:: 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 Association of Mizar Users

begin

defpred S₁[ Nat] means for R being finite Subset of st R <> {} & card R = $1 holds
( R is bounded_above & upper_bound R in R & R is bounded_below & lower_bound R in R );

Lm1: S₁[ 0 ]
;

Lm2: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof end;

Lm3: for n being Element of NAT holds S₁[n]
from NAT_1:sch 1(Lm1, Lm2);

theorem Th1: :: GOBOARD2:1

for R being finite Subset of st R <> {} holds
( R is bounded_above & upper_bound R in R & R is bounded_below & lower_bound R in R )

proof end;

begin

theorem :: GOBOARD2:2

canceled;

theorem Th3: :: GOBOARD2:3

for n being Nat
for f being FinSequence holds
( 1 <= n & n + 1 <= len f iff ( n in dom f & n + 1 in dom f ) )

proof end;

theorem :: GOBOARD2:4

for n being Nat
for f being FinSequence holds
( 1 <= n & n + 2 <= len f iff ( n in dom f & n + 1 in dom f & n + 2 in dom f ) )

proof end;

theorem Th5: :: GOBOARD2:5

for D being non empty set
for f1, f2 being FinSequence of D
for n being Element of NAT st 1 <= n & n <= len f2 holds
(f1 ^ f2) /. (n + (len f1)) = f2 /. n

proof end;

theorem :: GOBOARD2:6

for f being FinSequence of st ( for n, m being Element of 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 :: GOBOARD2:7

for f being FinSequence of
for i being Element of 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 Th8: :: GOBOARD2:8

for f being FinSequence of
for k being Element of NAT st k <> 0 & len f = k + 1 holds
L~ f = (L~ (f | k)) \/ (LSeg f,k)

proof end;

theorem :: GOBOARD2:9

for f being FinSequence of
for k being Element of 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 :: GOBOARD2:10

for f1, f2 being FinSequence of
for n, m being Element of 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 Th11: :: GOBOARD2:11

for f, g being FinSequence of holds L~ f c= L~ (f ^ g)

proof end;

theorem :: GOBOARD2:12

for f being FinSequence of
for i being Element of NAT st f is s.n.c. holds
f | i is s.n.c.

proof end;

theorem :: GOBOARD2:13

for f1, f2 being FinSequence of 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 Th14: :: GOBOARD2:14

for f being FinSequence of st f <> {} holds
X_axis f <> {}

proof end;

theorem Th15: :: GOBOARD2:15

for f being FinSequence of st f <> {} holds
Y_axis f <> {}

proof end;

registration

let f be non empty FinSequence of ;
cluster X_axis f -> non empty ;
coherence
not X_axis f is empty by Th14;
cluster Y_axis f -> non empty ;
coherence
not Y_axis f is empty by Th15;

end;

theorem Th16: :: GOBOARD2:16

for f being FinSequence of
for G being Go-board st f is special holds
for n being Element of NAT st n in dom f & n + 1 in dom f holds
for i, j, m, k being Element of 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 :: GOBOARD2:17

for f being FinSequence of
for G being Go-board st ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * i,j ) ) & f is special & ( for n being Element of NAT st n in dom f & n + 1 in dom f holds
f /. n <> f /. (n + 1) ) holds
ex g being FinSequence of 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;

theorem :: GOBOARD2:18

for v being FinSequence of REAL st v is increasing holds
for n, m being Element of NAT st n in dom v & m in dom v & n <= m holds
v . n <= v . m

proof end;

theorem Th19: :: GOBOARD2:19

for v being FinSequence of REAL st v is increasing holds
for n, m being Element of NAT st n in dom v & m in dom v & n <> m holds
v . n <> v . m

proof end;

theorem Th20: :: GOBOARD2:20

for v, v1 being FinSequence of REAL
for n being Element of NAT st v is increasing & v1 = v | (Seg n) holds
v1 is increasing

proof end;

defpred S₂[ Element of NAT ] means for v being FinSequence of REAL st card (rng v) = $1 holds
ex v1 being FinSequence of REAL st
( rng v1 = rng v & len v1 = card (rng v) & v1 is increasing );

Lm4: S₂[ 0 ]

proof end;

Lm5: for n being Element of NAT st S₂[n] holds
S₂[n + 1]

proof end;

Lm6: for n being Element of NAT holds S₂[n]
from NAT_1:sch 1(Lm4, Lm5);

theorem :: GOBOARD2:21

for v being FinSequence of REAL ex v1 being FinSequence of REAL st
( rng v1 = rng v & len v1 = card (rng v) & v1 is increasing ) by Lm6;

defpred S₃[ Element of NAT ] means for v1, v2 being FinSequence of REAL st len v1 = $1 & len v2 = $1 & rng v1 = rng v2 & v1 is increasing & v2 is increasing holds
v1 = v2;

Lm7: S₃[ 0 ]

proof end;

Lm8: for n being Element of NAT st S₃[n] holds
S₃[n + 1]

proof end;

Lm9: for n being Element of NAT holds S₃[n]
from NAT_1:sch 1(Lm7, Lm8);

theorem :: GOBOARD2:22

for v1, v2 being FinSequence of REAL st len v1 = len v2 & rng v1 = rng v2 & v1 is increasing & v2 is increasing holds
v1 = v2 by Lm9;

begin

definition

let v1, v2 be FinSequence of REAL ;
assume A1: v1 <> {} ;
func GoB v1,v2 -> Matrix of means :Def1: :: GOBOARD2:def 1
( len it = len v1 & width it = len v2 & ( for n, m being Element of NAT st [n,m] in Indices it holds
it * n,m = |[(v1 . n),(v2 . m)]| ) );
existence
ex b₁ being Matrix of st
( len b₁ = len v1 & width b₁ = len v2 & ( for n, m being Element of NAT st [n,m] in Indices b₁ holds
b₁ * n,m = |[(v1 . n),(v2 . m)]| ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of st len b₁ = len v1 & width b₁ = len v2 & ( for n, m being Element of 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 Element of 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 holds
( b₃ = GoB v1,v2 iff ( len b₃ = len v1 & width b₃ = len v2 & ( for n, m being Element of 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 v be FinSequence of REAL ;
func Incr v -> increasing FinSequence of REAL means :Def2: :: GOBOARD2:def 2
( rng it = rng v & len it = card (rng v) );
existence
ex b₁ being increasing FinSequence of REAL st
( rng b₁ = rng v & len b₁ = card (rng v) )

proof end;

uniqueness
for b₁, b₂ being increasing FinSequence of REAL st rng b₁ = rng v & len b₁ = card (rng v) & rng b₂ = rng v & len b₂ = card (rng v) holds
b₁ = b₂ by Lm9;

end;

:: deftheorem Def2 defines Incr GOBOARD2:def 2 :

for v being FinSequence of REAL
for b₂ being increasing FinSequence of REAL holds
( b₂ = Incr v iff ( rng b₂ = rng v & len b₂ = card (rng v) ) );

definition

let f be non empty FinSequence of ;
func GoB f -> Matrix of equals :: GOBOARD2:def 3
GoB (Incr (X_axis f)),(Incr (Y_axis f));
correctness
coherence
GoB (Incr (X_axis f)),(Incr (Y_axis f)) is Matrix of ;
;

end;

:: deftheorem defines GoB GOBOARD2:def 3 :

for f being non empty FinSequence of holds GoB f = GoB (Incr (X_axis f)),(Incr (Y_axis f));

theorem Th23: :: GOBOARD2:23

for v being FinSequence of REAL st v <> {} holds
Incr v <> {}

proof end;

registration

let v be non empty FinSequence of REAL ;
cluster Incr v -> non empty increasing ;
coherence
not Incr v is empty by Th23;

end;

registration

let f be non empty FinSequence of ;
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 Th24: :: GOBOARD2:24

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

proof end;

theorem :: GOBOARD2:25

for f being non empty FinSequence of
for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices (GoB f) & f /. n = (GoB f) * i,j )

proof end;

theorem :: GOBOARD2:26

for n being Element of NAT
for f being non empty FinSequence of st n in dom f & ( for m being Element of 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 :: GOBOARD2:27

for n being Element of NAT
for f being non empty FinSequence of st n in dom f & ( for m being Element of 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 :: GOBOARD2:28

for n being Element of NAT
for f being non empty FinSequence of st n in dom f & ( for m being Element of 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 :: GOBOARD2:29

for n being Element of NAT
for f being non empty FinSequence of st n in dom f & ( for m being Element of 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;