:: GOBOARD8 semantic presentation
theorem Th1: :: GOBOARD8:1
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len (GoB b1) & 1
<= b
4 & b
4 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b3 + 1),b
4 & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),
(b4 + 2) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),b
4 & b
1 /. b
2 = (GoB b1) * (b3 + 1),
(b4 + 2) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b3,b4) + ((GoB b1) * (b3 + 1),(b4 + 1)))),
((1 / 2) * (((GoB b1) * b3,(b4 + 1)) + ((GoB b1) * (b3 + 1),(b4 + 2)))) misses L~ b
1 ) )
theorem Th2: :: GOBOARD8:2
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & 1
<= b
4 & b
4 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b3 + 2),
(b4 + 1) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),
(b4 + 2) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b3 + 2),
(b4 + 1) & b
1 /. b
2 = (GoB b1) * (b3 + 1),
(b4 + 2) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b3,b4) + ((GoB b1) * (b3 + 1),(b4 + 1)))),
((1 / 2) * (((GoB b1) * b3,(b4 + 1)) + ((GoB b1) * (b3 + 1),(b4 + 2)))) misses L~ b
1 ) )
theorem Th3: :: GOBOARD8:3
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & 1
<= b
4 & b
4 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b3 + 2),
(b4 + 1) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),b
4 ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b3 + 2),
(b4 + 1) & b
1 /. b
2 = (GoB b1) * (b3 + 1),b
4 ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b3,b4) + ((GoB b1) * (b3 + 1),(b4 + 1)))),
((1 / 2) * (((GoB b1) * b3,(b4 + 1)) + ((GoB b1) * (b3 + 1),(b4 + 2)))) misses L~ b
1 ) )
theorem Th4: :: GOBOARD8:4
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len (GoB b1) & 1
<= b
4 & b
4 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * b
3,
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * b
3,b
4 & b
1 /. (b2 + 2) = (GoB b1) * b
3,
(b4 + 2) ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
3,b
4 & b
1 /. b
2 = (GoB b1) * b
3,
(b4 + 2) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b3,b4) + ((GoB b1) * (b3 + 1),(b4 + 1)))),
((1 / 2) * (((GoB b1) * b3,(b4 + 1)) + ((GoB b1) * (b3 + 1),(b4 + 2)))) misses L~ b
1 ) )
theorem Th5: :: GOBOARD8:5
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & 1
<= b
4 & b
4 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * b
3,
(b4 + 1) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),
(b4 + 2) ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
3,
(b4 + 1) & b
1 /. b
2 = (GoB b1) * (b3 + 1),
(b4 + 2) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * (b3 + 1),b4) + ((GoB b1) * (b3 + 2),(b4 + 1)))),
((1 / 2) * (((GoB b1) * (b3 + 1),(b4 + 1)) + ((GoB b1) * (b3 + 2),(b4 + 2)))) misses L~ b
1 ) )
theorem Th6: :: GOBOARD8:6
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & 1
<= b
4 & b
4 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * b
3,
(b4 + 1) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),b
4 ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
3,
(b4 + 1) & b
1 /. b
2 = (GoB b1) * (b3 + 1),b
4 ) ) implies
LSeg ((1 / 2) * (((GoB b1) * (b3 + 1),b4) + ((GoB b1) * (b3 + 2),(b4 + 1)))),
((1 / 2) * (((GoB b1) * (b3 + 1),(b4 + 1)) + ((GoB b1) * (b3 + 2),(b4 + 2)))) misses L~ b
1 ) )
theorem Th7: :: GOBOARD8:7
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),1 & ( ( b
1 /. b
2 = (GoB b1) * (b3 + 2),1 & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),2 ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b3 + 2),1 & b
1 /. b
2 = (GoB b1) * (b3 + 1),2 ) ) implies
LSeg (((1 / 2) * (((GoB b1) * b3,1) + ((GoB b1) * (b3 + 1),1))) - |[0,1]|),
((1 / 2) * (((GoB b1) * b3,1) + ((GoB b1) * (b3 + 1),2))) misses L~ b
1 ) )
theorem Th8: :: GOBOARD8:8
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),1 & ( ( b
1 /. b
2 = (GoB b1) * b
3,1 & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),2 ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
3,1 & b
1 /. b
2 = (GoB b1) * (b3 + 1),2 ) ) implies
LSeg (((1 / 2) * (((GoB b1) * (b3 + 1),1) + ((GoB b1) * (b3 + 2),1))) - |[0,1]|),
((1 / 2) * (((GoB b1) * (b3 + 1),1) + ((GoB b1) * (b3 + 2),2))) misses L~ b
1 ) )
theorem Th9: :: GOBOARD8:9
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(width (GoB b1)) & ( ( b
1 /. b
2 = (GoB b1) * (b3 + 2),
(width (GoB b1)) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),
((width (GoB b1)) -' 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b3 + 2),
(width (GoB b1)) & b
1 /. b
2 = (GoB b1) * (b3 + 1),
((width (GoB b1)) -' 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b3,((width (GoB b1)) -' 1)) + ((GoB b1) * (b3 + 1),(width (GoB b1))))),
(((1 / 2) * (((GoB b1) * b3,(width (GoB b1))) + ((GoB b1) * (b3 + 1),(width (GoB b1))))) + |[0,1]|) misses L~ b
1 ) )
theorem Th10: :: GOBOARD8:10
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(width (GoB b1)) & ( ( b
1 /. b
2 = (GoB b1) * b
3,
(width (GoB b1)) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 1),
((width (GoB b1)) -' 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
3,
(width (GoB b1)) & b
1 /. b
2 = (GoB b1) * (b3 + 1),
((width (GoB b1)) -' 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * (b3 + 1),((width (GoB b1)) -' 1)) + ((GoB b1) * (b3 + 2),(width (GoB b1))))),
(((1 / 2) * (((GoB b1) * (b3 + 1),(width (GoB b1))) + ((GoB b1) * (b3 + 2),(width (GoB b1))))) + |[0,1]|) misses L~ b
1 ) )
theorem Th11: :: GOBOARD8:11
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
4 & b
4 + 1
<= width (GoB b1) & 1
<= b
3 & b
3 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b3 + 1),
(b4 + 1) & ( ( b
1 /. b
2 = (GoB b1) * b
3,
(b4 + 1) & b
1 /. (b2 + 2) = (GoB b1) * (b3 + 2),
(b4 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
3,
(b4 + 1) & b
1 /. b
2 = (GoB b1) * (b3 + 2),
(b4 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b3,b4) + ((GoB b1) * (b3 + 1),(b4 + 1)))),
((1 / 2) * (((GoB b1) * (b3 + 1),b4) + ((GoB b1) * (b3 + 2),(b4 + 1)))) misses L~ b
1 ) )
theorem Th12: :: GOBOARD8:12
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & 1
<= b
4 & b
4 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b4 + 1),
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b4 + 1),
(b3 + 2) & b
1 /. (b2 + 2) = (GoB b1) * (b4 + 2),
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b4 + 1),
(b3 + 2) & b
1 /. b
2 = (GoB b1) * (b4 + 2),
(b3 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b4,b3) + ((GoB b1) * (b4 + 1),(b3 + 1)))),
((1 / 2) * (((GoB b1) * (b4 + 1),b3) + ((GoB b1) * (b4 + 2),(b3 + 1)))) misses L~ b
1 ) )
theorem Th13: :: GOBOARD8:13
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & 1
<= b
4 & b
4 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b4 + 1),
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b4 + 1),
(b3 + 2) & b
1 /. (b2 + 2) = (GoB b1) * b
4,
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b4 + 1),
(b3 + 2) & b
1 /. b
2 = (GoB b1) * b
4,
(b3 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b4,b3) + ((GoB b1) * (b4 + 1),(b3 + 1)))),
((1 / 2) * (((GoB b1) * (b4 + 1),b3) + ((GoB b1) * (b4 + 2),(b3 + 1)))) misses L~ b
1 ) )
theorem Th14: :: GOBOARD8:14
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= width (GoB b1) & 1
<= b
4 & b
4 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b4 + 1),b
3 & ( ( b
1 /. b
2 = (GoB b1) * b
4,b
3 & b
1 /. (b2 + 2) = (GoB b1) * (b4 + 2),b
3 ) or ( b
1 /. (b2 + 2) = (GoB b1) * b
4,b
3 & b
1 /. b
2 = (GoB b1) * (b4 + 2),b
3 ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b4,b3) + ((GoB b1) * (b4 + 1),(b3 + 1)))),
((1 / 2) * (((GoB b1) * (b4 + 1),b3) + ((GoB b1) * (b4 + 2),(b3 + 1)))) misses L~ b
1 ) )
theorem Th15: :: GOBOARD8:15
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & 1
<= b
4 & b
4 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b4 + 1),
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b4 + 1),b
3 & b
1 /. (b2 + 2) = (GoB b1) * (b4 + 2),
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b4 + 1),b
3 & b
1 /. b
2 = (GoB b1) * (b4 + 2),
(b3 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b4,(b3 + 1)) + ((GoB b1) * (b4 + 1),(b3 + 2)))),
((1 / 2) * (((GoB b1) * (b4 + 1),(b3 + 1)) + ((GoB b1) * (b4 + 2),(b3 + 2)))) misses L~ b
1 ) )
theorem Th16: :: GOBOARD8:16
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & 1
<= b
4 & b
4 + 2
<= len (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (b4 + 1),
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (b4 + 1),b
3 & b
1 /. (b2 + 2) = (GoB b1) * b
4,
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (b4 + 1),b
3 & b
1 /. b
2 = (GoB b1) * b
4,
(b3 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * b4,(b3 + 1)) + ((GoB b1) * (b4 + 1),(b3 + 2)))),
((1 / 2) * (((GoB b1) * (b4 + 1),(b3 + 1)) + ((GoB b1) * (b4 + 2),(b3 + 2)))) misses L~ b
1 ) )
theorem Th17: :: GOBOARD8:17
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * 1,
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * 1,
(b3 + 2) & b
1 /. (b2 + 2) = (GoB b1) * 2,
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * 1,
(b3 + 2) & b
1 /. b
2 = (GoB b1) * 2,
(b3 + 1) ) ) implies
LSeg (((1 / 2) * (((GoB b1) * 1,b3) + ((GoB b1) * 1,(b3 + 1)))) - |[1,0]|),
((1 / 2) * (((GoB b1) * 1,b3) + ((GoB b1) * 2,(b3 + 1)))) misses L~ b
1 ) )
theorem Th18: :: GOBOARD8:18
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * 1,
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * 1,b
3 & b
1 /. (b2 + 2) = (GoB b1) * 2,
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * 1,b
3 & b
1 /. b
2 = (GoB b1) * 2,
(b3 + 1) ) ) implies
LSeg (((1 / 2) * (((GoB b1) * 1,(b3 + 1)) + ((GoB b1) * 1,(b3 + 2)))) - |[1,0]|),
((1 / 2) * (((GoB b1) * 1,(b3 + 1)) + ((GoB b1) * 2,(b3 + 2)))) misses L~ b
1 ) )
theorem Th19: :: GOBOARD8:19
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (len (GoB b1)),
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (len (GoB b1)),
(b3 + 2) & b
1 /. (b2 + 2) = (GoB b1) * ((len (GoB b1)) -' 1),
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (len (GoB b1)),
(b3 + 2) & b
1 /. b
2 = (GoB b1) * ((len (GoB b1)) -' 1),
(b3 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * ((len (GoB b1)) -' 1),b3) + ((GoB b1) * (len (GoB b1)),(b3 + 1)))),
(((1 / 2) * (((GoB b1) * (len (GoB b1)),b3) + ((GoB b1) * (len (GoB b1)),(b3 + 1)))) + |[1,0]|) misses L~ b
1 ) )
theorem Th20: :: GOBOARD8:20
for b
1 being non
constant standard special_circular_sequencefor b
2 being
Nat holds
( 1
<= b
2 & b
2 + 2
<= len b
1 implies for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 2
<= width (GoB b1) & b
1 /. (b2 + 1) = (GoB b1) * (len (GoB b1)),
(b3 + 1) & ( ( b
1 /. b
2 = (GoB b1) * (len (GoB b1)),b
3 & b
1 /. (b2 + 2) = (GoB b1) * ((len (GoB b1)) -' 1),
(b3 + 1) ) or ( b
1 /. (b2 + 2) = (GoB b1) * (len (GoB b1)),b
3 & b
1 /. b
2 = (GoB b1) * ((len (GoB b1)) -' 1),
(b3 + 1) ) ) implies
LSeg ((1 / 2) * (((GoB b1) * ((len (GoB b1)) -' 1),(b3 + 1)) + ((GoB b1) * (len (GoB b1)),(b3 + 2)))),
(((1 / 2) * (((GoB b1) * (len (GoB b1)),(b3 + 1)) + ((GoB b1) * (len (GoB b1)),(b3 + 2)))) + |[1,0]|) misses L~ b
1 ) )
theorem Th21: :: GOBOARD8:21
theorem Th22: :: GOBOARD8:22
theorem Th23: :: GOBOARD8:23
theorem Th24: :: GOBOARD8:24
theorem Th25: :: GOBOARD8:25
theorem Th26: :: GOBOARD8:26
theorem Th27: :: GOBOARD8:27
theorem Th28: :: GOBOARD8:28
theorem Th29: :: GOBOARD8:29
theorem Th30: :: GOBOARD8:30
theorem Th31: :: GOBOARD8:31
theorem Th32: :: GOBOARD8:32
theorem Th33: :: GOBOARD8:33
theorem Th34: :: GOBOARD8:34
theorem Th35: :: GOBOARD8:35
theorem Th36: :: GOBOARD8:36
theorem Th37: :: GOBOARD8:37
theorem Th38: :: GOBOARD8:38