TPTP Problem File: DAT245^1.p
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%------------------------------------------------------------------------------
% File : DAT245^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Infinite streams (sequences/lists) 108
% 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__108.p [Bla16]
% Status : Theorem
% Rating : 1.00 v8.1.0, 0.75 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 341 ( 132 unt; 50 typ; 0 def)
% Number of atoms : 796 ( 258 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3890 ( 79 ~; 9 |; 56 &;3424 @)
% ( 0 <=>; 322 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 187 ( 187 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 49 usr; 5 con; 0-7 aty)
% Number of variables : 1030 ( 51 ^; 914 !; 18 ?;1030 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:45.745
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
stream170649215stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (46)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
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_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osmember,type,
stream1586597341member:
!>[A: $tType] : ( A > ( stream170649215stream @ A ) > $o ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_OSCons,type,
stream641971652_SCons:
!>[A: $tType] : ( A > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocase__stream,type,
stream1342653232stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream170649215stream @ A ) > B ) > ( stream170649215stream @ A ) > B ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_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_Osset,type,
stream30925839e_sset:
!>[A: $tType] : ( ( stream170649215stream @ A ) > ( set @ 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_A,type,
a2: set @ a ).
thf(sy_v_s,type,
s: stream170649215stream @ a ).
thf(sy_v_sa____,type,
sa: stream170649215stream @ a ).
%----Relevant facts (256)
thf(fact_0_streams,axiom,
ord_less_eq @ ( set @ a ) @ ( stream30925839e_sset @ a @ sa ) @ a2 ).
% streams
thf(fact_1_assms,axiom,
ord_less_eq @ ( set @ a ) @ ( stream30925839e_sset @ a @ s ) @ a2 ).
% assms
thf(fact_2_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_3_stream_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A2: stream170649215stream @ A,A1: A] :
( ( member @ A @ X @ ( stream30925839e_sset @ A @ A2 ) )
=> ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A2 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_4_stream_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A2: stream170649215stream @ A] : ( member @ A @ A1 @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A2 ) ) ) ).
% stream.set_intros(1)
thf(fact_5_streams_Ocases,axiom,
! [A: $tType,A3: stream170649215stream @ A,A4: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ A3 @ ( stream2015131171treams @ A @ A4 ) )
=> ~ ! [A5: A,S: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ A5 @ S ) )
=> ( ( member @ A @ A5 @ A4 )
=> ~ ( member @ ( stream170649215stream @ A ) @ S @ ( stream2015131171treams @ A @ A4 ) ) ) ) ) ).
% streams.cases
thf(fact_6_streams_Osimps,axiom,
! [A: $tType,A3: stream170649215stream @ A,A4: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ A3 @ ( stream2015131171treams @ A @ A4 ) )
= ( ? [A6: A,S2: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ A6 @ S2 ) )
& ( member @ A @ A6 @ A4 )
& ( member @ ( stream170649215stream @ A ) @ S2 @ ( stream2015131171treams @ A @ A4 ) ) ) ) ) ).
% streams.simps
thf(fact_7_stream_Oexhaust,axiom,
! [A: $tType,Y: stream170649215stream @ A] :
~ ! [X12: A,X22: stream170649215stream @ A] :
( Y
!= ( stream641971652_SCons @ A @ X12 @ X22 ) ) ).
% stream.exhaust
thf(fact_8_streams__Stream,axiom,
! [A: $tType,X: A,S3: stream170649215stream @ A,A4: set @ A] :
( ( member @ ( stream170649215stream @ A ) @ ( stream641971652_SCons @ A @ X @ S3 ) @ ( stream2015131171treams @ A @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( member @ ( stream170649215stream @ A ) @ S3 @ ( stream2015131171treams @ A @ A4 ) ) ) ) ).
% streams_Stream
thf(fact_9_stream_Oset__cases,axiom,
! [A: $tType,E: A,A3: stream170649215stream @ A] :
( ( member @ A @ E @ ( stream30925839e_sset @ A @ A3 ) )
=> ( ! [Z2: stream170649215stream @ A] :
( A3
!= ( stream641971652_SCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( stream30925839e_sset @ A @ Z2 ) ) ) ) ) ).
% stream.set_cases
thf(fact_10_streams_Ocoinduct,axiom,
! [A: $tType,X3: ( stream170649215stream @ A ) > $o,X: stream170649215stream @ A,A4: set @ A] :
( ( X3 @ X )
=> ( ! [X4: stream170649215stream @ A] :
( ( X3 @ X4 )
=> ? [A7: A,S4: stream170649215stream @ A] :
( ( X4
= ( stream641971652_SCons @ A @ A7 @ S4 ) )
& ( member @ A @ A7 @ A4 )
& ( ( X3 @ S4 )
| ( member @ ( stream170649215stream @ A ) @ S4 @ ( stream2015131171treams @ A @ A4 ) ) ) ) )
=> ( member @ ( stream170649215stream @ A ) @ X @ ( stream2015131171treams @ A @ A4 ) ) ) ) ).
% streams.coinduct
thf(fact_11_stream_Oset__induct,axiom,
! [A: $tType,X: A,A3: stream170649215stream @ A,P: A > ( stream170649215stream @ A ) > $o] :
( ( member @ A @ X @ ( stream30925839e_sset @ A @ A3 ) )
=> ( ! [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 @ A3 ) ) ) ) ).
% stream.set_induct
thf(fact_12_smember__code,axiom,
! [A: $tType,X: A,Y: A,S3: stream170649215stream @ A] :
( ( stream1586597341member @ A @ X @ ( stream641971652_SCons @ A @ Y @ S3 ) )
= ( ( X != Y )
=> ( stream1586597341member @ A @ X @ S3 ) ) ) ).
% smember_code
thf(fact_13_subsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ A @ X4 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% subsetI
thf(fact_14_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% subset_antisym
thf(fact_15_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_16_Stream__Mirabelle__hbrgyiwlrc_Osmember__def,axiom,
! [A: $tType] :
( ( stream1586597341member @ A )
= ( ^ [X5: A,S2: stream170649215stream @ A] : ( member @ A @ X5 @ ( stream30925839e_sset @ A @ S2 ) ) ) ) ).
% Stream_Mirabelle_hbrgyiwlrc.smember_def
thf(fact_17_set__mp,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_18_in__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_19_subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_20_subsetCE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetCE
thf(fact_21_equalityE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% equalityE
thf(fact_22_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B3: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( member @ A @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_23_equalityD1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% equalityD1
thf(fact_24_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
=> ( A3 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_25_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_26_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B4: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_27_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_28_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_29_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_30_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_31_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( A3 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_32_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_33_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_34_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_35_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_36_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_37_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_38_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_39_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z3: A] : Y3 = Z3 )
= ( ^ [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_40_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B4: B,C2: B] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_42_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_43_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_44_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A4 ) )
= A4 ) ).
% 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_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G2 @ X4 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_50_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_51_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_52_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_53_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ~ ( member @ A @ C2 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_54_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z3: set @ A] : Y3 = Z3 )
= ( ^ [A8: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_55_subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_56_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_57_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_58_rev__subsetD,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_59_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B3: set @ A] :
! [T: A] :
( ( member @ A @ T @ A8 )
=> ( member @ A @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_60_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_61_equalityD2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% equalityD2
thf(fact_62_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_63_stream_Opred__inject,axiom,
! [A: $tType,P: A > $o,A3: A,Aa: stream170649215stream @ A] :
( ( stream1153105665stream @ A @ P @ ( stream641971652_SCons @ A @ A3 @ Aa ) )
= ( ( P @ A3 )
& ( stream1153105665stream @ A @ P @ Aa ) ) ) ).
% stream.pred_inject
thf(fact_64_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% antimono_def
thf(fact_65_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ Y5 ) @ ( F @ X4 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_66_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_67_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_68_streamsp_Ocoinduct,axiom,
! [A: $tType,X3: ( stream170649215stream @ A ) > $o,X: stream170649215stream @ A,A4: A > $o] :
( ( X3 @ X )
=> ( ! [X4: stream170649215stream @ A] :
( ( X3 @ X4 )
=> ? [A7: A,S4: stream170649215stream @ A] :
( ( X4
= ( stream641971652_SCons @ A @ A7 @ S4 ) )
& ( A4 @ A7 )
& ( ( X3 @ S4 )
| ( stream86250253reamsp @ A @ A4 @ S4 ) ) ) )
=> ( stream86250253reamsp @ A @ A4 @ X ) ) ) ).
% streamsp.coinduct
thf(fact_69_streamsp_Osimps,axiom,
! [A: $tType] :
( ( stream86250253reamsp @ A )
= ( ^ [A8: A > $o,A6: stream170649215stream @ A] :
? [B6: A,S2: stream170649215stream @ A] :
( ( A6
= ( stream641971652_SCons @ A @ B6 @ S2 ) )
& ( A8 @ B6 )
& ( stream86250253reamsp @ A @ A8 @ S2 ) ) ) ) ).
% streamsp.simps
thf(fact_70_streamsp_Ocases,axiom,
! [A: $tType,A4: A > $o,A3: stream170649215stream @ A] :
( ( stream86250253reamsp @ A @ A4 @ A3 )
=> ~ ! [A5: A,S: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ A5 @ S ) )
=> ( ( A4 @ A5 )
=> ~ ( stream86250253reamsp @ A @ A4 @ S ) ) ) ) ).
% streamsp.cases
thf(fact_71_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_72_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B4: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_73_insert__iff,axiom,
! [A: $tType,A3: A,B4: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
= ( ( A3 = B4 )
| ( member @ A @ A3 @ A4 ) ) ) ).
% insert_iff
thf(fact_74_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_75_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_subset
thf(fact_76_insertE,axiom,
! [A: $tType,A3: A,B4: A,A4: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B4 @ A4 ) )
=> ( ( A3 != B4 )
=> ( member @ A @ A3 @ A4 ) ) ) ).
% insertE
thf(fact_77_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_78_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B4: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert @ A @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_79_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B7: set @ A] :
( ( A4
= ( insert @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_80_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B2 ) )
= ( A4 = B2 ) ) ) ) ).
% insert_ident
thf(fact_81_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert @ A @ A3 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_82_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set @ A,B4: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ B4 @ B2 )
=> ( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A4 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set @ A] :
( ( A4
= ( insert @ A @ B4 @ C4 ) )
& ~ ( member @ A @ B4 @ C4 )
& ( B2
= ( insert @ A @ A3 @ C4 ) )
& ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_83_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_84_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ? [B7: set @ A] :
( ( A4
= ( insert @ A @ A3 @ B7 ) )
& ~ ( member @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_85_insert__mono,axiom,
! [A: $tType,C3: set @ A,D: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C3 ) @ ( insert @ A @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_86_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% subset_insert
thf(fact_87_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_88_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_89_stream_Opred__cong,axiom,
! [A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( stream30925839e_sset @ A @ Ya ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( stream1153105665stream @ A @ P @ X )
= ( stream1153105665stream @ A @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_90_stream_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: stream170649215stream @ A,Pa: A > $o] :
( ( stream1153105665stream @ A @ P @ X )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( stream30925839e_sset @ A @ X ) )
=> ( ( P @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( stream1153105665stream @ A @ Pa @ X ) ) ) ).
% stream.pred_mono_strong
thf(fact_91_insert__subsetI,axiom,
! [A: $tType,X: A,A4: set @ A,X3: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X3 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_92_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set @ A,B4: A] :
( ( ( insert @ A @ A3 @ A4 )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_93_singleton__insert__inj__eq,axiom,
! [A: $tType,B4: A,A3: A,A4: set @ A] :
( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A4 ) )
= ( ( A3 = B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_94_stream_Opred__set,axiom,
! [A: $tType] :
( ( stream1153105665stream @ A )
= ( ^ [P2: A > $o,X5: stream170649215stream @ A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( stream30925839e_sset @ A @ X5 ) )
=> ( P2 @ Y4 ) ) ) ) ).
% stream.pred_set
thf(fact_95_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_96_subset__singleton__iff,axiom,
! [A: $tType,X3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X3
= ( bot_bot @ ( set @ A ) ) )
| ( X3
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_97_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_98_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X5: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_99_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X5: A] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_100_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_101_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_102_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_103_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_104_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_105_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_106_ball__empty,axiom,
! [A: $tType,P: A > $o,X6: A] :
( ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X6 ) ) ).
% ball_empty
thf(fact_107_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_108_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( stream660621732stream @ C @ A )
= ( stream660621732stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_109_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_110_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_111_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y5: A] :
~ ( member @ A @ Y5 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_112_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_113_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P2: A > $o] :
! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( P2 @ X5 ) ) ) ) ).
% Ball_def
thf(fact_114_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_115_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_116_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_117_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_118_singleton__inject,axiom,
! [A: $tType,A3: A,B4: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B4 ) ) ).
% singleton_inject
thf(fact_119_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert @ A @ A3 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_120_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B4: A,C2: A,D3: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B4 = D3 ) )
| ( ( A3 = D3 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_121_singleton__iff,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_122_singletonD,axiom,
! [A: $tType,B4: A,A3: A] :
( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_123_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_124_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_125_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A8 @ ( collect @ A @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_126_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: set @ A] :
( A8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_127_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X4: A] :
( A4
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_128_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_129_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
( A8
= ( insert @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_130_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y5: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y5 @ A4 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_131_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
? [X5: A] :
( A8
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_132_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A4: A] : ( pairwise @ A @ P @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_133_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B4 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_134_ball__reg,axiom,
! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( P @ X4 ) )
=> ! [X6: A] :
( ( member @ A @ X6 @ R )
=> ( Q @ X6 ) ) ) ) ).
% ball_reg
thf(fact_135_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_136_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A4 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_137_Compl__iff,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C2 @ A4 ) ) ) ).
% Compl_iff
thf(fact_138_ComplI,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_139_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_140_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_141_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_142_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S5: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ S5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S5 )
=> ( ( X5 != Y4 )
=> ( R2 @ X5 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_143_ComplD,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C2 @ A4 ) ) ).
% ComplD
thf(fact_144_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_145_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_146_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S6: set @ A,T2: set @ A] :
( ( pairwise @ A @ P @ S6 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S6 )
=> ( pairwise @ A @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_147_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X: A,S3: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X @ S3 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S3 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise @ A @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_148_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B4 ) ) ) ).
% neg_le_iff_le
thf(fact_149_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_150_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A8: A > B,X5: A] : ( uminus_uminus @ B @ ( A8 @ X5 ) ) ) ) ) ).
% uminus_apply
thf(fact_151_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_152_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_153_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_154_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B4 ) )
= ( A3 = B4 ) ) ) ).
% neg_equal_iff_equal
thf(fact_155_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ( uminus_uminus @ A @ A3 )
= B4 )
= ( ( uminus_uminus @ A @ B4 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_156_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( A3
= ( uminus_uminus @ A @ B4 ) )
= ( B4
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_157_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A8: A > B,X5: A] : ( uminus_uminus @ B @ ( A8 @ X5 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_158_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_159_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B4 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_160_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B4 ) )
= ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_161_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_162_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_163_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_164_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_165_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R4: A > A > $o,F2: B > A,X5: B,Y4: B] : ( R4 @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_166_Compl__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_167_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_idemp
thf(fact_168_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_169_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% DiffI
thf(fact_170_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B4 ) )
= ( minus_minus @ A @ B4 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_171_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_172_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_173_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_174_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_175_insert__Diff1,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_176_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_177_insert__Diff__single,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A4 ) ) ).
% insert_Diff_single
thf(fact_178_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B4 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_179_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% diff_left_mono
thf(fact_180_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_181_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B4 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_182_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,D: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_183_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ A4 ) ).
% Diff_subset
thf(fact_184_double__diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C3 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_185_insert__Diff__if,axiom,
! [A: $tType,X: A,B2: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B2 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_186_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( member @ A @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_187_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_188_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_189_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_190_insert__Diff,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_191_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A3: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_192_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_193_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X: A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert @ A @ X @ C3 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_194_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_195_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_196_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X5: A,A8: set @ A] : ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_197_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B2 ) )
= ( ( ( member @ A @ X @ B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) )
& ( ~ ( member @ A @ X @ B2 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_198_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_199_IntI,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% IntI
thf(fact_200_Int__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C2 @ A4 )
& ( member @ A @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_201_member__remove,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A4 ) )
= ( ( member @ A @ X @ A4 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_202_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_203_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B4 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B4 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_204_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B4 ) ) ) ).
% neg_less_iff_less
thf(fact_205_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_206_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_207_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_208_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_209_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_210_insert__inter__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A4 ) @ ( insert @ A @ A3 @ B2 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_211_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C3: set @ A,B2: set @ A] :
( ( member @ A @ A3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C3 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_212_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C3: set @ A,B2: set @ A] :
( ~ ( member @ A @ A3 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_213_psubsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% psubsetI
thf(fact_214_compl__inf__bot,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% compl_inf_bot
thf(fact_215_inf__compl__bot,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot
thf(fact_216_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_217_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_218_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_219_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A4 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_220_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A4: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A4 ) @ B2 ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_221_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A3: A,A4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_222_disjoint__insert_I2_J,axiom,
! [A: $tType,A4: set @ A,B4: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ ( insert @ A @ B4 @ B2 ) ) )
= ( ~ ( member @ A @ B4 @ A4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_223_Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_224_Compl__disjoint,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_225_Compl__disjoint2,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_226_Diff__Compl,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_Compl
thf(fact_227_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_228_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_229_Diff__Int__distrib,axiom,
! [A: $tType,C3: set @ A,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C3 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_230_Diff__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_231_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_232_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_233_Int__emptyI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ~ ( member @ A @ X4 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_234_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_235_Int__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_236_disjoint__iff__not__equal,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ B2 )
=> ( X5 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_237_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_238_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_239_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_240_IntE,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ~ ( member @ A @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_241_IntD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% IntD1
thf(fact_242_IntD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
=> ( member @ A @ C2 @ B2 ) ) ).
% IntD2
thf(fact_243_psubsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_244_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_245_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_246_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B3: set @ A] : ( inf_inf @ ( set @ A ) @ B3 @ A8 ) ) ) ).
% Int_commute
thf(fact_247_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_248_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B2 ) ) ).
% Int_left_absorb
thf(fact_249_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_250_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B4: B,C2: B] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_251_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B4: A,F: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_252_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_253_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_254_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% lt_ex
thf(fact_255_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% gt_ex
%----Type constructors (31)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A9: $tType,A10: $tType] :
( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
=> ( bounded_lattice @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A9: $tType,A10: $tType] :
( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
=> ( bounded_lattice_bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_inf @ A10 @ ( type2 @ A10 ) )
=> ( semilattice_inf @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A9: $tType,A10: $tType] :
( ( boolean_algebra @ A10 @ ( type2 @ A10 ) )
=> ( boolean_algebra @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A9: $tType,A10: $tType] :
( ( order_bot @ A10 @ ( type2 @ A10 ) )
=> ( order_bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 @ ( type2 @ A10 ) )
=> ( preorder @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 @ ( type2 @ A10 ) )
=> ( order @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 @ ( type2 @ A10 ) )
=> ( ord @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 @ ( type2 @ A10 ) )
=> ( bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A10: $tType] :
( ( uminus @ A10 @ ( type2 @ A10 ) )
=> ( uminus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_3,axiom,
! [A9: $tType] : ( bounded_lattice_bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_4,axiom,
! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A9: $tType] : ( order_bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_11,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_12,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_13,axiom,
semilattice_inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_14,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_15,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_19,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_20,axiom,
uminus @ $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,
? [A7: a,S4: stream170649215stream @ a] :
( ( sa
= ( stream641971652_SCons @ a @ A7 @ S4 ) )
& ( member @ a @ A7 @ a2 )
& ( ? [Sa: stream170649215stream @ a] :
( ( S4 = Sa )
& ( ord_less_eq @ ( set @ a ) @ ( stream30925839e_sset @ a @ Sa ) @ a2 ) )
| ( member @ ( stream170649215stream @ a ) @ S4 @ ( stream2015131171treams @ a @ a2 ) ) ) ) ).
%------------------------------------------------------------------------------