) ) ) ).
% ltake.code
thf(fact_227_ltake_Octr_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1802687541_ltake @ A @ ( coinductive_epred @ N ) @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltake.ctr(2)
thf(fact_228_ltake__epred__ltl,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( coinductive_epred @ N ) @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1047225960le_ltl @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake_epred_ltl
thf(fact_229_epred__0,axiom,
( ( coinductive_epred @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% epred_0
thf(fact_230_epred__inject,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X
!= ( zero_zero @ extended_enat ) )
=> ( ( Y
!= ( zero_zero @ extended_enat ) )
=> ( ( ( coinductive_epred @ X )
= ( coinductive_epred @ Y ) )
= ( X = Y ) ) ) ) ).
% epred_inject
thf(fact_231_enat__coinduct,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N3: extended_enat] :
( ( P @ M2 @ N3 )
=> ( ( ( M2
= ( zero_zero @ extended_enat ) )
= ( N3
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N3
!= ( zero_zero @ extended_enat ) )
=> ( ( P @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N3 ) )
| ( ( coinductive_epred @ M2 )
= ( coinductive_epred @ N3 ) ) ) ) ) ) )
=> ( M = N ) ) ) ).
% enat_coinduct
thf(fact_232_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_233_enat__coinduct2,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N3: extended_enat] :
( ( P @ M2 @ N3 )
=> ( ( M2
= ( zero_zero @ extended_enat ) )
= ( N3
= ( zero_zero @ extended_enat ) ) ) )
=> ( ! [M2: extended_enat,N3: extended_enat] :
( ( P @ M2 @ N3 )
=> ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N3
!= ( zero_zero @ extended_enat ) )
=> ( ( P @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N3 ) )
| ( ( coinductive_epred @ M2 )
= ( coinductive_epred @ N3 ) ) ) ) ) )
=> ( M = N ) ) ) ) ).
% enat_coinduct2
thf(fact_234_ltake_Osimps_I4_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1802687541_ltake @ A @ ( coinductive_epred @ N ) @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% ltake.simps(4)
thf(fact_235_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_236_co_Oenat_Ocase__eq__if,axiom,
! [A: $tType] :
( ( coindu440805660e_enat @ A )
= ( ^ [F12: A,F22: extended_enat > A,Enat3: extended_enat] :
( if @ A
@ ( Enat3
= ( zero_zero @ extended_enat ) )
@ F12
@ ( F22 @ ( coinductive_epred @ Enat3 ) ) ) ) ) ).
% co.enat.case_eq_if
thf(fact_237_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_238_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_239_enat__cocase__0,axiom,
! [A: $tType,Z: A,S2: extended_enat > A] :
( ( coindu440805660e_enat @ A @ Z @ S2 @ ( zero_zero @ extended_enat ) )
= Z ) ).
% enat_cocase_0
thf(fact_240_co_Oenat_Ocase_I1_J,axiom,
! [A: $tType,F1: A,F2: extended_enat > A] :
( ( coindu440805660e_enat @ A @ F1 @ F2 @ ( zero_zero @ extended_enat ) )
= F1 ) ).
% co.enat.case(1)
%----Type constructors (83)
thf(tcon_Coinductive__List__Mirabelle__kmikjhschf_Ollist___Code__Evaluation_Oterm__of,axiom,
! [A7: $tType] :
( ( typerep2 @ A7 @ ( type2 @ A7 ) )
=> ( code_term_of @ ( coindu1593790203_llist @ A7 ) @ ( type2 @ ( coindu1593790203_llist @ A7 ) ) ) ) ).
thf(tcon_Coinductive__List__Mirabelle__kmikjhschf_Ollist___HOL_Oequal,axiom,
! [A7: $tType] : ( cl_HOL_Oequal @ ( coindu1593790203_llist @ A7 ) @ ( type2 @ ( coindu1593790203_llist @ A7 ) ) ) ).
thf(tcon_Record_Otuple__isomorphism___Code__Evaluation_Oterm__of_1,axiom,
! [A7: $tType,A8: $tType,A9: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) )
& ( typerep2 @ A9 @ ( type2 @ A9 ) ) )
=> ( code_term_of @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) @ ( type2 @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) ) ) ) ).
thf(tcon_Record_Otuple__isomorphism___HOL_Oequal_2,axiom,
! [A7: $tType,A8: $tType,A9: $tType] : ( cl_HOL_Oequal @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) @ ( type2 @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) ) ) ).
thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_3,axiom,
code_term_of @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_4,axiom,
cl_HOL_Oequal @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
code_term_of @ code_term @ ( type2 @ code_term ) ).
thf(tcon_Code__Evaluation_Oterm___HOL_Oequal_6,axiom,
cl_HOL_Oequal @ code_term @ ( type2 @ code_term ) ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_7,axiom,
code_term_of @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_8,axiom,
cl_HOL_Oequal @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_9,axiom,
! [A7: $tType,A8: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) ) )
=> ( code_term_of @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_10,axiom,
! [A7: $tType,A8: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_11,axiom,
code_term_of @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___HOL_Oequal_12,axiom,
cl_HOL_Oequal @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_13,axiom,
code_term_of @ typerep @ ( type2 @ typerep ) ).
thf(tcon_Typerep_Otyperep___HOL_Oequal_14,axiom,
cl_HOL_Oequal @ typerep @ ( type2 @ typerep ) ).
thf(tcon_String_Oliteral___Code__Evaluation_Oterm__of_15,axiom,
code_term_of @ literal @ ( type2 @ literal ) ).
thf(tcon_String_Oliteral___HOL_Oequal_16,axiom,
cl_HOL_Oequal @ literal @ ( type2 @ literal ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_17,axiom,
! [A7: $tType] :
( ( typerep2 @ A7 @ ( type2 @ A7 ) )
=> ( code_term_of @ ( option @ A7 ) @ ( type2 @ ( option @ A7 ) ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_18,axiom,
! [A7: $tType] : ( cl_HOL_Oequal @ ( option @ A7 ) @ ( type2 @ ( option @ A7 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_19,axiom,
! [A7: $tType,A8: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) ) )
=> ( code_term_of @ ( sum_sum @ A7 @ A8 ) @ ( type2 @ ( sum_sum @ A7 @ A8 ) ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_20,axiom,
! [A7: $tType,A8: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A7 @ A8 ) @ ( type2 @ ( sum_sum @ A7 @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_21,axiom,
code_term_of @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___HOL_Oequal_22,axiom,
cl_HOL_Oequal @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_23,axiom,
! [A7: $tType] :
( ( typerep2 @ A7 @ ( type2 @ A7 ) )
=> ( code_term_of @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_24,axiom,
! [A7: $tType] :
( ( cl_HOL_Oequal @ A7 @ ( type2 @ A7 ) )
=> ( cl_HOL_Oequal @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_25,axiom,
code_term_of @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___HOL_Oequal_26,axiom,
cl_HOL_Oequal @ nat @ ( type2 @ nat ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_27,axiom,
! [A7: $tType,A8: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) ) )
=> ( code_term_of @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___HOL_Oequal_28,axiom,
! [A7: $tType,A8: $tType] :
( ( ( enum @ A7 @ ( type2 @ A7 ) )
& ( cl_HOL_Oequal @ A8 @ ( type2 @ A8 ) ) )
=> ( cl_HOL_Oequal @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Typerep_Otyperep,axiom,
! [A7: $tType,A8: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) ) )
=> ( typerep2 @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Enum_Oenum,axiom,
! [A7: $tType,A8: $tType] :
( ( ( enum @ A7 @ ( type2 @ A7 ) )
& ( enum @ A8 @ ( type2 @ A8 ) ) )
=> ( enum @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Typerep_Otyperep_29,axiom,
typerep2 @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_30,axiom,
! [A7: $tType] :
( ( typerep2 @ A7 @ ( type2 @ A7 ) )
=> ( typerep2 @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_31,axiom,
! [A7: $tType] :
( ( enum @ A7 @ ( type2 @ A7 ) )
=> ( enum @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
thf(tcon_HOL_Obool___Typerep_Otyperep_32,axiom,
typerep2 @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Enum_Oenum_33,axiom,
enum @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_34,axiom,
! [A7: $tType,A8: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) ) )
=> ( typerep2 @ ( sum_sum @ A7 @ A8 ) @ ( type2 @ ( sum_sum @ A7 @ A8 ) ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_35,axiom,
! [A7: $tType,A8: $tType] :
( ( ( enum @ A7 @ ( type2 @ A7 ) )
& ( enum @ A8 @ ( type2 @ A8 ) ) )
=> ( enum @ ( sum_sum @ A7 @ A8 ) @ ( type2 @ ( sum_sum @ A7 @ A8 ) ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_36,axiom,
! [A7: $tType] :
( ( typerep2 @ A7 @ ( type2 @ A7 ) )
=> ( typerep2 @ ( option @ A7 ) @ ( type2 @ ( option @ A7 ) ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_37,axiom,
! [A7: $tType] :
( ( enum @ A7 @ ( type2 @ A7 ) )
=> ( enum @ ( option @ A7 ) @ ( type2 @ ( option @ A7 ) ) ) ) ).
thf(tcon_String_Oliteral___Typerep_Otyperep_38,axiom,
typerep2 @ literal @ ( type2 @ literal ) ).
thf(tcon_Typerep_Otyperep___Typerep_Otyperep_39,axiom,
typerep2 @ typerep @ ( type2 @ typerep ) ).
thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_40,axiom,
typerep2 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_41,axiom,
! [A7: $tType,A8: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) ) )
=> ( typerep2 @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_42,axiom,
! [A7: $tType,A8: $tType] :
( ( ( enum @ A7 @ ( type2 @ A7 ) )
& ( enum @ A8 @ ( type2 @ A8 ) ) )
=> ( enum @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_43,axiom,
typerep2 @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___Enum_Oenum_44,axiom,
enum @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_45,axiom,
typerep2 @ code_term @ ( type2 @ code_term ) ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_46,axiom,
typerep2 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Record_Otuple__isomorphism___Typerep_Otyperep_47,axiom,
! [A7: $tType,A8: $tType,A9: $tType] :
( ( ( typerep2 @ A7 @ ( type2 @ A7 ) )
& ( typerep2 @ A8 @ ( type2 @ A8 ) )
& ( typerep2 @ A9 @ ( type2 @ A9 ) ) )
=> ( typerep2 @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) @ ( type2 @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) ) ) ) ).
thf(tcon_Coinductive__List__Mirabelle__kmikjhschf_Ollist___Typerep_Otyperep_48,axiom,
! [A7: $tType] :
( ( typerep2 @ A7 @ ( type2 @ A7 ) )
=> ( typerep2 @ ( coindu1593790203_llist @ A7 ) @ ( type2 @ ( coindu1593790203_llist @ A7 ) ) ) ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A7: $tType,A8: $tType] :
( ( ( code_term_of @ A7 @ ( type2 @ A7 ) )
& ( cl_HOL_Oequal @ A7 @ ( type2 @ A7 ) )
& ( quickcheck_random @ A8 @ ( type2 @ A8 ) ) )
=> ( quickcheck_random @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Divides_Osemiring__numeral__div,axiom,
semiring_numeral_div @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_49,axiom,
quickcheck_random @ 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___Nat_Osize,axiom,
size @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_50,axiom,
! [A7: $tType] :
( ( quickcheck_random @ A7 @ ( type2 @ A7 ) )
=> ( quickcheck_random @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_51,axiom,
quickcheck_random @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Nat_Osize_52,axiom,
size @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_53,axiom,
! [A7: $tType,A8: $tType] :
( ( ( quickcheck_random @ A7 @ ( type2 @ A7 ) )
& ( quickcheck_random @ A8 @ ( type2 @ A8 ) ) )
=> ( quickcheck_random @ ( sum_sum @ A7 @ A8 ) @ ( type2 @ ( sum_sum @ A7 @ A8 ) ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_54,axiom,
! [A7: $tType,A8: $tType] : ( size @ ( sum_sum @ A7 @ A8 ) @ ( type2 @ ( sum_sum @ A7 @ A8 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_55,axiom,
! [A7: $tType] :
( ( quickcheck_random @ A7 @ ( type2 @ A7 ) )
=> ( quickcheck_random @ ( option @ A7 ) @ ( type2 @ ( option @ A7 ) ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_56,axiom,
! [A7: $tType] : ( size @ ( option @ A7 ) @ ( type2 @ ( option @ A7 ) ) ) ).
thf(tcon_String_Oliteral___Quickcheck__Random_Orandom_57,axiom,
quickcheck_random @ literal @ ( type2 @ literal ) ).
thf(tcon_String_Oliteral___Nat_Osize_58,axiom,
size @ literal @ ( type2 @ literal ) ).
thf(tcon_Typerep_Otyperep___Nat_Osize_59,axiom,
size @ typerep @ ( type2 @ typerep ) ).
thf(tcon_Extended__Nat_Oenat___Quickcheck__Random_Orandom_60,axiom,
quickcheck_random @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_61,axiom,
semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_62,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Nat_Osize_63,axiom,
size @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_64,axiom,
! [A7: $tType,A8: $tType] :
( ( ( quickcheck_random @ A7 @ ( type2 @ A7 ) )
& ( quickcheck_random @ A8 @ ( type2 @ A8 ) ) )
=> ( quickcheck_random @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_65,axiom,
! [A7: $tType,A8: $tType] : ( size @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_66,axiom,
quickcheck_random @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___Nat_Osize_67,axiom,
size @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_68,axiom,
quickcheck_random @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_69,axiom,
semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_70,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_71,axiom,
size @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Record_Otuple__isomorphism___Quickcheck__Random_Orandom_72,axiom,
! [A7: $tType,A8: $tType,A9: $tType] :
( ( ( code_term_of @ A7 @ ( type2 @ A7 ) )
& ( cl_HOL_Oequal @ A7 @ ( type2 @ A7 ) )
& ( quickcheck_random @ A7 @ ( type2 @ A7 ) )
& ( quickcheck_random @ A8 @ ( type2 @ A8 ) )
& ( quickcheck_random @ A9 @ ( type2 @ A9 ) ) )
=> ( quickcheck_random @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) @ ( type2 @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) ) ) ) ).
thf(tcon_Record_Otuple__isomorphism___Nat_Osize_73,axiom,
! [A7: $tType,A8: $tType,A9: $tType] : ( size @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) @ ( type2 @ ( tuple_isomorphism @ A7 @ A8 @ A9 ) ) ) ).
thf(tcon_Coinductive__List__Mirabelle__kmikjhschf_Ollist___Quickcheck__Random_Orandom_74,axiom,
! [A7: $tType] :
( ( quickcheck_random @ A7 @ ( type2 @ A7 ) )
=> ( quickcheck_random @ ( coindu1593790203_llist @ A7 ) @ ( type2 @ ( coindu1593790203_llist @ A7 ) ) ) ) ).
%----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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
~ ( coindu335574135_lnull @ a @ xs ) ).
thf(conj_1,conjecture,
( ( coindu1046438764le_lhd @ a @ xs )
= ( coindu749330388e_lnth @ a @ xs @ ( zero_zero @ nat ) ) ) ).
%------------------------------------------------------------------------------