:: Gauges
:: by Czes\law Byli\'nski
::
:: Copyright (c) 1999-2021 Association of Mizar Users

theorem :: JORDAN8:1
for D being set
for G being Matrix of D holds <*> D is_sequence_on G ;

theorem :: JORDAN8:2
for m being Nat
for D being non empty set
for f being FinSequence of D
for G being Matrix of D st f is_sequence_on G holds
f /^ m is_sequence_on G
proof end;

theorem Th3: :: JORDAN8:3
for k being Nat
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
ex i1, j1, i2, j2 being Nat st
( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) & ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) )
proof end;

theorem :: JORDAN8:4
for G being Go-board
for f being non empty FinSequence of () st f is_sequence_on G holds
( f is standard & f is special )
proof end;

theorem :: JORDAN8:5
for G being Go-board
for f being non empty FinSequence of () st len f >= 2 & f is_sequence_on G holds
not f is constant
proof end;

theorem :: JORDAN8:6
for G being Go-board
for p being Point of ()
for f being non empty FinSequence of () st f is_sequence_on G & ex i, j being Nat st
( [i,j] in Indices G & p = G * (i,j) ) & ( for i1, j1, i2, j2 being Nat st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. (len f) = G * (i1,j1) & p = G * (i2,j2) holds
|.(i2 - i1).| + |.(j2 - j1).| = 1 ) holds
f ^ <*p*> is_sequence_on G
proof end;

theorem :: JORDAN8:7
for i, j, k being Nat
for G being Go-board st i + k < len G & 1 <= j & j < width G & cell (G,i,j) meets cell (G,(i + k),j) holds
k <= 1
proof end;

theorem Th8: :: JORDAN8:8
for C being non empty compact Subset of () holds
( C is vertical iff E-bound C <= W-bound C )
proof end;

theorem Th9: :: JORDAN8:9
for C being non empty compact Subset of () holds
( C is horizontal iff N-bound C <= S-bound C )
proof end;

definition
let C be Subset of ();
let n be natural Number ;
deffunc H1( natural Number , natural Number ) -> Element of the carrier of () = |[(() + (((() - ()) / (2 |^ n)) * ($1 - 2))),(() + (((() - ()) / (2 |^ n)) * ($2 - 2)))]|;
func Gauge (C,n) -> Matrix of () means :Def1: :: JORDAN8:def 1
( len it = (2 |^ n) + 3 & len it = width it & ( for i, j being Nat st [i,j] in Indices it holds
it * (i,j) = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * (j - 2)))]| ) );
existence
ex b1 being Matrix of () st
( len b1 = (2 |^ n) + 3 & len b1 = width b1 & ( for i, j being Nat st [i,j] in Indices b1 holds
b1 * (i,j) = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * (j - 2)))]| ) )
proof end;
uniqueness
for b1, b2 being Matrix of () st len b1 = (2 |^ n) + 3 & len b1 = width b1 & ( for i, j being Nat st [i,j] in Indices b1 holds
b1 * (i,j) = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * (j - 2)))]| ) & len b2 = (2 |^ n) + 3 & len b2 = width b2 & ( for i, j being Nat st [i,j] in Indices b2 holds
b2 * (i,j) = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * (j - 2)))]| ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Gauge JORDAN8:def 1 :
for C being Subset of ()
for n being natural Number
for b3 being Matrix of () holds
( b3 = Gauge (C,n) iff ( len b3 = (2 |^ n) + 3 & len b3 = width b3 & ( for i, j being Nat st [i,j] in Indices b3 holds
b3 * (i,j) = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * (j - 2)))]| ) ) );

registration
let C be non empty Subset of ();
let n be Nat;
coherence
( not Gauge (C,n) is empty-yielding & Gauge (C,n) is X_equal-in-line & Gauge (C,n) is Y_equal-in-column )
proof end;
end;

registration
let C be non empty compact non horizontal non vertical Subset of ();
let n be Nat;
coherence
( Gauge (C,n) is Y_increasing-in-line & Gauge (C,n) is X_increasing-in-column )
proof end;
end;

theorem Th10: :: JORDAN8:10
for T being non empty Subset of ()
for n being Nat holds len (Gauge (T,n)) >= 4
proof end;

theorem :: JORDAN8:11
for j, n being Nat
for T being non empty Subset of () st 1 <= j & j <= len (Gauge (T,n)) holds
((Gauge (T,n)) * (2,j)) 1 = W-bound T
proof end;

theorem :: JORDAN8:12
for j, n being Nat
for T being non empty Subset of () st 1 <= j & j <= len (Gauge (T,n)) holds
((Gauge (T,n)) * (((len (Gauge (T,n))) -' 1),j)) 1 = E-bound T
proof end;

theorem :: JORDAN8:13
for i, n being Nat
for T being non empty Subset of () st 1 <= i & i <= len (Gauge (T,n)) holds
((Gauge (T,n)) * (i,2)) 2 = S-bound T
proof end;

theorem :: JORDAN8:14
for i, n being Nat
for T being non empty Subset of () st 1 <= i & i <= len (Gauge (T,n)) holds
((Gauge (T,n)) * (i,((len (Gauge (T,n))) -' 1))) 2 = N-bound T
proof end;

theorem :: JORDAN8:15
for C being non empty compact non horizontal non vertical Subset of ()
for i, n being Nat st i <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) misses C
proof end;

theorem :: JORDAN8:16
for C being non empty compact non horizontal non vertical Subset of ()
for j, n being Nat st j <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),(len (Gauge (C,n))),j) misses C
proof end;

theorem :: JORDAN8:17
for C being non empty compact non horizontal non vertical Subset of ()
for i, n being Nat st i <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),i,0) misses C
proof end;

theorem :: JORDAN8:18
for C being non empty compact non horizontal non vertical Subset of ()
for j, n being Nat st j <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),0,j) misses C
proof end;