:: Brouwer Fixed Point Theorem for Disks on the Plane
:: by Artur Korni{\l}owicz and Yasunari Shidama
::
:: Received February 22, 2005
:: Copyright (c) 2005-2025 Association of Mizar Users


set T2 = TOP-REAL 2;

definition
let S, T be non empty TopSpace;
func DiffElems (S,T) -> Subset of [:S,T:] equals :: BROUWER:def 1
 { [s,t] where s is Point of S, t is Point of T : s <> t } ;
coherence
{ [s,t] where s is Point of S, t is Point of T : s <> t } is Subset of [:S,T:]
proof end;
end;

:: deftheorem defines DiffElems BROUWER:def 1 :
for S, T being non empty TopSpace holds DiffElems (S,T) = { [s,t] where s is Point of S, t is Point of T : s <> t } ;

theorem :: BROUWER:1
for S, T being non empty TopSpace
for x being set holds
( x in DiffElems (S,T) iff ex s being Point of S ex t being Point of T st
( x = [s,t] & s <> t ) ) ;

registration
let S be non trivial TopSpace;
let T be non empty TopSpace;
cluster DiffElems (S,T) -> non empty ;
coherence
not DiffElems (S,T) is empty
proof end;
end;

registration
let S be non empty TopSpace;
let T be non trivial TopSpace;
cluster DiffElems (S,T) -> non empty ;
coherence
not DiffElems (S,T) is empty
proof end;
end;

theorem Th2: :: BROUWER:2
for n being Element of NAT
for x being Point of (TOP-REAL n) holds cl_Ball (x,0) = {x}
proof end;

definition
let n be Nat;
let x be Point of (TOP-REAL n);
let r be Real;
func Tdisk (x,r) -> SubSpace of TOP-REAL n equals :: BROUWER:def 2
(TOP-REAL n) | (cl_Ball (x,r));
coherence
(TOP-REAL n) | (cl_Ball (x,r)) is SubSpace of TOP-REAL n ;
end;

:: deftheorem defines Tdisk BROUWER:def 2 :
for n being Nat
for x being Point of (TOP-REAL n)
for r being Real holds Tdisk (x,r) = (TOP-REAL n) | (cl_Ball (x,r));

registration
let n be Nat;
let x be Point of (TOP-REAL n);
let r be non negative Real;
cluster Tdisk (x,r) -> non empty ;
coherence
not Tdisk (x,r) is empty ;
end;

theorem Th3: :: BROUWER:3
for n being Element of NAT
for r being Real
for x being Point of (TOP-REAL n) holds the carrier of (Tdisk (x,r)) = cl_Ball (x,r)
proof end;

registration
let n be Element of NAT ;
let x be Point of (TOP-REAL n);
let r be Real;
cluster Tdisk (x,r) -> convex ;
coherence
Tdisk (x,r) is convex by Th3;
end;

theorem Th4: :: BROUWER:4
for n being Element of NAT
for r being non negative Real
for s, t, x being Point of (TOP-REAL n) st s <> t & s is Point of (Tdisk (x,r)) & s is not Point of (Tcircle (x,r)) holds
ex e being Point of (Tcircle (x,r)) st {e} = (halfline (s,t)) /\ (Sphere (x,r))
proof end;

theorem Th5: :: BROUWER:5
for n being Element of NAT
for r being non negative Real
for s, t, x being Point of (TOP-REAL n) st s <> t & s in the carrier of (Tcircle (x,r)) & t is Point of (Tdisk (x,r)) holds
ex e being Point of (Tcircle (x,r)) st
( e <> s & {s,e} = (halfline (s,t)) /\ (Sphere (x,r)) )
proof end;

definition
let n be non zero Element of NAT ;
let o be Point of (TOP-REAL n);
let s, t be Point of (TOP-REAL n);
let r be non negative Real;
assume that
A1: s is Point of (Tdisk (o,r)) and
A2: t is Point of (Tdisk (o,r)) and
A3: s <> t ;
func HC (s,t,o,r) -> Point of (TOP-REAL n) means :Def3: :: BROUWER:def 3
( it in (halfline (s,t)) /\ (Sphere (o,r)) & it <> s );
existence
ex b1 being Point of (TOP-REAL n) st
( b1 in (halfline (s,t)) /\ (Sphere (o,r)) & b1 <> s )
proof end;
uniqueness
for b1, b2 being Point of (TOP-REAL n) st b1 in (halfline (s,t)) /\ (Sphere (o,r)) & b1 <> s & b2 in (halfline (s,t)) /\ (Sphere (o,r)) & b2 <> s holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines HC BROUWER:def 3 :
for n being non zero Element of NAT
for o, s, t being Point of (TOP-REAL n)
for r being non negative Real st s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t holds
for b6 being Point of (TOP-REAL n) holds
( b6 = HC (s,t,o,r) iff ( b6 in (halfline (s,t)) /\ (Sphere (o,r)) & b6 <> s ) );

theorem Th6: :: BROUWER:6
for r being non negative Real
for n being non zero Element of NAT
for s, t, o being Point of (TOP-REAL n) st s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t holds
HC (s,t,o,r) is Point of (Tcircle (o,r))
proof end;

theorem :: BROUWER:7
for a being Real
for r being non negative Real
for n being non zero Element of NAT
for s, t, o being Point of (TOP-REAL n)
for S, T, O being Element of REAL n st S = s & T = t & O = o & s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t & a = ((- |((t - s),(s - o))|) + (sqrt ((|((t - s),(s - o))| ^2) - ((Sum (sqr (T - S))) * ((Sum (sqr (S - O))) - (r ^2)))))) / (Sum (sqr (T - S))) holds
HC (s,t,o,r) = ((1 - a) * s) + (a * t)
proof end;

theorem Th8: :: BROUWER:8
for a being Real
for r being non negative Real
for r1, r2, s1, s2 being Real
for s, t, o being Point of (TOP-REAL 2) st s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t & r1 = (t `1) - (s `1) & r2 = (t `2) - (s `2) & s1 = (s `1) - (o `1) & s2 = (s `2) - (o `2) & a = ((- ((s1 * r1) + (s2 * r2))) + (sqrt ((((s1 * r1) + (s2 * r2)) ^2) - (((r1 ^2) + (r2 ^2)) * (((s1 ^2) + (s2 ^2)) - (r ^2)))))) / ((r1 ^2) + (r2 ^2)) holds
HC (s,t,o,r) = |[((s `1) + (a * r1)),((s `2) + (a * r2))]|
proof end;

definition
let n be non zero Element of NAT ;
let o be Point of (TOP-REAL n);
let r be non negative Real;
let x be Point of (Tdisk (o,r));
let f be Function of (Tdisk (o,r)),(Tdisk (o,r));
assume A1: not x is_a_fixpoint_of f ;
func HC (x,f) -> Point of (Tcircle (o,r)) means :Def4: :: BROUWER:def 4
 ex y, z being Point of (TOP-REAL n) st
( y = x & z = f . x & it = HC (z,y,o,r) );
existence
ex b1 being Point of (Tcircle (o,r)) ex y, z being Point of (TOP-REAL n) st
( y = x & z = f . x & b1 = HC (z,y,o,r) )
proof end;
uniqueness
for b1, b2 being Point of (Tcircle (o,r)) st ex y, z being Point of (TOP-REAL n) st
( y = x & z = f . x & b1 = HC (z,y,o,r) ) & ex y, z being Point of (TOP-REAL n) st
( y = x & z = f . x & b2 = HC (z,y,o,r) ) holds
b1 = b2 ;
end;

:: deftheorem Def4 defines HC BROUWER:def 4 :
for n being non zero Element of NAT
for o being Point of (TOP-REAL n)
for r being non negative Real
for x being Point of (Tdisk (o,r))
for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) st not x is_a_fixpoint_of f holds
for b6 being Point of (Tcircle (o,r)) holds
( b6 = HC (x,f) iff ex y, z being Point of (TOP-REAL n) st
( y = x & z = f . x & b6 = HC (z,y,o,r) ) );

theorem Th9: :: BROUWER:9
for r being non negative Real
for n being non zero Element of NAT
for o being Point of (TOP-REAL n)
for x being Point of (Tdisk (o,r))
for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) st not x is_a_fixpoint_of f & x is Point of (Tcircle (o,r)) holds
HC (x,f) = x
proof end;

theorem Th10: :: BROUWER:10
for r being positive Real
for o being Point of (TOP-REAL 2)
for Y being non empty SubSpace of Tdisk (o,r) st Y = Tcircle (o,r) holds
not Y is_a_retract_of Tdisk (o,r)
proof end;

definition
let n be non zero Element of NAT ;
let r be non negative Real;
let o be Point of (TOP-REAL n);
let f be Function of (Tdisk (o,r)),(Tdisk (o,r));
func BR-map f -> Function of (Tdisk (o,r)),(Tcircle (o,r)) means :Def5: :: BROUWER:def 5
 for x being Point of (Tdisk (o,r)) holds it . x = HC (x,f);
existence
ex b1 being Function of (Tdisk (o,r)),(Tcircle (o,r)) st
for x being Point of (Tdisk (o,r)) holds b1 . x = HC (x,f)
proof end;
uniqueness
for b1, b2 being Function of (Tdisk (o,r)),(Tcircle (o,r)) st ( for x being Point of (Tdisk (o,r)) holds b1 . x = HC (x,f) ) & ( for x being Point of (Tdisk (o,r)) holds b2 . x = HC (x,f) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines BR-map BROUWER:def 5 :
for n being non zero Element of NAT
for r being non negative Real
for o being Point of (TOP-REAL n)
for f being Function of (Tdisk (o,r)),(Tdisk (o,r))
for b5 being Function of (Tdisk (o,r)),(Tcircle (o,r)) holds
( b5 = BR-map f iff for x being Point of (Tdisk (o,r)) holds b5 . x = HC (x,f) );

theorem Th11: :: BROUWER:11
for r being non negative Real
for n being non zero Element of NAT
for o being Point of (TOP-REAL n)
for x being Point of (Tdisk (o,r))
for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) st not x is_a_fixpoint_of f & x is Point of (Tcircle (o,r)) holds
(BR-map f) . x = x
proof end;

theorem :: BROUWER:12
for r being non negative Real
for n being non zero Element of NAT
for o being Point of (TOP-REAL n)
for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) st f is without_fixpoints holds
(BR-map f) | (Sphere (o,r)) = id (Tcircle (o,r))
proof end;

Lm1: now :: thesis: for T being non empty TopSpace
for a being Element of REAL holds the carrier of T --> a is continuous
let T be non empty TopSpace; :: thesis: for a being Element of REAL holds the carrier of T --> a is continuous
let a be Element of REAL ; :: thesis: the carrier of T --> a is continuous
set c = the carrier of T;
set f = the carrier of T --> a;
thus the carrier of T --> a is continuous :: thesis: verum
proof
let Y be Subset of REAL; :: according to PSCOMP_1:def 3 :: thesis: ( not Y is closed or ( the carrier of T --> a) " Y is closed )
A1: dom ( the carrier of T --> a) = the carrier of T by FUNCT_2:def 1;
assume Y is closed ; :: thesis: ( the carrier of T --> a) " Y is closed
A2: rng ( the carrier of T --> a) = {a} by FUNCOP_1:8;
per cases ( a in Y or not a in Y ) ;
suppose a in Y ; :: thesis: ( the carrier of T --> a) " Y is closed
then A3: rng ( the carrier of T --> a) c= Y by A2, ZFMISC_1:31;
( the carrier of T --> a) " Y = ( the carrier of T --> a) " ((rng ( the carrier of T --> a)) /\ Y) by RELAT_1:133
.= ( the carrier of T --> a) " (rng ( the carrier of T --> a)) by A3, XBOOLE_1:28
.= [#] T by A1, RELAT_1:134 ;
hence ( the carrier of T --> a) " Y is closed ; :: thesis: verum
end;
suppose not a in Y ; :: thesis: ( the carrier of T --> a) " Y is closed
then A4: rng ( the carrier of T --> a) misses Y by A2, ZFMISC_1:50;
( the carrier of T --> a) " Y = ( the carrier of T --> a) " ((rng ( the carrier of T --> a)) /\ Y) by RELAT_1:133
.= ( the carrier of T --> a) " {} by A4
.= {} T ;
hence ( the carrier of T --> a) " Y is closed ; :: thesis: verum
end;
end;
end;
end;

theorem Th13: :: BROUWER:13
for r being positive Real
for o being Point of (TOP-REAL 2)
for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) st f is without_fixpoints holds
BR-map f is continuous
proof end;

:: WP: Brouwer Fixed Point Theorem
theorem Th14: :: BROUWER:14
for r being non negative Real
for o being Point of (TOP-REAL 2)
for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) holds f is with_fixpoint
proof end;

:: WP: Brouwer Fixed Point Theorem for Disks on the Plane
theorem :: BROUWER:15
for r being non negative Real
for o being Point of (TOP-REAL 2)
for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) ex x being Point of (Tdisk (o,r)) st f . x = x
proof end;