TPTP Problem File: DAT248^1.p
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%------------------------------------------------------------------------------
% File : DAT248^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Infinite streams (sequences/lists) 250
% 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__250.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 340 ( 126 unt; 56 typ; 0 def)
% Number of atoms : 776 ( 259 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3704 ( 68 ~; 20 |; 53 &;3198 @)
% ( 0 <=>; 365 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 201 ( 201 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 4 con; 0-7 aty)
% Number of variables : 971 ( 32 ^; 872 !; 16 ?; 971 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:35.379
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
stream170649215stream: $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 (51)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[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_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere1490568538miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord20386208strict:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_OLeastM,type,
hilbert_LeastM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $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_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osdrop,type,
stream135081970_sdrop:
!>[A: $tType] : ( nat > ( stream170649215stream @ 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_Osmember,type,
stream1586597341member:
!>[A: $tType] : ( A > ( stream170649215stream @ A ) > $o ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osnth,type,
stream370371455e_snth:
!>[A: $tType] : ( ( stream170649215stream @ A ) > nat > 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_Ocorec__stream,type,
stream660621732stream:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream170649215stream @ A ) ) > ( C > C ) > C > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Opred__stream,type,
stream1153105665stream:
!>[A: $tType] : ( ( A > $o ) > ( stream170649215stream @ A ) > $o ) ).
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_Osset,type,
stream30925839e_sset:
!>[A: $tType] : ( ( stream170649215stream @ A ) > ( set @ A ) ) ).
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_Stream__Mirabelle__hbrgyiwlrc_Ostreamsp,type,
stream86250253reamsp:
!>[A: $tType] : ( ( A > $o ) > ( stream170649215stream @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > $o ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_s,type,
s: stream170649215stream @ a ).
%----Relevant facts (253)
thf(fact_0_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_1_stream_Oexhaust,axiom,
! [A: $tType,Y: stream170649215stream @ A] :
~ ! [X12: A,X22: stream170649215stream @ A] :
( Y
!= ( stream641971652_SCons @ A @ X12 @ X22 ) ) ).
% stream.exhaust
thf(fact_2_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_3_stream_Opred__inject,axiom,
! [A: $tType,P: A > $o,A2: A,Aa2: stream170649215stream @ A] :
( ( stream1153105665stream @ A @ P @ ( stream641971652_SCons @ A @ A2 @ Aa2 ) )
= ( ( P @ A2 )
& ( stream1153105665stream @ A @ P @ Aa2 ) ) ) ).
% stream.pred_inject
thf(fact_4_stream_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( stream170649215stream @ A ) > B,X1: A,X2: stream170649215stream @ A] :
( ( stream1342653232stream @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% stream.case
thf(fact_5_streamsp_Ocases,axiom,
! [A: $tType,A3: A > $o,A2: stream170649215stream @ A] :
( ( stream86250253reamsp @ A @ A3 @ A2 )
=> ~ ! [A4: A,S2: stream170649215stream @ A] :
( ( A2
= ( stream641971652_SCons @ A @ A4 @ S2 ) )
=> ( ( A3 @ A4 )
=> ~ ( stream86250253reamsp @ A @ A3 @ S2 ) ) ) ) ).
% streamsp.cases
thf(fact_6_streamsp_Osimps,axiom,
! [A: $tType] :
( ( stream86250253reamsp @ A )
= ( ^ [A5: A > $o,A6: stream170649215stream @ A] :
? [B2: A,S3: stream170649215stream @ A] :
( ( A6
= ( stream641971652_SCons @ A @ B2 @ S3 ) )
& ( A5 @ B2 )
& ( stream86250253reamsp @ A @ A5 @ S3 ) ) ) ) ).
% streamsp.simps
thf(fact_7_streamsp_Ocoinduct,axiom,
! [A: $tType,X3: ( stream170649215stream @ A ) > $o,X: stream170649215stream @ A,A3: A > $o] :
( ( X3 @ X )
=> ( ! [X4: stream170649215stream @ A] :
( ( X3 @ X4 )
=> ? [A7: A,S4: stream170649215stream @ A] :
( ( X4
= ( stream641971652_SCons @ A @ A7 @ S4 ) )
& ( A3 @ A7 )
& ( ( X3 @ S4 )
| ( stream86250253reamsp @ A @ A3 @ S4 ) ) ) )
=> ( stream86250253reamsp @ A @ A3 @ X ) ) ) ).
% streamsp.coinduct
thf(fact_8_stream_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( stream660621732stream @ C @ A )
= ( ^ [G1: C > A,Q2: C > $o,G21: C > ( stream170649215stream @ A ),G22: C > C,A6: C] : ( stream641971652_SCons @ A @ ( G1 @ A6 ) @ ( if @ ( stream170649215stream @ A ) @ ( Q2 @ A6 ) @ ( G21 @ A6 ) @ ( stream660621732stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ ( G22 @ A6 ) ) ) ) ) ) ).
% stream.corec_code
thf(fact_9_streams_Ocases,axiom,
! [A: $tType,A2: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ A2 @ ( stream2015131171treams @ A @ A3 ) )
=> ~ ! [A4: A,S2: stream170649215stream @ A] :
( ( A2
= ( stream641971652_SCons @ A @ A4 @ S2 ) )
=> ( ( member @ A @ A4 @ A3 )
=> ~ ( member @ ( stream170649215stream @ A ) @ S2 @ ( stream2015131171treams @ A @ A3 ) ) ) ) ) ).
% streams.cases
thf(fact_10_streams_Osimps,axiom,
! [A: $tType,A2: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ A2 @ ( stream2015131171treams @ A @ A3 ) )
= ( ? [A6: A,S3: stream170649215stream @ A] :
( ( A2
= ( stream641971652_SCons @ A @ A6 @ S3 ) )
& ( member @ A @ A6 @ A3 )
& ( member @ ( stream170649215stream @ A ) @ S3 @ ( stream2015131171treams @ A @ A3 ) ) ) ) ) ).
% streams.simps
thf(fact_11_streams__Stream,axiom,
! [A: $tType,X: A,S: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ ( stream641971652_SCons @ A @ X @ S ) @ ( stream2015131171treams @ A @ A3 ) )
= ( ( member @ A @ X @ A3 )
& ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) ) ) ) ).
% streams_Stream
thf(fact_12_streams_Ocoinduct,axiom,
! [A: $tType,X3: ( stream170649215stream @ A ) > $o,X: stream170649215stream @ A,A3: set @ A] :
( ( X3 @ X )
=> ( ! [X4: stream170649215stream @ A] :
( ( X3 @ X4 )
=> ? [A7: A,S4: stream170649215stream @ A] :
( ( X4
= ( stream641971652_SCons @ A @ A7 @ S4 ) )
& ( member @ A @ A7 @ A3 )
& ( ( X3 @ S4 )
| ( member @ ( stream170649215stream @ A ) @ S4 @ ( stream2015131171treams @ A @ A3 ) ) ) ) )
=> ( member @ ( stream170649215stream @ A ) @ X @ ( stream2015131171treams @ A @ A3 ) ) ) ) ).
% streams.coinduct
thf(fact_13_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_14_stream_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,G12: C > A,Q22: C > $o,G212: C > ( stream170649215stream @ A ),G222: C > C,A2: C] :
( ( stream_Mirabelle_shd @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A2 ) )
= ( G12 @ A2 ) ) ).
% stream.corec_sel(1)
thf(fact_15_streams__shd,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
=> ( member @ A @ ( stream_Mirabelle_shd @ A @ S ) @ A3 ) ) ).
% streams_shd
thf(fact_16_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( stream660621732stream @ C @ A )
= ( stream660621732stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_17_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_18_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_19_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_20_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_21_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( stream1342653232stream @ A @ B )
= ( ^ [F2: A > ( stream170649215stream @ A ) > B,Stream2: stream170649215stream @ A] : ( F2 @ ( stream_Mirabelle_shd @ A @ Stream2 ) @ ( stream_Mirabelle_stl @ A @ Stream2 ) ) ) ) ).
% stream.case_eq_if
thf(fact_22_streamsE,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
=> ~ ( ( member @ A @ ( stream_Mirabelle_shd @ A @ S ) @ A3 )
=> ~ ( member @ ( stream170649215stream @ A ) @ ( stream_Mirabelle_stl @ A @ S ) @ ( stream2015131171treams @ A @ A3 ) ) ) ) ).
% streamsE
thf(fact_23_in__streams,axiom,
! [A: $tType,S: stream170649215stream @ A,S5: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ ( stream_Mirabelle_stl @ A @ S ) @ ( stream2015131171treams @ A @ S5 ) )
=> ( ( member @ A @ ( stream_Mirabelle_shd @ A @ S ) @ S5 )
=> ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ S5 ) ) ) ) ).
% in_streams
thf(fact_24_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_25_Stream__Mirabelle__hbrgyiwlrc_Osmember__def,axiom,
! [A: $tType] :
( ( stream1586597341member @ A )
= ( ^ [X5: A,S3: stream170649215stream @ A] : ( member @ A @ X5 @ ( stream30925839e_sset @ A @ S3 ) ) ) ) ).
% Stream_Mirabelle_hbrgyiwlrc.smember_def
thf(fact_26_snth__in,axiom,
! [A: $tType,S: stream170649215stream @ A,X3: set @ A,N: nat] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ X3 ) )
=> ( member @ A @ ( stream370371455e_snth @ A @ S @ N ) @ X3 ) ) ).
% snth_in
thf(fact_27_streams__iff__snth,axiom,
! [A: $tType,S: stream170649215stream @ A,X3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ X3 ) )
= ( ! [N2: nat] : ( member @ A @ ( stream370371455e_snth @ A @ S @ N2 ) @ X3 ) ) ) ).
% streams_iff_snth
thf(fact_28_snth__sset,axiom,
! [A: $tType,S: stream170649215stream @ A,N: nat] : ( member @ A @ ( stream370371455e_snth @ A @ S @ N ) @ ( stream30925839e_sset @ A @ S ) ) ).
% snth_sset
thf(fact_29_stl__sset,axiom,
! [A: $tType,X: A,A2: stream170649215stream @ A] :
( ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream_Mirabelle_stl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( stream30925839e_sset @ A @ A2 ) ) ) ).
% stl_sset
thf(fact_30_sset__induct,axiom,
! [A: $tType,Y: A,S: stream170649215stream @ A,P: A > ( stream170649215stream @ A ) > $o] :
( ( member @ A @ Y @ ( stream30925839e_sset @ A @ S ) )
=> ( ! [S2: stream170649215stream @ A] : ( P @ ( stream_Mirabelle_shd @ A @ S2 ) @ S2 )
=> ( ! [S2: stream170649215stream @ A,Y3: A] :
( ( member @ A @ Y3 @ ( stream30925839e_sset @ A @ ( stream_Mirabelle_stl @ A @ S2 ) ) )
=> ( ( P @ Y3 @ ( stream_Mirabelle_stl @ A @ S2 ) )
=> ( P @ Y3 @ S2 ) ) )
=> ( P @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_31_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_32_stream_Ocoinduct__strong,axiom,
! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream3: stream170649215stream @ A] :
( ( R @ Stream @ Stream3 )
=> ( ! [Stream4: stream170649215stream @ A,Stream5: stream170649215stream @ A] :
( ( R @ Stream4 @ Stream5 )
=> ( ( ( stream_Mirabelle_shd @ A @ Stream4 )
= ( stream_Mirabelle_shd @ A @ Stream5 ) )
& ( ( R @ ( stream_Mirabelle_stl @ A @ Stream4 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) )
| ( ( stream_Mirabelle_stl @ A @ Stream4 )
= ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) ) )
=> ( Stream = Stream3 ) ) ) ).
% stream.coinduct_strong
thf(fact_33_stream_Ocoinduct,axiom,
! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream3: stream170649215stream @ A] :
( ( R @ Stream @ Stream3 )
=> ( ! [Stream4: stream170649215stream @ A,Stream5: stream170649215stream @ A] :
( ( R @ Stream4 @ Stream5 )
=> ( ( ( stream_Mirabelle_shd @ A @ Stream4 )
= ( stream_Mirabelle_shd @ A @ Stream5 ) )
& ( R @ ( stream_Mirabelle_stl @ A @ Stream4 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) )
=> ( Stream = Stream3 ) ) ) ).
% stream.coinduct
thf(fact_34_stream_Oexpand,axiom,
! [A: $tType,Stream: stream170649215stream @ A,Stream3: stream170649215stream @ A] :
( ( ( ( stream_Mirabelle_shd @ A @ Stream )
= ( stream_Mirabelle_shd @ A @ Stream3 ) )
& ( ( stream_Mirabelle_stl @ A @ Stream )
= ( stream_Mirabelle_stl @ A @ Stream3 ) ) )
=> ( Stream = Stream3 ) ) ).
% stream.expand
thf(fact_35_streams__stl,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
=> ( member @ ( stream170649215stream @ A ) @ ( stream_Mirabelle_stl @ A @ S ) @ ( stream2015131171treams @ A @ A3 ) ) ) ).
% streams_stl
thf(fact_36_stream_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: stream170649215stream @ A,A1: A] :
( ( member @ A @ X @ ( stream30925839e_sset @ A @ A22 ) )
=> ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_37_stream_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: stream170649215stream @ A] : ( member @ A @ A1 @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A22 ) ) ) ).
% stream.set_intros(1)
thf(fact_38_stream_Oset__cases,axiom,
! [A: $tType,E: A,A2: stream170649215stream @ A] :
( ( member @ A @ E @ ( stream30925839e_sset @ A @ A2 ) )
=> ( ! [Z2: stream170649215stream @ A] :
( A2
!= ( stream641971652_SCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: stream170649215stream @ A] :
( ( A2
= ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( stream30925839e_sset @ A @ Z2 ) ) ) ) ) ).
% stream.set_cases
thf(fact_39_stream_Oset__induct,axiom,
! [A: $tType,X: A,A2: stream170649215stream @ A,P: A > ( stream170649215stream @ A ) > $o] :
( ( member @ A @ X @ ( stream30925839e_sset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: stream170649215stream @ A] : ( P @ Z1 @ ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: stream170649215stream @ A,Xa: A] :
( ( member @ A @ Xa @ ( stream30925839e_sset @ A @ Z2 ) )
=> ( ( P @ Xa @ Z2 )
=> ( P @ Xa @ ( stream641971652_SCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_40_shd__sset,axiom,
! [A: $tType,A2: stream170649215stream @ A] : ( member @ A @ ( stream_Mirabelle_shd @ A @ A2 ) @ ( stream30925839e_sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_41_stream_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: stream170649215stream @ A,Pa: A > $o] :
( ( stream1153105665stream @ A @ P @ X )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( stream1153105665stream @ A @ Pa @ X ) ) ) ).
% stream.pred_mono_strong
thf(fact_42_stream_Opred__cong,axiom,
! [A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( stream1153105665stream @ A @ P @ X )
= ( stream1153105665stream @ A @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_43_stream_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,Q22: C > $o,A2: C,G12: C > A,G212: C > ( stream170649215stream @ A ),G222: C > C] :
( ( ( Q22 @ A2 )
=> ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A2 ) )
= ( G212 @ A2 ) ) )
& ( ~ ( Q22 @ A2 )
=> ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A2 ) )
= ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ ( G222 @ A2 ) ) ) ) ) ).
% stream.corec_sel(2)
thf(fact_44_sdrop__simps_I1_J,axiom,
! [A: $tType,N: nat,S: stream170649215stream @ A] :
( ( stream_Mirabelle_shd @ A @ ( stream135081970_sdrop @ A @ N @ S ) )
= ( stream370371455e_snth @ A @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_smap__ctr,axiom,
! [B: $tType,A: $tType,F: B > A,S: stream170649215stream @ B,X: A,S6: stream170649215stream @ A] :
( ( ( stream2128578057e_smap @ B @ A @ F @ S )
= ( stream641971652_SCons @ A @ X @ S6 ) )
= ( ( ( F @ ( stream_Mirabelle_shd @ B @ S ) )
= X )
& ( ( stream2128578057e_smap @ B @ A @ F @ ( stream_Mirabelle_stl @ B @ S ) )
= S6 ) ) ) ).
% smap_ctr
thf(fact_49_stream_Oset,axiom,
! [A: $tType,X1: A,X2: stream170649215stream @ A] :
( ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= ( insert @ A @ X1 @ ( stream30925839e_sset @ A @ X2 ) ) ) ).
% stream.set
thf(fact_50_snth_Osimps_I1_J,axiom,
! [A: $tType,S: stream170649215stream @ A] :
( ( stream370371455e_snth @ A @ S @ ( zero_zero @ nat ) )
= ( stream_Mirabelle_shd @ A @ S ) ) ).
% snth.simps(1)
thf(fact_51_streams__iff__sset,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
= ( ord_less_eq @ ( set @ A ) @ ( stream30925839e_sset @ A @ S ) @ A3 ) ) ).
% streams_iff_sset
thf(fact_52_streams__sset,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( stream30925839e_sset @ A @ S ) @ A3 ) ) ).
% streams_sset
thf(fact_53_sset__streams,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( stream30925839e_sset @ A @ S ) @ A3 )
=> ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) ) ) ).
% sset_streams
thf(fact_54_snth_Osimps_I2_J,axiom,
! [A: $tType,S: stream170649215stream @ A,N: nat] :
( ( stream370371455e_snth @ A @ S @ ( suc @ N ) )
= ( stream370371455e_snth @ A @ ( stream_Mirabelle_stl @ A @ S ) @ N ) ) ).
% snth.simps(2)
thf(fact_55_snth__Stream,axiom,
! [A: $tType,X: A,S: stream170649215stream @ A,I: nat] :
( ( stream370371455e_snth @ A @ ( stream641971652_SCons @ A @ X @ S ) @ ( suc @ I ) )
= ( stream370371455e_snth @ A @ S @ I ) ) ).
% snth_Stream
thf(fact_56_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_57_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_58_snth__smap,axiom,
! [A: $tType,B: $tType,F: B > A,S: stream170649215stream @ B,N: nat] :
( ( stream370371455e_snth @ A @ ( stream2128578057e_smap @ B @ A @ F @ S ) @ N )
= ( F @ ( stream370371455e_snth @ B @ S @ N ) ) ) ).
% snth_smap
thf(fact_59_sdrop__smap,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,S: stream170649215stream @ B] :
( ( stream135081970_sdrop @ A @ N @ ( stream2128578057e_smap @ B @ A @ F @ S ) )
= ( stream2128578057e_smap @ B @ A @ F @ ( stream135081970_sdrop @ B @ N @ S ) ) ) ).
% sdrop_smap
thf(fact_60_sdrop__simps_I2_J,axiom,
! [A: $tType,N: nat,S: stream170649215stream @ A] :
( ( stream_Mirabelle_stl @ A @ ( stream135081970_sdrop @ A @ N @ S ) )
= ( stream135081970_sdrop @ A @ ( suc @ N ) @ S ) ) ).
% sdrop_simps(2)
thf(fact_61_sdrop_Osimps_I1_J,axiom,
! [A: $tType,S: stream170649215stream @ A] :
( ( stream135081970_sdrop @ A @ ( zero_zero @ nat ) @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_62_streams__mono2,axiom,
! [A: $tType,S5: set @ A,T: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S5 @ T )
=> ( ord_less_eq @ ( set @ ( stream170649215stream @ A ) ) @ ( stream2015131171treams @ A @ S5 ) @ ( stream2015131171treams @ A @ T ) ) ) ).
% streams_mono2
thf(fact_63_sdrop_Osimps_I2_J,axiom,
! [A: $tType,N: nat,S: stream170649215stream @ A] :
( ( stream135081970_sdrop @ A @ ( suc @ N ) @ S )
= ( stream135081970_sdrop @ A @ N @ ( stream_Mirabelle_stl @ A @ S ) ) ) ).
% sdrop.simps(2)
thf(fact_64_streams__mono,axiom,
! [A: $tType,S: stream170649215stream @ A,A3: set @ A,B3: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ B3 ) ) ) ) ).
% streams_mono
thf(fact_65_stream_Omap,axiom,
! [B: $tType,A: $tType,F: A > B,X1: A,X2: stream170649215stream @ A] :
( ( stream2128578057e_smap @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
= ( stream641971652_SCons @ B @ ( F @ X1 ) @ ( stream2128578057e_smap @ A @ B @ F @ X2 ) ) ) ).
% stream.map
thf(fact_66_stream_Omap__cong,axiom,
! [B: $tType,A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( stream2128578057e_smap @ A @ B @ F @ X )
= ( stream2128578057e_smap @ A @ B @ G @ Ya ) ) ) ) ).
% stream.map_cong
thf(fact_67_stream_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: stream170649215stream @ A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( stream2128578057e_smap @ A @ B @ F @ X )
= ( stream2128578057e_smap @ A @ B @ G @ X ) ) ) ).
% stream.map_cong0
thf(fact_68_stream_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: stream170649215stream @ A,Xa2: stream170649215stream @ A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
=> ( ( member @ A @ Za @ ( stream30925839e_sset @ A @ Xa2 ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( stream2128578057e_smap @ A @ B @ F @ X )
= ( stream2128578057e_smap @ A @ B @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% stream.inj_map_strong
thf(fact_69_smap__streams,axiom,
! [A: $tType,B: $tType,S: stream170649215stream @ A,A3: set @ A,F: A > B,B3: set @ B] :
( ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A3 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ B @ ( F @ X4 ) @ B3 ) )
=> ( member @ ( stream170649215stream @ B ) @ ( stream2128578057e_smap @ A @ B @ F @ S ) @ ( stream2015131171treams @ B @ B3 ) ) ) ) ).
% smap_streams
thf(fact_70_smap__alt,axiom,
! [A: $tType,B: $tType,F: B > A,S: stream170649215stream @ B,S6: stream170649215stream @ A] :
( ( ( stream2128578057e_smap @ B @ A @ F @ S )
= S6 )
= ( ! [N2: nat] :
( ( F @ ( stream370371455e_snth @ B @ S @ N2 ) )
= ( stream370371455e_snth @ A @ S6 @ N2 ) ) ) ) ).
% smap_alt
thf(fact_71_sdrop__stl,axiom,
! [A: $tType,N: nat,S: stream170649215stream @ A] :
( ( stream135081970_sdrop @ A @ N @ ( stream_Mirabelle_stl @ A @ S ) )
= ( stream_Mirabelle_stl @ A @ ( stream135081970_sdrop @ A @ N @ S ) ) ) ).
% sdrop_stl
thf(fact_72_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( ( member @ A @ X @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_73_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_74_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B4: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B4 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% insertCI
thf(fact_75_insert__iff,axiom,
! [A: $tType,A2: A,B4: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B4 @ A3 ) )
= ( ( A2 = B4 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_76_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_77_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_78_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_79_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_80_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_81_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_82_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_83_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_84_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_85_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_86_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_87_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_88_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_89_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_90_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_91_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_92_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_93_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_94_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_95_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ~ ( member @ A @ C2 @ A3 ) ) ) ).
% contra_subsetD
thf(fact_96_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_97_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_98_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_99_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_100_rev__subsetD,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% rev_subsetD
thf(fact_101_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_102_set__rev__mp,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% set_rev_mp
thf(fact_103_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_104_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_105_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ A @ X5 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_106_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_107_subsetCE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetCE
thf(fact_108_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_109_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_110_set__mp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% set_mp
thf(fact_111_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_112_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_113_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B6: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B6 ) )
& ~ ( member @ A @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_114_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_115_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B4: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B4 @ B3 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B4 @ B3 ) )
= ( ( ( A2 = B4 )
=> ( A3 = B3 ) )
& ( ( A2 != B4 )
=> ? [C4: set @ A] :
( ( A3
= ( insert @ A @ B4 @ C4 ) )
& ~ ( member @ A @ B4 @ C4 )
& ( B3
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_116_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_117_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ~ ( member @ A @ X @ B3 )
=> ( ( ( insert @ A @ X @ A3 )
= ( insert @ A @ X @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_118_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ~ ! [B6: set @ A] :
( ( A3
= ( insert @ A @ X @ B6 ) )
=> ( member @ A @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_119_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B4: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% insertI2
thf(fact_120_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_121_insertE,axiom,
! [A: $tType,A2: A,B4: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B4 @ A3 ) )
=> ( ( A2 != B4 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_122_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_123_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_124_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_125_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_126_subset__insertI,axiom,
! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_127_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_128_insert__mono,axiom,
! [A: $tType,C3: set @ A,D: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_129_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_130_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_131_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_132_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_133_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_134_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_135_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_136_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_137_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_138_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_139_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_140_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_141_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_142_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_143_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X13: A] : ( P @ ( zero_zero @ nat ) @ X13 )
=> ( ! [X4: A,N3: nat] :
( ( P @ N3 @ X4 )
=> ? [Y5: A] :
( ( P @ ( suc @ N3 ) @ Y5 )
& ( Q @ N3 @ X4 @ Y5 ) ) )
=> ? [F3: nat > A] :
! [N5: nat] :
( ( P @ N5 @ ( F3 @ N5 ) )
& ( Q @ N5 @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_144_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_145_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X3 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_146_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_147_antimono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) ) ) ) ) ).
% antimono_iff_le_Suc
thf(fact_148_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_149_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_150_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_151_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_152_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_153_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_154_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_155_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_156_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_157_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
=> ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% antimono_def
thf(fact_158_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_159_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_160_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_161_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_162_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_163_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_164_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B4: B,C2: B] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_165_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_166_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_167_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_168_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_169_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_170_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_171_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_172_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_173_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_174_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( A2 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_175_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_176_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_177_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% order_trans
thf(fact_178_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_179_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B4: A] :
( ! [A4: A,B7: A] :
( ( ord_less_eq @ A @ A4 @ B7 )
=> ( P @ A4 @ B7 ) )
=> ( ! [A4: A,B7: A] :
( ( P @ B7 @ A4 )
=> ( P @ A4 @ B7 ) )
=> ( P @ A2 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_180_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_181_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A] :
( ( ord_less_eq @ A @ B4 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_182_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_183_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_184_LeastMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y3 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M @ X4 ) @ ( M @ Y5 ) ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% LeastMI2
thf(fact_185_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_186_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_187_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_188_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_189_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_190_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_191_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_192_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_193_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_194_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_195_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_196_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_197_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_198_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B4 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_199_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B4 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_200_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A6: A,B2: A] : ( times_times @ A @ B2 @ A6 ) ) ) ) ).
% mult.commute
thf(fact_201_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C2: A] :
( ( times_times @ A @ B4 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_202_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_203_LeastM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% LeastM_nat_lemma
thf(fact_204_LeastM__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% LeastM_nat_le
thf(fact_205_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B4: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B4 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B4 ) ) ) ) ).
% mult_cancel_right
thf(fact_206_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B4: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B4 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B4 ) ) ) ) ).
% mult_cancel_left
thf(fact_207_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_208_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_209_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ( times_times @ A @ A2 @ B4 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B4
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_210_LeastM__natI,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) ) ).
% LeastM_natI
thf(fact_211_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ( times_times @ A @ A2 @ B4 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B4
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_212_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ( times_times @ A @ A2 @ B4 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B4
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_213_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_214_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% mult_left_cancel
thf(fact_215_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B4 @ C2 ) )
= ( A2 = B4 ) ) ) ) ).
% mult_right_cancel
thf(fact_216_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere1490568538miring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_217_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B4 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_218_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B4 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_219_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_220_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_221_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B4 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_222_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B4 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_223_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B4 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_224_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_225_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_226_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% mult_left_mono
thf(fact_227_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B4 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_228_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_229_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B4 ) ) ) ) ).
% split_mult_pos_le
thf(fact_230_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_231_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_232_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_233_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_234_semiring__normalization__rules_I10_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(10)
thf(fact_235_semiring__normalization__rules_I9_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(9)
thf(fact_236_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_237_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_238_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_239_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_240_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_241_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_242_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_243_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_244_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_245_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_246_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_247_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_248_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_249_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_250_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord20386208strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_251_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord20386208strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B4 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_252_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord20386208strict @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B4: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B4 ) ) ) ) ).
% mult_right_le_imp_le
%----Type constructors (27)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ( 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 ) ) ) ) ).
%------------------------------------------------------------------------------