TPTP Problem File: DAT244^1.p
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%------------------------------------------------------------------------------
% File : DAT244^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Infinite streams (sequences/lists) 29
% 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__29.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 332 ( 138 unt; 47 typ; 0 def)
% Number of atoms : 707 ( 301 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3749 ( 64 ~; 7 |; 37 &;3320 @)
% ( 0 <=>; 321 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 407 ( 407 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 5 con; 0-7 aty)
% Number of variables : 1224 ( 62 ^;1102 !; 11 ?;1224 :)
% ( 49 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:28.127
%------------------------------------------------------------------------------
%----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 (43)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[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_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
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_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
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_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
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_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_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_Ostream_Ostream__all2,type,
stream686525009m_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_s,type,
s: stream170649215stream @ a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
%----Relevant facts (256)
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_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_2_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_3_stream_Oexhaust,axiom,
! [A: $tType,Y: stream170649215stream @ A] :
~ ! [X12: A,X22: stream170649215stream @ A] :
( Y
!= ( stream641971652_SCons @ A @ X12 @ X22 ) ) ).
% stream.exhaust
thf(fact_4_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_5_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_6_Stream__Mirabelle__hbrgyiwlrc_Osmember__def,axiom,
! [A: $tType] :
( ( stream1586597341member @ A )
= ( ^ [X3: A,S: stream170649215stream @ A] : ( member @ A @ X3 @ ( stream30925839e_sset @ A @ S ) ) ) ) ).
% Stream_Mirabelle_hbrgyiwlrc.smember_def
thf(fact_7_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_8_stream_Opred__inject,axiom,
! [A: $tType,P: A > $o,A3: A,Aa2: stream170649215stream @ A] :
( ( stream1153105665stream @ A @ P @ ( stream641971652_SCons @ A @ A3 @ Aa2 ) )
= ( ( P @ A3 )
& ( stream1153105665stream @ A @ P @ Aa2 ) ) ) ).
% stream.pred_inject
thf(fact_9_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_10_stream_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( stream660621732stream @ C @ A )
= ( ^ [G1: C > A,Q2: C > $o,G21: C > ( stream170649215stream @ A ),G22: C > C,A4: C] : ( stream641971652_SCons @ A @ ( G1 @ A4 ) @ ( if @ ( stream170649215stream @ A ) @ ( Q2 @ A4 ) @ ( G21 @ A4 ) @ ( stream660621732stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ ( G22 @ A4 ) ) ) ) ) ) ).
% stream.corec_code
thf(fact_11_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_12_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_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_Oset__sel_I1_J,axiom,
! [A: $tType,A3: stream170649215stream @ A] : ( member @ A @ ( stream_Mirabelle_shd @ A @ A3 ) @ ( stream30925839e_sset @ A @ A3 ) ) ).
% stream.set_sel(1)
thf(fact_15_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_16_stream_Oset__sel_I2_J,axiom,
! [A: $tType,X: A,A3: stream170649215stream @ A] :
( ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream_Mirabelle_stl @ A @ A3 ) ) )
=> ( member @ A @ X @ ( stream30925839e_sset @ A @ A3 ) ) ) ).
% stream.set_sel(2)
thf(fact_17_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_18_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_19_stream_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,F: A > B,A3: stream170649215stream @ A] :
( ( stream_Mirabelle_stl @ B @ ( stream2128578057e_smap @ A @ B @ F @ A3 ) )
= ( stream2128578057e_smap @ A @ B @ F @ ( stream_Mirabelle_stl @ A @ A3 ) ) ) ).
% stream.map_sel(2)
thf(fact_20_stream_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,F: A > B,A3: stream170649215stream @ A] :
( ( stream_Mirabelle_shd @ B @ ( stream2128578057e_smap @ A @ B @ F @ A3 ) )
= ( F @ ( stream_Mirabelle_shd @ A @ A3 ) ) ) ).
% stream.map_sel(1)
thf(fact_21_stream_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,Q22: C > $o,A3: C,G12: C > A,G212: C > ( stream170649215stream @ A ),G222: C > C] :
( ( ( Q22 @ A3 )
=> ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
= ( G212 @ A3 ) ) )
& ( ~ ( Q22 @ A3 )
=> ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
= ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ).
% stream.corec_sel(2)
thf(fact_22_stream_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,G12: C > A,Q22: C > $o,G212: C > ( stream170649215stream @ A ),G222: C > C,A3: C] :
( ( stream_Mirabelle_shd @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
= ( G12 @ A3 ) ) ).
% stream.corec_sel(1)
thf(fact_23_stream_Oexpand,axiom,
! [A: $tType,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
( ( ( ( stream_Mirabelle_shd @ A @ Stream )
= ( stream_Mirabelle_shd @ A @ Stream2 ) )
& ( ( stream_Mirabelle_stl @ A @ Stream )
= ( stream_Mirabelle_stl @ A @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_24_stream_Ocoinduct,axiom,
! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
= ( stream_Mirabelle_shd @ A @ Stream4 ) )
& ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_25_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( stream1342653232stream @ A @ B )
= ( ^ [F2: A > ( stream170649215stream @ A ) > B,Stream5: stream170649215stream @ A] : ( F2 @ ( stream_Mirabelle_shd @ A @ Stream5 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_26_stream_Ocoinduct__strong,axiom,
! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
= ( stream_Mirabelle_shd @ A @ Stream4 ) )
& ( ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) )
| ( ( stream_Mirabelle_stl @ A @ Stream3 )
= ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_27_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( stream660621732stream @ C @ A )
= ( stream660621732stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_28_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_29_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_30_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_31_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_32_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_33_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_34_insertCI,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_35_insert__iff,axiom,
! [A: $tType,A3: A,B3: A,A5: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A5 ) )
= ( ( A3 = B3 )
| ( member @ A @ A3 @ A5 ) ) ) ).
% insert_iff
thf(fact_36_insert__absorb2,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A5 ) )
= ( insert @ A @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_37_insertE,axiom,
! [A: $tType,A3: A,B3: A,A5: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B3 @ A5 ) )
=> ( ( A3 != B3 )
=> ( member @ A @ A3 @ A5 ) ) ) ).
% insertE
thf(fact_38_insertI1,axiom,
! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% insertI1
thf(fact_39_insertI2,axiom,
! [A: $tType,A3: A,B2: set @ A,B3: A] :
( ( member @ A @ A3 @ B2 )
=> ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_40_Set_Oset__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ~ ! [B4: set @ A] :
( ( A5
= ( insert @ A @ X @ B4 ) )
=> ( member @ A @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_41_insert__ident,axiom,
! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ~ ( member @ A @ X @ B2 )
=> ( ( ( insert @ A @ X @ A5 )
= ( insert @ A @ X @ B2 ) )
= ( A5 = B2 ) ) ) ) ).
% insert_ident
thf(fact_42_insert__absorb,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( insert @ A @ A3 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_43_insert__eq__iff,axiom,
! [A: $tType,A3: A,A5: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A3 @ A5 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A3 @ A5 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A3 = B3 )
=> ( A5 = B2 ) )
& ( ( A3 != B3 )
=> ? [C2: set @ A] :
( ( A5
= ( insert @ A @ B3 @ C2 ) )
& ~ ( member @ A @ B3 @ C2 )
& ( B2
= ( insert @ A @ A3 @ C2 ) )
& ~ ( member @ A @ A3 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_44_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ? [B4: set @ A] :
( ( A5
= ( insert @ A @ A3 @ B4 ) )
& ~ ( member @ A @ A3 @ B4 ) ) ) ).
% mk_disjoint_insert
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,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
= A5 ) ).
% 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,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A5 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_50_stream_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o,X: stream170649215stream @ A,Y: stream170649215stream @ B,R: A > B > $o] :
( ( P @ X @ Y )
=> ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ B] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( stream_Mirabelle_shd @ A @ Stream3 ) @ ( stream_Mirabelle_shd @ B @ Stream4 ) )
& ( P @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ B @ Stream4 ) ) ) )
=> ( stream686525009m_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_51_stream_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( stream686525009m_all2 @ A @ B )
= ( ^ [R2: A > B > $o,A4: stream170649215stream @ A,B5: stream170649215stream @ B] :
( ( R2 @ ( stream_Mirabelle_shd @ A @ A4 ) @ ( stream_Mirabelle_shd @ B @ B5 ) )
& ( stream686525009m_all2 @ A @ B @ R2 @ ( stream_Mirabelle_stl @ A @ A4 ) @ ( stream_Mirabelle_stl @ B @ B5 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_52_stream_Opred__set,axiom,
! [A: $tType] :
( ( stream1153105665stream @ A )
= ( ^ [P2: A > $o,X3: stream170649215stream @ A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( stream30925839e_sset @ A @ X3 ) )
=> ( P2 @ Y3 ) ) ) ) ).
% stream.pred_set
thf(fact_53_stream_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: stream170649215stream @ A] :
( ( stream1153105665stream @ B @ Q @ ( stream2128578057e_smap @ A @ B @ F @ X ) )
= ( stream1153105665stream @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% stream.pred_map
thf(fact_54_stream_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: stream170649215stream @ A] :
( ( stream30925839e_sset @ B @ ( stream2128578057e_smap @ A @ B @ F @ V ) )
= ( image @ A @ B @ F @ ( stream30925839e_sset @ A @ V ) ) ) ).
% stream.set_map
thf(fact_55_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X: A,S2: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X @ S2 ) )
= ( ! [Y3: A] :
( ( ( member @ A @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R3 @ X @ Y3 )
& ( R3 @ Y3 @ X ) ) )
& ( pairwise @ A @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_56_stream_Orel__inject,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: A,X2: stream170649215stream @ A,Y1: B,Y2: stream170649215stream @ B] :
( ( stream686525009m_all2 @ A @ B @ R @ ( stream641971652_SCons @ A @ X1 @ X2 ) @ ( stream641971652_SCons @ B @ Y1 @ Y2 ) )
= ( ( R @ X1 @ Y1 )
& ( stream686525009m_all2 @ A @ B @ R @ X2 @ Y2 ) ) ) ).
% stream.rel_inject
thf(fact_57_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_58_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A5: set @ B] :
( ( B3
= ( F @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_59_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_60_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_61_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_62_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_63_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( image @ B @ A @ F @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_64_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_65_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_66_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A3: B,B2: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A3 @ B2 ) )
= ( insert @ A @ ( F @ A3 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% image_insert
thf(fact_67_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) )
= ( image @ A @ B @ F @ A5 ) ) ) ).
% insert_image
thf(fact_68_ball__empty,axiom,
! [A: $tType,P: A > $o,X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_69_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_70_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,B3: B,F: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( B3
= ( F @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_71_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S3: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S3 )
=> ( ( X3 != Y3 )
=> ( R2 @ X3 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_72_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A5 ) )
=> ( P @ X4 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_73_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_74_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_75_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A5 ) )
& ( P @ X5 ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_76_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F: B > A,A5: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A5 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_77_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_78_equals0D,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A5 ) ) ).
% equals0D
thf(fact_79_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P2: A > $o] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( P2 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_80_imageI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) ) ) ).
% imageI
thf(fact_81_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_82_stream_Orel__eq,axiom,
! [A: $tType] :
( ( stream686525009m_all2 @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [Y5: stream170649215stream @ A,Z4: stream170649215stream @ A] : Y5 = Z4 ) ) ).
% stream.rel_eq
thf(fact_83_stream_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: stream170649215stream @ B] :
( ! [X4: B] : ( Ra @ X4 @ X4 )
=> ( stream686525009m_all2 @ B @ B @ Ra @ X @ X ) ) ).
% stream.rel_refl
thf(fact_84_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A5: A] : ( pairwise @ A @ P @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_85_stream_Omap__o__corec,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A,Ga: C > $o,Gb: C > ( stream170649215stream @ A ),Gc: C > C] :
( ( comp @ ( stream170649215stream @ A ) @ ( stream170649215stream @ B ) @ C @ ( stream2128578057e_smap @ A @ B @ F ) @ ( stream660621732stream @ C @ A @ G @ Ga @ Gb @ Gc ) )
= ( stream660621732stream @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ Ga @ ( comp @ ( stream170649215stream @ A ) @ ( stream170649215stream @ B ) @ C @ ( stream2128578057e_smap @ A @ B @ F ) @ Gb ) @ Gc ) ) ).
% stream.map_o_corec
thf(fact_86_stream_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: stream170649215stream @ A] :
( ( stream2128578057e_smap @ B @ C @ G @ ( stream2128578057e_smap @ A @ B @ F @ V ) )
= ( stream2128578057e_smap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% stream.map_comp
thf(fact_87_singletonD,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_88_singleton__iff,axiom,
! [A: $tType,B3: A,A3: A] :
( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_89_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B3: A,C3: A,D: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C3 )
& ( B3 = D ) )
| ( ( A3 = D )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_90_insert__not__empty,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert @ A @ A3 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_91_singleton__inject,axiom,
! [A: $tType,A3: A,B3: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_92_stream_Orel__refl__strong,axiom,
! [A: $tType,X: stream170649215stream @ A,Ra: A > A > $o] :
( ! [Z: A] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( stream686525009m_all2 @ A @ A @ Ra @ X @ X ) ) ).
% stream.rel_refl_strong
thf(fact_93_stream_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: stream170649215stream @ A,Y: stream170649215stream @ B,Ra: A > B > $o] :
( ( stream686525009m_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
=> ( ( member @ B @ Yb @ ( stream30925839e_sset @ B @ Y ) )
=> ( ( R @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( stream686525009m_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_94_stream_Orel__cong,axiom,
! [A: $tType,B: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,Y: stream170649215stream @ B,Xa2: stream170649215stream @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( stream30925839e_sset @ B @ Xa2 ) )
=> ( ( R @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( stream686525009m_all2 @ A @ B @ R @ X @ Y )
= ( stream686525009m_all2 @ A @ B @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% stream.rel_cong
thf(fact_95_stream_Orel__intros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: A,Y1: B,X2: stream170649215stream @ A,Y2: stream170649215stream @ B] :
( ( R @ X1 @ Y1 )
=> ( ( stream686525009m_all2 @ A @ B @ R @ X2 @ Y2 )
=> ( stream686525009m_all2 @ A @ B @ R @ ( stream641971652_SCons @ A @ X1 @ X2 ) @ ( stream641971652_SCons @ B @ Y1 @ Y2 ) ) ) ) ).
% stream.rel_intros
thf(fact_96_stream_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A3: stream170649215stream @ A,B3: stream170649215stream @ B] :
( ( stream686525009m_all2 @ A @ B @ R @ A3 @ B3 )
=> ~ ! [X1a: A,X2a: stream170649215stream @ A] :
( ( A3
= ( stream641971652_SCons @ A @ X1a @ X2a ) )
=> ! [Y1a: B,Y2a: stream170649215stream @ B] :
( ( B3
= ( stream641971652_SCons @ B @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream686525009m_all2 @ A @ B @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_97_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,G: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A5 ) )
=> ( G @ X3 ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( comp @ A @ $o @ B @ G @ F @ X3 ) ) ) ) ).
% Ball_image_comp
thf(fact_98_not__False__in__image__Ball,axiom,
! [A: $tType,P: A > $o,A5: set @ A] :
( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P @ A5 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( P @ X3 ) ) ) ) ).
% not_False_in_image_Ball
thf(fact_99_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_100_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_101_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_102_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_103_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R3: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R3 ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_104_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_105_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_106_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( X4 = Y4 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_107_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A5: set @ B,B2: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A5 = B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D3 @ X4 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C4 @ A5 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_108_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A5: set @ B,B2: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A5 = B2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B2 )
=> ( ( C4 @ X4 )
= ( D3 @ X4 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C4 @ A5 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_109_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_110_comp__apply__eq,axiom,
! [B: $tType,D2: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D2 > A,K: C > D2] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D2 @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_111_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= C3 )
=> ( ( A3 @ ( B3 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_112_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D2: $tType,A: $tType,A3: C > B,B3: A > C,C3: D2 > B,D: A > D2] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D2 @ B @ A @ C3 @ D ) )
=> ! [V2: A] :
( ( A3 @ ( B3 @ V2 ) )
= ( C3 @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_113_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D2: $tType,A: $tType,A3: C > B,B3: A > C,C3: D2 > B,D: A > D2,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D2 @ B @ A @ C3 @ D ) )
=> ( ( A3 @ ( B3 @ V ) )
= ( C3 @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_114_comp__assoc,axiom,
! [B: $tType,D2: $tType,C: $tType,A: $tType,F: D2 > B,G: C > D2,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D2 @ B @ C @ F @ G ) @ H )
= ( comp @ D2 @ B @ A @ F @ ( comp @ C @ D2 @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_115_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_116_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X4: A] :
( A5
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_117_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X3: A] :
( A6
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_118_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B,X: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F @ A5 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_119_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% Sup.SUP_image
thf(fact_120_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% Inf.INF_image
thf(fact_121_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D2: $tType,C: $tType,F: B > A,A5: set @ B,G: C > A,B2: set @ C,H: A > D2] :
( ( ( image @ B @ A @ F @ A5 )
= ( image @ C @ A @ G @ B2 ) )
=> ( ( image @ B @ D2 @ ( comp @ A @ D2 @ B @ H @ F ) @ A5 )
= ( image @ C @ D2 @ ( comp @ A @ D2 @ C @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_122_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_123_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_124_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 ) )
=> ! [X5: A] :
( ( member @ A @ X5 @ R )
=> ( Q @ X5 ) ) ) ) ).
% ball_reg
thf(fact_125_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E2: $tType,D2: $tType,A: $tType,G: C > B,H: A > C,R1: D2 > B,R22: A > D2,F: B > E2,L: D2 > E2] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D2 @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E2 @ D2 @ F @ R1 )
= L )
=> ( ( comp @ C @ E2 @ A @ ( comp @ B @ E2 @ C @ F @ G ) @ H )
= ( comp @ D2 @ E2 @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_126_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D2: $tType,E2: $tType,F: C > B,G: A > C,L1: D2 > B,L2: A > D2,H: E2 > A,R3: E2 > D2] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D2 @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D2 @ E2 @ L2 @ H )
= R3 )
=> ( ( comp @ C @ B @ E2 @ F @ ( comp @ A @ C @ E2 @ G @ H ) )
= ( comp @ D2 @ B @ E2 @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_127_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A3 @ A5 )
=> ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
= ( G @ A3 ) ) ) ).
% override_on_apply_in
thf(fact_128_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A3 @ A5 )
=> ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
= ( F @ A3 ) ) ) ).
% override_on_apply_notin
thf(fact_129_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F2: A > B,G2: A > B,A6: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A6 ) @ ( G2 @ A4 ) @ ( F2 @ A4 ) ) ) ) ).
% override_on_def
thf(fact_130_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D2: $tType,F: C > B,G: A > C,L: A > B,H: D2 > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D2 @ F @ ( comp @ A @ C @ D2 @ G @ H ) )
= ( comp @ A @ B @ D2 @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_131_rewriteR__comp__comp,axiom,
! [C: $tType,D2: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D2] :
( ( ( comp @ C @ B @ A @ G @ H )
= R3 )
=> ( ( comp @ C @ D2 @ A @ ( comp @ B @ D2 @ C @ F @ G ) @ H )
= ( comp @ B @ D2 @ A @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_132_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_133_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_134_comp__cong,axiom,
! [C: $tType,B: $tType,D2: $tType,A: $tType,E2: $tType,F: B > A,G: C > B,X: C,F3: D2 > A,G3: E2 > D2,X6: E2] :
( ( ( F @ ( G @ X ) )
= ( F3 @ ( G3 @ X6 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D2 @ A @ E2 @ F3 @ G3 @ X6 ) ) ) ).
% comp_cong
thf(fact_135_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D2: $tType,G: B > C,F: A > B,V: D2 > A] :
( ( comp @ B @ C @ D2 @ G @ ( comp @ A @ B @ D2 @ F @ V ) )
= ( comp @ A @ C @ D2 @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_136_type__copy__map__cong0,axiom,
! [B: $tType,D2: $tType,E2: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D2 > A,H: C > D2,F: A > E2] :
( ( ( M @ ( G @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp @ B @ E2 @ C @ ( comp @ A @ E2 @ B @ F @ M ) @ G @ X )
= ( comp @ D2 @ E2 @ C @ ( comp @ A @ E2 @ D2 @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_137_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F: B > A,A3: A,X: B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_138_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A5: set @ A,B3: A] :
( ( ( insert @ A @ A3 @ A5 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_139_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A3: A,A5: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A5 ) )
= ( ( A3 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_140_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_141_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A5 )
=> ( A5 = B2 ) ) ) ).
% subset_antisym
thf(fact_142_subsetI,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% subsetI
thf(fact_143_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X3: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_144_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_145_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_146_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_147_insert__subset,axiom,
! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B2 )
= ( ( member @ A @ X @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% insert_subset
thf(fact_148_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_149_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_150_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_151_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_152_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_153_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A7: A,B6: A] :
( ( ord_less_eq @ A @ A7 @ B6 )
=> ( P @ A7 @ B6 ) )
=> ( ! [A7: A,B6: A] :
( ( P @ B6 @ A7 )
=> ( P @ A7 @ B6 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_154_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_155_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_156_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_157_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_158_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_159_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_160_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_161_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_162_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_163_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_164_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_165_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_166_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_167_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_168_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_169_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_171_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_172_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_173_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_174_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_175_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_176_contra__subsetD,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ~ ( member @ A @ C3 @ B2 )
=> ~ ( member @ A @ C3 @ A5 ) ) ) ).
% contra_subsetD
thf(fact_177_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4 )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_178_subset__trans,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_179_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_180_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_181_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_182_subset__UNIV,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_183_rev__subsetD,axiom,
! [A: $tType,C3: A,A5: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_184_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_185_set__rev__mp,axiom,
! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_186_equalityD2,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( A5 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% equalityD2
thf(fact_187_equalityD1,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( A5 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% equalityD1
thf(fact_188_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_189_equalityE,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( A5 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% equalityE
thf(fact_190_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X4: A] : ( member @ A @ X4 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_191_subsetCE,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetCE
thf(fact_192_subsetD,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_193_in__mono,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_194_set__mp,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_195_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_196_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_197_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_198_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_199_subset__insert,axiom,
! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% subset_insert
thf(fact_200_insert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% insert_mono
thf(fact_201_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_202_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X4: B] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_203_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X4: B] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_204_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X4: A] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_205_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y3: A] :
? [X3: B] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_206_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_207_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
( ( B3
= ( F @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_208_image__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B2: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% image_mono
thf(fact_209_image__subsetI,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: A > B,B2: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ B2 ) ) ).
% image_subsetI
thf(fact_210_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A5 ) )
=> ~ ! [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
=> ( B2
!= ( image @ B @ A @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_211_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) @ B2 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( member @ A @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_212_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A5 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
& ( B2
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_213_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_214_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_215_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_216_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S4: set @ A,T2: set @ A] :
( ( pairwise @ A @ P @ S4 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S4 )
=> ( pairwise @ A @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_217_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_218_subset__singleton__iff,axiom,
! [A: $tType,X7: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
| ( X7
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_219_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_220_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D2: $tType,X: D2 > A,Xa2: D2 > A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( image @ D2 @ A @ X @ ( top_top @ ( set @ D2 ) ) ) )
=> ( ( member @ A @ Za @ ( image @ D2 @ A @ Xa2 @ ( top_top @ ( set @ D2 ) ) ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D2 @ F @ X )
= ( comp @ A @ B @ D2 @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_221_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D2: $tType,X: D2 > A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( image @ D2 @ A @ X @ ( top_top @ ( set @ D2 ) ) ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( comp @ A @ B @ D2 @ F @ X )
= ( comp @ A @ B @ D2 @ G @ X ) ) ) ).
% fun.map_cong0
thf(fact_222_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D2: $tType,X: D2 > A,Ya: D2 > A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( image @ D2 @ A @ Ya @ ( top_top @ ( set @ D2 ) ) ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( comp @ A @ B @ D2 @ F @ X )
= ( comp @ A @ B @ D2 @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_223_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D2: $tType,F: A > B,V: D2 > A] :
( ( image @ D2 @ B @ ( comp @ A @ B @ D2 @ F @ V ) @ ( top_top @ ( set @ D2 ) ) )
= ( image @ A @ B @ F @ ( image @ D2 @ A @ V @ ( top_top @ ( set @ D2 ) ) ) ) ) ).
% fun.set_map
thf(fact_224_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > A,X7: set @ B,G12: A > C,G23: A > C] :
( ( ( image @ B @ A @ F @ X7 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ X7 )
=> ( ( comp @ A @ C @ B @ G12 @ F @ X4 )
= ( comp @ A @ C @ B @ G23 @ F @ X4 ) ) )
=> ( G12 = G23 ) ) ) ).
% surj_fun_eq
thf(fact_225_insert__subsetI,axiom,
! [A: $tType,X: A,A5: set @ A,X7: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X7 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_226_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_227_subset__Compl__singleton,axiom,
! [A: $tType,A5: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_228_ComplI,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ~ ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% ComplI
thf(fact_229_Compl__iff,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% Compl_iff
thf(fact_230_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A5 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A5 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_231_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_232_Compl__anti__mono,axiom,
! [A: $tType,A5: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_233_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_234_stream_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o ) @ ( stream686525009m_all2 @ A @ B @ R ) @ ( stream686525009m_all2 @ A @ B @ Ra ) ) ) ).
% stream.rel_mono
thf(fact_235_ComplD,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
=> ~ ( member @ A @ C3 @ A5 ) ) ).
% ComplD
thf(fact_236_double__complement,axiom,
! [A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_237_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_238_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_239_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_240_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_241_compl__top__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% compl_top_eq
thf(fact_242_compl__bot__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% compl_bot_eq
thf(fact_243_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X4: A,Y4: B] :
( ( P @ X4 @ Y4 )
=> ( Q @ X4 @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_244_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_245_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_246_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_247_refl__ge__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X4: A] : ( R @ X4 @ X4 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R ) ) ).
% refl_ge_eq
thf(fact_248_ge__eq__refl,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_249_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_250_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_251_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_252_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_253_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P2: A > $o,X3: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P2 @ Y3 ) ) ) ) ).
% ID.pred_set
thf(fact_254_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_255_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A5: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A5 ) ) ).
% inj_uminus
%----Type constructors (25)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
=> ( boolean_algebra @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 @ ( type2 @ A9 ) )
=> ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
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_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( 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_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
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_Otop_6,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $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,
( ( member @ a @ x @ ( stream30925839e_sset @ a @ ( stream641971652_SCons @ a @ y @ s ) ) )
!= ( ~ ( ( x != y )
=> ( member @ a @ x @ ( stream30925839e_sset @ a @ s ) ) ) ) ) ).
%------------------------------------------------------------------------------