:: The Incompleteness of the Lattice of Substitutions
:: by Adam Grabowski
::
:: Received July 17, 2000
:: Copyright (c) 2000-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, ABIAN, RELAT_1, CARD_1, ARYTM_3, ORDERS_2, SUBSET_1,
TARSKI, ZFMISC_1, FUNCT_1, XXREAL_0, FINSET_1, XBOOLE_0, FINSEQ_1,
MCART_1, LATTICES, LATTICE3, PBOOLE, EQREL_1, FINSUB_1, PARTFUN1,
HEYTING1, SUBSTLAT, NORMFORM, REALSET1, STRUCT_0, BINOP_1, RELAT_2,
ORDINAL4, CAT_1, YELLOW_0, WELLORD1, HEYTING2, REWRITE1, HEYTING3,
XCMPLX_0, NAT_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, RELAT_1,
FUNCT_1, ORDERS_2, MEMBERED, XCMPLX_0, NAT_1, FINSEQ_1, BINOP_1,
FINSET_1, FINSUB_1, PARTFUN1, LATTICES, DOMAIN_1, REALSET1, STRUCT_0,
SUBSTLAT, XTUPLE_0, MCART_1, WELLORD1, YELLOW_0, ABIAN, XXREAL_2,
LATTICE3, HEYTING2, XXREAL_0;
constructors BINOP_1, TOLER_1, REALSET1, ABIAN, LATTICE3, YELLOW_0, HEYTING2,
VALUED_1, XXREAL_2, RELSET_1, XTUPLE_0, XXREAL_1;
registrations XBOOLE_0, SUBSET_1, ORDINAL1, RELSET_1, PARTFUN1, FINSET_1,
FINSUB_1, XXREAL_0, NAT_1, MEMBERED, FINSEQ_1, REALSET1, ABIAN, STRUCT_0,
LATTICES, LATTICE3, YELLOW_0, YELLOW_1, SUBSTLAT, XXREAL_2, RELAT_1,
HEYTING2, XTUPLE_0, XCMPLX_0;
requirements SUBSET, BOOLE, NUMERALS, REAL, ARITHM;
begin :: Preliminaries
scheme :: HEYTING3:sch 1
SSubsetUniq { R() -> 1-sorted, P[object] } :
for A1, A2 being Subset of R() st (
for x being object holds x in A1 iff P[x]) &
(for x being object holds x in A2 iff P[
x]) holds A1 = A2;
registration
let A, x be set;
cluster [:A, {x}:] -> Function-like;
end;
theorem :: HEYTING3:1
for X being finite non empty Subset of NAT holds ex n being
Element of NAT st X c= Seg n \/ {0};
theorem :: HEYTING3:2
for X being finite Subset of NAT holds ex k being odd Element of NAT
st not k in X;
theorem :: HEYTING3:3
for k being Element of NAT, X being finite non empty Subset of [:
NAT,{k}:] holds ex n being non zero Element of NAT st X c= [:Seg n \/ {0},{k}
:];
theorem :: HEYTING3:4
for m being Element of NAT, X being finite non empty Subset of [:
NAT,{m}:] holds ex k being non zero Element of NAT st not [2*k+1,m] in X;
theorem :: HEYTING3:5
for m being Element of NAT, X being finite Subset of [:NAT,{m}:] ex k
being Element of NAT st for l being Element of NAT st l >= k holds not [l,m] in
X;
theorem :: HEYTING3:6
for L being upper-bounded Lattice holds Top L = Top LattPOSet L;
theorem :: HEYTING3:7
for L being lower-bounded Lattice holds Bottom L = Bottom LattPOSet L;
begin :: Poset of Substitutions
theorem :: HEYTING3:8
for V being set, C being finite set, A, B being Element of Fin PFuncs
(V, C) st A = {} & B <> {} holds B =>> A = {};
theorem :: HEYTING3:9
for V, V9, C, C9 being set st V c= V9 & C c= C9 holds
SubstitutionSet (V, C) c= SubstitutionSet (V9, C9);
theorem :: HEYTING3:10
for V, V9, C, C9 being set, A being Element of Fin PFuncs (V, C)
, B being Element of Fin PFuncs (V9, C9) st V c= V9 & C c= C9 & A = B holds mi
A = mi B;
theorem :: HEYTING3:11
for V, V9, C, C9 being set st V c= V9 & C c= C9 holds the L_join of
SubstLatt (V, C) = (the L_join of SubstLatt (V9, C9))||the carrier of SubstLatt
(V, C);
definition
let V, C be set;
func SubstPoset (V, C) -> RelStr equals
:: HEYTING3:def 1
LattPOSet SubstLatt (V, C);
end;
registration
let V, C be set;
cluster SubstPoset (V, C) -> with_suprema with_infima;
end;
registration
let V, C be set;
cluster SubstPoset (V, C) -> reflexive antisymmetric transitive;
end;
theorem :: HEYTING3:12
for V, C being set, a, b being Element of SubstPoset (V, C)
holds a <= b iff for x being set st x in a ex y being set st y in b & y c= x;
theorem :: HEYTING3:13
for V, V9, C, C9 being set st V c= V9 & C c= C9 holds SubstPoset (V, C
) is full SubRelStr of SubstPoset (V9, C9);
definition
let n, k be Nat;
func PFArt (n, k) -> Element of PFuncs (NAT, {k}) means
:: HEYTING3:def 2
for x being object
holds x in it iff ( ex m being odd Element of NAT st m <= 2*n & [m,k] = x )
or [2*n,k] = x;
end;
registration
let n, k be Element of NAT;
cluster PFArt (n, k) -> finite;
end;
definition
let n, k be Nat;
func PFCrt (n, k) -> Element of PFuncs (NAT, {k}) means
:: HEYTING3:def 3
for x being object
holds x in it iff ex m being odd Element of NAT st m <= 2*n + 1 & [m,k] = x;
end;
registration
let n,k be Element of NAT;
cluster PFCrt (n,k) -> finite;
end;
theorem :: HEYTING3:14
for n, k being Element of NAT holds [2*n+1,k] in PFCrt (n,k);
theorem :: HEYTING3:15
for n, k being Element of NAT holds PFCrt (n,k) misses {[2*n+3,k ]};
theorem :: HEYTING3:16
for n, k being Element of NAT holds PFCrt (n+1,k) = PFCrt (n,k)
\/ {[2*n+3,k]};
theorem :: HEYTING3:17
for n, k being Element of NAT holds PFCrt (n,k) c< PFCrt (n+1,k);
registration
let n, k be Element of NAT;
cluster PFArt (n, k) -> non empty;
end;
theorem :: HEYTING3:18
for n, m, k being Element of NAT holds not PFArt (n, m) c= PFCrt (k, m);
theorem :: HEYTING3:19
for n, m, k being Element of NAT st n <= k holds PFCrt (n,m) c= PFCrt (k,m);
theorem :: HEYTING3:20
for n being Element of NAT holds PFArt (1,n) = { [1,n], [2,n] };
definition
let n, k be Nat;
func PFBrt (n,k) -> Element of Fin PFuncs (NAT, {k}) means
:: HEYTING3:def 4
for x being object
holds x in it iff ( ex m being non zero Element of NAT st m <= n & x
= PFArt (m,k) ) or x = PFCrt (n,k);
end;
theorem :: HEYTING3:21
for n, k being Element of NAT, x being set st x in PFBrt (n+1,k) holds
ex y being set st y in PFBrt (n,k) & y c= x;
theorem :: HEYTING3:22
for n, k being Element of NAT holds not PFCrt (n,k) in PFBrt (n+1,k);
theorem :: HEYTING3:23
for n, m, k being Element of NAT st PFArt (n,m) c= PFArt (k,m) holds n = k;
theorem :: HEYTING3:24
for n, m, k being Element of NAT holds PFCrt (n,m) c= PFArt (k,m ) iff n < k;
begin :: Incompleteness
theorem :: HEYTING3:25
for n, k being Element of NAT holds PFBrt (n,k) is Element of
SubstPoset (NAT, {k});
definition
let k be Element of NAT;
func PFDrt k -> Subset of SubstPoset (NAT, {k}) means
:: HEYTING3:def 5
for x being object
holds x in it iff ex n being non zero Element of NAT st x = PFBrt (n,k);
end;
theorem :: HEYTING3:26
for k being Element of NAT holds PFBrt (1,k) = { PFArt (1,k), PFCrt (1 ,k) };
theorem :: HEYTING3:27
for k being Element of NAT holds PFBrt (1,k) <> {{}};
registration
let k be Element of NAT;
cluster PFBrt (1,k) -> non empty;
end;
theorem :: HEYTING3:28
for n, k being Element of NAT holds { PFArt (n,k) } is Element
of SubstPoset (NAT, {k});
theorem :: HEYTING3:29
for k being Element of NAT, V, X being set, a being Element of
SubstPoset (V, {k}) st X in a holds X is finite Subset of [:V, {k}:];
theorem :: HEYTING3:30
for m being Element of NAT, a being Element of SubstPoset (NAT,
{m}) st PFDrt m is_>=_than a holds for X being non empty set st X in a holds
not ( for n being Element of NAT st [n,m] in X holds n is odd );
theorem :: HEYTING3:31
for k being Element of NAT, a, b being Element of SubstPoset (
NAT, {k}), X being Subset of SubstPoset (NAT, {k}) st a is_<=_than X & b
is_<=_than X holds a "\/" b is_<=_than X;
registration
let k be Element of NAT;
cluster non empty for Element of SubstPoset (NAT, {k});
end;
theorem :: HEYTING3:32
for n being Element of NAT, a being Element of SubstPoset (NAT,
{n}) st {} in a holds a = {{}};
theorem :: HEYTING3:33
for k being Element of NAT, a being non empty Element of
SubstPoset (NAT, {k}) st a <> {{}} ex f being finite Function st f in a & f <>
{};
theorem :: HEYTING3:34
for k being Element of NAT, a being non empty Element of
SubstPoset (NAT, {k}), a9 being Element of Fin PFuncs (NAT, {k}) st a <> {{}} &
a = a9 holds Involved a9 is finite non empty Subset of NAT;
theorem :: HEYTING3:35
for k being Element of NAT, a being Element of SubstPoset (NAT,
{k}), a9 being Element of Fin PFuncs (NAT, {k}), B being finite non empty
Subset of NAT st B = Involved a9 & a9 = a holds for X being set st X in a for l
being Element of NAT st l > (max B) + 1 holds not [l,k] in X;
theorem :: HEYTING3:36
for k being Element of NAT holds Top SubstPoset (NAT, {k}) = {{} };
theorem :: HEYTING3:37
for k being Element of NAT holds Bottom SubstPoset (NAT, {k}) = {};
theorem :: HEYTING3:38
for k being Element of NAT, a, b being Element of SubstPoset (
NAT, {k}) st a <= b & a = {{}} holds b = {{}};
theorem :: HEYTING3:39
for k being Element of NAT, a, b being Element of SubstPoset (
NAT, {k}) st a <= b & b = {} holds a = {};
theorem :: HEYTING3:40
for m being Element of NAT, a being Element of SubstPoset (NAT,
{m}) st PFDrt m is_>=_than a holds a <> {{}};
registration
let m be Element of NAT;
cluster SubstPoset (NAT, {m}) -> non complete;
end;
registration
let m be Element of NAT;
cluster SubstLatt (NAT, {m}) -> non complete;
end;