:: 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;