TPTP Problem File: DAT125^1.p
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%------------------------------------------------------------------------------
% File : DAT125^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 2490
% 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__2490.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 345 ( 146 unt; 50 typ; 0 def)
% Number of atoms : 810 ( 209 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 4278 ( 65 ~; 13 |; 29 &;3861 @)
% ( 0 <=>; 310 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 451 ( 451 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 3 con; 0-7 aty)
% Number of variables : 1297 ( 140 ^;1084 !; 22 ?;1297 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:56:46.397
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $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_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_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple1141879883l_ccpo:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix,type,
coindu1571841457prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop,type,
coindu191418589_ldrop:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OldropWhile,type,
coindu438612276pWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn,type,
coindu531130065ldropn:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfilter,type,
coindu1889960678filter:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
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_Olzip,type,
coindu847746867e_lzip:
!>[A: $tType,B: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > ( coindu1593790203_llist @ ( product_prod @ A @ 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_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple939513234o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
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_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
thf(sy_c_Partial__Function_Omk__less,type,
partial_mk_less:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
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_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
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 @ b ).
%----Relevant facts (255)
thf(fact_0_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_1_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_2_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_3_monotone__lzip1,axiom,
! [B: $tType,A: $tType,Ys: coindu1593790203_llist @ B] :
( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ ( product_prod @ A @ B ) ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ ( product_prod @ A @ B ) )
@ ^ [Xs2: coindu1593790203_llist @ A] : ( coindu847746867e_lzip @ A @ B @ Xs2 @ Ys ) ) ).
% monotone_lzip1
thf(fact_4_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_5_llist_Omono2mono,axiom,
! [B: $tType,A: $tType,C: $tType,Ordb: B > B > $o,F: B > ( coindu1593790203_llist @ A ),Orda: C > C > $o,T2: C > B] :
( ( comple1396247847notone @ B @ ( coindu1593790203_llist @ A ) @ Ordb @ ( coindu1696667936prefix @ A ) @ F )
=> ( ( comple1396247847notone @ C @ B @ Orda @ Ordb @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A )
@ ^ [X2: C] : ( F @ ( T2 @ X2 ) ) ) ) ) ).
% llist.mono2mono
thf(fact_6_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_7_mono2mono__lzip1,axiom,
! [B2: $tType,A2: $tType,C: $tType,Orda: C > C > $o,T2: C > ( coindu1593790203_llist @ A2 ),Ys2: coindu1593790203_llist @ B2] :
( ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A2 ) @ Orda @ ( coindu1696667936prefix @ A2 ) @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ ( product_prod @ A2 @ B2 ) ) @ Orda @ ( coindu1696667936prefix @ ( product_prod @ A2 @ B2 ) )
@ ^ [X2: C] : ( coindu847746867e_lzip @ A2 @ B2 @ ( T2 @ X2 ) @ Ys2 ) ) ) ).
% mono2mono_lzip1
thf(fact_8_llist_Oconst__mono,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,C2: coindu1593790203_llist @ A] :
( comple1396247847notone @ B @ ( coindu1593790203_llist @ A ) @ Ord @ ( coindu1696667936prefix @ A )
@ ^ [F2: B] : C2 ) ).
% llist.const_mono
thf(fact_9_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_10_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_11_llist_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType,Bound: coindu1593790203_llist @ A,G: ( coindu1593790203_llist @ A ) > B,Bot: B,F: ( coindu1593790203_llist @ A ) > B,Ord: B > B > $o] :
( ! [X3: coindu1593790203_llist @ A] :
( ( ( coindu1696667936prefix @ A @ X3 @ Bound )
=> ( ( G @ X3 )
= Bot ) )
& ( ~ ( coindu1696667936prefix @ A @ X3 @ Bound )
=> ( ( G @ X3 )
= ( F @ X3 ) ) ) )
=> ( ! [X3: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y2 )
=> ( ~ ( coindu1696667936prefix @ A @ X3 @ Bound )
=> ( Ord @ ( F @ X3 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X3: coindu1593790203_llist @ A] :
( ~ ( coindu1696667936prefix @ A @ X3 @ Bound )
=> ( Ord @ Bot @ ( F @ X3 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ Ord @ G ) ) ) ) ) ).
% llist.monotone_if_bot
thf(fact_12_if__mono,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F3: A > B,G2: A > B,C2: $o] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F3 )
=> ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ G2 )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( if @ B @ C2 @ ( F3 @ F2 ) @ ( G2 @ F2 ) ) ) ) ) ).
% if_mono
thf(fact_13_let__mono,axiom,
! [A: $tType,C: $tType,B: $tType,Orda: B > B > $o,Ordb: C > C > $o,B3: B > A > C,T2: A] :
( ! [X3: A] :
( comple1396247847notone @ B @ C @ Orda @ Ordb
@ ^ [F2: B] : ( B3 @ F2 @ X3 ) )
=> ( comple1396247847notone @ B @ C @ Orda @ Ordb
@ ^ [F2: B] : ( B3 @ F2 @ T2 ) ) ) ).
% let_mono
thf(fact_14_monotone__id_H,axiom,
! [A: $tType,Ord: A > A > $o] :
( comple1396247847notone @ A @ A @ Ord @ Ord
@ ^ [X2: A] : X2 ) ).
% monotone_id'
thf(fact_15_monotoneD,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F: A > B,X: A,Y: A] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_16_monotoneI,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F: A > B] :
( ! [X3: A,Y2: A] :
( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_17_monotone__def,axiom,
! [B: $tType,A: $tType] :
( ( comple1396247847notone @ A @ B )
= ( ^ [Orda2: A > A > $o,Ordb2: B > B > $o,F2: A > B] :
! [X2: A,Y3: A] :
( ( Orda2 @ X2 @ Y3 )
=> ( Ordb2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% monotone_def
thf(fact_18_monotone__ldropn_H,axiom,
! [A: $tType,N: nat] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu531130065ldropn @ A @ N ) ) ).
% monotone_ldropn'
thf(fact_19_mono2mono__ldrop2,axiom,
! [A: $tType,C: $tType,Orda: C > C > $o,T2: C > ( coindu1593790203_llist @ A ),N2: extended_enat] :
( ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A ) @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A )
@ ^ [X2: C] : ( coindu191418589_ldrop @ A @ N2 @ ( T2 @ X2 ) ) ) ) ).
% mono2mono_ldrop2
thf(fact_20_mono2mono__ldropn,axiom,
! [A: $tType,C: $tType,Orda: C > C > $o,T2: C > ( coindu1593790203_llist @ A ),N2: nat] :
( ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A ) @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A )
@ ^ [X2: C] : ( coindu531130065ldropn @ A @ N2 @ ( T2 @ X2 ) ) ) ) ).
% mono2mono_ldropn
thf(fact_21_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu1571841457prefix @ A )
= ( coindu1696667936prefix @ A ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_def
thf(fact_22_monotone__ldrop2,axiom,
! [A: $tType,N: extended_enat] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu191418589_ldrop @ A @ N ) ) ).
% monotone_ldrop2
thf(fact_23_mono2mono__LCons,axiom,
! [A: $tType,C: $tType,Orda: C > C > $o,T2: C > ( coindu1593790203_llist @ A ),X4: A] :
( ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A ) @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A )
@ ^ [X2: C] : ( coindu1121789889_LCons @ A @ X4 @ ( T2 @ X2 ) ) ) ) ).
% mono2mono_LCons
thf(fact_24_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_25_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_26_ldropn__lzip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu531130065ldropn @ ( product_prod @ A @ B ) @ N @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( coindu847746867e_lzip @ A @ B @ ( coindu531130065ldropn @ A @ N @ Xs ) @ ( coindu531130065ldropn @ B @ N @ Ys ) ) ) ).
% ldropn_lzip
thf(fact_27_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys3: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys3 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys3 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_28_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_29_monotone__LCons,axiom,
! [A: $tType,X: A] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X ) ) ).
% monotone_LCons
thf(fact_30_mk__less__def,axiom,
! [A: $tType] :
( ( partial_mk_less @ A )
= ( ^ [R: A > A > $o,X2: A,Y3: A] :
( ( R @ X2 @ Y3 )
& ~ ( R @ Y3 @ X2 ) ) ) ) ).
% mk_less_def
thf(fact_31_img__ord__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( partial_img_ord @ A @ C @ B )
= ( ^ [F2: A > C,Ord2: C > C > B,X2: A,Y3: A] : ( Ord2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ).
% img_ord_def
thf(fact_32_ldropn__LCons,axiom,
! [A: $tType,N: nat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ N @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( case_nat @ ( coindu1593790203_llist @ A ) @ ( coindu1121789889_LCons @ A @ X @ Xs )
@ ^ [N3: nat] : ( coindu531130065ldropn @ A @ N3 @ Xs )
@ N ) ) ).
% ldropn_LCons
thf(fact_33_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_34_ldrop__LCons,axiom,
! [A: $tType,N: extended_enat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu440805660e_enat @ ( coindu1593790203_llist @ A ) @ ( coindu1121789889_LCons @ A @ X @ Xs )
@ ^ [N3: extended_enat] : ( coindu191418589_ldrop @ A @ N3 @ Xs )
@ N ) ) ).
% ldrop_LCons
thf(fact_35_lzip__simps_I3_J,axiom,
! [C: $tType,B: $tType,X: C,Xs: coindu1593790203_llist @ C,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu847746867e_lzip @ C @ B @ ( coindu1121789889_LCons @ C @ X @ Xs ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( coindu1121789889_LCons @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X @ Y ) @ ( coindu847746867e_lzip @ C @ B @ Xs @ Ys ) ) ) ).
% lzip_simps(3)
thf(fact_36_ldropn__Suc__LCons,axiom,
! [A: $tType,N: nat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( suc @ N ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu531130065ldropn @ A @ N @ Xs ) ) ).
% ldropn_Suc_LCons
thf(fact_37_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu574146665prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_38_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1571841457prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs3 ) )
& ( coindu1571841457prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(3)
thf(fact_39_ldrop__eSuc__LCons,axiom,
! [B: $tType,N: extended_enat,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ B @ ( extended_eSuc @ N ) @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu191418589_ldrop @ B @ N @ Xs ) ) ).
% ldrop_eSuc_LCons
thf(fact_40_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_41_ldropn__LNil,axiom,
! [A: $tType,N: nat] :
( ( coindu531130065ldropn @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldropn_LNil
thf(fact_42_ldrop__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldrop_LNil
thf(fact_43_lzip__simps_I2_J,axiom,
! [D: $tType,C: $tType,Xs: coindu1593790203_llist @ C] :
( ( coindu847746867e_lzip @ C @ D @ Xs @ ( coindu1598213697e_LNil @ D ) )
= ( coindu1598213697e_LNil @ ( product_prod @ C @ D ) ) ) ).
% lzip_simps(2)
thf(fact_44_lzip__simps_I1_J,axiom,
! [B: $tType,A: $tType,Ys: coindu1593790203_llist @ B] :
( ( coindu847746867e_lzip @ A @ B @ ( coindu1598213697e_LNil @ A ) @ Ys )
= ( coindu1598213697e_LNil @ ( product_prod @ A @ B ) ) ) ).
% lzip_simps(1)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_50_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coindu1593790203_llist @ B] : ( coindu574146665prefix @ B @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_51_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
~ ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1598213697e_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_52_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs4: coindu1593790203_llist @ A] :
( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs4 )
=> ( P @ Ys4 ) )
=> ( P @ Xs4 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_53_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_54_lzip__eq__LNil__conv,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ( coindu847746867e_lzip @ A @ B @ Xs @ Ys )
= ( coindu1598213697e_LNil @ ( product_prod @ A @ B ) ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( Ys
= ( coindu1598213697e_LNil @ B ) ) ) ) ).
% lzip_eq_LNil_conv
thf(fact_55_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_56_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X2: A,Xs3: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) ) ) ) ).
% neq_LNil_conv
thf(fact_57_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_58_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_59_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1571841457prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ A ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(1)
thf(fact_60_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1571841457prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(2)
thf(fact_61_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs2: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys5 )
& ( Xs2 != Ys5 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_62_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_63_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,Ys6: coindu1593790203_llist @ A,X3: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys6 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs4 @ Ys6 ) ) ) ) ) ).
% lprefix.cases
thf(fact_64_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Xs2 ) )
| ? [Xs2: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys5 ) )
& ( coindu1696667936prefix @ A @ Xs2 @ Ys5 ) ) ) ) ) ).
% lprefix.simps
thf(fact_65_lprefix_Ocoinduct,axiom,
! [A: $tType,X5: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X5 @ X @ Xa )
=> ( ! [X3: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X5 @ 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 ) )
& ( ( X5 @ Xs5 @ Ys4 )
| ( coindu1696667936prefix @ A @ Xs5 @ Ys4 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_66_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 ) )
| ? [Xs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs3 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_67_lzip__eq__LCons__conv,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,Z: product_prod @ A @ B,Zs: coindu1593790203_llist @ ( product_prod @ A @ B )] :
( ( ( coindu847746867e_lzip @ A @ B @ Xs @ Ys )
= ( coindu1121789889_LCons @ ( product_prod @ A @ B ) @ Z @ Zs ) )
= ( ? [X2: A,Xs3: coindu1593790203_llist @ A,Y3: B,Ys3: coindu1593790203_llist @ B] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( Ys
= ( coindu1121789889_LCons @ B @ Y3 @ Ys3 ) )
& ( Z
= ( product_Pair @ A @ B @ X2 @ Y3 ) )
& ( Zs
= ( coindu847746867e_lzip @ A @ B @ Xs3 @ Ys3 ) ) ) ) ) ).
% lzip_eq_LCons_conv
thf(fact_68_co_Oenat_Oinject,axiom,
! [X23: extended_enat,Y23: extended_enat] :
( ( ( extended_eSuc @ X23 )
= ( extended_eSuc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% co.enat.inject
thf(fact_69_eSuc__inject,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( extended_eSuc @ M )
= ( extended_eSuc @ N ) )
= ( M = N ) ) ).
% eSuc_inject
thf(fact_70_enat__cocase__mono,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,Zero: A > B,Esuc: A > extended_enat > B,X: extended_enat] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ Zero )
=> ( ! [N4: extended_enat] :
( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( Esuc @ F2 @ N4 ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( coindu440805660e_enat @ B @ ( Zero @ F2 ) @ ( Esuc @ F2 ) @ X ) ) ) ) ).
% enat_cocase_mono
thf(fact_71_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_72_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_73_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
= ( ( A3 = A5 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_74_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y23: B] :
( ( ( product_Pair @ A @ B @ X1 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_75_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y2: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_76_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_77_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ~ ( ( A3 = A5 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_78_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_79_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_80_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,B5: B,C3: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_81_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,B5: B,C3: 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 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_82_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_83_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,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_84_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,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_85_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,B5: B,C3: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_86_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,B5: B,C3: 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 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_87_prod__induct7,axiom,
! [G3: $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 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_88_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_89_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_90_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_91_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_92_co_Oenat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: extended_enat > A,Enat: extended_enat] :
( ( H @ ( coindu440805660e_enat @ A @ F1 @ F22 @ Enat ) )
= ( coindu440805660e_enat @ B @ ( H @ F1 )
@ ^ [X2: extended_enat] : ( H @ ( F22 @ X2 ) )
@ Enat ) ) ).
% co.enat.case_distrib
thf(fact_93_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X2: nat] : ( H @ ( F22 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_94_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X23: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X23 ) )
= ( F22 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_95_co_Oenat_Ocase_I2_J,axiom,
! [A: $tType,F1: A,F22: extended_enat > A,X23: extended_enat] :
( ( coindu440805660e_enat @ A @ F1 @ F22 @ ( extended_eSuc @ X23 ) )
= ( F22 @ X23 ) ) ).
% co.enat.case(2)
thf(fact_96_enat__cocase__eSuc,axiom,
! [A: $tType,Z: A,S: extended_enat > A,N: extended_enat] :
( ( coindu440805660e_enat @ A @ Z @ S @ ( extended_eSuc @ N ) )
= ( S @ N ) ) ).
% enat_cocase_eSuc
thf(fact_97_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_98_monotone__enat__le__lprefix__case,axiom,
! [A: $tType,F: extended_enat > extended_enat > ( coindu1593790203_llist @ A )] :
( ( comple1396247847notone @ extended_enat @ ( coindu1593790203_llist @ A ) @ ( ord_less_eq @ extended_enat ) @ ( coindu1696667936prefix @ A )
@ ^ [X2: extended_enat] : ( F @ X2 @ ( extended_eSuc @ X2 ) ) )
=> ( comple1396247847notone @ extended_enat @ ( coindu1593790203_llist @ A ) @ ( ord_less_eq @ extended_enat ) @ ( coindu1696667936prefix @ A )
@ ^ [X2: extended_enat] :
( coindu440805660e_enat @ ( coindu1593790203_llist @ A ) @ ( coindu1598213697e_LNil @ A )
@ ^ [X6: extended_enat] : ( F @ X6 @ X2 )
@ X2 ) ) ) ).
% monotone_enat_le_lprefix_case
thf(fact_99_monotone__lprefix__case,axiom,
! [B: $tType,A: $tType,F: A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B )] :
( ! [X3: A] :
( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ B ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ B )
@ ^ [Xs2: coindu1593790203_llist @ A] : ( F @ X3 @ Xs2 @ ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) ) )
=> ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ B ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ B )
@ ^ [Xs2: coindu1593790203_llist @ A] :
( coindu882539134_llist @ ( coindu1593790203_llist @ B ) @ A @ ( coindu1598213697e_LNil @ B )
@ ^ [X2: A,Xs3: coindu1593790203_llist @ A] : ( F @ X2 @ Xs3 @ Xs2 )
@ Xs2 ) ) ) ).
% monotone_lprefix_case
thf(fact_100_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_101_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C2 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_102_ldrop_Osimps,axiom,
! [A: $tType] :
( ( coindu191418589_ldrop @ A )
= ( ^ [N5: extended_enat,Xs2: coindu1593790203_llist @ A] :
( coindu440805660e_enat @ ( coindu1593790203_llist @ A ) @ Xs2
@ ^ [N3: extended_enat] :
( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A] : ( coindu191418589_ldrop @ A @ N3 )
@ Xs2 )
@ N5 ) ) ) ).
% ldrop.simps
thf(fact_103_eSuc__ile__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% eSuc_ile_mono
thf(fact_104_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ord_less_eq @ A @ ( F @ N6 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_105_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N6: nat] :
( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N6 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_106_llist_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( H @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( coindu882539134_llist @ C @ A @ ( H @ F1 )
@ ^ [X12: A,X24: coindu1593790203_llist @ A] : ( H @ ( F22 @ X12 @ X24 ) )
@ Llist ) ) ).
% llist.case_distrib
thf(fact_107_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% lfp.leq_refl
thf(fact_108_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ Y )
=> ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% gfp.leq_trans
thf(fact_109_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% lfp.leq_trans
thf(fact_110_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( X = Y ) ) ) ) ).
% gfp.leq_antisym
thf(fact_111_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% lfp.leq_antisym
thf(fact_112_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_113_lfp_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Bound: A,G: A > B,Bot: B,F: A > B,Ord: B > B > $o] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ X3 @ Bound )
=> ( ( G @ X3 )
= Bot ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Bound )
=> ( ( G @ X3 )
= ( F @ X3 ) ) ) )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ~ ( ord_less_eq @ A @ X3 @ Bound )
=> ( Ord @ ( F @ X3 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Bound )
=> ( Ord @ Bot @ ( F @ X3 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ A @ B @ ( ord_less_eq @ A ) @ Ord @ G ) ) ) ) ) ) ).
% lfp.monotone_if_bot
thf(fact_114_monotone__eSuc,axiom,
comple1396247847notone @ extended_enat @ extended_enat @ ( ord_less_eq @ extended_enat ) @ ( ord_less_eq @ extended_enat ) @ extended_eSuc ).
% monotone_eSuc
thf(fact_115_eSuc__le__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ X ) @ Y )
= ( ? [Y4: extended_enat] :
( ( Y
= ( extended_eSuc @ Y4 ) )
& ( ord_less_eq @ extended_enat @ X @ Y4 ) ) ) ) ).
% eSuc_le_iff
thf(fact_116_ile__eSuc,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% ile_eSuc
thf(fact_117_gfp_Omono2mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Ordb: B > B > $o,F: B > A,Orda: C > C > $o,T2: C > B] :
( ( comple1396247847notone @ B @ A @ Ordb
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ F )
=> ( ( comple1396247847notone @ C @ B @ Orda @ Ordb @ T2 )
=> ( comple1396247847notone @ C @ A @ Orda
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [X2: C] : ( F @ ( T2 @ X2 ) ) ) ) ) ) ).
% gfp.mono2mono
thf(fact_118_lfp_Omono2mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Ordb: B > B > $o,F: B > A,Orda: C > C > $o,T2: C > B] :
( ( comple1396247847notone @ B @ A @ Ordb @ ( ord_less_eq @ A ) @ F )
=> ( ( comple1396247847notone @ C @ B @ Orda @ Ordb @ T2 )
=> ( comple1396247847notone @ C @ A @ Orda @ ( ord_less_eq @ A )
@ ^ [X2: C] : ( F @ ( T2 @ X2 ) ) ) ) ) ) ).
% lfp.mono2mono
thf(fact_119_gfp_Oconst__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Ord: B > B > $o,C2: A] :
( comple1396247847notone @ B @ A @ Ord
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ ^ [F2: B] : C2 ) ) ).
% gfp.const_mono
thf(fact_120_lfp_Oconst__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Ord: B > B > $o,C2: A] :
( comple1396247847notone @ B @ A @ Ord @ ( ord_less_eq @ A )
@ ^ [F2: B] : C2 ) ) ).
% lfp.const_mono
thf(fact_121_gfp_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Bound: A,G: A > B,Bot: B,F: A > B,Ord: B > B > $o] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ Bound @ X3 )
=> ( ( G @ X3 )
= Bot ) )
& ( ~ ( ord_less_eq @ A @ Bound @ X3 )
=> ( ( G @ X3 )
= ( F @ X3 ) ) ) )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ~ ( ord_less_eq @ A @ Bound @ X3 )
=> ( Ord @ ( F @ X3 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X3: A] :
( ~ ( ord_less_eq @ A @ Bound @ X3 )
=> ( Ord @ Bot @ ( F @ X3 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ A @ B
@ ^ [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X2 )
@ Ord
@ G ) ) ) ) ) ) ).
% gfp.monotone_if_bot
thf(fact_122_monotone__const,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [Ord: B > B > $o,C2: A] :
( comple1396247847notone @ B @ A @ Ord @ ( ord_less_eq @ A )
@ ^ [Uu: B] : C2 ) ) ).
% monotone_const
thf(fact_123_monotone__if__bot,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [Bound: A,Ord: B > B > $o,F: A > B,Bot: B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ~ ( ord_less_eq @ A @ X3 @ Bound )
=> ( Ord @ ( F @ X3 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Bound )
=> ( Ord @ Bot @ ( F @ X3 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ A @ B @ ( ord_less_eq @ A ) @ Ord
@ ^ [X2: A] : ( if @ B @ ( ord_less_eq @ A @ X2 @ Bound ) @ Bot @ ( F @ X2 ) ) ) ) ) ) ) ).
% monotone_if_bot
thf(fact_124_mono2mono__eSuc,axiom,
! [C: $tType,Orda: C > C > $o,T2: C > extended_enat] :
( ( comple1396247847notone @ C @ extended_enat @ Orda @ ( ord_less_eq @ extended_enat ) @ T2 )
=> ( comple1396247847notone @ C @ extended_enat @ Orda @ ( ord_less_eq @ extended_enat )
@ ^ [X2: C] : ( extended_eSuc @ ( T2 @ X2 ) ) ) ) ).
% mono2mono_eSuc
thf(fact_125_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_126_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_127_llist__case__mono,axiom,
! [C: $tType,B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,Lnil: A > B,Lcons: A > C > ( coindu1593790203_llist @ C ) > B,X: coindu1593790203_llist @ C] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ Lnil )
=> ( ! [X3: C,Xs4: coindu1593790203_llist @ C] :
( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( Lcons @ F2 @ X3 @ Xs4 ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( coindu882539134_llist @ B @ C @ ( Lnil @ F2 ) @ ( Lcons @ F2 ) @ X ) ) ) ) ).
% llist_case_mono
thf(fact_128_monotone__enat__cocase,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,F: extended_enat > extended_enat > B,A3: B] :
( ! [N4: A] :
( comple1396247847notone @ extended_enat @ B @ ( ord_less_eq @ extended_enat ) @ Ord
@ ^ [O: extended_enat] : ( F @ O @ ( extended_eSuc @ O ) ) )
=> ( ! [N4: extended_enat] : ( Ord @ A3 @ ( F @ N4 @ ( extended_eSuc @ N4 ) ) )
=> ( ( Ord @ A3 @ A3 )
=> ( comple1396247847notone @ extended_enat @ B @ ( ord_less_eq @ extended_enat ) @ Ord
@ ^ [N5: extended_enat] :
( coindu440805660e_enat @ B @ A3
@ ^ [N3: extended_enat] : ( F @ N3 @ N5 )
@ N5 ) ) ) ) ) ).
% monotone_enat_cocase
thf(fact_129_monotone__enat__le__case,axiom,
! [A: $tType,Ord: A > A > $o,F: extended_enat > extended_enat > A,Bot: A] :
( ( comple1396247847notone @ extended_enat @ A @ ( ord_less_eq @ extended_enat ) @ Ord
@ ^ [X2: extended_enat] : ( F @ X2 @ ( extended_eSuc @ X2 ) ) )
=> ( ! [X3: extended_enat] : ( Ord @ Bot @ ( F @ X3 @ ( extended_eSuc @ X3 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ extended_enat @ A @ ( ord_less_eq @ extended_enat ) @ Ord
@ ^ [X2: extended_enat] :
( coindu440805660e_enat @ A @ Bot
@ ^ [X6: extended_enat] : ( F @ X6 @ X2 )
@ X2 ) ) ) ) ) ).
% monotone_enat_le_case
thf(fact_130_ldropn__Suc,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( suc @ N ) @ Xs )
= ( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A] : ( coindu531130065ldropn @ A @ N )
@ Xs ) ) ).
% ldropn_Suc
thf(fact_131_ldrop__eSuc,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ ( extended_eSuc @ N ) @ Xs )
= ( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A] : ( coindu191418589_ldrop @ A @ N )
@ Xs ) ) ).
% ldrop_eSuc
thf(fact_132_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_133_lfilter_Osimps,axiom,
! [A: $tType] :
( ( coindu1889960678filter @ A )
= ( ^ [P3: A > $o] :
( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A,Xs3: coindu1593790203_llist @ A] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ X2 ) @ ( coindu1121789889_LCons @ A @ X2 @ ( coindu1889960678filter @ A @ P3 @ Xs3 ) ) @ ( coindu1889960678filter @ A @ P3 @ Xs3 ) ) ) ) ) ).
% lfilter.simps
thf(fact_134_ldropWhile_Osimps,axiom,
! [A: $tType] :
( ( coindu438612276pWhile @ A )
= ( ^ [P3: A > $o,Xs2: coindu1593790203_llist @ A] :
( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A,Xs3: coindu1593790203_llist @ A] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ X2 ) @ ( coindu438612276pWhile @ A @ P3 @ Xs3 ) @ Xs2 )
@ Xs2 ) ) ) ).
% ldropWhile.simps
thf(fact_135_monotone__lprefix__case__lfp,axiom,
! [D: $tType,C: $tType] :
( ( order_bot @ D @ ( type2 @ D ) )
=> ! [F: C > ( coindu1593790203_llist @ C ) > ( coindu1593790203_llist @ C ) > D] :
( ! [X3: C] :
( comple1396247847notone @ ( coindu1593790203_llist @ C ) @ D @ ( coindu1696667936prefix @ C ) @ ( ord_less_eq @ D )
@ ^ [Xs2: coindu1593790203_llist @ C] : ( F @ X3 @ Xs2 @ ( coindu1121789889_LCons @ C @ X3 @ Xs2 ) ) )
=> ( comple1396247847notone @ ( coindu1593790203_llist @ C ) @ D @ ( coindu1696667936prefix @ C ) @ ( ord_less_eq @ D )
@ ( coindu882539134_llist @ D @ C @ ( bot_bot @ D )
@ ^ [X2: C,Xs2: coindu1593790203_llist @ C] : ( F @ X2 @ Xs2 @ ( coindu1121789889_LCons @ C @ X2 @ Xs2 ) ) ) ) ) ) ).
% monotone_lprefix_case_lfp
thf(fact_136_lfilter_Omono,axiom,
! [A: $tType,P: A > $o,X: coindu1593790203_llist @ A] :
( comple1396247847notone @ ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) @ ( coindu1593790203_llist @ A ) @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) ) @ ( coindu1696667936prefix @ A )
@ ^ [Lfilter: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A )] :
( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A,Xs3: coindu1593790203_llist @ A] : ( if @ ( coindu1593790203_llist @ A ) @ ( P @ X2 ) @ ( coindu1121789889_LCons @ A @ X2 @ ( Lfilter @ Xs3 ) ) @ ( Lfilter @ Xs3 ) )
@ X ) ) ).
% lfilter.mono
thf(fact_137_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_138_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_139_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_140_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_141_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_142_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_143_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_144_monotone__if__fun,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Orda: B > B > $o,Ordb: D > D > $o,F3: ( A > B ) > C > D,G2: ( A > B ) > C > D,C2: C > $o] :
( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ F3 )
=> ( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ G2 )
=> ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb )
@ ^ [F2: A > B,N5: C] : ( if @ D @ ( C2 @ N5 ) @ ( F3 @ F2 @ N5 ) @ ( G2 @ F2 @ N5 ) ) ) ) ) ).
% monotone_if_fun
thf(fact_145_call__mono,axiom,
! [B: $tType,A: $tType,Ord: B > B > $o,T2: A] :
( comple1396247847notone @ ( A > B ) @ B @ ( partial_fun_ord @ B @ B @ A @ Ord ) @ Ord
@ ^ [F2: A > B] : ( F2 @ T2 ) ) ).
% call_mono
thf(fact_146_monotone__fun__apply__fun,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Ord: C > C > $o,T2: A,G: D > B] :
( comple1396247847notone @ ( A > B > C ) @ ( D > C ) @ ( partial_fun_ord @ ( B > C ) @ ( B > C ) @ A @ ( partial_fun_ord @ C @ C @ B @ Ord ) ) @ ( partial_fun_ord @ C @ C @ D @ Ord )
@ ^ [F2: A > B > C,N5: D] : ( F2 @ T2 @ ( G @ N5 ) ) ) ).
% monotone_fun_apply_fun
thf(fact_147_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_148_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_149_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_150_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_151_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_152_llist__lift_Oleq__refl,axiom,
! [A: $tType,B: $tType,X: B > ( coindu1593790203_llist @ A )] : ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ X @ X ) ).
% llist_lift.leq_refl
thf(fact_153_llist__lift_Oleq__trans,axiom,
! [A: $tType,B: $tType,X: B > ( coindu1593790203_llist @ A ),Y: B > ( coindu1593790203_llist @ A ),Z: B > ( coindu1593790203_llist @ A )] :
( ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ X @ Y )
=> ( ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ Y @ Z )
=> ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ X @ Z ) ) ) ).
% llist_lift.leq_trans
thf(fact_154_llist__lift_Oleq__antisym,axiom,
! [A: $tType,B: $tType,X: B > ( coindu1593790203_llist @ A ),Y: B > ( coindu1593790203_llist @ A )] :
( ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ X @ Y )
=> ( ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ Y @ X )
=> ( X = Y ) ) ) ).
% llist_lift.leq_antisym
thf(fact_155_llist__lift_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType,Ba: $tType,Bound: Ba > ( coindu1593790203_llist @ A ),G: ( Ba > ( coindu1593790203_llist @ A ) ) > B,Bot: B,F: ( Ba > ( coindu1593790203_llist @ A ) ) > B,Ord: B > B > $o] :
( ! [X3: Ba > ( coindu1593790203_llist @ A )] :
( ( ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) @ X3 @ Bound )
=> ( ( G @ X3 )
= Bot ) )
& ( ~ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) @ X3 @ Bound )
=> ( ( G @ X3 )
= ( F @ X3 ) ) ) )
=> ( ! [X3: Ba > ( coindu1593790203_llist @ A ),Y2: Ba > ( coindu1593790203_llist @ A )] :
( ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) @ X3 @ Y2 )
=> ( ~ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) @ X3 @ Bound )
=> ( Ord @ ( F @ X3 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X3: Ba > ( coindu1593790203_llist @ A )] :
( ~ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) @ X3 @ Bound )
=> ( Ord @ Bot @ ( F @ X3 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ ( Ba > ( coindu1593790203_llist @ A ) ) @ B @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) ) @ Ord @ G ) ) ) ) ) ).
% llist_lift.monotone_if_bot
thf(fact_156_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_157_fun__ord__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( partial_fun_ord @ A @ B @ C )
= ( ^ [Ord2: A > B > $o,F2: C > A,G5: C > B] :
! [X2: C] : ( Ord2 @ ( F2 @ X2 ) @ ( G5 @ X2 ) ) ) ) ).
% fun_ord_def
thf(fact_158_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ).
% subrelI
thf(fact_159_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N4 )
=> ( P @ M2 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_160_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_161_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_162_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_163_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_164_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_165_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_166_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_167_monotone__fun__ord__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Orda: A > A > $o,Ordb: C > C > $o,F: A > B > C] :
( ( comple1396247847notone @ A @ ( B > C ) @ Orda @ ( partial_fun_ord @ C @ C @ B @ Ordb ) @ F )
= ( ! [X2: B] :
( comple1396247847notone @ A @ C @ Orda @ Ordb
@ ^ [Y3: A] : ( F @ Y3 @ X2 ) ) ) ) ).
% monotone_fun_ord_apply
thf(fact_168_monotone__applyI,axiom,
! [B: $tType,A: $tType,C: $tType,Orda: A > A > $o,Ordb: B > B > $o,F3: A > B,X: C] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F3 )
=> ( comple1396247847notone @ ( C > A ) @ B @ ( partial_fun_ord @ A @ A @ C @ Orda ) @ Ordb
@ ^ [F2: C > A] : ( F3 @ ( F2 @ X ) ) ) ) ).
% monotone_applyI
thf(fact_169_llist__lift_Oconst__mono,axiom,
! [Ba: $tType,A: $tType,B: $tType,Ord: B > B > $o,C2: Ba > ( coindu1593790203_llist @ A )] :
( comple1396247847notone @ B @ ( Ba > ( coindu1593790203_llist @ A ) ) @ Ord @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) )
@ ^ [F2: B] : C2 ) ).
% llist_lift.const_mono
thf(fact_170_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_171_llist__lift_Omono2mono,axiom,
! [B: $tType,A: $tType,Ba: $tType,C: $tType,Ordb: B > B > $o,F: B > Ba > ( coindu1593790203_llist @ A ),Orda: C > C > $o,T2: C > B] :
( ( comple1396247847notone @ B @ ( Ba > ( coindu1593790203_llist @ A ) ) @ Ordb @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) ) @ F )
=> ( ( comple1396247847notone @ C @ B @ Orda @ Ordb @ T2 )
=> ( comple1396247847notone @ C @ ( Ba > ( coindu1593790203_llist @ A ) ) @ Orda @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Ba @ ( coindu1696667936prefix @ A ) )
@ ^ [X2: C] : ( F @ ( T2 @ X2 ) ) ) ) ) ).
% llist_lift.mono2mono
thf(fact_172_LCons__mono,axiom,
! [C: $tType,B: $tType,A: $tType,A4: ( A > ( coindu1593790203_llist @ B ) ) > ( coindu1593790203_llist @ C ),X: C] :
( ( comple1396247847notone @ ( A > ( coindu1593790203_llist @ B ) ) @ ( coindu1593790203_llist @ C ) @ ( partial_fun_ord @ ( coindu1593790203_llist @ B ) @ ( coindu1593790203_llist @ B ) @ A @ ( coindu1696667936prefix @ B ) ) @ ( coindu1696667936prefix @ C ) @ A4 )
=> ( comple1396247847notone @ ( A > ( coindu1593790203_llist @ B ) ) @ ( coindu1593790203_llist @ C ) @ ( partial_fun_ord @ ( coindu1593790203_llist @ B ) @ ( coindu1593790203_llist @ B ) @ A @ ( coindu1696667936prefix @ B ) ) @ ( coindu1696667936prefix @ C )
@ ^ [F2: A > ( coindu1593790203_llist @ B )] : ( coindu1121789889_LCons @ C @ X @ ( A4 @ F2 ) ) ) ) ).
% LCons_mono
thf(fact_173_Coinductive__Nat_OeSuc__mono,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: ( A > B ) > extended_enat] :
( ( comple1396247847notone @ ( A > B ) @ extended_enat @ ( partial_fun_ord @ B @ B @ A @ ( ord_less_eq @ B ) ) @ ( ord_less_eq @ extended_enat ) @ F )
=> ( comple1396247847notone @ ( A > B ) @ extended_enat @ ( partial_fun_ord @ B @ B @ A @ ( ord_less_eq @ B ) ) @ ( ord_less_eq @ extended_enat )
@ ^ [X2: A > B] : ( extended_eSuc @ ( F @ X2 ) ) ) ) ) ).
% Coinductive_Nat.eSuc_mono
thf(fact_174_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_175_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_176_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_177_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G5: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G5 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_178_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_180_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_181_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_182_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_183_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_184_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_185_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_186_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_187_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_188_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_189_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_190_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_191_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_192_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_193_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_194_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_195_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_196_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_197_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_198_ldropWhile_Omono,axiom,
! [A: $tType,P: A > $o,X: coindu1593790203_llist @ A] :
( comple1396247847notone @ ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) @ ( coindu1593790203_llist @ A ) @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) ) @ ( coindu1696667936prefix @ A )
@ ^ [LdropWhile: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A )] :
( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A,Xs3: coindu1593790203_llist @ A] : ( if @ ( coindu1593790203_llist @ A ) @ ( P @ X2 ) @ ( LdropWhile @ Xs3 ) @ X )
@ X ) ) ).
% ldropWhile.mono
thf(fact_199_lmap__mono,axiom,
! [B: $tType,A: $tType,F: A > B,Xs: coindu1593790203_llist @ A] :
( comple1396247847notone @ ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) ) @ ( coindu1593790203_llist @ B ) @ ( partial_fun_ord @ ( coindu1593790203_llist @ B ) @ ( coindu1593790203_llist @ B ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ B ) ) @ ( coindu1696667936prefix @ B )
@ ^ [Lmap: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B )] :
( coindu882539134_llist @ ( coindu1593790203_llist @ B ) @ A @ ( coindu1598213697e_LNil @ B )
@ ^ [X2: A,Xs2: coindu1593790203_llist @ A] : ( coindu1121789889_LCons @ B @ ( F @ X2 ) @ ( Lmap @ Xs2 ) )
@ Xs ) ) ).
% lmap_mono
thf(fact_200_fixp__mono,axiom,
! [A: $tType] :
( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
=> ! [F: A > A,G: A > A] :
( ( partial_fun_ord @ A @ A @ A @ ( ord_less_eq @ A ) @ F @ G )
=> ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
=> ( ord_less_eq @ A @ ( comple939513234o_fixp @ A @ F ) @ ( comple939513234o_fixp @ A @ G ) ) ) ) ) ) ).
% fixp_mono
thf(fact_201_mono2mono__ltake1,axiom,
! [A: $tType,C: $tType,Orda: C > C > $o,T2: C > extended_enat,Xs6: coindu1593790203_llist @ A] :
( ( comple1396247847notone @ C @ extended_enat @ Orda @ ( ord_less_eq @ extended_enat ) @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A )
@ ^ [X2: C] : ( coindu1802687541_ltake @ A @ ( T2 @ X2 ) @ Xs6 ) ) ) ).
% mono2mono_ltake1
thf(fact_202_ltake__eSuc,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( extended_eSuc @ N ) @ Xs )
= ( coindu882539134_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu1598213697e_LNil @ A )
@ ^ [X2: A,Xs3: coindu1593790203_llist @ A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1802687541_ltake @ A @ N @ Xs3 ) )
@ Xs ) ) ).
% ltake_eSuc
thf(fact_203_mono2mono__llength,axiom,
! [A2: $tType,C: $tType,Orda: C > C > $o,T2: C > ( coindu1593790203_llist @ A2 )] :
( ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A2 ) @ Orda @ ( coindu1696667936prefix @ A2 ) @ T2 )
=> ( comple1396247847notone @ C @ extended_enat @ Orda @ ( ord_less_eq @ extended_enat )
@ ^ [X2: C] : ( coindu1018505716length @ A2 @ ( T2 @ X2 ) ) ) ) ).
% mono2mono_llength
thf(fact_204_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y2: B] :
( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_205_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_206_ltake__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_LNil
thf(fact_207_llength__LCons,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1018505716length @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( extended_eSuc @ ( coindu1018505716length @ B @ Xs ) ) ) ).
% llength_LCons
thf(fact_208_ltake__eSuc__LCons,axiom,
! [A: $tType,N: extended_enat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( extended_eSuc @ N ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake_eSuc_LCons
thf(fact_209_lprefix__ltake__same,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M @ Xs ) )
= ( ( ord_less_eq @ extended_enat @ N @ M )
| ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M ) ) ) ).
% lprefix_ltake_same
thf(fact_210_ldrop__eq__LNil,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu191418589_ldrop @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% ldrop_eq_LNil
thf(fact_211_ltake__eq__ltake__antimono,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1802687541_ltake @ A @ N @ Ys ) )
=> ( ( ord_less_eq @ extended_enat @ M @ N )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% ltake_eq_ltake_antimono
thf(fact_212_ltake__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= Xs ) ) ).
% ltake_all
thf(fact_213_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_214_pred__subset__eq,axiom,
! [A: $tType,R2: set @ A,S2: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R2 )
@ ^ [X2: A] : ( member @ A @ X2 @ S2 ) )
= ( ord_less_eq @ ( set @ A ) @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_215_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_216_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_217_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_218_ltake__lzip,axiom,
! [A: $tType,B: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu1802687541_ltake @ ( product_prod @ A @ B ) @ N @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( coindu847746867e_lzip @ A @ B @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ B @ N @ Ys ) ) ) ).
% ltake_lzip
thf(fact_219_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_220_monotone__llength,axiom,
! [A: $tType] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ extended_enat @ ( coindu1696667936prefix @ A ) @ ( ord_less_eq @ extended_enat ) @ ( coindu1018505716length @ A ) ) ).
% monotone_llength
thf(fact_221_lprefix__llength__le,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% lprefix_llength_le
thf(fact_222_monotone__ltake2,axiom,
! [A: $tType,N: extended_enat] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1802687541_ltake @ A @ N ) ) ).
% monotone_ltake2
thf(fact_223_monotone__ldropn__aux,axiom,
! [A: $tType] :
( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( nat > ( coindu1593790203_llist @ A ) ) @ ( coindu1696667936prefix @ A ) @ ( partial_fun_ord @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ nat @ ( coindu1696667936prefix @ A ) )
@ ^ [Xs2: coindu1593790203_llist @ A,N5: nat] : ( coindu531130065ldropn @ A @ N5 @ Xs2 ) ) ).
% monotone_ldropn_aux
thf(fact_224_mono2mono__ltake2,axiom,
! [A: $tType,C: $tType,Orda: C > C > $o,T2: C > ( coindu1593790203_llist @ A ),N2: extended_enat] :
( ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A ) @ T2 )
=> ( comple1396247847notone @ C @ ( coindu1593790203_llist @ A ) @ Orda @ ( coindu1696667936prefix @ A )
@ ^ [X2: C] : ( coindu1802687541_ltake @ A @ N2 @ ( T2 @ X2 ) ) ) ) ).
% mono2mono_ltake2
thf(fact_225_monotone__fun__eSuc,axiom,
! [A: $tType,X: A] :
( comple1396247847notone @ ( A > extended_enat ) @ extended_enat
@ ( partial_fun_ord @ extended_enat @ extended_enat @ A
@ ^ [Y3: extended_enat,X2: extended_enat] : ( ord_less_eq @ extended_enat @ X2 @ Y3 ) )
@ ^ [Y3: extended_enat,X2: extended_enat] : ( ord_less_eq @ extended_enat @ X2 @ Y3 )
@ ^ [F2: A > extended_enat] : ( extended_eSuc @ ( F2 @ X ) ) ) ).
% monotone_fun_eSuc
thf(fact_226_ldrop__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
=> ( ( coindu191418589_ldrop @ A @ M @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldrop_all
thf(fact_227_fixp__unfold,axiom,
! [A: $tType] :
( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
=> ! [F: A > A] :
( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( comple939513234o_fixp @ A @ F )
= ( F @ ( comple939513234o_fixp @ A @ F ) ) ) ) ) ).
% fixp_unfold
thf(fact_228_fixp__lowerbound,axiom,
! [A: $tType] :
( ( comple1141879883l_ccpo @ A @ ( type2 @ A ) )
=> ! [F: A > A,Z: A] :
( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( ord_less_eq @ A @ ( F @ Z ) @ Z )
=> ( ord_less_eq @ A @ ( comple939513234o_fixp @ A @ F ) @ Z ) ) ) ) ).
% fixp_lowerbound
thf(fact_229_ltake__LCons,axiom,
! [A: $tType,N: extended_enat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu440805660e_enat @ ( coindu1593790203_llist @ A ) @ ( coindu1598213697e_LNil @ A )
@ ^ [N3: extended_enat] : ( coindu1121789889_LCons @ A @ X @ ( coindu1802687541_ltake @ A @ N3 @ Xs ) )
@ N ) ) ).
% ltake_LCons
thf(fact_230_monotone__ltake1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( comple1396247847notone @ extended_enat @ ( coindu1593790203_llist @ A ) @ ( ord_less_eq @ extended_enat ) @ ( coindu1696667936prefix @ A )
@ ^ [N5: extended_enat] : ( coindu1802687541_ltake @ A @ N5 @ Xs ) ) ).
% monotone_ltake1
thf(fact_231_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_232_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_233_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_234_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_235_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_236_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_237_subsetI,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ).
% subsetI
thf(fact_238_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( A4 = B6 ) ) ) ).
% subset_antisym
thf(fact_239_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_240_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_241_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_242_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_243_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_244_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_245_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_246_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_247_set__mp,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_mp
thf(fact_248_in__mono,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_249_subsetD,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetD
thf(fact_250_subsetCE,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetCE
thf(fact_251_equalityE,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( A4 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% equalityE
thf(fact_252_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( member @ A @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_253_equalityD1,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( A4 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ).
% equalityD1
thf(fact_254_equalityD2,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( A4 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ).
% equalityD2
%----Type constructors (36)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A2: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( comple187826305attice @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A2: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( comple1141879883l_ccpo @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A2: $tType,A8: $tType] :
( ( order_bot @ A8 @ ( type2 @ A8 ) )
=> ( order_bot @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A2: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A2: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A2: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A2: $tType,A8: $tType] :
( ( bot @ A8 @ ( type2 @ A8 ) )
=> ( bot @ ( A2 > A8 ) @ ( type2 @ ( A2 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_6,axiom,
! [A2: $tType] : ( comple187826305attice @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_7,axiom,
! [A2: $tType] : ( comple1141879883l_ccpo @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_8,axiom,
! [A2: $tType] : ( order_bot @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
! [A2: $tType] : ( preorder @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_10,axiom,
! [A2: $tType] : ( order @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
! [A2: $tType] : ( ord @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_12,axiom,
! [A2: $tType] : ( bot @ ( set @ A2 ) @ ( type2 @ ( set @ A2 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_13,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_14,axiom,
comple1141879883l_ccpo @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_15,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_17,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_21,axiom,
comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_22,axiom,
comple1141879883l_ccpo @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_23,axiom,
order_bot @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_24,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_25,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_26,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_27,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_28,axiom,
bot @ extended_enat @ ( type2 @ extended_enat ) ).
%----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 (1)
thf(conj_0,conjecture,
comple1396247847notone @ ( coindu1593790203_llist @ a ) @ ( coindu1593790203_llist @ ( product_prod @ b @ a ) ) @ ( coindu1696667936prefix @ a ) @ ( coindu1696667936prefix @ ( product_prod @ b @ a ) ) @ ( coindu847746867e_lzip @ b @ a @ xs ) ).
%------------------------------------------------------------------------------