:: Introduction to Go-Board - Part II. Go-Board Determined by Finite Sequence of point from ${\calE}^2_{\rm T}$
:: by Jaros{\l}aw Kotowicz and Yatsuka Nakamura
::
:: Received August 24, 1992
:: Copyright (c) 1992-2021 Association of Mizar Users

theorem :: GOBOARD2:1

for f being FinSequence of (TOP-REAL 2) st ( for n, m being Nat st m > n + 1 & n in dom f & n + 1 in dom f & m in dom f & m + 1 in dom f holds
LSeg (f,n) misses LSeg (f,m) ) holds
f is s.n.c.

proof end;

theorem :: GOBOARD2:2

for f being FinSequence of (TOP-REAL 2)
for i being Nat st f is unfolded & f is s.n.c. & f is one-to-one & f /. (len f) in LSeg (f,i) & i in dom f & i + 1 in dom f holds
i + 1 = len f

proof end;

theorem Th3: :: GOBOARD2:3

for f being FinSequence of (TOP-REAL 2)
for k being Nat st k <> 0 & len f = k + 1 holds
L~ f = (L~ (f | k)) \/ (LSeg (f,k))

proof end;

theorem :: GOBOARD2:4

for f being FinSequence of (TOP-REAL 2)
for k being Nat st 1 < k & len f = k + 1 & f is unfolded & f is s.n.c. holds
(L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)}

proof end;

theorem :: GOBOARD2:5

for f1, f2 being FinSequence of (TOP-REAL 2)
for n, m being Nat st len f1 < n & n + 1 <= len (f1 ^ f2) & m + (len f1) = n holds
LSeg ((f1 ^ f2),n) = LSeg (f2,m)

proof end;

theorem Th6: :: GOBOARD2:6

for f, g being FinSequence of (TOP-REAL 2) holds L~ f c= L~ (f ^ g)

proof end;

theorem :: GOBOARD2:7

for f being FinSequence of (TOP-REAL 2)
for i being Nat st f is s.n.c. holds
f | i is s.n.c.

proof end;

theorem :: GOBOARD2:8

for f1, f2 being FinSequence of (TOP-REAL 2) st f1 is special & f2 is special & ( (f1 /. (len f1)) `1 = (f2 /. 1) `1 or (f1 /. (len f1)) `2 = (f2 /. 1) `2 ) holds
f1 ^ f2 is special

proof end;

theorem Th9: :: GOBOARD2:9

for f being FinSequence of (TOP-REAL 2) st f <> {} holds
X_axis f <> {}

proof end;

theorem Th10: :: GOBOARD2:10

for f being FinSequence of (TOP-REAL 2) st f <> {} holds
Y_axis f <> {}

proof end;

registration

let f be non empty FinSequence of (TOP-REAL 2);

cluster X_axis f -> non empty ;

coherence
not X_axis f is empty by Th9;

cluster Y_axis f -> non empty ;

coherence
not Y_axis f is empty by Th10;

end;

theorem Th11: :: GOBOARD2:11

for f being FinSequence of (TOP-REAL 2)
for G being Go-board st f is special holds
for n being Nat st n in dom f & n + 1 in dom f holds
for i, j, m, k being Nat st [i,j] in Indices G & [m,k] in Indices G & f /. n = G * (i,j) & f /. (n + 1) = G * (m,k) & not i = m holds
k = j

proof end;

theorem :: GOBOARD2:12

for f being FinSequence of (TOP-REAL 2)
for G being Go-board st ( for n being Nat st n in dom f holds
ex i, j being Nat st
( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is special & ( for n being Nat st n in dom f & n + 1 in dom f holds
f /. n <> f /. (n + 1) ) holds
ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & L~ f = L~ g & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g )

proof end;

definition

let v1, v2 be FinSequence of REAL ;
assume A1: v1 <> {} ;

func GoB (v1,v2) -> Matrix of (TOP-REAL 2) means :Def1: :: GOBOARD2:def 1
( len it = len v1 & width it = len v2 & ( for n, m being Nat st [n,m] in Indices it holds
it * (n,m) = |[(v1 . n),(v2 . m)]| ) );

existence
ex b₁ being Matrix of (TOP-REAL 2) st
( len b₁ = len v1 & width b₁ = len v2 & ( for n, m being Nat st [n,m] in Indices b₁ holds
b₁ * (n,m) = |[(v1 . n),(v2 . m)]| ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of (TOP-REAL 2) st len b₁ = len v1 & width b₁ = len v2 & ( for n, m being Nat st [n,m] in Indices b₁ holds
b₁ * (n,m) = |[(v1 . n),(v2 . m)]| ) & len b₂ = len v1 & width b₂ = len v2 & ( for n, m being Nat st [n,m] in Indices b₂ holds
b₂ * (n,m) = |[(v1 . n),(v2 . m)]| ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines GoB GOBOARD2:def 1 :
for v1, v2 being FinSequence of REAL st v1 <> {} holds
for b₃ being Matrix of (TOP-REAL 2) holds
( b₃ = GoB (v1,v2) iff ( len b₃ = len v1 & width b₃ = len v2 & ( for n, m being Nat st [n,m] in Indices b₃ holds
b₃ * (n,m) = |[(v1 . n),(v2 . m)]| ) ) );

registration

let v1, v2 be non empty FinSequence of REAL ;

cluster GoB (v1,v2) -> V3() X_equal-in-line Y_equal-in-column ;

coherence
( not GoB (v1,v2) is empty-yielding & GoB (v1,v2) is X_equal-in-line & GoB (v1,v2) is Y_equal-in-column )

proof end;

end;

registration

let v1, v2 be non empty increasing FinSequence of REAL ;

cluster GoB (v1,v2) -> Y_increasing-in-line X_increasing-in-column ;

coherence
( GoB (v1,v2) is Y_increasing-in-line & GoB (v1,v2) is X_increasing-in-column )

proof end;

end;

definition

let f be non empty FinSequence of (TOP-REAL 2);

func GoB f -> Matrix of (TOP-REAL 2) equals :: GOBOARD2:def 2
GoB ((Incr (X_axis f)),(Incr (Y_axis f)));

correctness
coherence
GoB ((Incr (X_axis f)),(Incr (Y_axis f))) is Matrix of (TOP-REAL 2);
;

end;

:: deftheorem defines GoB GOBOARD2:def 2 :
for f being non empty FinSequence of (TOP-REAL 2) holds GoB f = GoB ((Incr (X_axis f)),(Incr (Y_axis f)));

registration

let f be non empty FinSequence of (TOP-REAL 2);

cluster GoB f -> V3() X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ;

coherence
( not GoB f is empty-yielding & GoB f is X_equal-in-line & GoB f is Y_equal-in-column & GoB f is Y_increasing-in-line & GoB f is X_increasing-in-column ) ;

end;

theorem Th13: :: GOBOARD2:13

for f being non empty FinSequence of (TOP-REAL 2) holds
( len (GoB f) = card (rng (X_axis f)) & width (GoB f) = card (rng (Y_axis f)) )

proof end;

theorem :: GOBOARD2:14

for f being non empty FinSequence of (TOP-REAL 2)
for n being Nat st n in dom f holds
ex i, j being Nat st
( [i,j] in Indices (GoB f) & f /. n = (GoB f) * (i,j) )

proof end;

theorem :: GOBOARD2:15

for n being Nat
for f being non empty FinSequence of (TOP-REAL 2) st n in dom f & ( for m being Nat st m in dom f holds
(X_axis f) . n <= (X_axis f) . m ) holds
f /. n in rng (Line ((GoB f),1))

proof end;

theorem :: GOBOARD2:16

for n being Nat
for f being non empty FinSequence of (TOP-REAL 2) st n in dom f & ( for m being Nat st m in dom f holds
(X_axis f) . m <= (X_axis f) . n ) holds
f /. n in rng (Line ((GoB f),(len (GoB f))))

proof end;

theorem :: GOBOARD2:17

for n being Nat
for f being non empty FinSequence of (TOP-REAL 2) st n in dom f & ( for m being Nat st m in dom f holds
(Y_axis f) . n <= (Y_axis f) . m ) holds
f /. n in rng (Col ((GoB f),1))

proof end;

theorem :: GOBOARD2:18

for n being Nat
for f being non empty FinSequence of (TOP-REAL 2) st n in dom f & ( for m being Nat st m in dom f holds
(Y_axis f) . m <= (Y_axis f) . n ) holds
f /. n in rng (Col ((GoB f),(width (GoB f))))

proof end;