KNASTER: Fix-points in complete lattices

:: Fix-points in complete lattices
:: by Piotr Rudnicki and Andrzej Trybulec
::
:: Received September 16, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users

theorem :: KNASTER:1

for f, g, h being Function st h = f \/ g & dom f misses dom g holds
( h is one-to-one iff ( f is one-to-one & g is one-to-one & rng f misses rng g ) )

proof end;

theorem Th2: :: KNASTER:2

for x, X being set
for n being Element of NAT
for f being Function of X,X st x is_a_fixpoint_of iter (f,n) holds
f . x is_a_fixpoint_of iter (f,n)

proof end;

theorem :: KNASTER:3

for x, X being set
for f being Function of X,X st ex n being Element of NAT st
( x is_a_fixpoint_of iter (f,n) & ( for y being set st y is_a_fixpoint_of iter (f,n) holds
x = y ) ) holds
x is_a_fixpoint_of f

proof end;

definition

let A, B be non empty set ;
let f be Function of A,B;

:: original: c=-monotone
redefine attr f is c=-monotone means :Def1: :: KNASTER:def 1
for x, y being Element of A st x c= y holds
f . x c= f . y;

compatibility

proof end;

end;

:: deftheorem Def1 defines c=-monotone KNASTER:def 1 :

registration

let A be set ;
let B be non empty set ;

cluster Relation-like A -defined B -valued Function-like quasi_total c=-monotone for Element of bool [:A,B:];

existence

proof end;

end;

definition

let X be set ;
let f be V224() Function of (bool X),(bool X);

func lfp (X,f) -> Subset of X equals :: KNASTER:def 2
meet { h where h is Subset of X : f . h c= h } ;

coherence

proof end;

func gfp (X,f) -> Subset of X equals :: KNASTER:def 3
union { h where h is Subset of X : h c= f . h } ;

coherence

proof end;

end;

:: deftheorem defines lfp KNASTER:def 2 :

:: deftheorem defines gfp KNASTER:def 3 :

theorem Th4: :: KNASTER:4

for X being set
for f being V224() Function of (bool X),(bool X) holds lfp (X,f) is_a_fixpoint_of f

proof end;

theorem Th5: :: KNASTER:5

for X being set
for f being V224() Function of (bool X),(bool X) holds gfp (X,f) is_a_fixpoint_of f

proof end;

theorem Th6: :: KNASTER:6

for X being set
for f being V224() Function of (bool X),(bool X)
for S being Subset of X st f . S c= S holds
lfp (X,f) c= S

proof end;

theorem Th7: :: KNASTER:7

for X being set
for f being V224() Function of (bool X),(bool X)
for S being Subset of X st S c= f . S holds
S c= gfp (X,f)

proof end;

theorem Th8: :: KNASTER:8

for X being set
for f being V224() Function of (bool X),(bool X)
for S being Subset of X st S is_a_fixpoint_of f holds
( lfp (X,f) c= S & S c= gfp (X,f) ) by Th6, Th7;

Knaster Theorem

scheme :: KNASTER:sch 1
Knaster{ F₁() -> set , F₂( object ) -> set } :

ex D being set st
( F₂(D) = D & D c= F₁() )

provided

A1: for X, Y being set st X c= Y holds
F₂(X) c= F₂(Y) and
A2: F₂(F₁()) c= F₁()

proof end;

:: Banach decomposition

theorem Th9: :: KNASTER:9

for X, Y being non empty set
for f being Function of X,Y
for g being Function of Y,X ex Xa, Xb, Ya, Yb being set st
( Xa misses Xb & Ya misses Yb & Xa \/ Xb = X & Ya \/ Yb = Y & f .: Xa = Ya & g .: Yb = Xb )

proof end;

Schroeder Bernstein theorem

theorem Th10: :: KNASTER:10

for X, Y being non empty set
for f being Function of X,Y
for g being Function of Y,X st f is one-to-one & g is one-to-one holds
ex h being Function of X,Y st h is bijective

proof end;

theorem Th11: :: KNASTER:11

for X, Y being non empty set
for f being Function of X,Y st f is bijective holds
X,Y are_equipotent

proof end;

theorem :: KNASTER:12

for X, Y being non empty set
for f being Function of X,Y
for g being Function of Y,X st f is one-to-one & g is one-to-one holds
X,Y are_equipotent

proof end;

definition

let L be Lattice;
let f be Function of the carrier of L, the carrier of L;
let x be Element of L;
let O be Ordinal;

func (f,O) +. x -> set means :Def4: :: KNASTER:def 4
ex L0 being Sequence st
( it = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = f . (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = "\/" ((rng (L0 | C)),L) ) );

correctness

proof end;

func (f,O) -. x -> set means :Def5: :: KNASTER:def 5
ex L0 being Sequence st
( it = last L0 & dom L0 = succ O & L0 . 0 = x & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = f . (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = "/\" ((rng (L0 | C)),L) ) );

correctness

proof end;

end;

:: deftheorem Def4 defines +. KNASTER:def 4 :

:: deftheorem Def5 defines -. KNASTER:def 5 :

theorem Th13: :: KNASTER:13

for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L holds (f,0) +. x = x

proof end;

theorem Th14: :: KNASTER:14

for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L holds (f,0) -. x = x

proof end;

theorem Th15: :: KNASTER:15

for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L
for O being Ordinal holds (f,(succ O)) +. x = f . ((f,O) +. x)

proof end;

theorem Th16: :: KNASTER:16

for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L
for O being Ordinal holds (f,(succ O)) -. x = f . ((f,O) -. x)

proof end;

theorem Th17: :: KNASTER:17

for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L
for O being Ordinal
for T being Sequence st O <> 0 & O is limit_ordinal & dom T = O & ( for A being Ordinal st A in O holds
T . A = (f,A) +. x ) holds
(f,O) +. x = "\/" ((rng T),L)

proof end;

theorem Th18: :: KNASTER:18

proof end;

theorem :: KNASTER:19

for n being Element of NAT
for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L holds (iter (f,n)) . x = (f,n) +. x

proof end;

theorem :: KNASTER:20

for n being Element of NAT
for L being Lattice
for f being Function of the carrier of L, the carrier of L
for x being Element of L holds (iter (f,n)) . x = (f,n) -. x

proof end;

definition

let L be Lattice;
let f be UnOp of the carrier of L;
let a be Element of L;
let O be Ordinal;
:: original: +.
redefine func (f,O) +. a -> Element of L;
coherence

proof end;

end;

definition

let L be Lattice;
let f be UnOp of the carrier of L;
let a be Element of L;
let O be Ordinal;
:: original: -.
redefine func (f,O) -. a -> Element of L;
coherence

proof end;

end;

definition

let L be non empty LattStr ;
let P be Subset of L;

attr P is with_suprema means :Def6: :: KNASTER:def 6
for x, y being Element of L st x in P & y in P holds
ex z being Element of L st
( z in P & x [= z & y [= z & ( for z9 being Element of L st z9 in P & x [= z9 & y [= z9 holds
z [= z9 ) );

attr P is with_infima means :Def7: :: KNASTER:def 7
for x, y being Element of L st x in P & y in P holds
ex z being Element of L st
( z in P & z [= x & z [= y & ( for z9 being Element of L st z9 in P & z9 [= x & z9 [= y holds
z9 [= z ) );

end;

:: deftheorem Def6 defines with_suprema KNASTER:def 6 :

:: deftheorem Def7 defines with_infima KNASTER:def 7 :

registration

let L be Lattice;

cluster non empty with_suprema with_infima for Element of bool the carrier of L;

existence

proof end;

end;

definition

let L be Lattice;
let P be non empty with_suprema with_infima Subset of L;

func latt P -> strict Lattice means :Def8: :: KNASTER:def 8
( the carrier of it = P & ( for x, y being Element of it ex x9, y9 being Element of L st
( x = x9 & y = y9 & ( x [= y implies x9 [= y9 ) & ( x9 [= y9 implies x [= y ) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines latt KNASTER:def 8 :

registration

cluster non empty Lattice-like complete -> bounded for LattStr ;

coherence

proof end;

end;

theorem Th21: :: KNASTER:21

for L being complete Lattice
for f being monotone UnOp of L ex a being Element of L st a is_a_fixpoint_of f

proof end;

theorem Th22: :: KNASTER:22

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a [= f . a holds
for O being Ordinal holds a [= (f,O) +. a

proof end;

theorem Th23: :: KNASTER:23

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st f . a [= a holds
for O being Ordinal holds (f,O) -. a [= a

proof end;

theorem Th24: :: KNASTER:24

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a [= f . a holds
for O1, O2 being Ordinal st O1 c= O2 holds
(f,O1) +. a [= (f,O2) +. a

proof end;

theorem Th25: :: KNASTER:25

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st f . a [= a holds
for O1, O2 being Ordinal st O1 c= O2 holds
(f,O2) -. a [= (f,O1) -. a

proof end;

theorem Th26: :: KNASTER:26

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a [= f . a holds
for O1, O2 being Ordinal st O1 c< O2 & not (f,O2) +. a is_a_fixpoint_of f holds
(f,O1) +. a <> (f,O2) +. a

proof end;

theorem Th27: :: KNASTER:27

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st f . a [= a holds
for O1, O2 being Ordinal st O1 c< O2 & not (f,O2) -. a is_a_fixpoint_of f holds
(f,O1) -. a <> (f,O2) -. a

proof end;

theorem Th28: :: KNASTER:28

for O1 being Ordinal
for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a [= f . a & (f,O1) +. a is_a_fixpoint_of f holds
for O2 being Ordinal st O1 c= O2 holds
(f,O1) +. a = (f,O2) +. a

proof end;

theorem Th29: :: KNASTER:29

for O1 being Ordinal
for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st f . a [= a & (f,O1) -. a is_a_fixpoint_of f holds
for O2 being Ordinal st O1 c= O2 holds
(f,O1) -. a = (f,O2) -. a

proof end;

Lm1: for O1, O2 being Ordinal holds
( O1 c< O2 or O1 = O2 or O2 c< O1 )

proof end;

theorem Th30: :: KNASTER:30

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a [= f . a holds
ex O being Ordinal st
( card O c= card the carrier of L & (f,O) +. a is_a_fixpoint_of f )

proof end;

theorem Th31: :: KNASTER:31

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st f . a [= a holds
ex O being Ordinal st
( card O c= card the carrier of L & (f,O) -. a is_a_fixpoint_of f )

proof end;

theorem Th32: :: KNASTER:32

for L being complete Lattice
for f being monotone UnOp of L
for a, b being Element of L st a is_a_fixpoint_of f & b is_a_fixpoint_of f holds
ex O being Ordinal st
( card O c= card the carrier of L & (f,O) +. (a "\/" b) is_a_fixpoint_of f & a [= (f,O) +. (a "\/" b) & b [= (f,O) +. (a "\/" b) )

proof end;

theorem Th33: :: KNASTER:33

for L being complete Lattice
for f being monotone UnOp of L
for a, b being Element of L st a is_a_fixpoint_of f & b is_a_fixpoint_of f holds
ex O being Ordinal st
( card O c= card the carrier of L & (f,O) -. (a "/\" b) is_a_fixpoint_of f & (f,O) -. (a "/\" b) [= a & (f,O) -. (a "/\" b) [= b )

proof end;

theorem Th34: :: KNASTER:34

for L being complete Lattice
for f being monotone UnOp of L
for a, b being Element of L st a [= b & b is_a_fixpoint_of f holds
for O2 being Ordinal holds (f,O2) +. a [= b

proof end;

theorem Th35: :: KNASTER:35

for L being complete Lattice
for f being monotone UnOp of L
for a, b being Element of L st b [= a & b is_a_fixpoint_of f holds
for O2 being Ordinal holds b [= (f,O2) -. a

proof end;

definition

let L be complete Lattice;
let f be UnOp of L;
assume A1: f is monotone ;

func FixPoints f -> strict Lattice means :Def9: :: KNASTER:def 9
ex P being non empty with_suprema with_infima Subset of L st
( P = { x where x is Element of L : x is_a_fixpoint_of f } & it = latt P );

existence

proof end;

uniqueness ;

end;

:: deftheorem Def9 defines FixPoints KNASTER:def 9 :

theorem Th36: :: KNASTER:36

for L being complete Lattice
for f being monotone UnOp of L holds the carrier of (FixPoints f) = { x where x is Element of L : x is_a_fixpoint_of f }

proof end;

theorem Th37: :: KNASTER:37

for L being complete Lattice
for f being monotone UnOp of L holds the carrier of (FixPoints f) c= the carrier of L

proof end;

theorem Th38: :: KNASTER:38

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L holds
( a in the carrier of (FixPoints f) iff a is_a_fixpoint_of f )

proof end;

theorem Th39: :: KNASTER:39

for L being complete Lattice
for f being monotone UnOp of L
for x, y being Element of (FixPoints f)
for a, b being Element of L st x = a & y = b holds
( x [= y iff a [= b )

proof end;

theorem :: KNASTER:40

for L being complete Lattice
for f being monotone UnOp of L holds FixPoints f is complete

proof end;

definition

let L be complete Lattice;
let f be monotone UnOp of L;

func lfp f -> Element of L equals :: KNASTER:def 10
(f,(nextcard the carrier of L)) +. (Bottom L);

coherence ;

func gfp f -> Element of L equals :: KNASTER:def 11
(f,(nextcard the carrier of L)) -. (Top L);

coherence ;

end;

:: deftheorem defines lfp KNASTER:def 10 :

:: deftheorem defines gfp KNASTER:def 11 :

theorem Th41: :: KNASTER:41

for L being complete Lattice
for f being monotone UnOp of L holds
( lfp f is_a_fixpoint_of f & ex O being Ordinal st
( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f ) )

proof end;

theorem Th42: :: KNASTER:42

for L being complete Lattice
for f being monotone UnOp of L holds
( gfp f is_a_fixpoint_of f & ex O being Ordinal st
( card O c= card the carrier of L & (f,O) -. (Top L) = gfp f ) )

proof end;

theorem Th43: :: KNASTER:43

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a is_a_fixpoint_of f holds
( lfp f [= a & a [= gfp f )

proof end;

theorem :: KNASTER:44

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st f . a [= a holds
lfp f [= a

proof end;

theorem :: KNASTER:45

for L being complete Lattice
for f being monotone UnOp of L
for a being Element of L st a [= f . a holds
a [= gfp f

proof end;

theorem Th46: :: KNASTER:46

for A being non empty set
for f being UnOp of (BooleLatt A) holds
( f is monotone iff f is V224() )

proof end;

theorem :: KNASTER:47

for A being non empty set
for f being monotone UnOp of (BooleLatt A) ex g being V224() Function of (bool A),(bool A) st lfp (A,g) = lfp f

proof end;

theorem :: KNASTER:48

for A being non empty set
for f being monotone UnOp of (BooleLatt A) ex g being V224() Function of (bool A),(bool A) st gfp (A,g) = gfp f

proof end;