TPTP Problem File: DAT226^1.p
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%------------------------------------------------------------------------------
% File : DAT226^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Tllist 793
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : tllist__793.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 368 ( 173 unt; 77 typ; 0 def)
% Number of atoms : 567 ( 288 equ; 0 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 3045 ( 91 ~; 6 |; 27 &;2693 @)
% ( 0 <=>; 228 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 271 ( 271 >; 0 *; 0 +; 0 <<)
% Number of symbols : 75 ( 72 usr; 5 con; 0-10 aty)
% Number of variables : 904 ( 50 ^; 756 !; 19 ?; 904 :)
% ( 79 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:53:55.673
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist,type,
tLList446370796tllist: $tType > $tType > $tType ).
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (68)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olsublist,type,
coinductive_lsublist:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ nat ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
coindu780009021istsum:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Obool_Osize__bool,type,
size_bool: $o > nat ).
thf(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
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_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Ounit_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_TLList__Mirabelle__qhjoikztpd_Olappendt,type,
tLList98099029ppendt:
!>[A: $tType,B: $tType] : ( ( coinductive_llist @ A ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otconcat,type,
tLList365522113concat:
!>[B: $tType,A: $tType] : ( B > ( tLList446370796tllist @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otconcat_H,type,
tLList1329345514concat:
!>[B: $tType,A: $tType] : ( ( product_unit > B ) > ( tLList446370796tllist @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otdropn,type,
tLList1881248882tdropn:
!>[A: $tType,B: $tType] : ( nat > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otfilter,type,
tLList1813626245filter:
!>[B: $tType,A: $tType] : ( B > ( A > $o ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otfilter_H,type,
tLList1122525870filter:
!>[B: $tType,A: $tType] : ( ( product_unit > B ) > ( A > $o ) > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_OTCons,type,
tLList1992840728_TCons:
!>[A: $tType,B: $tType] : ( A > ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_OTNil,type,
tLList57907754e_TNil:
!>[B: $tType,A: $tType] : ( B > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocase__tllist,type,
tLList200813139tllist:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > ( tLList446370796tllist @ A @ B ) > C ) > ( tLList446370796tllist @ A @ B ) > C ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocorec__tllist,type,
tLList1614408749tllist:
!>[E: $tType,B: $tType,A: $tType] : ( ( E > $o ) > ( E > B ) > ( E > A ) > ( E > $o ) > ( E > ( tLList446370796tllist @ A @ B ) ) > ( E > E ) > E > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Opred__tllist,type,
tLList11265572tllist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( tLList446370796tllist @ A @ B ) > $o ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oset2__tllist,type,
tLList457864275tllist:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Otset,type,
tLList1719791291e_tset:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ottl,type,
tLList_Mirabelle_ttl:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( tLList446370796tllist @ A @ B ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otnth,type,
tLList2115636725e_tnth:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > nat > A ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ounfold__tllist,type,
tLList636831159tllist:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > $o ) > ( A > B ) > ( A > C ) > ( A > A ) > A > ( tLList446370796tllist @ C @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_xs,type,
xs: tLList446370796tllist @ a @ b ).
%----Relevant facts (252)
thf(fact_0_tllist_Oinject_I1_J,axiom,
! [A: $tType,B: $tType,X1: B,Y1: B] :
( ( ( tLList57907754e_TNil @ B @ A @ X1 )
= ( tLList57907754e_TNil @ B @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tllist.inject(1)
thf(fact_1_tdropn__TNil,axiom,
! [A: $tType,B: $tType,N: nat,B2: B] :
( ( tLList1881248882tdropn @ A @ B @ N @ ( tLList57907754e_TNil @ B @ A @ B2 ) )
= ( tLList57907754e_TNil @ B @ A @ B2 ) ) ).
% tdropn_TNil
thf(fact_2_tllist_Osimps_I5_J,axiom,
! [A: $tType,C: $tType,B: $tType,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,X1: B] :
( ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( F1 @ X1 ) ) ).
% tllist.simps(5)
thf(fact_3_tllist_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,Tllist: tLList446370796tllist @ A @ B] :
( ( H @ ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ Tllist ) )
= ( tLList200813139tllist @ B @ D @ A
@ ^ [X: B] : ( H @ ( F1 @ X ) )
@ ^ [X12: A,X2: tLList446370796tllist @ A @ B] : ( H @ ( F2 @ X12 @ X2 ) )
@ Tllist ) ) ).
% tllist.case_distrib
thf(fact_4_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_5_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_6_ttl__def,axiom,
! [B: $tType,A: $tType] :
( ( tLList_Mirabelle_ttl @ A @ B )
= ( tLList200813139tllist @ B @ ( tLList446370796tllist @ A @ B ) @ A @ ( tLList57907754e_TNil @ B @ A )
@ ^ [X21: A,X222: tLList446370796tllist @ A @ B] : X222 ) ) ).
% ttl_def
thf(fact_7_tconcat__TNil,axiom,
! [A: $tType,B: $tType,B2: B,B3: B] :
( ( tLList365522113concat @ B @ A @ B2 @ ( tLList57907754e_TNil @ B @ ( coinductive_llist @ A ) @ B3 ) )
= ( tLList57907754e_TNil @ B @ A @ B3 ) ) ).
% tconcat_TNil
thf(fact_8_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_9_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_10_tconcat_H__code_I1_J,axiom,
! [A: $tType,B: $tType,B2: product_unit > B,B3: B] :
( ( tLList1329345514concat @ B @ A @ B2 @ ( tLList57907754e_TNil @ B @ ( coinductive_llist @ A ) @ B3 ) )
= ( tLList57907754e_TNil @ B @ A @ B3 ) ) ).
% tconcat'_code(1)
thf(fact_11_tdropn__Suc__TCons,axiom,
! [B: $tType,A: $tType,N: nat,X3: A,Xs: tLList446370796tllist @ A @ B] :
( ( tLList1881248882tdropn @ A @ B @ ( suc @ N ) @ ( tLList1992840728_TCons @ A @ B @ X3 @ Xs ) )
= ( tLList1881248882tdropn @ A @ B @ N @ Xs ) ) ).
% tdropn_Suc_TCons
thf(fact_12_tfilter_H__code_I1_J,axiom,
! [A: $tType,B: $tType,B3: product_unit > B,P: A > $o,B2: B] :
( ( tLList1122525870filter @ B @ A @ B3 @ P @ ( tLList57907754e_TNil @ B @ A @ B2 ) )
= ( tLList57907754e_TNil @ B @ A @ B2 ) ) ).
% tfilter'_code(1)
thf(fact_13_tfilter__TNil,axiom,
! [A: $tType,B: $tType,B3: B,P: A > $o,B2: B] :
( ( tLList1813626245filter @ B @ A @ B3 @ P @ ( tLList57907754e_TNil @ B @ A @ B2 ) )
= ( tLList57907754e_TNil @ B @ A @ B2 ) ) ).
% tfilter_TNil
thf(fact_14_tllist_Opred__inject_I1_J,axiom,
! [A: $tType,B: $tType,P1: A > $o,P2: B > $o,A2: B] :
( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ ( tLList57907754e_TNil @ B @ A @ A2 ) )
= ( P2 @ A2 ) ) ).
% tllist.pred_inject(1)
thf(fact_15_tllist_Oinject_I2_J,axiom,
! [B: $tType,A: $tType,X212: A,X223: tLList446370796tllist @ A @ B,Y21: A,Y22: tLList446370796tllist @ A @ B] :
( ( ( tLList1992840728_TCons @ A @ B @ X212 @ X223 )
= ( tLList1992840728_TCons @ A @ B @ Y21 @ Y22 ) )
= ( ( X212 = Y21 )
& ( X223 = Y22 ) ) ) ).
% tllist.inject(2)
thf(fact_16_tllist_Opred__inject_I2_J,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,A2: A,Aa: tLList446370796tllist @ A @ B] :
( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ ( tLList1992840728_TCons @ A @ B @ A2 @ Aa ) )
= ( ( P1 @ A2 )
& ( tLList11265572tllist @ A @ B @ P1 @ P2 @ Aa ) ) ) ).
% tllist.pred_inject(2)
thf(fact_17_tfilter__TCons,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,B2: B,Tr: tLList446370796tllist @ A @ B] :
( ( ( P @ A2 )
=> ( ( tLList1813626245filter @ B @ A @ B2 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1992840728_TCons @ A @ B @ A2 @ ( tLList1813626245filter @ B @ A @ B2 @ P @ Tr ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( tLList1813626245filter @ B @ A @ B2 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1813626245filter @ B @ A @ B2 @ P @ Tr ) ) ) ) ).
% tfilter_TCons
thf(fact_18_tllist_Osel_I5_J,axiom,
! [B: $tType,A: $tType,X212: A,X223: tLList446370796tllist @ A @ B] :
( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X212 @ X223 ) )
= X223 ) ).
% tllist.sel(5)
thf(fact_19_tfilter_H__code_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,B3: product_unit > B,Tr: tLList446370796tllist @ A @ B] :
( ( ( P @ A2 )
=> ( ( tLList1122525870filter @ B @ A @ B3 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1992840728_TCons @ A @ B @ A2 @ ( tLList1122525870filter @ B @ A @ B3 @ P @ Tr ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( tLList1122525870filter @ B @ A @ B3 @ P @ ( tLList1992840728_TCons @ A @ B @ A2 @ Tr ) )
= ( tLList1122525870filter @ B @ A @ B3 @ P @ Tr ) ) ) ) ).
% tfilter'_code(2)
thf(fact_20_tfilter__code,axiom,
! [B: $tType,A: $tType] :
( ( tLList1813626245filter @ A @ B )
= ( ^ [B4: A] :
( tLList1122525870filter @ A @ B
@ ^ [Uu: product_unit] : B4 ) ) ) ).
% tfilter_code
thf(fact_21_tconcat__code,axiom,
! [B: $tType,A: $tType] :
( ( tLList365522113concat @ A @ B )
= ( ^ [B4: A] :
( tLList1329345514concat @ A @ B
@ ^ [Uu: product_unit] : B4 ) ) ) ).
% tconcat_code
thf(fact_22_tllist_Opred__True,axiom,
! [B: $tType,A: $tType] :
( ( tLList11265572tllist @ A @ B
@ ^ [Uu: A] : $true
@ ^ [Uu: B] : $true )
= ( ^ [Uu: tLList446370796tllist @ A @ B] : $true ) ) ).
% tllist.pred_True
thf(fact_23_tllist_Oexhaust,axiom,
! [B: $tType,A: $tType,Y: tLList446370796tllist @ A @ B] :
( ! [X13: B] :
( Y
!= ( tLList57907754e_TNil @ B @ A @ X13 ) )
=> ~ ! [X213: A,X224: tLList446370796tllist @ A @ B] :
( Y
!= ( tLList1992840728_TCons @ A @ B @ X213 @ X224 ) ) ) ).
% tllist.exhaust
thf(fact_24_tllist_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X1: B,X212: A,X223: tLList446370796tllist @ A @ B] :
( ( tLList57907754e_TNil @ B @ A @ X1 )
!= ( tLList1992840728_TCons @ A @ B @ X212 @ X223 ) ) ).
% tllist.distinct(1)
thf(fact_25_tllist_Osel_I4_J,axiom,
! [A: $tType,B: $tType,X1: B] :
( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList57907754e_TNil @ B @ A @ X1 ) )
= ( tLList57907754e_TNil @ B @ A @ X1 ) ) ).
% tllist.sel(4)
thf(fact_26_tllist_Osimps_I6_J,axiom,
! [C: $tType,B: $tType,A: $tType,F1: B > C,F2: A > ( tLList446370796tllist @ A @ B ) > C,X212: A,X223: tLList446370796tllist @ A @ B] :
( ( tLList200813139tllist @ B @ C @ A @ F1 @ F2 @ ( tLList1992840728_TCons @ A @ B @ X212 @ X223 ) )
= ( F2 @ X212 @ X223 ) ) ).
% tllist.simps(6)
thf(fact_27_tnth__Suc__TCons,axiom,
! [A: $tType,B: $tType,X3: A,Xs: tLList446370796tllist @ A @ B,N: nat] :
( ( tLList2115636725e_tnth @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X3 @ Xs ) @ ( suc @ N ) )
= ( tLList2115636725e_tnth @ A @ B @ Xs @ N ) ) ).
% tnth_Suc_TCons
thf(fact_28_tconcat__TCons,axiom,
! [B: $tType,A: $tType,B2: B,A2: coinductive_llist @ A,Tr: tLList446370796tllist @ ( coinductive_llist @ A ) @ B] :
( ( tLList365522113concat @ B @ A @ B2 @ ( tLList1992840728_TCons @ ( coinductive_llist @ A ) @ B @ A2 @ Tr ) )
= ( tLList98099029ppendt @ A @ B @ A2 @ ( tLList365522113concat @ B @ A @ B2 @ Tr ) ) ) ).
% tconcat_TCons
thf(fact_29_ttl__unfold__tllist,axiom,
! [B: $tType,A: $tType,C: $tType,IS_TNIL: C > $o,A2: C,TNIL: C > B,THD: C > A,TTL: C > C] :
( ( ( IS_TNIL @ A2 )
=> ( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ A2 ) )
= ( tLList57907754e_TNil @ B @ A @ ( TNIL @ A2 ) ) ) )
& ( ~ ( IS_TNIL @ A2 )
=> ( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ A2 ) )
= ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ ( TTL @ A2 ) ) ) ) ) ).
% ttl_unfold_tllist
thf(fact_30_unfold__tllist_Ocode,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( tLList636831159tllist @ A @ B @ C )
= ( ^ [P3: A > $o,G1: A > B,G21: A > C,G22: A > A,A3: A] : ( if @ ( tLList446370796tllist @ C @ B ) @ ( P3 @ A3 ) @ ( tLList57907754e_TNil @ B @ C @ ( G1 @ A3 ) ) @ ( tLList1992840728_TCons @ C @ B @ ( G21 @ A3 ) @ ( tLList636831159tllist @ A @ B @ C @ P3 @ G1 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% unfold_tllist.code
thf(fact_31_tllist_Ocorec__code,axiom,
! [B: $tType,A: $tType,E: $tType] :
( ( tLList1614408749tllist @ E @ B @ A )
= ( ^ [P3: E > $o,G1: E > B,G21: E > A,Q22: E > $o,G221: E > ( tLList446370796tllist @ A @ B ),G222: E > E,A3: E] : ( if @ ( tLList446370796tllist @ A @ B ) @ ( P3 @ A3 ) @ ( tLList57907754e_TNil @ B @ A @ ( G1 @ A3 ) ) @ ( tLList1992840728_TCons @ A @ B @ ( G21 @ A3 ) @ ( if @ ( tLList446370796tllist @ A @ B ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( tLList1614408749tllist @ E @ B @ A @ P3 @ G1 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% tllist.corec_code
thf(fact_32_tconcat_H__def,axiom,
! [A: $tType,B: $tType] :
( ( tLList1329345514concat @ B @ A )
= ( ^ [B4: product_unit > B] : ( tLList365522113concat @ B @ A @ ( B4 @ product_Unity ) ) ) ) ).
% tconcat'_def
thf(fact_33_tfilter_H__def,axiom,
! [A: $tType,B: $tType] :
( ( tLList1122525870filter @ B @ A )
= ( ^ [B4: product_unit > B] : ( tLList1813626245filter @ B @ A @ ( B4 @ product_Unity ) ) ) ) ).
% tfilter'_def
thf(fact_34_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri532925092at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_35_tconcat_H__code_I2_J,axiom,
! [B: $tType,C: $tType,B2: product_unit > B,A2: coinductive_llist @ C,Tr: tLList446370796tllist @ ( coinductive_llist @ C ) @ B] :
( ( tLList1329345514concat @ B @ C @ B2 @ ( tLList1992840728_TCons @ ( coinductive_llist @ C ) @ B @ A2 @ Tr ) )
= ( tLList98099029ppendt @ C @ B @ A2 @ ( tLList1329345514concat @ B @ C @ B2 @ Tr ) ) ) ).
% tconcat'_code(2)
thf(fact_36_tllist_Oset__induct_I2_J,axiom,
! [A: $tType,Aa2: $tType,X3: A,A2: tLList446370796tllist @ Aa2 @ A,P: A > ( tLList446370796tllist @ Aa2 @ A ) > $o] :
( ( member @ A @ X3 @ ( tLList457864275tllist @ Aa2 @ A @ A2 ) )
=> ( ! [Z: A] : ( P @ Z @ ( tLList57907754e_TNil @ A @ Aa2 @ Z ) )
=> ( ! [Z1: Aa2,Z2: tLList446370796tllist @ Aa2 @ A,Xb: A] :
( ( member @ A @ Xb @ ( tLList457864275tllist @ Aa2 @ A @ Z2 ) )
=> ( ( P @ Xb @ Z2 )
=> ( P @ Xb @ ( tLList1992840728_TCons @ Aa2 @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% tllist.set_induct(2)
thf(fact_37_TNil__eq__unfold__tllist,axiom,
! [A: $tType,B: $tType,C: $tType,B2: B,IS_TNIL: C > $o,TNIL: C > B,THD: C > A,TTL: C > C,A2: C] :
( ( ( tLList57907754e_TNil @ B @ A @ B2 )
= ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ A2 ) )
= ( ( IS_TNIL @ A2 )
& ( B2
= ( TNIL @ A2 ) ) ) ) ).
% TNil_eq_unfold_tllist
thf(fact_38_unfold__tllist__eq__TNil,axiom,
! [A: $tType,B: $tType,C: $tType,IS_TNIL: C > $o,TNIL: C > B,THD: C > A,TTL: C > C,A2: C,B2: B] :
( ( ( tLList636831159tllist @ C @ B @ A @ IS_TNIL @ TNIL @ THD @ TTL @ A2 )
= ( tLList57907754e_TNil @ B @ A @ B2 ) )
= ( ( IS_TNIL @ A2 )
& ( B2
= ( TNIL @ A2 ) ) ) ) ).
% unfold_tllist_eq_TNil
thf(fact_39_tllist_Oset__intros_I3_J,axiom,
! [A: $tType,B: $tType,Aa: B] : ( member @ B @ Aa @ ( tLList457864275tllist @ A @ B @ ( tLList57907754e_TNil @ B @ A @ Aa ) ) ) ).
% tllist.set_intros(3)
thf(fact_40_tllist_Oset__intros_I4_J,axiom,
! [B: $tType,A: $tType,Xc: B,A2a: tLList446370796tllist @ A @ B,A1a: A] :
( ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ A2a ) )
=> ( member @ B @ Xc @ ( tLList457864275tllist @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1a @ A2a ) ) ) ) ).
% tllist.set_intros(4)
thf(fact_41_tllist_Osimps_I23_J,axiom,
! [B: $tType,A: $tType,X212: A,X223: tLList446370796tllist @ A @ B] :
( ( tLList457864275tllist @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X212 @ X223 ) )
= ( tLList457864275tllist @ A @ B @ X223 ) ) ).
% tllist.simps(23)
thf(fact_42_tllist_Ocorec_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ( P4 @ A2 )
=> ( ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( tLList57907754e_TNil @ B @ A @ ( G12 @ A2 ) ) ) ) ).
% tllist.corec(1)
thf(fact_43_unfold__tllist_Octr_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,P4: A > $o,A2: A,G12: A > B,G212: A > C,G223: A > A] :
( ( P4 @ A2 )
=> ( ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ A2 )
= ( tLList57907754e_TNil @ B @ C @ ( G12 @ A2 ) ) ) ) ).
% unfold_tllist.ctr(1)
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_unfold__tllist_Octr_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,P4: A > $o,A2: A,G12: A > B,G212: A > C,G223: A > A] :
( ~ ( P4 @ A2 )
=> ( ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ A2 )
= ( tLList1992840728_TCons @ C @ B @ ( G212 @ A2 ) @ ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ ( G223 @ A2 ) ) ) ) ) ).
% unfold_tllist.ctr(2)
thf(fact_49_tllist_Ocorec_I2_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,G12: E > B,G212: E > A,Q222: E > $o,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ( ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( tLList1992840728_TCons @ A @ B @ ( G212 @ A2 ) @ ( if @ ( tLList446370796tllist @ A @ B ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% tllist.corec(2)
thf(fact_50_unfold__tllist_Osimps_I5_J,axiom,
! [B: $tType,C: $tType,A: $tType,P4: A > $o,A2: A,G12: A > B,G212: A > C,G223: A > A] :
( ~ ( P4 @ A2 )
=> ( ( tLList_Mirabelle_ttl @ C @ B @ ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ A2 ) )
= ( tLList636831159tllist @ A @ B @ C @ P4 @ G12 @ G212 @ G223 @ ( G223 @ A2 ) ) ) ) ).
% unfold_tllist.simps(5)
thf(fact_51_tllist_Ocorec__sel_I3_J,axiom,
! [B: $tType,A: $tType,E: $tType,P4: E > $o,A2: E,Q222: E > $o,G12: E > B,G212: E > A,G2212: E > ( tLList446370796tllist @ A @ B ),G2222: E > E] :
( ~ ( P4 @ A2 )
=> ( ( ( Q222 @ A2 )
=> ( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( G2212 @ A2 ) ) )
& ( ~ ( Q222 @ A2 )
=> ( ( tLList_Mirabelle_ttl @ A @ B @ ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( tLList1614408749tllist @ E @ B @ A @ P4 @ G12 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% tllist.corec_sel(3)
thf(fact_52_tllist_Oset__cases_I2_J,axiom,
! [B: $tType,A: $tType,E2: B,A2: tLList446370796tllist @ A @ B] :
( ( member @ B @ E2 @ ( tLList457864275tllist @ A @ B @ A2 ) )
=> ( ( A2
!= ( tLList57907754e_TNil @ B @ A @ E2 ) )
=> ~ ! [Z1: A,Z2: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) )
=> ~ ( member @ B @ E2 @ ( tLList457864275tllist @ A @ B @ Z2 ) ) ) ) ) ).
% tllist.set_cases(2)
thf(fact_53_unit__abs__eta__conv,axiom,
! [A: $tType,F: product_unit > A] :
( ( ^ [U: product_unit] : ( F @ product_Unity ) )
= F ) ).
% unit_abs_eta_conv
thf(fact_54_tnth__TCons,axiom,
! [A: $tType,B: $tType,X3: A,Xs: tLList446370796tllist @ A @ B,N: nat] :
( ( tLList2115636725e_tnth @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X3 @ Xs ) @ N )
= ( case_nat @ A @ X3 @ ( tLList2115636725e_tnth @ A @ B @ Xs ) @ N ) ) ).
% tnth_TCons
thf(fact_55_eq__sym__Unity__conv,axiom,
! [X3: $o] :
( ( X3
= ( product_Unity = product_Unity ) )
= X3 ) ).
% eq_sym_Unity_conv
thf(fact_56_unit__all__impI,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( ( P @ product_Unity )
=> ( Q @ product_Unity ) )
=> ! [X5: product_unit] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ).
% unit_all_impI
thf(fact_57_old_Ounit_Oinducts,axiom,
! [P: product_unit > $o,Unit: product_unit] :
( ( P @ product_Unity )
=> ( P @ Unit ) ) ).
% old.unit.inducts
thf(fact_58_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
thf(fact_59_tnth__0,axiom,
! [B: $tType,A: $tType,X3: A,Xs: tLList446370796tllist @ A @ B] :
( ( tLList2115636725e_tnth @ A @ B @ ( tLList1992840728_TCons @ A @ B @ X3 @ Xs ) @ ( zero_zero @ nat ) )
= X3 ) ).
% tnth_0
thf(fact_60_lappendt__LCons,axiom,
! [B: $tType,A: $tType,X3: A,Xs: coinductive_llist @ A,Tr: tLList446370796tllist @ A @ B] :
( ( tLList98099029ppendt @ A @ B @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Tr )
= ( tLList1992840728_TCons @ A @ B @ X3 @ ( tLList98099029ppendt @ A @ B @ Xs @ Tr ) ) ) ).
% lappendt_LCons
thf(fact_61_tllist_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,X3: tLList446370796tllist @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ X3 )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ X3 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ X3 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( tLList11265572tllist @ A @ B @ P1a @ P2a @ X3 ) ) ) ) ).
% tllist.pred_mono_strong
thf(fact_62_tdropn__0,axiom,
! [B: $tType,A: $tType,Xs: tLList446370796tllist @ A @ B] :
( ( tLList1881248882tdropn @ A @ B @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% tdropn_0
thf(fact_63_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B @ ( type2 @ B ) )
=> ! [F: ( A > B ) > C,G: C] :
( ( F
= ( ^ [X: A > B] : G ) )
=> ( ( F
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_64_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_65_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_66_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: nat > A] :
( ( case_nat @ A @ F1 @ F2 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_67_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F2: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F2 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X: nat] : ( H @ ( F2 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_68_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_69_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_70_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_71_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_72_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_73_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_74_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_75_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_76_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_77_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_78_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_79_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_80_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_81_tllist_Oset__cases_I1_J,axiom,
! [B: $tType,A: $tType,E2: A,A2: tLList446370796tllist @ A @ B] :
( ( member @ A @ E2 @ ( tLList1719791291e_tset @ A @ B @ A2 ) )
=> ( ! [Z2: tLList446370796tllist @ A @ B] :
( A2
!= ( tLList1992840728_TCons @ A @ B @ E2 @ Z2 ) )
=> ~ ! [Z1: A,Z2: tLList446370796tllist @ A @ B] :
( ( A2
= ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E2 @ ( tLList1719791291e_tset @ A @ B @ Z2 ) ) ) ) ) ).
% tllist.set_cases(1)
thf(fact_82_tllist_Oset__induct_I1_J,axiom,
! [B: $tType,A: $tType,X3: A,A2: tLList446370796tllist @ A @ B,P: A > ( tLList446370796tllist @ A @ B ) > $o] :
( ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ A2 ) )
=> ( ! [Z1: A,Z2: tLList446370796tllist @ A @ B] : ( P @ Z1 @ ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: tLList446370796tllist @ A @ B,Xa: A] :
( ( member @ A @ Xa @ ( tLList1719791291e_tset @ A @ B @ Z2 ) )
=> ( ( P @ Xa @ Z2 )
=> ( P @ Xa @ ( tLList1992840728_TCons @ A @ B @ Z1 @ Z2 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% tllist.set_induct(1)
thf(fact_83_tllist_Oset__intros_I2_J,axiom,
! [B: $tType,A: $tType,X3: A,A22: tLList446370796tllist @ A @ B,A1: A] :
( ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ A22 ) )
=> ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1 @ A22 ) ) ) ) ).
% tllist.set_intros(2)
thf(fact_84_tllist_Oset__intros_I1_J,axiom,
! [B: $tType,A: $tType,A1: A,A22: tLList446370796tllist @ A @ B] : ( member @ A @ A1 @ ( tLList1719791291e_tset @ A @ B @ ( tLList1992840728_TCons @ A @ B @ A1 @ A22 ) ) ) ).
% tllist.set_intros(1)
thf(fact_85_in__tset__ttlD,axiom,
! [B: $tType,A: $tType,X3: A,Xs: tLList446370796tllist @ A @ B] :
( ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ ( tLList_Mirabelle_ttl @ A @ B @ Xs ) ) )
=> ( member @ A @ X3 @ ( tLList1719791291e_tset @ A @ B @ Xs ) ) ) ).
% in_tset_ttlD
thf(fact_86_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F2: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F2 @ ( suc @ X22 ) )
= ( F2 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_87_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_88_tllist_Opred__cong,axiom,
! [B: $tType,A: $tType,X3: tLList446370796tllist @ A @ B,Ya: tLList446370796tllist @ A @ B,P1: A > $o,P1a: A > $o,P2: B > $o,P2a: B > $o] :
( ( X3 = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( tLList1719791291e_tset @ A @ B @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( tLList457864275tllist @ A @ B @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( tLList11265572tllist @ A @ B @ P1 @ P2 @ X3 )
= ( tLList11265572tllist @ A @ B @ P1a @ P2a @ Ya ) ) ) ) ) ).
% tllist.pred_cong
thf(fact_89_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_90_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_91_llist_Oinject,axiom,
! [A: $tType,X212: A,X223: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X212 @ X223 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X212 = Y21 )
& ( X223 = Y22 ) ) ) ).
% llist.inject
thf(fact_92_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F2: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F2 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_93_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F2: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F2 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_94_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X3
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_95_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_96_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X2: nat] : X2 ) ) ).
% pred_def
thf(fact_97_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X14: nat] : ( P @ X14 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_98_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X14: A] : ( P @ ( zero_zero @ nat ) @ X14 )
=> ( ! [X4: A,N2: nat] :
( ( P @ N2 @ X4 )
=> ? [Y4: A] :
( ( P @ ( suc @ N2 ) @ Y4 )
& ( Q @ N2 @ X4 @ Y4 ) ) )
=> ? [F3: nat > A] :
! [N3: nat] :
( ( P @ N3 @ ( F3 @ N3 ) )
& ( Q @ N3 @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_99_unit_Osize_I1_J,axiom,
( ( product_size_unit @ product_Unity )
= ( zero_zero @ nat ) ) ).
% unit.size(1)
thf(fact_100_lsublist__LCons,axiom,
! [A: $tType,A4: set @ nat,X3: A,Xs: coinductive_llist @ A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( coinductive_lsublist @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ A4 )
= ( coinductive_LCons @ A @ X3
@ ( coinductive_lsublist @ A @ Xs
@ ( collect @ nat
@ ^ [N4: nat] : ( member @ nat @ ( suc @ N4 ) @ A4 ) ) ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( coinductive_lsublist @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ A4 )
= ( coinductive_lsublist @ A @ Xs
@ ( collect @ nat
@ ^ [N4: nat] : ( member @ nat @ ( suc @ N4 ) @ A4 ) ) ) ) ) ) ).
% lsublist_LCons
thf(fact_101_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_102_size__unit__def,axiom,
( product_size_unit
= ( product_rec_unit @ nat @ ( zero_zero @ nat ) ) ) ).
% size_unit_def
thf(fact_103_unit_Osize_I2_J,axiom,
( ( size_size @ product_unit @ product_Unity )
= ( zero_zero @ nat ) ) ).
% unit.size(2)
thf(fact_104_bool_Osize_I2_J,axiom,
( ( size_bool @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(2)
thf(fact_105_bool_Osize_I1_J,axiom,
( ( size_bool @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(1)
thf(fact_106_lsublist__singleton,axiom,
! [A: $tType,A4: set @ nat,X3: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( coinductive_lsublist @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) @ A4 )
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( coinductive_lsublist @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) @ A4 )
= ( coinductive_LNil @ A ) ) ) ) ).
% lsublist_singleton
thf(fact_107_diff__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K2: nat] : K2
@ ( minus_minus @ nat @ M2 @ N ) ) ) ).
% diff_Suc
thf(fact_108_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_109_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_110_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_111_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_112_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_113_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_114_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_115_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_116_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_117_lsublist__LNil,axiom,
! [A: $tType,A4: set @ nat] :
( ( coinductive_lsublist @ A @ ( coinductive_LNil @ A ) @ A4 )
= ( coinductive_LNil @ A ) ) ).
% lsublist_LNil
thf(fact_118_lappendt__LNil,axiom,
! [B: $tType,A: $tType,Tr: tLList446370796tllist @ A @ B] :
( ( tLList98099029ppendt @ A @ B @ ( coinductive_LNil @ A ) @ Tr )
= Tr ) ).
% lappendt_LNil
thf(fact_119_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_120_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M2 )
= ( zero_zero @ nat ) )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_121_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_122_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_123_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_124_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_125_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A3: A,B4: A] :
( ( minus_minus @ A @ A3 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_126_llist_Odistinct_I1_J,axiom,
! [A: $tType,X212: A,X223: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X212 @ X223 ) ) ).
% llist.distinct(1)
thf(fact_127_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X213: A,X224: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X213 @ X224 ) ) ) ).
% llist.exhaust
thf(fact_128_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_129_size__unit__overloaded__def,axiom,
( ( size_size @ product_unit )
= ( product_rec_unit @ nat @ ( zero_zero @ nat ) ) ) ).
% size_unit_overloaded_def
thf(fact_130_size__bool__def,axiom,
( size_bool
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_def
thf(fact_131_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_132_llistsum__LNil,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ( ( coindu780009021istsum @ A @ ( coinductive_LNil @ A ) )
= ( zero_zero @ A ) ) ) ).
% llistsum_LNil
thf(fact_133_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_134_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_135_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_136_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_137_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X3 = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_138_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_139_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_140_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_141_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_142_size__bool__overloaded__def,axiom,
( ( size_size @ $o )
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_overloaded_def
thf(fact_143_llast__singleton,axiom,
! [A: $tType,X3: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
= X3 ) ).
% llast_singleton
thf(fact_144_Suc__pred,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% Suc_pred
thf(fact_145_bool_Osize_I4_J,axiom,
( ( size_size @ $o @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(4)
thf(fact_146_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_147_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_148_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_149_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_150_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_151_llast__LCons2,axiom,
! [A: $tType,X3: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_152_bool_Osize_I3_J,axiom,
( ( size_size @ $o @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(3)
thf(fact_153_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_154_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_155_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_156_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_157_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_158_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_159_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_160_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_161_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_162_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_163_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_164_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_165_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_166_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_167_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less @ nat @ M2 @ N )
| ( ord_less @ nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_168_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_169_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_170_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_171_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_172_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_173_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_174_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_175_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ord_less @ nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_176_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_177_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_178_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_179_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_180_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_181_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_182_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_183_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_184_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_185_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_186_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_187_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_188_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_189_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_190_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_191_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_192_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_193_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_194_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_195_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X3: A] :
( ! [X4: A] :
( ( ( V @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_196_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_197_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_198_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_199_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_200_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_201_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_202_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_203_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A3 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_204_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_205_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_206_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_207_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_208_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_209_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_210_size__bool,axiom,
( ( size_size @ $o )
= ( ^ [B4: $o] : ( zero_zero @ nat ) ) ) ).
% size_bool
thf(fact_211_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_212_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_213_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_214_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_215_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M2: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( semiring_1_of_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% of_nat_eq_iff
thf(fact_216_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_217_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_218_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_219_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_220_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_iff
thf(fact_221_of__nat__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% of_nat_1
thf(fact_222_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_223_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_224_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [N2: nat] : ( ord_less @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_225_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_226_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_227_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_228_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_229_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_230_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_231_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_232_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_233_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_234_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_235_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_236_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X: A,F4: nat > A,N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ X
@ ( F4 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_237_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_238_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_239_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_240_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_241_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_242_transfer__int__nat__numerals_I1_J,axiom,
( ( zero_zero @ int )
= ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% transfer_int_nat_numerals(1)
thf(fact_243_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( semiri532925092at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N4
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_244_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_245_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_246_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_247_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_248_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_249_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_250_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_251_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
%----Type constructors (35)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oorder_1,axiom,
order @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int @ ( type2 @ int ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_2,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_3,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_4,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_5,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_7,axiom,
zero_less_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
semiring_char_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_9,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_10,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero_13,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone_14,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder_15,axiom,
! [A5: $tType] : ( order @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_17,axiom,
order @ product_unit @ ( type2 @ product_unit ) ).
%----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,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( tLList1881248882tdropn @ a @ b @ ( suc @ n ) @ xs )
= ( tLList200813139tllist @ b @ ( tLList446370796tllist @ a @ b ) @ a @ ( tLList57907754e_TNil @ b @ a )
@ ^ [X: a] : ( tLList1881248882tdropn @ a @ b @ n )
@ xs ) ) ).
%------------------------------------------------------------------------------