GOBRD13: Some Properties of Cells on Go Board

:: Some Properties of Cells on Go Board
:: by Czes{\l}aw Byli\'nski
::
:: Received April 23, 1999
:: Copyright (c) 1999 Association of Mizar Users

begin

definition

let E be non empty set ;
let S be FinSequence-DOMAIN of the carrier of (TOP-REAL 2);
let F be Function of E,S;
let e be Element of E;
:: original: .
redefine func F . e -> FinSequence of (TOP-REAL 2);
coherence

proof end;

end;

definition

let F be Function;
func Values F -> set equals :: GOBRD13:def 1
Union (rngs F);
correctness ;

end;

:: deftheorem defines Values GOBRD13:def 1 :

theorem Th1: :: GOBRD13:1

for i being Element of NAT
for D being non empty set
for M being FinSequence of D * holds M . i is FinSequence of D

proof end;

theorem :: GOBRD13:2

canceled;

theorem Th3: :: GOBRD13:3

for D being non empty set
for M being FinSequence of D * holds Values M = union { (rng f) where f is Element of D * : f in rng M }

proof end;

registration

let D be non empty set ;
let M be FinSequence of D * ;
cluster Values M -> finite ;
coherence

proof end;

end;

theorem Th4: :: GOBRD13:4

for i being Element of NAT
for D being non empty set
for f being FinSequence of D
for M being Matrix of D st i in dom M & M . i = f holds
len f = width M

proof end;

theorem Th5: :: GOBRD13:5

for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D
for M being Matrix of D st i in dom M & M . i = f & j in dom f holds
[i,j] in Indices M

proof end;

theorem Th6: :: GOBRD13:6

for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D
for M being Matrix of D st [i,j] in Indices M & M . i = f holds
( len f = width M & j in dom f )

proof end;

theorem Th7: :: GOBRD13:7

for D being non empty set
for M being Matrix of D holds Values M = { (M * (i,j)) where i, j is Element of NAT : [i,j] in Indices M }

proof end;

theorem :: GOBRD13:8

for D being non empty set
for M being Matrix of D holds card (Values M) <= (len M) * (width M)

proof end;

theorem :: GOBRD13:9

for f being FinSequence of (TOP-REAL 2)
for G being Matrix of (TOP-REAL 2) st f is_sequence_on G holds
rng f c= Values G

proof end;

theorem Th10: :: GOBRD13:10

for i1, j1, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (1,j2) holds
i1 = 1

proof end;

theorem Th11: :: GOBRD13:11

for i1, j1, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * ((len G2),j2) holds
i1 = len G1

proof end;

theorem Th12: :: GOBRD13:12

for i1, j1, i2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,1) holds
j1 = 1

proof end;

theorem Th13: :: GOBRD13:13

for i1, j1, i2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,(width G2)) holds
j1 = width G1

proof end;

theorem Th14: :: GOBRD13:14

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 < len G1 & 1 <= j1 & j1 <= width G1 & 1 <= i2 & i2 < len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1

proof end;

theorem Th15: :: GOBRD13:15

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st G1 * ((i1 -' 1),j1) in Values G2 & 1 < i1 & i1 <= len G1 & 1 <= j1 & j1 <= width G1 & 1 < i2 & i2 <= len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1

proof end;

theorem Th16: :: GOBRD13:16

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st G1 * (i1,(j1 + 1)) in Values G2 & 1 <= i1 & i1 <= len G1 & 1 <= j1 & j1 < width G1 & 1 <= i2 & i2 <= len G2 & 1 <= j2 & j2 < width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * (i2,(j2 + 1))) `2 <= (G1 * (i1,(j1 + 1))) `2

proof end;

theorem Th17: :: GOBRD13:17

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 <= len G1 & 1 < j1 & j1 <= width G1 & 1 <= i2 & i2 <= len G2 & 1 < j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G1 * (i1,(j1 -' 1))) `2 <= (G2 * (i2,(j2 -' 1))) `2

proof end;

Lm1: for i, j being Element of NAT
for D being non empty set
for M being Matrix of D st 1 <= i & i <= len M & 1 <= j & j <= width M holds
M * (i,j) in Values M

proof end;

theorem Th18: :: GOBRD13:18

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,i2,j2) c= cell (G1,i1,j1)

proof end;

theorem Th19: :: GOBRD13:19

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,(i2 -' 1),j2) c= cell (G1,(i1 -' 1),j1)

proof end;

theorem Th20: :: GOBRD13:20

for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,i2,(j2 -' 1)) c= cell (G1,i1,(j1 -' 1))

proof end;

Lm2: for i, j being Element of NAT
for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Element of NAT st
( k in dom f & (f /. k) `1 = ((GoB f) * (i,j)) `1 )

proof end;

Lm3: for i, j being Element of NAT
for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Element of NAT st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )

proof end;

theorem Th21: :: GOBRD13:21

for G being Go-board
for f being standard special_circular_sequence st f is_sequence_on G holds
Values (GoB f) c= Values G

proof end;

definition

let f be FinSequence of (TOP-REAL 2);
let G be Go-board;
let k be Element of NAT ;
assume ( 1 <= k & k + 1 <= len f & f is_sequence_on G ) ;
then consider i1, j1, i2, j2 being Element of NAT such that
A1: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) ) and
A2: ( [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) ) and
A3: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by JORDAN8:6;
func right_cell (f,k,G) -> Subset of (TOP-REAL 2) means :Def2: :: GOBRD13:def 2
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,(i1 -' 1),j2) );
existence

proof end;

uniqueness

proof end;

func left_cell (f,k,G) -> Subset of (TOP-REAL 2) means :Def3: :: GOBRD13:def 3
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,i1,j2) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines right_cell GOBRD13:def 2 :

:: deftheorem Def3 defines left_cell GOBRD13:def 3 :

theorem Th22: :: GOBRD13:22

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
left_cell (f,k,G) = cell (G,(i -' 1),j)

proof end;

theorem Th23: :: GOBRD13:23

proof end;

theorem Th24: :: GOBRD13:24

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
left_cell (f,k,G) = cell (G,i,j)

proof end;

theorem Th25: :: GOBRD13:25

proof end;

theorem Th26: :: GOBRD13:26

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,(j -' 1))

proof end;

theorem Th27: :: GOBRD13:27

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,i,j)

proof end;

theorem Th28: :: GOBRD13:28

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,j)

proof end;

theorem Th29: :: GOBRD13:29

proof end;

theorem :: GOBRD13:30

for k being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
(left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)

proof end;

theorem :: GOBRD13:31

for k being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
right_cell (f,k,G) is closed

proof end;

theorem :: GOBRD13:32

for k, n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )

proof end;

theorem :: GOBRD13:33

for k, n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len (f /^ n) & n <= len f & f is_sequence_on G holds
( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )

proof end;

theorem :: GOBRD13:34

for n being Element of NAT
for G being Go-board
for f being standard special_circular_sequence st 1 <= n & n + 1 <= len f & f is_sequence_on G holds
( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )

proof end;

definition

let f be FinSequence of (TOP-REAL 2);
let G be Go-board;
let k be Element of NAT ;
assume ( 1 <= k & k + 1 <= len f & f is_sequence_on G ) ;
then consider i1, j1, i2, j2 being Element of NAT such that
A1: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) ) and
A2: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by JORDAN8:6;
func front_right_cell (f,k,G) -> Subset of (TOP-REAL 2) means :Def4: :: GOBRD13:def 4
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,(i2 -' 1),(j2 -' 1)) );
existence

proof end;

uniqueness

proof end;

func front_left_cell (f,k,G) -> Subset of (TOP-REAL 2) means :Def5: :: GOBRD13:def 5
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,i2,(j2 -' 1)) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines front_right_cell GOBRD13:def 4 :

:: deftheorem Def5 defines front_left_cell GOBRD13:def 5 :

theorem :: GOBRD13:35

proof end;

theorem :: GOBRD13:36

proof end;

theorem :: GOBRD13:37

proof end;

theorem :: GOBRD13:38

proof end;

theorem :: GOBRD13:39

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))

proof end;

theorem :: GOBRD13:40

for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),j)

proof end;

theorem :: GOBRD13:41

proof end;

theorem :: GOBRD13:42

proof end;

theorem :: GOBRD13:43

for k, n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )

proof end;

definition

let D be set ;
let f be FinSequence of D;
let G be Matrix of D;
let k be Element of NAT ;
pred f turns_right k,G means :Def6: :: GOBRD13:def 6
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) );
pred f turns_left k,G means :Def7: :: GOBRD13:def 7
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) );
pred f goes_straight k,G means :Def8: :: GOBRD13:def 8
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) & not ( i1 + 1 = i2 & j1 = j2 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) & not ( i1 = i2 + 1 & j1 = j2 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) );

end;

:: deftheorem Def6 defines turns_right GOBRD13:def 6 :

:: deftheorem Def7 defines turns_left GOBRD13:def 7 :

:: deftheorem Def8 defines goes_straight GOBRD13:def 8 :

theorem :: GOBRD13:44

for k, n being Element of NAT
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_right k,G holds
f turns_right k,G

proof end;

theorem :: GOBRD13:45

for k, n being Element of NAT
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_left k,G holds
f turns_left k,G

proof end;

theorem :: GOBRD13:46

for k, n being Element of NAT
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n goes_straight k,G holds
f goes_straight k,G

proof end;

theorem :: GOBRD13:47

for k being Element of NAT
for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_right k -' 1,G & f2 turns_right k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)

proof end;

theorem :: GOBRD13:48

for k being Element of NAT
for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)

proof end;

theorem :: GOBRD13:49

for k being Element of NAT
for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 goes_straight k -' 1,G & f2 goes_straight k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)

proof end;

theorem :: GOBRD13:50

for i, j being Element of NAT
for D being non empty set
for M being Matrix of D st 1 <= i & i <= len M & 1 <= j & j <= width M holds
M * (i,j) in Values M by Lm1;