:: GOBOARD7 semantic presentation
theorem Th1: :: GOBOARD7:1
for b
1, b
2, b
3 being
Real holds
not (
abs (b1 - b2) > b
3 & not b
1 + b
3 < b
2 & not b
2 + b
3 < b
1 )
theorem Th2: :: GOBOARD7:2
for b
1, b
2 being
Real holds
(
abs (b1 - b2) = 0 iff b
1 = b
2 )
theorem Th3: :: GOBOARD7:3
for b
1 being
Natfor b
2, b
3, b
4 being
Point of
(TOP-REAL b1) holds
( b
2 + b
3 = b
4 + b
3 implies b
2 = b
4 )
theorem Th4: :: GOBOARD7:4
theorem Th5: :: GOBOARD7:5
theorem Th6: :: GOBOARD7:6
theorem Th7: :: GOBOARD7:7
theorem Th8: :: GOBOARD7:8
theorem Th9: :: GOBOARD7:9
theorem Th10: :: GOBOARD7:10
theorem Th11: :: GOBOARD7:11
for b
1, b
2 being
Natfor b
3 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
3 & 1
<= b
2 & b
2 + 1
<= width b
3 implies
(1 / 2) * ((b3 * b1,b2) + (b3 * (b1 + 1),(b2 + 1))) = (1 / 2) * ((b3 * b1,(b2 + 1)) + (b3 * (b1 + 1),b2)) )
theorem Th12: :: GOBOARD7:12
theorem Th13: :: GOBOARD7:13
theorem Th14: :: GOBOARD7:14
theorem Th15: :: GOBOARD7:15
theorem Th16: :: GOBOARD7:16
theorem Th17: :: GOBOARD7:17
theorem Th18: :: GOBOARD7:18
theorem Th19: :: GOBOARD7:19
theorem Th20: :: GOBOARD7:20
theorem Th21: :: GOBOARD7:21
theorem Th22: :: GOBOARD7:22
theorem Th23: :: GOBOARD7:23
theorem Th24: :: GOBOARD7:24
for b
1 being
Go-boardfor b
2, b
3, b
4, b
5 being
Nat holds
not ( 1
<= b
2 & b
2 <= len b
1 & 1
<= b
3 & b
3 + 1
<= width b
1 & 1
<= b
4 & b
4 <= len b
1 & 1
<= b
5 & b
5 + 1
<= width b
1 &
LSeg (b1 * b2,b3),
(b1 * b2,(b3 + 1)) meets LSeg (b1 * b4,b5),
(b1 * b4,(b5 + 1)) & not ( b
3 = b
5 &
LSeg (b1 * b2,b3),
(b1 * b2,(b3 + 1)) = LSeg (b1 * b4,b5),
(b1 * b4,(b5 + 1)) ) & not ( b
3 = b
5 + 1 &
(LSeg (b1 * b2,b3),(b1 * b2,(b3 + 1))) /\ (LSeg (b1 * b4,b5),(b1 * b4,(b5 + 1))) = {(b1 * b2,b3)} ) & not ( b
3 + 1
= b
5 &
(LSeg (b1 * b2,b3),(b1 * b2,(b3 + 1))) /\ (LSeg (b1 * b4,b5),(b1 * b4,(b5 + 1))) = {(b1 * b4,b5)} ) )
theorem Th25: :: GOBOARD7:25
for b
1 being
Go-boardfor b
2, b
3, b
4, b
5 being
Nat holds
not ( 1
<= b
2 & b
2 + 1
<= len b
1 & 1
<= b
3 & b
3 <= width b
1 & 1
<= b
4 & b
4 + 1
<= len b
1 & 1
<= b
5 & b
5 <= width b
1 &
LSeg (b1 * b2,b3),
(b1 * (b2 + 1),b3) meets LSeg (b1 * b4,b5),
(b1 * (b4 + 1),b5) & not ( b
2 = b
4 &
LSeg (b1 * b2,b3),
(b1 * (b2 + 1),b3) = LSeg (b1 * b4,b5),
(b1 * (b4 + 1),b5) ) & not ( b
2 = b
4 + 1 &
(LSeg (b1 * b2,b3),(b1 * (b2 + 1),b3)) /\ (LSeg (b1 * b4,b5),(b1 * (b4 + 1),b5)) = {(b1 * b2,b3)} ) & not ( b
2 + 1
= b
4 &
(LSeg (b1 * b2,b3),(b1 * (b2 + 1),b3)) /\ (LSeg (b1 * b4,b5),(b1 * (b4 + 1),b5)) = {(b1 * b4,b5)} ) )
theorem Th26: :: GOBOARD7:26
for b
1 being
Go-boardfor b
2, b
3, b
4, b
5 being
Nat holds
not ( 1
<= b
2 & b
2 <= len b
1 & 1
<= b
3 & b
3 + 1
<= width b
1 & 1
<= b
4 & b
4 + 1
<= len b
1 & 1
<= b
5 & b
5 <= width b
1 &
LSeg (b1 * b2,b3),
(b1 * b2,(b3 + 1)) meets LSeg (b1 * b4,b5),
(b1 * (b4 + 1),b5) & not ( b
3 = b
5 &
(LSeg (b1 * b2,b3),(b1 * b2,(b3 + 1))) /\ (LSeg (b1 * b4,b5),(b1 * (b4 + 1),b5)) = {(b1 * b2,b3)} ) & not ( b
3 + 1
= b
5 &
(LSeg (b1 * b2,b3),(b1 * b2,(b3 + 1))) /\ (LSeg (b1 * b4,b5),(b1 * (b4 + 1),b5)) = {(b1 * b2,(b3 + 1))} ) )
Lemma21:
1 - (1 / 2) = 1 / 2
;
theorem Th27: :: GOBOARD7:27
theorem Th28: :: GOBOARD7:28
theorem Th29: :: GOBOARD7:29
theorem Th30: :: GOBOARD7:30
Lemma26:
for b1 being non constant standard special_circular_sequence holds
len b1 > 1
theorem Th31: :: GOBOARD7:31
theorem Th32: :: GOBOARD7:32
theorem Th33: :: GOBOARD7:33
theorem Th34: :: GOBOARD7:34
theorem Th35: :: GOBOARD7:35
theorem Th36: :: GOBOARD7:36
theorem Th37: :: GOBOARD7:37
theorem Th38: :: GOBOARD7:38
theorem Th39: :: GOBOARD7:39
theorem Th40: :: GOBOARD7:40
theorem Th41: :: GOBOARD7:41
theorem Th42: :: GOBOARD7:42
theorem Th43: :: GOBOARD7:43
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,
(b2 + 1) & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * (b1 + 1),b
2 ) )
theorem Th44: :: GOBOARD7:44
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 <= len (GoB b4) & 1
<= b
2 & b
2 + 1
< width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * b1,(b2 + 2)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * b1,(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,
(b2 + 2) & b
4 /. (b3 + 1) = (GoB b4) * b
1,
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * b
1,b
2 ) )
theorem Th45: :: GOBOARD7:45
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * b1,(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * (b1 + 1),
(b2 + 1) & b
4 /. (b3 + 1) = (GoB b4) * b
1,
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * b
1,b
2 ) )
theorem Th46: :: GOBOARD7:46
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * (b1 + 1),b
2 & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * b
1,
(b2 + 1) ) )
theorem Th47: :: GOBOARD7:47
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
< len (GoB b4) & 1
<= b
2 & b
2 <= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 2),b2) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * (b1 + 1),b2) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * (b1 + 2),b
2 & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),b
2 & b
4 /. (b3 + 2) = (GoB b4) * b
1,b
2 ) )
theorem Th48: :: GOBOARD7:48
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * (b1 + 1),b2) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * (b1 + 1),
(b2 + 1) & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),b
2 & b
4 /. (b3 + 2) = (GoB b4) * b
1,b
2 ) )
theorem Th49: :: GOBOARD7:49
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * (b1 + 1),b
2 & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * b
1,
(b2 + 1) ) )
theorem Th50: :: GOBOARD7:50
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 <= len (GoB b4) & 1
<= b
2 & b
2 + 1
< width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * b1,(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * b1,(b2 + 2)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,b
2 & b
4 /. (b3 + 1) = (GoB b4) * b
1,
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * b
1,
(b2 + 2) ) )
theorem Th51: :: GOBOARD7:51
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * b1,(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,b
2 & b
4 /. (b3 + 1) = (GoB b4) * b
1,
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * (b1 + 1),
(b2 + 1) ) )
theorem Th52: :: GOBOARD7:52
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,(b2 + 1)),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,b
3 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,
(b2 + 1) & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),
(b2 + 1) & b
4 /. (b3 + 2) = (GoB b4) * (b1 + 1),b
2 ) )
theorem Th53: :: GOBOARD7:53
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
< len (GoB b4) & 1
<= b
2 & b
2 <= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * (b1 + 1),b2) = LSeg b
4,b
3 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 2),b2) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,b
2 & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),b
2 & b
4 /. (b3 + 2) = (GoB b4) * (b1 + 2),b
2 ) )
theorem Th54: :: GOBOARD7:54
for b
1, b
2, b
3 being
Natfor b
4 being non
constant standard special_circular_sequence holds
( 1
<= b
1 & b
1 + 1
<= len (GoB b4) & 1
<= b
2 & b
2 + 1
<= width (GoB b4) & 1
<= b
3 & b
3 + 1
< len b
4 &
LSeg ((GoB b4) * b1,b2),
((GoB b4) * (b1 + 1),b2) = LSeg b
4,b
3 &
LSeg ((GoB b4) * (b1 + 1),b2),
((GoB b4) * (b1 + 1),(b2 + 1)) = LSeg b
4,
(b3 + 1) implies ( b
4 /. b
3 = (GoB b4) * b
1,b
2 & b
4 /. (b3 + 1) = (GoB b4) * (b1 + 1),b
2 & b
4 /. (b3 + 2) = (GoB b4) * (b1 + 1),
(b2 + 1) ) )
theorem Th55: :: GOBOARD7:55
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 <= len (GoB b3) & 1
<= b
2 & b
2 + 1
< width (GoB b3) &
LSeg ((GoB b3) * b1,b2),
((GoB b3) * b1,(b2 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * b1,(b2 + 1)),
((GoB b3) * b1,(b2 + 2)) c= L~ b
3 & not ( b
3 /. 1
= (GoB b3) * b
1,
(b2 + 1) & ( ( b
3 /. 2
= (GoB b3) * b
1,b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,
(b2 + 2) ) or ( b
3 /. 2
= (GoB b3) * b
1,
(b2 + 2) & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,b
2 ) ) ) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 + 1
< len b
3 & b
3 /. (b4 + 1) = (GoB b3) * b
1,
(b2 + 1) & ( ( b
3 /. b
4 = (GoB b3) * b
1,b
2 & b
3 /. (b4 + 2) = (GoB b3) * b
1,
(b2 + 2) ) or ( b
3 /. b
4 = (GoB b3) * b
1,
(b2 + 2) & b
3 /. (b4 + 2) = (GoB b3) * b
1,b
2 ) ) ) ) )
theorem Th56: :: GOBOARD7:56
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 + 1
<= len (GoB b3) & 1
<= b
2 & b
2 + 1
<= width (GoB b3) &
LSeg ((GoB b3) * b1,b2),
((GoB b3) * b1,(b2 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * b1,(b2 + 1)),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 & not ( b
3 /. 1
= (GoB b3) * b
1,
(b2 + 1) & ( ( b
3 /. 2
= (GoB b3) * b
1,b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * (b1 + 1),
(b2 + 1) ) or ( b
3 /. 2
= (GoB b3) * (b1 + 1),
(b2 + 1) & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,b
2 ) ) ) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 + 1
< len b
3 & b
3 /. (b4 + 1) = (GoB b3) * b
1,
(b2 + 1) & ( ( b
3 /. b
4 = (GoB b3) * b
1,b
2 & b
3 /. (b4 + 2) = (GoB b3) * (b1 + 1),
(b2 + 1) ) or ( b
3 /. b
4 = (GoB b3) * (b1 + 1),
(b2 + 1) & b
3 /. (b4 + 2) = (GoB b3) * b
1,b
2 ) ) ) ) )
theorem Th57: :: GOBOARD7:57
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 + 1
<= len (GoB b3) & 1
<= b
2 & b
2 + 1
<= width (GoB b3) &
LSeg ((GoB b3) * b1,(b2 + 1)),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * (b1 + 1),(b2 + 1)),
((GoB b3) * (b1 + 1),b2) c= L~ b
3 & not ( b
3 /. 1
= (GoB b3) * (b1 + 1),
(b2 + 1) & ( ( b
3 /. 2
= (GoB b3) * b
1,
(b2 + 1) & b
3 /. ((len b3) -' 1) = (GoB b3) * (b1 + 1),b
2 ) or ( b
3 /. 2
= (GoB b3) * (b1 + 1),b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,
(b2 + 1) ) ) ) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 + 1
< len b
3 & b
3 /. (b4 + 1) = (GoB b3) * (b1 + 1),
(b2 + 1) & ( ( b
3 /. b
4 = (GoB b3) * b
1,
(b2 + 1) & b
3 /. (b4 + 2) = (GoB b3) * (b1 + 1),b
2 ) or ( b
3 /. b
4 = (GoB b3) * (b1 + 1),b
2 & b
3 /. (b4 + 2) = (GoB b3) * b
1,
(b2 + 1) ) ) ) ) )
theorem Th58: :: GOBOARD7:58
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 + 1
< len (GoB b3) & 1
<= b
2 & b
2 <= width (GoB b3) &
LSeg ((GoB b3) * b1,b2),
((GoB b3) * (b1 + 1),b2) c= L~ b
3 &
LSeg ((GoB b3) * (b1 + 1),b2),
((GoB b3) * (b1 + 2),b2) c= L~ b
3 & not ( b
3 /. 1
= (GoB b3) * (b1 + 1),b
2 & ( ( b
3 /. 2
= (GoB b3) * b
1,b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * (b1 + 2),b
2 ) or ( b
3 /. 2
= (GoB b3) * (b1 + 2),b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,b
2 ) ) ) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 + 1
< len b
3 & b
3 /. (b4 + 1) = (GoB b3) * (b1 + 1),b
2 & ( ( b
3 /. b
4 = (GoB b3) * b
1,b
2 & b
3 /. (b4 + 2) = (GoB b3) * (b1 + 2),b
2 ) or ( b
3 /. b
4 = (GoB b3) * (b1 + 2),b
2 & b
3 /. (b4 + 2) = (GoB b3) * b
1,b
2 ) ) ) ) )
theorem Th59: :: GOBOARD7:59
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 + 1
<= len (GoB b3) & 1
<= b
2 & b
2 + 1
<= width (GoB b3) &
LSeg ((GoB b3) * b1,b2),
((GoB b3) * (b1 + 1),b2) c= L~ b
3 &
LSeg ((GoB b3) * (b1 + 1),b2),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 & not ( b
3 /. 1
= (GoB b3) * (b1 + 1),b
2 & ( ( b
3 /. 2
= (GoB b3) * b
1,b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * (b1 + 1),
(b2 + 1) ) or ( b
3 /. 2
= (GoB b3) * (b1 + 1),
(b2 + 1) & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,b
2 ) ) ) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 + 1
< len b
3 & b
3 /. (b4 + 1) = (GoB b3) * (b1 + 1),b
2 & ( ( b
3 /. b
4 = (GoB b3) * b
1,b
2 & b
3 /. (b4 + 2) = (GoB b3) * (b1 + 1),
(b2 + 1) ) or ( b
3 /. b
4 = (GoB b3) * (b1 + 1),
(b2 + 1) & b
3 /. (b4 + 2) = (GoB b3) * b
1,b
2 ) ) ) ) )
theorem Th60: :: GOBOARD7:60
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 + 1
<= len (GoB b3) & 1
<= b
2 & b
2 + 1
<= width (GoB b3) &
LSeg ((GoB b3) * (b1 + 1),b2),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * (b1 + 1),(b2 + 1)),
((GoB b3) * b1,(b2 + 1)) c= L~ b
3 & not ( b
3 /. 1
= (GoB b3) * (b1 + 1),
(b2 + 1) & ( ( b
3 /. 2
= (GoB b3) * (b1 + 1),b
2 & b
3 /. ((len b3) -' 1) = (GoB b3) * b
1,
(b2 + 1) ) or ( b
3 /. 2
= (GoB b3) * b
1,
(b2 + 1) & b
3 /. ((len b3) -' 1) = (GoB b3) * (b1 + 1),b
2 ) ) ) & ( for b
4 being
Nat holds
not ( 1
<= b
4 & b
4 + 1
< len b
3 & b
3 /. (b4 + 1) = (GoB b3) * (b1 + 1),
(b2 + 1) & ( ( b
3 /. b
4 = (GoB b3) * (b1 + 1),b
2 & b
3 /. (b4 + 2) = (GoB b3) * b
1,
(b2 + 1) ) or ( b
3 /. b
4 = (GoB b3) * b
1,
(b2 + 1) & b
3 /. (b4 + 2) = (GoB b3) * (b1 + 1),b
2 ) ) ) ) )
theorem Th61: :: GOBOARD7:61
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 < len (GoB b3) & 1
<= b
2 & b
2 + 1
< width (GoB b3) &
LSeg ((GoB b3) * b1,b2),
((GoB b3) * b1,(b2 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * b1,(b2 + 1)),
((GoB b3) * b1,(b2 + 2)) c= L~ b
3 &
LSeg ((GoB b3) * b1,(b2 + 1)),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 )
theorem Th62: :: GOBOARD7:62
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 < len (GoB b3) & 1
<= b
2 & b
2 + 1
< width (GoB b3) &
LSeg ((GoB b3) * (b1 + 1),b2),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * (b1 + 1),(b2 + 1)),
((GoB b3) * (b1 + 1),(b2 + 2)) c= L~ b
3 &
LSeg ((GoB b3) * b1,(b2 + 1)),
((GoB b3) * (b1 + 1),(b2 + 1)) c= L~ b
3 )
theorem Th63: :: GOBOARD7:63
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 < width (GoB b3) & 1
<= b
2 & b
2 + 1
< len (GoB b3) &
LSeg ((GoB b3) * b2,b1),
((GoB b3) * (b2 + 1),b1) c= L~ b
3 &
LSeg ((GoB b3) * (b2 + 1),b1),
((GoB b3) * (b2 + 2),b1) c= L~ b
3 &
LSeg ((GoB b3) * (b2 + 1),b1),
((GoB b3) * (b2 + 1),(b1 + 1)) c= L~ b
3 )
theorem Th64: :: GOBOARD7:64
for b
1, b
2 being
Natfor b
3 being non
constant standard special_circular_sequence holds
not ( 1
<= b
1 & b
1 < width (GoB b3) & 1
<= b
2 & b
2 + 1
< len (GoB b3) &
LSeg ((GoB b3) * b2,(b1 + 1)),
((GoB b3) * (b2 + 1),(b1 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * (b2 + 1),(b1 + 1)),
((GoB b3) * (b2 + 2),(b1 + 1)) c= L~ b
3 &
LSeg ((GoB b3) * (b2 + 1),b1),
((GoB b3) * (b2 + 1),(b1 + 1)) c= L~ b
3 )
theorem Th65: :: GOBOARD7:65
theorem Th66: :: GOBOARD7:66