TPTP Problem File: DAT249^1.p
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%------------------------------------------------------------------------------
% File : DAT249^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Infinite streams (sequences/lists) 301
% 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__301.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 332 ( 159 unt; 62 typ; 0 def)
% Number of atoms : 692 ( 495 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 4215 ( 139 ~; 29 |; 78 &;3683 @)
% ( 0 <=>; 286 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 251 ( 251 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 60 usr; 3 con; 0-5 aty)
% Number of variables : 1160 ( 33 ^; 993 !; 68 ?;1160 :)
% ( 66 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:50.225
%------------------------------------------------------------------------------
%----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 (56)
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_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
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_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
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_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( list @ 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_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_Ostake,type,
stream1746350922_stake:
!>[A: $tType] : ( nat > ( stream170649215stream @ A ) > ( list @ A ) ) ).
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_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_u,type,
u: 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_shift__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,S: stream170649215stream @ A] :
( ( stream1035003186_shift @ A @ ( append @ A @ Xs @ Ys ) @ S )
= ( stream1035003186_shift @ A @ Xs @ ( stream1035003186_shift @ A @ Ys @ S ) ) ) ).
% shift_append
thf(fact_3_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_4_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_5_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_6_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_7_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_8_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( gen_length @ A @ N @ ( nil @ A ) )
= N ) ).
% gen_length_code(1)
thf(fact_9_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_10_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_11_null__rec_I2_J,axiom,
! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% null_rec(2)
thf(fact_12_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_13_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_14_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_15_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_16_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_17_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_18_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_19_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_20_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_21_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_22_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_23_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_24_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_25_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_26_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_27_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_28_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps: list @ A,Xs2: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs2 ) )
& ( Ys
= ( append @ A @ Ps @ Ys2 ) )
& ( ( Xs2
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( ( hd @ A @ Xs2 )
!= ( hd @ A @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_29_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_30_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_31_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_32_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_33_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_34_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_35_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_36_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_37_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_38_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_39_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F1: B,F2: A > ( list @ A ) > B,List3: list @ A] :
( if @ B
@ ( List3
= ( nil @ A ) )
@ F1
@ ( F2 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_40_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( stream1342653232stream @ A @ B )
= ( ^ [F3: A > ( stream170649215stream @ A ) > B,Stream5: stream170649215stream @ A] : ( F3 @ ( stream_Mirabelle_shd @ A @ Stream5 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_41_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_42_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_43_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( stream1195056575_while @ A )
= ( ^ [P2: A > $o,S3: stream170649215stream @ A] : ( if @ ( stream170649215stream @ A ) @ ( P2 @ ( stream_Mirabelle_shd @ A @ S3 ) ) @ ( stream1195056575_while @ A @ P2 @ ( stream_Mirabelle_stl @ A @ S3 ) ) @ S3 ) ) ) ).
% sdrop_while.simps
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member2 @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_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_49_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_50_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_51_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F )
= ( append @ A @ ( F @ X3 ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_52_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_53_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_54_stream_Oinject,axiom,
! [A: $tType,X1: A,X23: stream170649215stream @ A,Y1: A,Y2: stream170649215stream @ A] :
( ( ( stream641971652_SCons @ A @ X1 @ X23 )
= ( stream641971652_SCons @ A @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% stream.inject
thf(fact_55_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_56_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% last_snoc
thf(fact_57_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F22: A > ( list @ A ) > B,X21: A,X22: list @ A] :
( ( case_list @ B @ A @ F12 @ F22 @ ( cons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_58_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_59_sdrop__while__SCons,axiom,
! [A: $tType,P: A > $o,A2: A,S: stream170649215stream @ A] :
( ( ( P @ A2 )
=> ( ( stream1195056575_while @ A @ P @ ( stream641971652_SCons @ A @ A2 @ S ) )
= ( stream1195056575_while @ A @ P @ S ) ) )
& ( ~ ( P @ A2 )
=> ( ( stream1195056575_while @ A @ P @ ( stream641971652_SCons @ A @ A2 @ S ) )
= ( stream641971652_SCons @ A @ A2 @ S ) ) ) ) ).
% sdrop_while_SCons
thf(fact_60_stream_Oexhaust,axiom,
! [A: $tType,Y: stream170649215stream @ A] :
~ ! [X12: A,X24: stream170649215stream @ A] :
( Y
!= ( stream641971652_SCons @ A @ X12 @ X24 ) ) ).
% stream.exhaust
thf(fact_61_stream_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( stream170649215stream @ A ) > B,X1: A,X23: stream170649215stream @ A] :
( ( stream1342653232stream @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X23 ) )
= ( F @ X1 @ X23 ) ) ).
% stream.case
thf(fact_62_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_63_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_64_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_65_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_66_shift_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,S: stream170649215stream @ A] :
( ( stream1035003186_shift @ A @ ( cons @ A @ X3 @ Xs ) @ S )
= ( stream641971652_SCons @ A @ X3 @ ( stream1035003186_shift @ A @ Xs @ S ) ) ) ).
% shift.simps(2)
thf(fact_67_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_68_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_69_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_70_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X24: list @ A] :
( ( P @ X24 )
=> ( P @ ( cons @ A @ X12 @ X24 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_71_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_72_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys4 ) )
=> ( ! [X2: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_73_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_74_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X3
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y4: A,Xs3: list @ A] :
( X3
!= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_75_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y4: A,Xs3: list @ A] :
( ( ( X2 = Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
=> ( ( ( X2 != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs3 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_76_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_77_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F4: A > B,X12: list @ B] : ( P @ F4 @ ( nil @ A ) @ X12 )
=> ( ! [F4: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F4 @ As @ ( cons @ B @ ( F4 @ A4 ) @ Bs ) )
=> ( P @ F4 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_78_append__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_79_Cons__eq__appendI,axiom,
! [A: $tType,X3: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_80_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_81_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_82_stream_Osel_I2_J,axiom,
! [A: $tType,X1: A,X23: stream170649215stream @ A] :
( ( stream_Mirabelle_stl @ A @ ( stream641971652_SCons @ A @ X1 @ X23 ) )
= X23 ) ).
% stream.sel(2)
thf(fact_83_stream_Osel_I1_J,axiom,
! [A: $tType,X1: A,X23: stream170649215stream @ A] :
( ( stream_Mirabelle_shd @ A @ ( stream641971652_SCons @ A @ X1 @ X23 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_84_splice_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X3 @ ( cons @ A @ Y @ ( splice @ A @ Xs @ Ys ) ) ) ) ).
% splice.simps(3)
thf(fact_85_stream_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
( ( P @ ( stream1342653232stream @ A @ B @ F @ Stream ) )
= ( ~ ( ( Stream
= ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
& ~ ( P @ ( F @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ) ).
% stream.split_sel_asm
thf(fact_86_stream_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
( ( P @ ( stream1342653232stream @ A @ B @ F @ Stream ) )
= ( ( Stream
= ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
=> ( P @ ( F @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ).
% stream.split_sel
thf(fact_87_null__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X3 @ Xs ) ) ).
% null_rec(1)
thf(fact_88_member__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_89_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F22: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F12 @ F22 @ ( nil @ A ) )
= F12 ) ).
% list.simps(4)
thf(fact_90_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F12 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F12 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_91_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F12 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F12 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_92_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_93_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X3: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X3 @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X3 @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_94_Cons__eq__append__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X3 @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_95_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y4: A] :
( Xs
!= ( append @ A @ Ys4 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_96_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_97_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X: A] :
( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_98_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X: A] :
( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_99_splice_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V: A,Va: list @ A] :
( ( X3
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X2: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y4: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y4 @ Ys4 ) )
=> ( Y
!= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ ( splice @ A @ Xs3 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% splice.elims
thf(fact_100_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_101_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_102_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs2: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Xs2 @ Ss ) )
& ( Ys
= ( append @ A @ Ys2 @ Ss ) )
& ( ( Xs2
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( ( last @ A @ Xs2 )
!= ( last @ A @ Ys2 ) ) ) ) ).
% longest_common_suffix
thf(fact_103_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_104_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_105_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X3: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X3 @ Xs ) )
= ( append @ A @ ( F @ X3 ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_106_cycle_Ocode,axiom,
! [A: $tType] :
( ( stream317748790_cycle @ A )
= ( ^ [Xs4: list @ A] : ( stream641971652_SCons @ A @ ( hd @ A @ Xs4 ) @ ( stream317748790_cycle @ A @ ( append @ A @ ( tl @ A @ Xs4 ) @ ( cons @ A @ ( hd @ A @ Xs4 ) @ ( nil @ A ) ) ) ) ) ) ) ).
% cycle.code
thf(fact_107_smember__code,axiom,
! [A: $tType,X3: A,Y: A,S: stream170649215stream @ A] :
( ( stream1586597341member @ A @ X3 @ ( stream641971652_SCons @ A @ Y @ S ) )
= ( ( X3 != Y )
=> ( stream1586597341member @ A @ X3 @ S ) ) ) ).
% smember_code
thf(fact_108_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_109_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_110_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_111_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_112_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_113_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_114_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_115_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_116_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_117_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_118_remdups__adj__Cons__alt,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_119_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
& ( ( X3 != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_120_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_121_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_122_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_123_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_124_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X3: A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_125_remdups__adj_Oelims,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X2: A] :
( ( X3
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Y4: A,Xs3: list @ A] :
( ( X3
= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs3 ) ) )
=> ~ ( ( ( X2 = Y4 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs3 ) ) ) )
& ( ( X2 != Y4 )
=> ( Y
= ( cons @ A @ X2 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_126_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_127_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_128_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X3: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_129_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_130_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_131_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_132_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A4: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_133_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_134_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_135_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_136_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_137_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_138_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_139_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_140_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A2: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A2 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A2 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_141_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_142_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X23 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X2: A,Y4: A,Xs3: list @ A,Ys4: list @ A] :
( ( Less @ X2 @ Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X2: A,Y4: A,Xs3: list @ A,Ys4: list @ A] :
( ~ ( Less @ X2 @ Y4 )
=> ( ~ ( Less @ Y4 @ X2 )
=> ( ( lexordp_eq @ A @ Less @ Xs3 @ Ys4 )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_143_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X: A,Y3: A,Xs4: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X @ Y3 ) )
| ? [X: A,Y3: A,Xs4: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X @ Y3 )
& ~ ( Less2 @ Y3 @ X )
& ( lexordp_eq @ A @ Less2 @ Xs4 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_144_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 ) )
=> ! [Ys4: list @ A] : A22 != Ys4 )
=> ( ! [X2: A] :
( ? [Xs3: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y4: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ~ ( Less @ X2 @ Y4 ) ) )
=> ~ ! [X2: A,Y4: A,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ( ~ ( Less @ X2 @ Y4 )
=> ( ~ ( Less @ Y4 @ X2 )
=> ~ ( lexordp_eq @ A @ Less @ Xs3 @ Ys4 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_145_smap__ctr,axiom,
! [B: $tType,A: $tType,F: B > A,S: stream170649215stream @ B,X3: A,S4: stream170649215stream @ A] :
( ( ( stream2128578057e_smap @ B @ A @ F @ S )
= ( stream641971652_SCons @ A @ X3 @ S4 ) )
= ( ( ( F @ ( stream_Mirabelle_shd @ B @ S ) )
= X3 )
& ( ( stream2128578057e_smap @ B @ A @ F @ ( stream_Mirabelle_stl @ B @ S ) )
= S4 ) ) ) ).
% smap_ctr
thf(fact_146_SuccI,axiom,
! [A: $tType,Kl: list @ A,K: A,Kl2: set @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 )
=> ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_147_SuccD,axiom,
! [A: $tType,K: A,Kl2: set @ ( list @ A ),Kl: list @ A] :
( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) )
=> ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_148_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_149_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_150_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_151_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_152_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_153_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_154_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_155_stream_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,F: A > B,A2: stream170649215stream @ A] :
( ( stream_Mirabelle_stl @ B @ ( stream2128578057e_smap @ A @ B @ F @ A2 ) )
= ( stream2128578057e_smap @ A @ B @ F @ ( stream_Mirabelle_stl @ A @ A2 ) ) ) ).
% stream.map_sel(2)
thf(fact_156_stream_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,F: A > B,A2: stream170649215stream @ A] :
( ( stream_Mirabelle_shd @ B @ ( stream2128578057e_smap @ A @ B @ F @ A2 ) )
= ( F @ ( stream_Mirabelle_shd @ A @ A2 ) ) ) ).
% stream.map_sel(1)
thf(fact_157_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_158_singleton__rev__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( cons @ A @ X3 @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% singleton_rev_conv
thf(fact_159_stream_Omap,axiom,
! [B: $tType,A: $tType,F: A > B,X1: A,X23: stream170649215stream @ A] :
( ( stream2128578057e_smap @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X23 ) )
= ( stream641971652_SCons @ B @ ( F @ X1 ) @ ( stream2128578057e_smap @ A @ B @ F @ X23 ) ) ) ).
% stream.map
thf(fact_160_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_161_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_162_rev_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_163_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% hd_rev
thf(fact_164_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% last_rev
thf(fact_165_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F3: A > B,As3: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F3 @ As3 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_166_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_167_empty__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl2 )
=> ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ ( nil @ A ) ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_168_ShiftD,axiom,
! [A: $tType,Kl: list @ A,Kl2: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ Kl @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_169_Succ__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K: A,Kl: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ @ A @ Kl2 @ ( cons @ A @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_170_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X1: list @ A,X23: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R2 @ X1 @ X23 )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Y4: B,Xs3: list @ A,Ys4: list @ B] :
( ( R2 @ X2 @ Y4 )
=> ( ( listrelp @ A @ B @ R2 @ Xs3 @ Ys4 )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_171_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 ) ) )
| ? [X: A,Y3: B,Xs4: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ B @ Y3 @ Ys3 ) )
& ( R3 @ X @ Y3 )
& ( listrelp @ A @ B @ R3 @ Xs4 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_172_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 ) ) )
=> ~ ! [X2: A,Y4: B,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( R2 @ X2 @ Y4 )
=> ~ ( listrelp @ A @ B @ R2 @ Xs3 @ Ys4 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_173_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R2 @ X3 @ Y )
=> ( ( listrelp @ A @ B @ R2 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R2 @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_174_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_175_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_176_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_177_rev__conv__fold,axiom,
! [A: $tType] :
( ( rev @ A )
= ( ^ [Xs4: list @ A] : ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs4 @ ( nil @ A ) ) ) ) ).
% rev_conv_fold
thf(fact_178_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq_simps(1)
thf(fact_179_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_180_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_181_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > A > A,X3: B,Xs: list @ B,S: A] :
( ( fold @ B @ A @ F @ ( cons @ B @ X3 @ Xs ) @ S )
= ( fold @ B @ A @ F @ Xs @ ( F @ X3 @ S ) ) ) ).
% fold_simps(2)
thf(fact_182_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > A > A,S: A] :
( ( fold @ B @ A @ F @ ( nil @ B ) @ S )
= S ) ).
% fold_simps(1)
thf(fact_183_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_184_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Zs )
=> ( ord_lexordp_eq @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_eq_trans
thf(fact_185_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
| ( ord_lexordp_eq @ A @ Ys @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_186_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_187_List_Ounion__def,axiom,
! [A: $tType] :
( ( union @ A )
= ( fold @ A @ ( list @ A ) @ ( insert @ A ) ) ) ).
% List.union_def
thf(fact_188_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X1 @ X23 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X2: A,Y4: A,Xs3: list @ A,Ys4: list @ A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X2: A,Y4: A,Xs3: list @ A,Ys4: list @ A] :
( ~ ( ord_less @ A @ X2 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X2 )
=> ( ( ord_lexordp_eq @ A @ Xs3 @ Ys4 )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_189_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X: A,Y3: A,Xs4: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X @ Y3 ) )
| ? [X: A,Y3: A,Xs4: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X )
& ( ord_lexordp_eq @ A @ Xs4 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_190_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_191_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_192_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_193_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 ) )
=> ! [Ys4: list @ A] : A22 != Ys4 )
=> ( ! [X2: A] :
( ? [Xs3: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y4: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ~ ( ord_less @ A @ X2 @ Y4 ) ) )
=> ~ ! [X2: A,Y4: A,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X2 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X2 )
=> ~ ( ord_lexordp_eq @ A @ Xs3 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_194_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs4: list @ A,Ys3: list @ A] :
( ? [X: A,Vs: list @ A] :
( Ys3
= ( append @ A @ Xs4 @ ( cons @ A @ X @ Vs ) ) )
| ? [Us2: list @ A,A5: A,B2: A,Vs: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A5 @ B2 )
& ( Xs4
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B2 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_195_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_196_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_197_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_198_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_199_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_200_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ~ ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_201_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ord_lexordp @ A @ Ys @ Zs )
=> ( ord_lexordp @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_trans
thf(fact_202_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
| ( Xs = Ys )
| ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_linear
thf(fact_203_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_204_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Us: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us @ Vs2 )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% lexordp_append_leftI
thf(fact_205_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs4: list @ A,Ys3: list @ A] :
( ( ord_lexordp_eq @ A @ Xs4 @ Ys3 )
& ~ ( ord_lexordp_eq @ A @ Ys3 @ Xs4 ) ) ) ) ) ).
% lexordp_conv_lexordp_eq
thf(fact_206_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [Xs4: list @ A,Ys3: list @ A] :
( ( Xs4 = Ys3 )
| ( ord_lexordp @ A @ Xs4 @ Ys3 ) ) ) ) ) ).
% lexordp_eq_conv_lexord
thf(fact_207_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ).
% lexordp_into_lexordp_eq
thf(fact_208_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp.Cons
thf(fact_209_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_210_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Y: A,Ys: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% lexordp.Nil
thf(fact_211_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Us: list @ A,Vs2: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 )
=> ( ord_lexordp @ A @ Us @ Vs2 ) ) ) ) ).
% lexordp_append_leftD
thf(fact_212_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ) ).
% lexordp_append_rightI
thf(fact_213_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ! [Y4: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y4 @ Ys4 ) )
=> ( ! [X2: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X2: A,Xs3: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs3 @ Ys4 )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ X2 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_214_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y4: A,Ys2: list @ A] :
( Ys
!= ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( Xs
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys2: list @ A] :
( Ys
= ( cons @ A @ Y4 @ Ys2 ) )
=> ~ ( ord_less @ A @ X2 @ Y4 ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys2 ) )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_215_lexordp_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ X1 @ X23 )
=> ( ! [Y4: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y4 @ Ys4 ) )
=> ( ! [X2: A,Y4: A,Xs3: list @ A,Ys4: list @ A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X2: A,Y4: A,Xs3: list @ A,Ys4: list @ A] :
( ~ ( ord_less @ A @ X2 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X2 )
=> ( ( ord_lexordp @ A @ Xs3 @ Ys4 )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ) ).
% lexordp.inducts
thf(fact_216_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X: A,Y3: A,Xs4: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X @ Y3 ) )
| ? [X: A,Y3: A,Xs4: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs4 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X )
& ( ord_lexordp @ A @ Xs4 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_217_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,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y4 @ Ys4 ) ) )
=> ( ! [X2: A] :
( ? [Xs3: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Y4: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ~ ( ord_less @ A @ X2 @ Y4 ) ) )
=> ~ ! [X2: A,Y4: A,Xs3: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs3 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X2 @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X2 )
=> ~ ( ord_lexordp @ A @ Xs3 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_218_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F3: B > A,As3: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F3 @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_219_replicate__append__same,axiom,
! [A: $tType,I: nat,X3: A] :
( ( append @ A @ ( replicate @ A @ I @ X3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( replicate @ A @ I @ X3 ) ) ) ).
% replicate_append_same
thf(fact_220_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_221_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_222_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_223_map__append,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F @ Xs ) @ ( map @ B @ A @ F @ Ys ) ) ) ).
% map_append
thf(fact_224_map__replicate,axiom,
! [A: $tType,B: $tType,F: B > A,N: nat,X3: B] :
( ( map @ B @ A @ F @ ( replicate @ B @ N @ X3 ) )
= ( replicate @ A @ N @ ( F @ X3 ) ) ) ).
% map_replicate
thf(fact_225_rev__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( rev @ A @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ N @ X3 ) ) ).
% rev_replicate
thf(fact_226_smap__shift,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,S: stream170649215stream @ B] :
( ( stream2128578057e_smap @ B @ A @ F @ ( stream1035003186_shift @ B @ Xs @ S ) )
= ( stream1035003186_shift @ A @ ( map @ B @ A @ F @ Xs ) @ ( stream2128578057e_smap @ B @ A @ F @ S ) ) ) ).
% smap_shift
thf(fact_227_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_228_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F @ A2 ) )
= ( F @ ( hd @ A @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_229_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_230_append__replicate__commute,axiom,
! [A: $tType,N: nat,X3: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( replicate @ A @ K @ X3 ) )
= ( append @ A @ ( replicate @ A @ K @ X3 ) @ ( replicate @ A @ N @ X3 ) ) ) ).
% append_replicate_commute
thf(fact_231_map__tl,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_tl
thf(fact_232_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map @ A @ B @ F @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_233_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F @ X21 ) @ ( map @ A @ B @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_234_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X3 @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z @ Zs2 ) )
& ( X3
= ( F @ Z ) )
& ( Xs
= ( map @ B @ A @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_235_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map @ B @ A @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_236_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X3 @ Xs )
= ( map @ B @ A @ F @ Ys ) )
= ( ? [Z2: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z2 @ Zs3 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_237_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z2: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map @ B @ A @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_238_map__butlast,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_239_rev__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( rev @ B @ Xs ) ) ) ).
% rev_map
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,A2: list @ A,F: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F @ A2 ) )
= ( map @ A @ B @ F @ ( tl @ A @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_242_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( append @ A @ ( replicate @ A @ N @ X3 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_243_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X3 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X3 ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_244_stake__smap,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,S: stream170649215stream @ B] :
( ( stream1746350922_stake @ A @ N @ ( stream2128578057e_smap @ B @ A @ F @ S ) )
= ( map @ B @ A @ F @ ( stream1746350922_stake @ B @ N @ S ) ) ) ).
% stake_smap
thf(fact_245_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X3: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X3 )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X3 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_246_replicate__empty,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( replicate @ A @ N @ X3 )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_247_empty__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X3 ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_248_stake__invert__Nil,axiom,
! [A: $tType,N: nat,S: stream170649215stream @ A] :
( ( ( stream1746350922_stake @ A @ N @ S )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% stake_invert_Nil
thf(fact_249_hd__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X3 ) )
= X3 ) ) ).
% hd_replicate
thf(fact_250_last__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X3 ) )
= X3 ) ) ).
% last_replicate
thf(fact_251_replicate__0,axiom,
! [A: $tType,X3: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X3 )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_252_stake_Osimps_I1_J,axiom,
! [A: $tType,S: stream170649215stream @ A] :
( ( stream1746350922_stake @ A @ ( zero_zero @ nat ) @ S )
= ( nil @ A ) ) ).
% stake.simps(1)
thf(fact_253_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_254_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
%----Type constructors (10)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( u
!= ( nil @ a ) ) ).
thf(conj_1,conjecture,
( ( stream317748790_cycle @ a @ u )
= ( stream1035003186_shift @ a @ u @ ( stream317748790_cycle @ a @ u ) ) ) ).
%------------------------------------------------------------------------------