:: Manysorted Sets
:: by Andrzej Trybulec
::
:: Received July 7, 1993
:: Copyright (c) 1993-2021 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 FUNCT_1, RELAT_1, SETFAM_1, XBOOLE_0, SUBSET_1, PARTFUN1,
FUNCOP_1, TARSKI, FUNCT_2, ZFMISC_1, MEMBER_1, FUNCT_4, PBOOLE, NAT_1,
SETLIM_2;
notations TARSKI, ZFMISC_1, XBOOLE_0, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1,
ORDINAL1, PARTFUN1, FUNCT_2, FUNCT_4, FUNCOP_1;
constructors SETFAM_1, FUNCOP_1, ORDINAL1, RELSET_1, PARTFUN1, FUNCT_4;
registrations XBOOLE_0, FUNCT_1, ORDINAL1, RELSET_1, FUNCOP_1, FUNCT_4,
RELAT_1;
requirements BOOLE, SUBSET;
begin
reserve i,j,e,u for object;
theorem :: PBOOLE:1
for f being Function st f is non-empty holds rng f is
with_non-empty_elements;
theorem :: PBOOLE:2
for f being Function holds f is empty-yielding iff f = {} or rng f = {
{} };
reserve I for set; :: of indices
registration
let I;
cluster total for I-defined Function;
end;
definition
let I;
mode ManySortedSet of I is total I-defined Function;
end;
reserve x,X,Y,Z,V for ManySortedSet of I;
scheme :: PBOOLE:sch 1
KuratowskiFunction{A()-> set, F(object) -> set}:
ex f being ManySortedSet of A() st
for e being object st e in A() holds f.e in F(e)
provided
for e being object st e in A() holds F(e) <> {};
definition
let I,X,Y;
pred X in Y means
:: PBOOLE:def 1
for i being object st i in I holds X.i in Y.i;
pred X c= Y means
:: PBOOLE:def 2
for i being object st i in I holds X.i c= Y.i;
reflexivity;
end;
definition
let I be non empty set,X,Y be ManySortedSet of I;
redefine pred X in Y;
asymmetry;
end;
scheme :: PBOOLE:sch 2
PSeparation { I()-> set, A() -> ManySortedSet of I(), P[object,object] } :
ex X being ManySortedSet of I() st
for i being object st i in I()
for e being object holds e in X.i iff e in A().i & P[i,e];
theorem :: PBOOLE:3
(for i being object st i in I holds X.i = Y.i) implies X = Y;
definition
let I;
func EmptyMS I -> ManySortedSet of I equals
:: PBOOLE:def 3
I --> {};
let X,Y;
func X (\/) Y -> ManySortedSet of I means
:: PBOOLE:def 4
for i being object st i in I holds it.i = X.i \/ Y.i;
commutativity;
idempotence;
func X (/\) Y -> ManySortedSet of I means
:: PBOOLE:def 5
for i being object st i in I holds it.i = X.i /\ Y.i;
commutativity;
idempotence;
func X (\) Y -> ManySortedSet of I means
:: PBOOLE:def 6
for i being object st i in I holds it.i = X.i \ Y.i;
pred X overlaps Y means
:: PBOOLE:def 7
for i being object st i in I holds X.i meets Y.i;
symmetry;
pred X misses Y means
:: PBOOLE:def 8
for i being object st i in I holds X.i misses Y.i;
symmetry;
end;
notation
let I;
let X,Y;
antonym X meets Y for X misses Y;
end;
definition
let I,X,Y;
func X (\+\) Y -> ManySortedSet of I equals
:: PBOOLE:def 9
(X (\) Y) (\/) (Y (\) X);
commutativity;
end;
theorem :: PBOOLE:4
for i st i in I holds (X (\+\) Y).i = X.i \+\ Y.i;
theorem :: PBOOLE:5
for i being object holds EmptyMS I.i = {};
theorem :: PBOOLE:6
(for i being object st i in I holds X.i = {}) implies X = EmptyMS I;
theorem :: PBOOLE:7
x in X or x in Y implies x in X (\/) Y;
theorem :: PBOOLE:8
x in X (/\) Y iff x in X & x in Y;
theorem :: PBOOLE:9
x in X & X c= Y implies x in Y;
theorem :: PBOOLE:10
x in X & x in Y implies X overlaps Y;
theorem :: PBOOLE:11
X overlaps Y implies ex x st x in X & x in Y;
theorem :: PBOOLE:12
x in X (\) Y implies x in X;
begin
definition
let I,X,Y;
redefine pred X = Y means
:: PBOOLE:def 10
for i being object st i in I holds X.i = Y.i;
end;
theorem :: PBOOLE:13
X c= Y & Y c= Z implies X c= Z;
theorem :: PBOOLE:14
X c= X (\/) Y;
theorem :: PBOOLE:15
X (/\) Y c= X;
theorem :: PBOOLE:16
X c= Z & Y c= Z implies X (\/) Y c= Z;
theorem :: PBOOLE:17
Z c= X & Z c= Y implies Z c= X (/\) Y;
theorem :: PBOOLE:18
X c= Y implies X (\/) Z c= Y (\/) Z;
theorem :: PBOOLE:19
X c= Y implies X (/\) Z c= Y (/\) Z;
theorem :: PBOOLE:20
X c= Y & Z c= V implies X (\/) Z c= Y (\/) V;
theorem :: PBOOLE:21
X c= Y & Z c= V implies X (/\) Z c= Y (/\) V;
theorem :: PBOOLE:22
X c= Y implies X (\/) Y = Y;
theorem :: PBOOLE:23
X c= Y implies X (/\) Y = X;
theorem :: PBOOLE:24
X (/\) Y c= X (\/) Z;
theorem :: PBOOLE:25
X c= Z implies X (\/) Y (/\) Z = (X (\/) Y) (/\) Z;
theorem :: PBOOLE:26
X = Y (\/) Z iff Y c= X & Z c= X & for V st Y c= V & Z c= V holds X c= V;
theorem :: PBOOLE:27
X = Y (/\) Z iff X c= Y & X c= Z & for V st V c= Y & V c= Z holds V c= X;
theorem :: PBOOLE:28
(X (\/) Y) (\/) Z = X (\/) (Y (\/) Z);
theorem :: PBOOLE:29
(X (/\) Y) (/\) Z = X (/\) (Y (/\) Z);
theorem :: PBOOLE:30
X (/\) (X (\/) Y) = X;
theorem :: PBOOLE:31
X (\/) (X (/\) Y) = X;
theorem :: PBOOLE:32
X (/\) (Y (\/) Z) = X (/\) Y (\/) X (/\) Z;
theorem :: PBOOLE:33
X (\/) Y (/\) Z = (X (\/) Y) (/\) (X (\/) Z);
theorem :: PBOOLE:34
(X (/\) Y) (\/) (X (/\) Z) = X implies X c= Y (\/) Z;
theorem :: PBOOLE:35
(X (\/) Y) (/\) (X (\/) Z) = X implies Y (/\) Z c= X;
theorem :: PBOOLE:36
(X (/\) Y) (\/) (Y (/\) Z) (\/) (Z (/\) X)
= (X (\/) Y) (/\) (Y (\/) Z) (/\) (Z (\/) X);
theorem :: PBOOLE:37
X (\/) Y c= Z implies X c= Z;
theorem :: PBOOLE:38
X c= Y (/\) Z implies X c= Y;
theorem :: PBOOLE:39
X (\/) Y (\/) Z = (X (\/) Z) (\/) (Y (\/) Z);
theorem :: PBOOLE:40
X (/\) Y (/\) Z = (X (/\) Z) (/\) (Y (/\) Z);
theorem :: PBOOLE:41
X (\/) (X (\/) Y) = X (\/) Y;
theorem :: PBOOLE:42
X (/\) (X (/\) Y) = X (/\) Y;
begin :: ManySortedSet with empty components
theorem :: PBOOLE:43
EmptyMS I c= X;
theorem :: PBOOLE:44
X c= EmptyMS I implies X = EmptyMS I;
theorem :: PBOOLE:45
X c= Y & X c= Z & Y (/\) Z = EmptyMS I implies X = EmptyMS I;
theorem :: PBOOLE:46
X c= Y & Y (/\) Z = EmptyMS I implies X (/\) Z = EmptyMS I;
theorem :: PBOOLE:47
X (\/) EmptyMS I = X & EmptyMS I (\/) X = X;
theorem :: PBOOLE:48
X (\/) Y = EmptyMS I implies X = EmptyMS I;
theorem :: PBOOLE:49
X (/\) EmptyMS I = EmptyMS I;
theorem :: PBOOLE:50
X c= (Y (\/) Z) & X (/\) Z = EmptyMS I implies X c= Y;
theorem :: PBOOLE:51
Y c= X & X (/\) Y = EmptyMS I implies Y = EmptyMS I;
begin :: Difference and symmetric difference
theorem :: PBOOLE:52
X (\) Y = EmptyMS I iff X c= Y;
theorem :: PBOOLE:53
X c= Y implies X (\) Z c= Y (\) Z;
theorem :: PBOOLE:54
X c= Y implies Z (\) Y c= Z (\) X;
theorem :: PBOOLE:55
X c= Y & Z c= V implies X (\) V c= Y (\) Z;
theorem :: PBOOLE:56
X (\) Y c= X;
theorem :: PBOOLE:57
X c= Y (\) X implies X = EmptyMS I;
theorem :: PBOOLE:58
X (\) X = EmptyMS I;
theorem :: PBOOLE:59
X (\) EmptyMS I = X;
theorem :: PBOOLE:60
EmptyMS I (\) X = EmptyMS I;
theorem :: PBOOLE:61
X (\) (X (\/) Y) = EmptyMS I;
theorem :: PBOOLE:62
X (/\) (Y (\) Z) = (X (/\) Y) (\) Z;
theorem :: PBOOLE:63
(X (\) Y) (/\) Y = EmptyMS I;
theorem :: PBOOLE:64
X (\) (Y (\) Z) = (X (\) Y) (\/) X (/\) Z;
theorem :: PBOOLE:65
(X (\) Y) (\/) X (/\) Y = X;
theorem :: PBOOLE:66
X c= Y implies Y = X (\/) (Y (\) X);
theorem :: PBOOLE:67
X (\/) (Y (\) X) = X (\/) Y;
theorem :: PBOOLE:68
X (\) (X (\) Y) = X (/\) Y;
theorem :: PBOOLE:69
X (\) (Y (/\) Z) = (X (\) Y) (\/) (X (\) Z);
theorem :: PBOOLE:70
X (\) X (/\) Y = X (\) Y;
theorem :: PBOOLE:71
X (/\) Y = EmptyMS I iff X (\) Y = X;
theorem :: PBOOLE:72
(X (\/) Y) (\) Z = (X (\) Z) (\/) (Y (\) Z);
theorem :: PBOOLE:73
(X (\) Y) (\) Z = X (\) (Y (\/) Z);
theorem :: PBOOLE:74
(X (/\) Y) (\) Z = (X (\) Z) (/\) (Y (\) Z);
theorem :: PBOOLE:75
(X (\/) Y) (\) Y = X (\) Y;
theorem :: PBOOLE:76
X c= Y (\/) Z implies X (\) Y c= Z & X (\) Z c= Y;
theorem :: PBOOLE:77
(X (\/) Y) (\) (X (/\) Y) = (X (\) Y) (\/) (Y (\) X);
theorem :: PBOOLE:78
(X (\) Y) (\) Y = X (\) Y;
theorem :: PBOOLE:79
X (\) (Y (\/) Z) = (X (\) Y) (/\) (X (\) Z);
theorem :: PBOOLE:80
X (\) Y = Y (\) X implies X = Y;
theorem :: PBOOLE:81
X (/\) (Y (\) Z) = X (/\) Y (\) X (/\) Z;
theorem :: PBOOLE:82
X (\) Y c= Z implies X c= Y (\/) Z;
theorem :: PBOOLE:83
X (\) Y c= X (\+\) Y;
theorem :: PBOOLE:84
X (\+\) EmptyMS I = X;
theorem :: PBOOLE:85
X (\+\) X = EmptyMS I;
theorem :: PBOOLE:86
X (\/) Y = (X (\+\) Y) (\/) X (/\) Y;
theorem :: PBOOLE:87
X (\+\) Y = (X (\/) Y) (\) X (/\) Y;
theorem :: PBOOLE:88
(X (\+\) Y) (\) Z = (X (\) (Y (\/) Z)) (\/) (Y (\) (X (\/) Z));
theorem :: PBOOLE:89
X (\) (Y (\+\) Z) = X (\) (Y (\/) Z) (\/) X (/\) Y (/\) Z;
theorem :: PBOOLE:90
(X (\+\) Y) (\+\) Z = X (\+\) (Y (\+\) Z);
theorem :: PBOOLE:91
X (\) Y c= Z & Y (\) X c= Z implies X (\+\) Y c= Z;
theorem :: PBOOLE:92
X (\/) Y = X (\+\) (Y (\) X);
theorem :: PBOOLE:93
X (/\) Y = X (\+\) (X (\) Y);
theorem :: PBOOLE:94
X (\) Y = X (\+\) (X (/\) Y);
theorem :: PBOOLE:95
Y (\) X = X (\+\) (X (\/) Y);
theorem :: PBOOLE:96
X (\/) Y = X (\+\) Y (\+\) X (/\) Y;
theorem :: PBOOLE:97
X (/\) Y = X (\+\) Y (\+\) (X (\/) Y);
begin :: Meeting and overlap(p?)ing
theorem :: PBOOLE:98
X overlaps Y or X overlaps Z implies X overlaps Y (\/) Z;
theorem :: PBOOLE:99
X overlaps Y & Y c= Z implies X overlaps Z;
theorem :: PBOOLE:100
X overlaps Y & X c= Z implies Z overlaps Y;
theorem :: PBOOLE:101
X c= Y & Z c= V & X overlaps Z implies Y overlaps V;
theorem :: PBOOLE:102
X overlaps Y (/\) Z implies X overlaps Y & X overlaps Z;
theorem :: PBOOLE:103
X overlaps Z & X c= V implies X overlaps Z (/\) V;
theorem :: PBOOLE:104
X overlaps Y (\) Z implies X overlaps Y;
theorem :: PBOOLE:105
Y does not overlap Z implies X (/\) Y does not overlap X (/\) Z;
theorem :: PBOOLE:106
X overlaps Y (\) Z implies Y overlaps X (\) Z;
theorem :: PBOOLE:107
X meets Y & Y c= Z implies X meets Z;
theorem :: PBOOLE:108
Y misses X (\) Y;
theorem :: PBOOLE:109
X (/\) Y misses X (\) Y;
theorem :: PBOOLE:110
X (/\) Y misses X (\+\) Y;
theorem :: PBOOLE:111
X misses Y implies X (/\) Y = EmptyMS I;
theorem :: PBOOLE:112
X <> EmptyMS I implies X meets X;
theorem :: PBOOLE:113
X c= Y & X c= Z & Y misses Z implies X = EmptyMS I;
theorem :: PBOOLE:114
Z (\/) V = X (\/) Y & X misses Z & Y misses V implies X = V & Y = Z;
theorem :: PBOOLE:115
X misses Y implies X (\) Y = X;
theorem :: PBOOLE:116
X misses Y implies (X (\/) Y) (\) Y = X;
theorem :: PBOOLE:117
X (\) Y = X implies X misses Y;
theorem :: PBOOLE:118
X (\) Y misses Y (\) X;
begin :: Roughly speaking
definition
let I,X,Y;
pred X [= Y means
:: PBOOLE:def 11
for x st x in X holds x in Y;
reflexivity;
end;
theorem :: PBOOLE:119
X c= Y implies X [= Y;
theorem :: PBOOLE:120
X [= Y & Y [= Z implies X [= Z;
begin :: Non empty set of sorts
theorem :: PBOOLE:121
EmptyMS {} in EmptyMS {};
theorem :: PBOOLE:122
for X being ManySortedSet of {} holds X = {};
reserve I for non empty set,
x,X,Y for ManySortedSet of I;
theorem :: PBOOLE:123
X overlaps Y implies X meets Y;
theorem :: PBOOLE:124
not ex x st x in EmptyMS I;
theorem :: PBOOLE:125
x in X & x in Y implies X (/\) Y <> EmptyMS I;
theorem :: PBOOLE:126
X does not overlap EmptyMS I;
theorem :: PBOOLE:127
X (/\) Y = EmptyMS I implies X does not overlap Y;
theorem :: PBOOLE:128
X overlaps X implies X <> EmptyMS I;
begin :: Empty and non-empty ManySortedSets
reserve I for set,
x,X,Y,Z for ManySortedSet of I;
definition
let I be set;
let X be ManySortedSet of I;
redefine attr X is empty-yielding means
:: PBOOLE:def 12
for i being object st i in I holds X.i is empty;
redefine attr X is non-empty means
:: PBOOLE:def 13
for i being object st i in I holds X.i is non empty;
end;
registration
let I be set;
cluster empty-yielding for ManySortedSet of I;
cluster non-empty for ManySortedSet of I;
end;
registration
let I be non empty set;
cluster non-empty -> non empty-yielding for ManySortedSet of I;
cluster empty-yielding -> non non-empty for ManySortedSet of I;
end;
theorem :: PBOOLE:129
X is empty-yielding iff X = EmptyMS I;
theorem :: PBOOLE:130
Y is empty-yielding & X c= Y implies X is empty-yielding;
theorem :: PBOOLE:131
X is non-empty & X c= Y implies Y is non-empty;
theorem :: PBOOLE:132
X is non-empty & X [= Y implies X c= Y;
theorem :: PBOOLE:133
X is non-empty & X [= Y implies Y is non-empty;
reserve X for non-empty ManySortedSet of I;
theorem :: PBOOLE:134
ex x st x in X;
theorem :: PBOOLE:135
(for x holds x in X iff x in Y) implies X = Y;
theorem :: PBOOLE:136
(for x holds x in X iff x in Y & x in Z) implies X = Y (/\) Z;
begin :: Addenda
scheme :: PBOOLE:sch 3
MSSEx { I() -> set, P[object,object] }:
ex f being ManySortedSet of I() st
for i being object st i in I() holds P[i,f.i]
provided
for i being object st i in I() ex j being object st P[i,j];
scheme :: PBOOLE:sch 4
MSSLambda { I() -> set, F(object) -> object }:
ex f being ManySortedSet of I() st
for i being object st i in I() holds f.i = F(i);
registration
let I be set;
cluster Function-yielding for ManySortedSet of I;
end;
definition
let I be set;
mode ManySortedFunction of I is Function-yielding ManySortedSet of I;
end;
theorem :: PBOOLE:137
not ex M being non-empty ManySortedSet of I st {} in rng M;
definition
let M be Function;
mode Component of M is Element of rng M;
end;
theorem :: PBOOLE:138
for I being non empty set for M being ManySortedSet of I, A being
Component of M ex i being object st i in I & A = M.i;
theorem :: PBOOLE:139
for M being ManySortedSet of I, i st i in I holds M.i is Component of M;
definition
let I;
let B be ManySortedSet of I;
mode Element of B -> ManySortedSet of I means
:: PBOOLE:def 14
for i being object st i in I holds it.i is Element of B.i;
end;
begin :: Many Sorted Functions
definition
let I;
let A be ManySortedSet of I, B be ManySortedSet of I;
mode ManySortedFunction of A,B -> ManySortedSet of I means
:: PBOOLE:def 15
for i st i in I holds it.i is Function of A.i, B.i;
end;
registration
let I;
let A be ManySortedSet of I, B be ManySortedSet of I;
cluster ->Function-yielding for ManySortedFunction of A,B;
end;
registration
let I be set;
let J be non empty set;
let O be Function of I,J;
let F be ManySortedSet of J;
cluster F*O -> total for I-defined Function;
end;
reserve D for non empty set,
n for Nat;
scheme :: PBOOLE:sch 5
LambdaDMS {D()->non empty set, F(object)->object}:
ex X be ManySortedSet of D() st
for d be Element of D() holds X.d = F(d);
registration
let J be non empty set, B be non-empty ManySortedSet of J, j be Element of J;
cluster B.j -> non empty;
end;
reserve X,Y for ManySortedSet of I;
definition
let I, X,Y;
func [|X,Y|] -> ManySortedSet of I means
:: PBOOLE:def 16
for i being object st i in I holds it.i = [:X.i,Y.i:];
end;
definition
let I, X, Y;
func (Funcs) (X,Y) -> ManySortedSet of I means
:: PBOOLE:def 17
for i being object st i in I holds it.i = Funcs(X.i,Y.i);
end;
definition
let I be set, M be ManySortedSet of I;
mode ManySortedSubset of M -> ManySortedSet of I means
:: PBOOLE:def 18
it c= M;
end;
registration
let I be set, M be non-empty ManySortedSet of I;
cluster non-empty for ManySortedSubset of M;
end;
definition
let F,G be Function-yielding Function;
func G**F -> Function means
:: PBOOLE:def 19
dom it = (dom F) /\ (dom G) &
for i be object st i in dom it holds it.i = (G.i)*(F.i);
end;
registration
let F,G be Function-yielding Function;
cluster G**F -> Function-yielding;
end;
definition
let I be set, A be ManySortedSet of I, F be ManySortedFunction of I;
func F.:.:A -> ManySortedSet of I means
:: PBOOLE:def 20
for i be set st i in I holds it.i = (F.i).:(A.i);
end;
registration
let I;
cluster EmptyMS I -> empty-yielding;
end;
scheme :: PBOOLE:sch 6
MSSExD { I() -> non empty set, P[object,object] }:
ex f being ManySortedSet of I()
st for i being Element of I() holds P[i,f.i]
provided
for i being Element of I() ex j being object st P[i,j];
registration
let A be non empty set;
cluster non-empty -> non empty-yielding for ManySortedSet of A;
end;
registration
let X be non empty set;
cluster -> non empty for ManySortedSet of X;
end;
theorem :: PBOOLE:140
for F, G, H be Function-yielding Function holds
(H ** G) ** F = H ** (G ** F);
registration
let I be set, f be non-empty ManySortedSet of I;
cluster total for I-defined f-compatible Function;
end;
theorem :: PBOOLE:141
for I being set, f being non-empty ManySortedSet of I
for p being f-compatible I-defined Function
ex s being f-compatible ManySortedSet of I st p c= s;
theorem :: PBOOLE:142
for I,A be set
for s,ss being ManySortedSet of I
holds (ss +* s | A) | A = s | A;
registration
let X be non empty set, Y be set;
cluster X-valued for ManySortedSet of Y;
end;
theorem :: PBOOLE:143
for I,Y being non empty set, M be Y-valued ManySortedSet of I,
x be Element of I
holds M.x = M/.x;
theorem :: PBOOLE:144
for f being Function, M being ManySortedSet of I
holds (f+*M)|I = M;
theorem :: PBOOLE:145
for I being set, Y being non empty set
for p being Y-valued I-defined Function
ex s being Y-valued ManySortedSet of I st p c= s;
theorem :: PBOOLE:146
X c= Y & Y c= X implies X = Y;
definition let I be non empty set, A,B be ManySortedSet of I;
redefine pred A = B means
:: PBOOLE:def 21
for i being Element of I holds A.i = B.i;
end;
registration
let X be with_non-empty_elements set;
cluster id X -> non-empty;
end;
scheme :: PBOOLE:sch 7
MSSLambda { I() -> set, F(object) -> object }:
ex f being ManySortedSet of I() st
for i being set st i in I() holds f.i = F(i);