TPTP Problem File: DAT251^1.p
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%------------------------------------------------------------------------------
% File : DAT251^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Infinite streams (sequences/lists) 443
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : stream__443.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 326 ( 150 unt; 59 typ; 0 def)
% Number of atoms : 698 ( 481 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 4460 ( 140 ~; 30 |; 80 &;3912 @)
% ( 0 <=>; 298 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 246 ( 246 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 3 con; 0-5 aty)
% Number of variables : 1171 ( 34 ^;1002 !; 72 ?;1171 :)
% ( 63 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:42:29.928
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
stream170649215stream: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (53)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
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_Stream__Mirabelle__hbrgyiwlrc_Ocycle,type,
stream317748790_cycle:
!>[A: $tType] : ( ( list @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Oflat,type,
stream942386729e_flat:
!>[A: $tType] : ( ( stream170649215stream @ ( list @ A ) ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osdrop__while,type,
stream1195056575_while:
!>[A: $tType] : ( ( A > $o ) > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Oshift,type,
stream1035003186_shift:
!>[A: $tType] : ( ( list @ A ) > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osmember,type,
stream1586597341member:
!>[A: $tType] : ( A > ( stream170649215stream @ A ) > $o ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_OSCons,type,
stream641971652_SCons:
!>[A: $tType] : ( A > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocase__stream,type,
stream1342653232stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream170649215stream @ A ) > B ) > ( stream170649215stream @ A ) > B ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Oshd,type,
stream_Mirabelle_shd:
!>[A: $tType] : ( ( stream170649215stream @ A ) > A ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osmap,type,
stream2128578057e_smap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream170649215stream @ A ) > ( stream170649215stream @ Aa ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostl,type,
stream_Mirabelle_stl:
!>[A: $tType] : ( ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostreams,type,
stream2015131171treams:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( stream170649215stream @ A ) ) ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_ws,type,
ws: stream170649215stream @ ( list @ a ) ).
%----Relevant facts (255)
thf(fact_0_shift__left__inj,axiom,
! [A: $tType,Xs: list @ A,S1: stream170649215stream @ A,S2: stream170649215stream @ A] :
( ( ( stream1035003186_shift @ A @ Xs @ S1 )
= ( stream1035003186_shift @ A @ Xs @ S2 ) )
= ( S1 = S2 ) ) ).
% shift_left_inj
thf(fact_1_shift_Osimps_I1_J,axiom,
! [A: $tType,S: stream170649215stream @ A] :
( ( stream1035003186_shift @ A @ ( nil @ A ) @ S )
= S ) ).
% shift.simps(1)
thf(fact_2_stream_Oexpand,axiom,
! [A: $tType,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
( ( ( ( stream_Mirabelle_shd @ A @ Stream )
= ( stream_Mirabelle_shd @ A @ Stream2 ) )
& ( ( stream_Mirabelle_stl @ A @ Stream )
= ( stream_Mirabelle_stl @ A @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_3_stream_Ocoinduct,axiom,
! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
= ( stream_Mirabelle_shd @ A @ Stream4 ) )
& ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_4_stream_Ocoinduct__strong,axiom,
! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
= ( stream_Mirabelle_shd @ A @ Stream4 ) )
& ( ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) )
| ( ( stream_Mirabelle_stl @ A @ Stream3 )
= ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_5_flat__Stream,axiom,
! [A: $tType,Xs: list @ A,Ws: stream170649215stream @ ( list @ A )] :
( ( Xs
!= ( nil @ A ) )
=> ( ( stream942386729e_flat @ A @ ( stream641971652_SCons @ ( list @ A ) @ Xs @ Ws ) )
= ( stream1035003186_shift @ A @ Xs @ ( stream942386729e_flat @ A @ Ws ) ) ) ) ).
% flat_Stream
thf(fact_6_shift__simps_I1_J,axiom,
! [A: $tType,Xs: list @ A,S: stream170649215stream @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( stream_Mirabelle_shd @ A @ ( stream1035003186_shift @ A @ Xs @ S ) )
= ( stream_Mirabelle_shd @ A @ S ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( stream_Mirabelle_shd @ A @ ( stream1035003186_shift @ A @ Xs @ S ) )
= ( hd @ A @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_7_flat_Osimps_I1_J,axiom,
! [A: $tType,Ws: stream170649215stream @ ( list @ A )] :
( ( stream_Mirabelle_shd @ A @ ( stream942386729e_flat @ A @ Ws ) )
= ( hd @ A @ ( stream_Mirabelle_shd @ ( list @ A ) @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_8_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( stream1342653232stream @ A @ B )
= ( ^ [F: A > ( stream170649215stream @ A ) > B,Stream5: stream170649215stream @ A] : ( F @ ( stream_Mirabelle_shd @ A @ Stream5 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_9_shift__simps_I2_J,axiom,
! [A: $tType,Xs: list @ A,S: stream170649215stream @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( stream_Mirabelle_stl @ A @ ( stream1035003186_shift @ A @ Xs @ S ) )
= ( stream_Mirabelle_stl @ A @ S ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( stream_Mirabelle_stl @ A @ ( stream1035003186_shift @ A @ Xs @ S ) )
= ( stream1035003186_shift @ A @ ( tl @ A @ Xs ) @ S ) ) ) ) ).
% shift_simps(2)
thf(fact_10_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( stream1195056575_while @ A )
= ( ^ [P: A > $o,S3: stream170649215stream @ A] : ( if @ ( stream170649215stream @ A ) @ ( P @ ( stream_Mirabelle_shd @ A @ S3 ) ) @ ( stream1195056575_while @ A @ P @ ( stream_Mirabelle_stl @ A @ S3 ) ) @ S3 ) ) ) ).
% sdrop_while.simps
thf(fact_11_flat_Osimps_I2_J,axiom,
! [A: $tType,Ws: stream170649215stream @ ( list @ A )] :
( ( stream_Mirabelle_stl @ A @ ( stream942386729e_flat @ A @ Ws ) )
= ( stream942386729e_flat @ A
@ ( if @ ( stream170649215stream @ ( list @ A ) )
@ ( ( tl @ A @ ( stream_Mirabelle_shd @ ( list @ A ) @ Ws ) )
= ( nil @ A ) )
@ ( stream_Mirabelle_stl @ ( list @ A ) @ Ws )
@ ( stream641971652_SCons @ ( list @ A ) @ ( tl @ A @ ( stream_Mirabelle_shd @ ( list @ A ) @ Ws ) ) @ ( stream_Mirabelle_stl @ ( list @ A ) @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_12_cycle__decomp,axiom,
! [A: $tType,U: list @ A] :
( ( U
!= ( nil @ A ) )
=> ( ( stream317748790_cycle @ A @ U )
= ( stream1035003186_shift @ A @ U @ ( stream317748790_cycle @ A @ U ) ) ) ) ).
% cycle_decomp
thf(fact_13_stream_Ocollapse,axiom,
! [A: $tType,Stream: stream170649215stream @ A] :
( ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_14_streamsE,axiom,
! [A: $tType,S: stream170649215stream @ A,A2: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A2 ) )
=> ~ ( ( member2 @ A @ ( stream_Mirabelle_shd @ A @ S ) @ A2 )
=> ~ ( member2 @ ( stream170649215stream @ A ) @ ( stream_Mirabelle_stl @ A @ S ) @ ( stream2015131171treams @ A @ A2 ) ) ) ) ).
% streamsE
thf(fact_15_in__streams,axiom,
! [A: $tType,S: stream170649215stream @ A,S4: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ ( stream_Mirabelle_stl @ A @ S ) @ ( stream2015131171treams @ A @ S4 ) )
=> ( ( member2 @ A @ ( stream_Mirabelle_shd @ A @ S ) @ S4 )
=> ( member2 @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ S4 ) ) ) ) ).
% in_streams
thf(fact_16_stream_Oinject,axiom,
! [A: $tType,X1: A,X2: stream170649215stream @ A,Y1: A,Y2: stream170649215stream @ A] :
( ( ( stream641971652_SCons @ A @ X1 @ X2 )
= ( stream641971652_SCons @ A @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% stream.inject
thf(fact_17_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_18_stream_Ocase,axiom,
! [B: $tType,A: $tType,F2: A > ( stream170649215stream @ A ) > B,X1: A,X2: stream170649215stream @ A] :
( ( stream1342653232stream @ A @ B @ F2 @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= ( F2 @ X1 @ X2 ) ) ).
% stream.case
thf(fact_19_streams_Ocases,axiom,
! [A: $tType,A3: stream170649215stream @ A,A2: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ A3 @ ( stream2015131171treams @ A @ A2 ) )
=> ~ ! [A4: A,S5: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ A4 @ S5 ) )
=> ( ( member2 @ A @ A4 @ A2 )
=> ~ ( member2 @ ( stream170649215stream @ A ) @ S5 @ ( stream2015131171treams @ A @ A2 ) ) ) ) ) ).
% streams.cases
thf(fact_20_streams_Osimps,axiom,
! [A: $tType,A3: stream170649215stream @ A,A2: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ A3 @ ( stream2015131171treams @ A @ A2 ) )
= ( ? [A5: A,S3: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ A5 @ S3 ) )
& ( member2 @ A @ A5 @ A2 )
& ( member2 @ ( stream170649215stream @ A ) @ S3 @ ( stream2015131171treams @ A @ A2 ) ) ) ) ) ).
% streams.simps
thf(fact_21_stream_Oexhaust,axiom,
! [A: $tType,Y: stream170649215stream @ A] :
~ ! [X12: A,X22: stream170649215stream @ A] :
( Y
!= ( stream641971652_SCons @ A @ X12 @ X22 ) ) ).
% stream.exhaust
thf(fact_22_streams__Stream,axiom,
! [A: $tType,X: A,S: stream170649215stream @ A,A2: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ ( stream641971652_SCons @ A @ X @ S ) @ ( stream2015131171treams @ A @ A2 ) )
= ( ( member2 @ A @ X @ A2 )
& ( member2 @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A2 ) ) ) ) ).
% streams_Stream
thf(fact_23_streams_Ocoinduct,axiom,
! [A: $tType,X3: ( stream170649215stream @ A ) > $o,X: stream170649215stream @ A,A2: set @ A] :
( ( X3 @ X )
=> ( ! [X4: stream170649215stream @ A] :
( ( X3 @ X4 )
=> ? [A6: A,S6: stream170649215stream @ A] :
( ( X4
= ( stream641971652_SCons @ A @ A6 @ S6 ) )
& ( member2 @ A @ A6 @ A2 )
& ( ( X3 @ S6 )
| ( member2 @ ( stream170649215stream @ A ) @ S6 @ ( stream2015131171treams @ A @ A2 ) ) ) ) )
=> ( member2 @ ( stream170649215stream @ A ) @ X @ ( stream2015131171treams @ A @ A2 ) ) ) ) ).
% streams.coinduct
thf(fact_24_sdrop__while__SCons,axiom,
! [A: $tType,P2: A > $o,A3: A,S: stream170649215stream @ A] :
( ( ( P2 @ A3 )
=> ( ( stream1195056575_while @ A @ P2 @ ( stream641971652_SCons @ A @ A3 @ S ) )
= ( stream1195056575_while @ A @ P2 @ S ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( stream1195056575_while @ A @ P2 @ ( stream641971652_SCons @ A @ A3 @ S ) )
= ( stream641971652_SCons @ A @ A3 @ S ) ) ) ) ).
% sdrop_while_SCons
thf(fact_25_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_26_cycle_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( stream_Mirabelle_shd @ A @ ( stream317748790_cycle @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% cycle.simps(1)
thf(fact_27_flat_Ocode,axiom,
! [A: $tType] :
( ( stream942386729e_flat @ A )
= ( ^ [Ws2: stream170649215stream @ ( list @ A )] :
( stream641971652_SCons @ A @ ( hd @ A @ ( stream_Mirabelle_shd @ ( list @ A ) @ Ws2 ) )
@ ( stream942386729e_flat @ A
@ ( if @ ( stream170649215stream @ ( list @ A ) )
@ ( ( tl @ A @ ( stream_Mirabelle_shd @ ( list @ A ) @ Ws2 ) )
= ( nil @ A ) )
@ ( stream_Mirabelle_stl @ ( list @ A ) @ Ws2 )
@ ( stream641971652_SCons @ ( list @ A ) @ ( tl @ A @ ( stream_Mirabelle_shd @ ( list @ A ) @ Ws2 ) ) @ ( stream_Mirabelle_stl @ ( list @ A ) @ Ws2 ) ) ) ) ) ) ) ).
% flat.code
thf(fact_28_stream_Osel_I2_J,axiom,
! [A: $tType,X1: A,X2: stream170649215stream @ A] :
( ( stream_Mirabelle_stl @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= X2 ) ).
% stream.sel(2)
thf(fact_29_stream_Osel_I1_J,axiom,
! [A: $tType,X1: A,X2: stream170649215stream @ A] :
( ( stream_Mirabelle_shd @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_30_stream_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F2: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
( ( P2 @ ( stream1342653232stream @ A @ B @ F2 @ Stream ) )
= ( ~ ( ( Stream
= ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
& ~ ( P2 @ ( F2 @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ) ).
% stream.split_sel_asm
thf(fact_31_stream_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F2: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
( ( P2 @ ( stream1342653232stream @ A @ B @ F2 @ Stream ) )
= ( ( Stream
= ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
=> ( P2 @ ( F2 @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ).
% stream.split_sel
thf(fact_32_streams__stl,axiom,
! [A: $tType,S: stream170649215stream @ A,A2: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A2 ) )
=> ( member2 @ ( stream170649215stream @ A ) @ ( stream_Mirabelle_stl @ A @ S ) @ ( stream2015131171treams @ A @ A2 ) ) ) ).
% streams_stl
thf(fact_33_streams__shd,axiom,
! [A: $tType,S: stream170649215stream @ A,A2: set @ A] :
( ( member2 @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A2 ) )
=> ( member2 @ A @ ( stream_Mirabelle_shd @ A @ S ) @ A2 ) ) ).
% streams_shd
thf(fact_34_stream_Oexhaust__sel,axiom,
! [A: $tType,Stream: stream170649215stream @ A] :
( Stream
= ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_35_smember__code,axiom,
! [A: $tType,X: A,Y: A,S: stream170649215stream @ A] :
( ( stream1586597341member @ A @ X @ ( stream641971652_SCons @ A @ Y @ S ) )
= ( ( X != Y )
=> ( stream1586597341member @ A @ X @ S ) ) ) ).
% smember_code
thf(fact_36_flat__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ws: stream170649215stream @ ( list @ A )] :
( ( stream942386729e_flat @ A @ ( stream641971652_SCons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ws ) )
= ( stream641971652_SCons @ A @ X
@ ( stream942386729e_flat @ A
@ ( if @ ( stream170649215stream @ ( list @ A ) )
@ ( Xs
= ( nil @ A ) )
@ Ws
@ ( stream641971652_SCons @ ( list @ A ) @ Xs @ Ws ) ) ) ) ) ).
% flat_Cons
thf(fact_37_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F1: B,F22: A > ( list @ A ) > B,List3: list @ A] :
( if @ B
@ ( List3
= ( nil @ A ) )
@ F1
@ ( F22 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_38_list__ex1__simps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
~ ( list_ex1 @ A @ P2 @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_39_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_40_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_41_smap__ctr,axiom,
! [B: $tType,A: $tType,F2: B > A,S: stream170649215stream @ B,X: A,S7: stream170649215stream @ A] :
( ( ( stream2128578057e_smap @ B @ A @ F2 @ S )
= ( stream641971652_SCons @ A @ X @ S7 ) )
= ( ( ( F2 @ ( stream_Mirabelle_shd @ B @ S ) )
= X )
& ( ( stream2128578057e_smap @ B @ A @ F2 @ ( stream_Mirabelle_stl @ B @ S ) )
= S7 ) ) ) ).
% smap_ctr
thf(fact_42_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F2 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_43_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: A > $o] :
( ( member2 @ A @ A3 @ ( collect @ A @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member2 @ A @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X4: A] :
( ( F2 @ X4 )
= ( G @ X4 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_49_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_50_stream_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: stream170649215stream @ A] :
( ( stream_Mirabelle_stl @ B @ ( stream2128578057e_smap @ A @ B @ F2 @ A3 ) )
= ( stream2128578057e_smap @ A @ B @ F2 @ ( stream_Mirabelle_stl @ A @ A3 ) ) ) ).
% stream.map_sel(2)
thf(fact_51_stream_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: stream170649215stream @ A] :
( ( stream_Mirabelle_shd @ B @ ( stream2128578057e_smap @ A @ B @ F2 @ A3 ) )
= ( F2 @ ( stream_Mirabelle_shd @ A @ A3 ) ) ) ).
% stream.map_sel(1)
thf(fact_52_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F23: A > ( list @ A ) > B,X21: A,X222: list @ A] :
( ( case_list @ B @ A @ F12 @ F23 @ ( cons @ A @ X21 @ X222 ) )
= ( F23 @ X21 @ X222 ) ) ).
% list.simps(5)
thf(fact_53_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_54_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_55_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X4: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_56_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_57_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_58_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_59_list_Oinducts,axiom,
! [A: $tType,P2: ( list @ A ) > $o,List: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X12: A,X22: list @ A] :
( ( P2 @ X22 )
=> ( P2 @ ( cons @ A @ X12 @ X22 ) ) )
=> ( P2 @ List ) ) ) ).
% list.inducts
thf(fact_60_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_61_list__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs2: list @ A] : ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys3: list @ B] : ( P2 @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys3 ) )
=> ( ! [X4: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_62_splice_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P2 @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P2 @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X4: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_63_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X4: A] :
( X
!= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Y4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_64_remdups__adj_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Y4: A,Xs2: list @ A] :
( ( ( X4 = Y4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) ) )
=> ( ( ( X4 != Y4 )
=> ( P2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P2 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_65_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P2: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_66_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P2: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F3: A > B,X12: list @ B] : ( P2 @ F3 @ ( nil @ A ) @ X12 )
=> ( ! [F3: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P2 @ F3 @ As @ ( cons @ B @ ( F3 @ A4 ) @ Bs ) )
=> ( P2 @ F3 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_67_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_68_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_69_stream_Omap,axiom,
! [B: $tType,A: $tType,F2: A > B,X1: A,X2: stream170649215stream @ A] :
( ( stream2128578057e_smap @ A @ B @ F2 @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= ( stream641971652_SCons @ B @ ( F2 @ X1 ) @ ( stream2128578057e_smap @ A @ B @ F2 @ X2 ) ) ) ).
% stream.map
thf(fact_70_smap__streams,axiom,
! [A: $tType,B: $tType,S: stream170649215stream @ A,A2: set @ A,F2: A > B,B2: set @ B] :
( ( member2 @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A2 ) )
=> ( ! [X4: A] :
( ( member2 @ A @ X4 @ A2 )
=> ( member2 @ B @ ( F2 @ X4 ) @ B2 ) )
=> ( member2 @ ( stream170649215stream @ B ) @ ( stream2128578057e_smap @ A @ B @ F2 @ S ) @ ( stream2015131171treams @ B @ B2 ) ) ) ) ).
% smap_streams
thf(fact_71_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F23: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F12 @ F23 @ ( nil @ A ) )
= F12 ) ).
% list.simps(4)
thf(fact_72_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F12: B,F23: A > ( list @ A ) > B,List: list @ A] :
( ( P2 @ ( case_list @ B @ A @ F12 @ F23 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P2 @ F12 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P2 @ ( F23 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_73_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F12: B,F23: A > ( list @ A ) > B,List: list @ A] :
( ( P2 @ ( case_list @ B @ A @ F12 @ F23 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P2 @ F12 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P2 @ ( F23 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_74_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X5: A] :
( Xs
= ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_75_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X5: A] :
( Xs
= ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_76_shift_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,S: stream170649215stream @ A] :
( ( stream1035003186_shift @ A @ ( cons @ A @ X @ Xs ) @ S )
= ( stream641971652_SCons @ A @ X @ ( stream1035003186_shift @ A @ Xs @ S ) ) ) ).
% shift.simps(2)
thf(fact_77_cycle__rotated,axiom,
! [A: $tType,V2: list @ A,U: list @ A,S: stream170649215stream @ A] :
( ( V2
!= ( nil @ A ) )
=> ( ( ( stream317748790_cycle @ A @ U )
= ( stream1035003186_shift @ A @ V2 @ S ) )
=> ( ( stream317748790_cycle @ A @ ( append @ A @ ( tl @ A @ U ) @ ( cons @ A @ ( hd @ A @ U ) @ ( nil @ A ) ) ) )
= ( stream1035003186_shift @ A @ ( tl @ A @ V2 ) @ S ) ) ) ) ).
% cycle_rotated
thf(fact_78_cycle_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( stream_Mirabelle_stl @ A @ ( stream317748790_cycle @ A @ Xs ) )
= ( stream317748790_cycle @ A @ ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% cycle.simps(2)
thf(fact_79_cycle_Ocode,axiom,
! [A: $tType] :
( ( stream317748790_cycle @ A )
= ( ^ [Xs3: list @ A] : ( stream641971652_SCons @ A @ ( hd @ A @ Xs3 ) @ ( stream317748790_cycle @ A @ ( append @ A @ ( tl @ A @ Xs3 ) @ ( cons @ A @ ( hd @ A @ Xs3 ) @ ( nil @ A ) ) ) ) ) ) ) ).
% cycle.code
thf(fact_80_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_81_cycle__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( stream317748790_cycle @ A @ ( cons @ A @ X @ Xs ) )
= ( stream641971652_SCons @ A @ X @ ( stream317748790_cycle @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ) ).
% cycle_Cons
thf(fact_82_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_83_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A4: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_84_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V: A,Va: list @ A] :
( ( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y4: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ Y4 @ Ys3 ) )
=> ( Y
!= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ ( splice @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% splice.elims
thf(fact_85_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_86_append__same__eq,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys2 @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_87_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_88_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_89_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys2 ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_90_self__append__conv2,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( append @ A @ Xs @ Ys2 ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_91_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Ys2 )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_92_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys2 ) )
= ( Ys2
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_93_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Xs )
= ( Ys2
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_94_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_95_shift__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,S: stream170649215stream @ A] :
( ( stream1035003186_shift @ A @ ( append @ A @ Xs @ Ys2 ) @ S )
= ( stream1035003186_shift @ A @ Xs @ ( stream1035003186_shift @ A @ Ys2 @ S ) ) ) ).
% shift_append
thf(fact_96_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq @ A @ Less @ Xs @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_97_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_98_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_99_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_100_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_101_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_102_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_103_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F2 )
= ( append @ A @ ( F2 @ X ) @ ( bind @ B @ A @ Xs @ F2 ) ) ) ).
% bind_simps(2)
thf(fact_104_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys2 )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys2
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_105_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_106_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys2: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_107_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_108_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_109_append__Nil,axiom,
! [A: $tType,Ys2: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_110_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs = Ys2 )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_111_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys2 )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_112_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_113_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_114_splice_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( cons @ A @ X @ ( cons @ A @ Y @ ( splice @ A @ Xs @ Ys2 ) ) ) ) ).
% splice.simps(3)
thf(fact_115_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% splice.simps(1)
thf(fact_116_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P2: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_117_append__eq__Cons__conv,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys2 @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_118_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys2 @ Zs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append @ A @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_119_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y4: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_120_rev__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_121_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( cons @ A @ A3 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F2 @ As2 @ ( cons @ B @ ( F2 @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_122_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F2: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_123_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Ys2 ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_124_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
? [Ps: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs4 ) )
& ( Ys2
= ( append @ A @ Ps @ Ys5 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_125_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys3: list @ A] : A22 != Ys3 )
=> ( ! [X4: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys3 ) )
=> ~ ( Less @ X4 @ Y4 ) ) )
=> ~ ! [X4: A,Y4: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys3 ) )
=> ( ~ ( Less @ X4 @ Y4 )
=> ( ~ ( Less @ Y4 @ X4 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_126_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys ) )
| ? [X5: A,Y3: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) )
& ( Less2 @ X5 @ Y3 ) )
| ? [X5: A,Y3: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) )
& ~ ( Less2 @ X5 @ Y3 )
& ~ ( Less2 @ Y3 @ X5 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_127_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X2: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X2 )
=> ( ! [X12: list @ A] : ( P2 @ ( nil @ A ) @ X12 )
=> ( ! [X4: A,Y4: A,Xs2: list @ A,Ys3: list @ A] :
( ( Less @ X4 @ Y4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ! [X4: A,Y4: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( Less @ X4 @ Y4 )
=> ( ~ ( Less @ Y4 @ X4 )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_128_splice_Osimps_I2_J,axiom,
! [A: $tType,V2: A,Va2: list @ A] :
( ( splice @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ).
% splice.simps(2)
thf(fact_129_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_130_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_131_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_132_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F2 @ X ) @ ( maps @ B @ A @ F2 @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_133_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_134_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_135_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% remdups_adj_Nil_iff
thf(fact_136_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_137_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_138_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_139_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_140_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_141_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_142_remdups__adj_Oelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X4: A] :
( ( X
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ~ ! [X4: A,Y4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ~ ( ( ( X4 = Y4 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X4 @ Xs2 ) ) ) )
& ( ( X4 != Y4 )
=> ( Y
= ( cons @ A @ X4 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_143_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_144_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( maps @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_145_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_146_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_147_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_148_SuccD,axiom,
! [A: $tType,K: A,Kl: set @ ( list @ A ),Kl2: list @ A] :
( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ Kl2 ) )
=> ( member2 @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl ) ) ).
% SuccD
thf(fact_149_SuccI,axiom,
! [A: $tType,Kl2: list @ A,K: A,Kl: set @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl )
=> ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_150_last__appendR,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ).
% last_appendR
thf(fact_151_last__appendL,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_152_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_153_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_154_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_155_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys2 )
= ( ( Ys2
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys2 )
= Xs )
& ( ( last @ A @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_156_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_157_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_158_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_159_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_160_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys2
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_161_butlast__append,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_162_last__append,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ) ).
% last_append
thf(fact_163_last__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( ( tl @ A @ Xs )
!= ( nil @ A ) ) )
=> ( ( last @ A @ ( tl @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_tl
thf(fact_164_BNF__Greatest__Fixpoint_Oshift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K2: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.shift_def
thf(fact_165_empty__Shift,axiom,
! [A: $tType,Kl: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl )
=> ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ ( nil @ A ) ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_166_Succ__Shift,axiom,
! [A: $tType,Kl: set @ ( list @ A ),K: A,Kl2: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ @ A @ Kl @ ( cons @ A @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_167_ShiftD,axiom,
! [A: $tType,Kl2: list @ A,Kl: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ Kl2 @ ( bNF_Greatest_Shift @ A @ Kl @ K ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_168_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys2 ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys2 ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_169_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X1: list @ A,X2: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R2 @ X1 @ X2 )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Y4: B,Xs2: list @ A,Ys3: list @ B] :
( ( R2 @ X4 @ Y4 )
=> ( ( listrelp @ A @ B @ R2 @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_170_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R3: A > B > $o,A12: list @ A,A23: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X5: A,Y3: B,Xs3: list @ A,Ys: list @ B] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y3 @ Ys ) )
& ( R3 @ X5 @ Y3 )
& ( listrelp @ A @ B @ R3 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_171_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys2 ) )
= ( Xs = Ys2 ) ) ).
% rev_is_rev_conv
thf(fact_172_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_173_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_174_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_175_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( rev @ A @ Ys2 ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_176_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_177_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_178_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% singleton_rev_conv
thf(fact_179_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_180_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( rev @ A @ Xs )
= Ys2 )
= ( Xs
= ( rev @ A @ Ys2 ) ) ) ).
% rev_swap
thf(fact_181_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( R2 @ X @ Y )
=> ( ( listrelp @ A @ B @ R2 @ Xs @ Ys2 )
=> ( listrelp @ A @ B @ R2 @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_182_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_183_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_184_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% last_rev
thf(fact_185_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% hd_rev
thf(fact_186_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: list @ A,A22: list @ B] :
( ( listrelp @ A @ B @ R2 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X4: A,Y4: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys3: list @ B] :
( ( A22
= ( cons @ B @ Y4 @ Ys3 ) )
=> ( ( R2 @ X4 @ Y4 )
=> ~ ( listrelp @ A @ B @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_187_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F: A > B,As3: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F @ As3 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_188_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_189_fold__Cons__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs )
= ( append @ A @ ( rev @ A @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_190_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq_simps(1)
thf(fact_191_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_192_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_193_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Zs )
=> ( ord_lexordp_eq @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_eq_trans
thf(fact_194_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
| ( ord_lexordp_eq @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_195_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Xs )
=> ( Xs = Ys2 ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_196_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq.Nil
thf(fact_197_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > A > A,S: A] :
( ( fold @ B @ A @ F2 @ ( nil @ B ) @ S )
= S ) ).
% fold_simps(1)
thf(fact_198_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > A > A,X: B,Xs: list @ B,S: A] :
( ( fold @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) @ S )
= ( fold @ B @ A @ F2 @ Xs @ ( F2 @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_199_rev__conv__fold,axiom,
! [A: $tType] :
( ( rev @ A )
= ( ^ [Xs3: list @ A] : ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs3 @ ( nil @ A ) ) ) ) ).
% rev_conv_fold
thf(fact_200_List_Ounion__def,axiom,
! [A: $tType] :
( ( union @ A )
= ( fold @ A @ ( list @ A ) @ ( insert @ A ) ) ) ).
% List.union_def
thf(fact_201_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X1: list @ A,X2: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X1 @ X2 )
=> ( ! [X12: list @ A] : ( P2 @ ( nil @ A ) @ X12 )
=> ( ! [X4: A,Y4: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ! [X4: A,Y4: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X4 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X4 )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_202_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_203_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_204_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_205_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys3: list @ A] : A22 != Ys3 )
=> ( ! [X4: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys3 ) )
=> ~ ( ord_less @ A @ X4 @ Y4 ) ) )
=> ~ ! [X4: A,Y4: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X4 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X4 )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_206_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys ) )
| ? [X5: A,Y3: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) )
& ( ord_less @ A @ X5 @ Y3 ) )
| ? [X5: A,Y3: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) )
& ~ ( ord_less @ A @ X5 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X5 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_207_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Us2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us2 @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_208_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ? [X5: A,Vs: list @ A] :
( Ys
= ( append @ A @ Xs3 @ ( cons @ A @ X5 @ Vs ) ) )
| ? [Us: list @ A,A5: A,B3: A,Vs: list @ A,Ws2: list @ A] :
( ( ord_less @ A @ A5 @ B3 )
& ( Xs3
= ( append @ A @ Us @ ( cons @ A @ A5 @ Vs ) ) )
& ( Ys
= ( append @ A @ Us @ ( cons @ A @ B3 @ Ws2 ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_209_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys2: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys2 )
= ( Ys2
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_210_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_211_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_212_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ! [X4: A] :
~ ( ord_less @ A @ X4 @ X4 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_213_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Us2: list @ A,Vs2: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 )
=> ( ord_lexordp @ A @ Us2 @ Vs2 ) ) ) ) ).
% lexordp_append_leftD
thf(fact_214_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_215_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_216_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Y: A,Ys2: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% lexordp.Nil
thf(fact_217_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys2: list @ A,Xs: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys2 ) ) ) ) ).
% lexordp_append_rightI
thf(fact_218_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ~ ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_219_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp @ A @ Ys2 @ Zs )
=> ( ord_lexordp @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_trans
thf(fact_220_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
| ( Xs = Ys2 )
| ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_linear
thf(fact_221_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_222_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Us2: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us2 @ Vs2 )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% lexordp_append_leftI
thf(fact_223_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs3 @ Ys )
& ~ ( ord_lexordp_eq @ A @ Ys @ Xs3 ) ) ) ) ) ).
% lexordp_conv_lexordp_eq
thf(fact_224_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( Xs3 = Ys )
| ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ).
% lexordp_eq_conv_lexord
thf(fact_225_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ).
% lexordp_into_lexordp_eq
thf(fact_226_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y4: A,Ys3: list @ A] : ( P2 @ ( nil @ A ) @ ( cons @ A @ Y4 @ Ys3 ) )
=> ( ! [X4: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ! [X4: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ X4 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_227_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y4: A,Ys5: list @ A] :
( Ys2
!= ( cons @ A @ Y4 @ Ys5 ) ) )
=> ( ! [X4: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Y4: A] :
( ? [Ys5: list @ A] :
( Ys2
= ( cons @ A @ Y4 @ Ys5 ) )
=> ~ ( ord_less @ A @ X4 @ Y4 ) ) )
=> ~ ! [X4: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ X4 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_228_lexordp_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X1: list @ A,X2: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ X1 @ X2 )
=> ( ! [Y4: A,Ys3: list @ A] : ( P2 @ ( nil @ A ) @ ( cons @ A @ Y4 @ Ys3 ) )
=> ( ! [X4: A,Y4: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ! [X4: A,Y4: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X4 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X4 )
=> ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ) ) ).
% lexordp.inducts
thf(fact_229_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) ) )
| ? [X5: A,Y3: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) )
& ( ord_less @ A @ X5 @ Y3 ) )
| ? [X5: A,Y3: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys ) )
& ~ ( ord_less @ A @ X5 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X5 )
& ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_230_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y4: A,Ys3: list @ A] :
( A22
!= ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ! [X4: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys3 ) )
=> ~ ( ord_less @ A @ X4 @ Y4 ) ) )
=> ~ ! [X4: A,Y4: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X4 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X4 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_231_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F: B > A,As3: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_232_replicate__append__same,axiom,
! [A: $tType,I: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_233_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F2 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_234_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_235_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: list @ A] :
( ( ( map @ A @ B @ F2 @ A3 )
= ( nil @ B ) )
= ( A3
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_236_map__append,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys2: list @ B] :
( ( map @ B @ A @ F2 @ ( append @ B @ Xs @ Ys2 ) )
= ( append @ A @ ( map @ B @ A @ F2 @ Xs ) @ ( map @ B @ A @ F2 @ Ys2 ) ) ) ).
% map_append
thf(fact_237_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N: nat,X: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N @ X ) )
= ( replicate @ A @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_238_rev__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( rev @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ).
% rev_replicate
thf(fact_239_smap__shift,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,S: stream170649215stream @ B] :
( ( stream2128578057e_smap @ B @ A @ F2 @ ( stream1035003186_shift @ B @ Xs @ S ) )
= ( stream1035003186_shift @ A @ ( map @ B @ A @ F2 @ Xs ) @ ( stream2128578057e_smap @ B @ A @ F2 @ S ) ) ) ).
% smap_shift
thf(fact_240_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_241_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F2: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F2 @ A3 ) )
= ( map @ A @ B @ F2 @ ( tl @ A @ A3 ) ) ) ) ).
% list.map_sel(2)
thf(fact_242_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( F2 @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_243_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F2: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F2 @ A3 ) )
= ( F2 @ ( hd @ A @ A3 ) ) ) ) ).
% list.map_sel(1)
thf(fact_244_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F2 @ Xs ) )
= ( F2 @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_245_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_246_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_247_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys2: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys2 ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Ys2
= ( cons @ B @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_248_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y: A,Ys2: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys2 ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map @ B @ A @ F2 @ Zs2 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_249_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys2: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys2 ) )
= ( ? [Z2: B,Zs3: list @ B] :
( ( Ys2
= ( cons @ B @ Z2 @ Zs3 ) )
& ( X
= ( F2 @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_250_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y: A,Ys2: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys2 ) )
= ( ? [Z2: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs3 ) )
& ( ( F2 @ Z2 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs3 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_251_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map @ A @ B @ F2 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_252_map__tl,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% map_tl
thf(fact_253_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K @ X ) )
= ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_254_map__butlast,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% map_butlast
%----Type constructors (7)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( ( stream_Mirabelle_shd @ ( list @ a ) @ ws )
!= ( nil @ a ) ) ).
thf(conj_1,conjecture,
( ( stream942386729e_flat @ a @ ws )
= ( stream1035003186_shift @ a @ ( stream_Mirabelle_shd @ ( list @ a ) @ ws ) @ ( stream942386729e_flat @ a @ ( stream_Mirabelle_stl @ ( list @ a ) @ ws ) ) ) ) ).
%------------------------------------------------------------------------------