:: GOBRD13 semantic presentation
:: deftheorem Def1 defines Values GOBRD13:def 1 :
theorem Th1: :: GOBRD13:1
theorem Th2: :: GOBRD13:2
canceled;
theorem Th3: :: GOBRD13:3
theorem Th4: :: GOBRD13:4
theorem Th5: :: GOBRD13:5
theorem Th6: :: GOBRD13:6
theorem Th7: :: GOBRD13:7
theorem Th8: :: GOBRD13:8
theorem Th9: :: GOBRD13:9
theorem Th10: :: GOBRD13:10
theorem Th11: :: GOBRD13:11
theorem Th12: :: GOBRD13:12
theorem Th13: :: GOBRD13:13
theorem Th14: :: GOBRD13:14
theorem Th15: :: GOBRD13:15
theorem Th16: :: GOBRD13:16
theorem Th17: :: GOBRD13:17
Lemma15:
for b1, b2 being Nat
for b3 being non empty set
for b4 being Matrix of b3 holds
( 1 <= b1 & b1 <= len b4 & 1 <= b2 & b2 <= width b4 implies b4 * b1,b2 in Values b4 )
theorem Th18: :: GOBRD13:18
for b
1, b
2, b
3, b
4 being
Natfor b
5, b
6 being
Go-board holds
(
Values b
5 c= Values b
6 &
[b1,b2] in Indices b
5 &
[b3,b4] in Indices b
6 & b
5 * b
1,b
2 = b
6 * b
3,b
4 implies
cell b
6,b
3,b
4 c= cell b
5,b
1,b
2 )
theorem Th19: :: GOBRD13:19
for b
1, b
2, b
3, b
4 being
Natfor b
5, b
6 being
Go-board holds
(
Values b
5 c= Values b
6 &
[b1,b2] in Indices b
5 &
[b3,b4] in Indices b
6 & b
5 * b
1,b
2 = b
6 * b
3,b
4 implies
cell b
6,
(b3 -' 1),b
4 c= cell b
5,
(b1 -' 1),b
2 )
theorem Th20: :: GOBRD13:20
for b
1, b
2, b
3, b
4 being
Natfor b
5, b
6 being
Go-board holds
(
Values b
5 c= Values b
6 &
[b1,b2] in Indices b
5 &
[b3,b4] in Indices b
6 & b
5 * b
1,b
2 = b
6 * b
3,b
4 implies
cell b
6,b
3,
(b4 -' 1) c= cell b
5,b
1,
(b2 -' 1) )
Lemma19:
for b1, b2 being Nat
for b3 being non empty FinSequence of (TOP-REAL 2) holds
not ( 1 <= b1 & b1 <= len (GoB b3) & 1 <= b2 & b2 <= width (GoB b3) & ( for b4 being Nat holds
not ( b4 in dom b3 & (b3 /. b4) `1 = ((GoB b3) * b1,b2) `1 ) ) )
Lemma20:
for b1, b2 being Nat
for b3 being non empty FinSequence of (TOP-REAL 2) holds
not ( 1 <= b1 & b1 <= len (GoB b3) & 1 <= b2 & b2 <= width (GoB b3) & ( for b4 being Nat holds
not ( b4 in dom b3 & (b3 /. b4) `2 = ((GoB b3) * b1,b2) `2 ) ) )
theorem Th21: :: GOBRD13:21
definition
let c
1 be
FinSequence of
(TOP-REAL 2);
let c
2 be
Go-board;
let c
3 be
Nat;
assume that E22:
( 1
<= c
3 & c
3 + 1
<= len c
1 )
and E23:
c
1 is_sequence_on c
2
;
consider c
4, c
5, c
6, c
7 being
Nat such that E24:
(
[c4,c5] in Indices c
2 & c
1 /. c
3 = c
2 * c
4,c
5 )
and E25:
(
[c6,c7] in Indices c
2 & c
1 /. (c3 + 1) = c
2 * c
6,c
7 )
and E26:
not ( not ( c
4 = c
6 & c
5 + 1
= c
7 ) & not ( c
4 + 1
= c
6 & c
5 = c
7 ) & not ( c
4 = c
6 + 1 & c
5 = c
7 ) & not ( c
4 = c
6 & c
5 = c
7 + 1 ) )
by E22, E23, JORDAN8:6;
func right_cell c
1,c
3,c
2 -> Subset of
(TOP-REAL 2) means :
Def2:
:: GOBRD13:def 2
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
2 &
[b3,b4] in Indices a
2 & a
1 /. a
3 = a
2 * b
1,b
2 & a
1 /. (a3 + 1) = a
2 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 & a
4 = cell a
2,b
1,b
2 ) & not ( b
1 + 1
= b
3 & b
2 = b
4 & a
4 = cell a
2,b
1,
(b2 -' 1) ) & not ( b
1 = b
3 + 1 & b
2 = b
4 & a
4 = cell a
2,b
3,b
4 ) & not ( b
1 = b
3 & b
2 = b
4 + 1 & a
4 = cell a
2,
(b1 -' 1),b
4 ) );
existence
ex b1 being Subset of (TOP-REAL 2) st
for b2, b3, b4, b5 being Nat holds
not ( [b2,b3] in Indices c2 & [b4,b5] in Indices c2 & c1 /. c3 = c2 * b2,b3 & c1 /. (c3 + 1) = c2 * b4,b5 & not ( b2 = b4 & b3 + 1 = b5 & b1 = cell c2,b2,b3 ) & not ( b2 + 1 = b4 & b3 = b5 & b1 = cell c2,b2,(b3 -' 1) ) & not ( b2 = b4 + 1 & b3 = b5 & b1 = cell c2,b4,b5 ) & not ( b2 = b4 & b3 = b5 + 1 & b1 = cell c2,(b2 -' 1),b5 ) )
uniqueness
for b1, b2 being Subset of (TOP-REAL 2) holds
( ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b1 = cell c2,b3,b4 ) & not ( b3 + 1 = b5 & b4 = b6 & b1 = cell c2,b3,(b4 -' 1) ) & not ( b3 = b5 + 1 & b4 = b6 & b1 = cell c2,b5,b6 ) & not ( b3 = b5 & b4 = b6 + 1 & b1 = cell c2,(b3 -' 1),b6 ) ) ) & ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b2 = cell c2,b3,b4 ) & not ( b3 + 1 = b5 & b4 = b6 & b2 = cell c2,b3,(b4 -' 1) ) & not ( b3 = b5 + 1 & b4 = b6 & b2 = cell c2,b5,b6 ) & not ( b3 = b5 & b4 = b6 + 1 & b2 = cell c2,(b3 -' 1),b6 ) ) ) implies b1 = b2 )
func left_cell c
1,c
3,c
2 -> Subset of
(TOP-REAL 2) means :
Def3:
:: GOBRD13:def 3
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
2 &
[b3,b4] in Indices a
2 & a
1 /. a
3 = a
2 * b
1,b
2 & a
1 /. (a3 + 1) = a
2 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 & a
4 = cell a
2,
(b1 -' 1),b
2 ) & not ( b
1 + 1
= b
3 & b
2 = b
4 & a
4 = cell a
2,b
1,b
2 ) & not ( b
1 = b
3 + 1 & b
2 = b
4 & a
4 = cell a
2,b
3,
(b4 -' 1) ) & not ( b
1 = b
3 & b
2 = b
4 + 1 & a
4 = cell a
2,b
1,b
4 ) );
existence
ex b1 being Subset of (TOP-REAL 2) st
for b2, b3, b4, b5 being Nat holds
not ( [b2,b3] in Indices c2 & [b4,b5] in Indices c2 & c1 /. c3 = c2 * b2,b3 & c1 /. (c3 + 1) = c2 * b4,b5 & not ( b2 = b4 & b3 + 1 = b5 & b1 = cell c2,(b2 -' 1),b3 ) & not ( b2 + 1 = b4 & b3 = b5 & b1 = cell c2,b2,b3 ) & not ( b2 = b4 + 1 & b3 = b5 & b1 = cell c2,b4,(b5 -' 1) ) & not ( b2 = b4 & b3 = b5 + 1 & b1 = cell c2,b2,b5 ) )
uniqueness
for b1, b2 being Subset of (TOP-REAL 2) holds
( ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b1 = cell c2,(b3 -' 1),b4 ) & not ( b3 + 1 = b5 & b4 = b6 & b1 = cell c2,b3,b4 ) & not ( b3 = b5 + 1 & b4 = b6 & b1 = cell c2,b5,(b6 -' 1) ) & not ( b3 = b5 & b4 = b6 + 1 & b1 = cell c2,b3,b6 ) ) ) & ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b2 = cell c2,(b3 -' 1),b4 ) & not ( b3 + 1 = b5 & b4 = b6 & b2 = cell c2,b3,b4 ) & not ( b3 = b5 + 1 & b4 = b6 & b2 = cell c2,b5,(b6 -' 1) ) & not ( b3 = b5 & b4 = b6 + 1 & b2 = cell c2,b3,b6 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines right_cell GOBRD13:def 2 :
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Go-boardfor b
3 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len b
1 & b
1 is_sequence_on b
2 implies for b
4 being
Subset of
(TOP-REAL 2) holds
( b
4 = right_cell b
1,b
3,b
2 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
2 &
[b7,b8] in Indices b
2 & b
1 /. b
3 = b
2 * b
5,b
6 & b
1 /. (b3 + 1) = b
2 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 & b
4 = cell b
2,b
5,b
6 ) & not ( b
5 + 1
= b
7 & b
6 = b
8 & b
4 = cell b
2,b
5,
(b6 -' 1) ) & not ( b
5 = b
7 + 1 & b
6 = b
8 & b
4 = cell b
2,b
7,b
8 ) & not ( b
5 = b
7 & b
6 = b
8 + 1 & b
4 = cell b
2,
(b5 -' 1),b
8 ) ) ) );
:: deftheorem Def3 defines left_cell GOBRD13:def 3 :
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Go-boardfor b
3 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len b
1 & b
1 is_sequence_on b
2 implies for b
4 being
Subset of
(TOP-REAL 2) holds
( b
4 = left_cell b
1,b
3,b
2 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
2 &
[b7,b8] in Indices b
2 & b
1 /. b
3 = b
2 * b
5,b
6 & b
1 /. (b3 + 1) = b
2 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 & b
4 = cell b
2,
(b5 -' 1),b
6 ) & not ( b
5 + 1
= b
7 & b
6 = b
8 & b
4 = cell b
2,b
5,b
6 ) & not ( b
5 = b
7 + 1 & b
6 = b
8 & b
4 = cell b
2,b
7,
(b8 -' 1) ) & not ( b
5 = b
7 & b
6 = b
8 + 1 & b
4 = cell b
2,b
5,b
8 ) ) ) );
theorem Th22: :: GOBRD13:22
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[b2,(b3 + 1)] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * b
2,
(b3 + 1) implies
left_cell b
4,b
1,b
5 = cell b
5,
(b2 -' 1),b
3 )
theorem Th23: :: GOBRD13:23
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[b2,(b3 + 1)] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * b
2,
(b3 + 1) implies
right_cell b
4,b
1,b
5 = cell b
5,b
2,b
3 )
theorem Th24: :: GOBRD13:24
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * (b2 + 1),b
3 implies
left_cell b
4,b
1,b
5 = cell b
5,b
2,b
3 )
theorem Th25: :: GOBRD13:25
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * (b2 + 1),b
3 implies
right_cell b
4,b
1,b
5 = cell b
5,b
2,
(b3 -' 1) )
theorem Th26: :: GOBRD13:26
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * (b2 + 1),b
3 & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
left_cell b
4,b
1,b
5 = cell b
5,b
2,
(b3 -' 1) )
theorem Th27: :: GOBRD13:27
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * (b2 + 1),b
3 & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
right_cell b
4,b
1,b
5 = cell b
5,b
2,b
3 )
theorem Th28: :: GOBRD13:28
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,(b3 + 1)] in Indices b
5 &
[b2,b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,
(b3 + 1) & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
left_cell b
4,b
1,b
5 = cell b
5,b
2,b
3 )
theorem Th29: :: GOBRD13:29
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,(b3 + 1)] in Indices b
5 &
[b2,b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,
(b3 + 1) & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
right_cell b
4,b
1,b
5 = cell b
5,
(b2 -' 1),b
3 )
theorem Th30: :: GOBRD13:30
theorem Th31: :: GOBRD13:31
theorem Th32: :: GOBRD13:32
theorem Th33: :: GOBRD13:33
theorem Th34: :: GOBRD13:34
definition
let c
1 be
FinSequence of
(TOP-REAL 2);
let c
2 be
Go-board;
let c
3 be
Nat;
assume that E32:
( 1
<= c
3 & c
3 + 1
<= len c
1 )
and E33:
c
1 is_sequence_on c
2
;
consider c
4, c
5, c
6, c
7 being
Nat such that E34:
(
[c4,c5] in Indices c
2 & c
1 /. c
3 = c
2 * c
4,c
5 )
and E35:
(
[c6,c7] in Indices c
2 & c
1 /. (c3 + 1) = c
2 * c
6,c
7 )
and E36:
not ( not ( c
4 = c
6 & c
5 + 1
= c
7 ) & not ( c
4 + 1
= c
6 & c
5 = c
7 ) & not ( c
4 = c
6 + 1 & c
5 = c
7 ) & not ( c
4 = c
6 & c
5 = c
7 + 1 ) )
by E32, E33, JORDAN8:6;
func front_right_cell c
1,c
3,c
2 -> Subset of
(TOP-REAL 2) means :
Def4:
:: GOBRD13:def 4
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
2 &
[b3,b4] in Indices a
2 & a
1 /. a
3 = a
2 * b
1,b
2 & a
1 /. (a3 + 1) = a
2 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 & a
4 = cell a
2,b
3,b
4 ) & not ( b
1 + 1
= b
3 & b
2 = b
4 & a
4 = cell a
2,b
3,
(b4 -' 1) ) & not ( b
1 = b
3 + 1 & b
2 = b
4 & a
4 = cell a
2,
(b3 -' 1),b
4 ) & not ( b
1 = b
3 & b
2 = b
4 + 1 & a
4 = cell a
2,
(b3 -' 1),
(b4 -' 1) ) );
existence
ex b1 being Subset of (TOP-REAL 2) st
for b2, b3, b4, b5 being Nat holds
not ( [b2,b3] in Indices c2 & [b4,b5] in Indices c2 & c1 /. c3 = c2 * b2,b3 & c1 /. (c3 + 1) = c2 * b4,b5 & not ( b2 = b4 & b3 + 1 = b5 & b1 = cell c2,b4,b5 ) & not ( b2 + 1 = b4 & b3 = b5 & b1 = cell c2,b4,(b5 -' 1) ) & not ( b2 = b4 + 1 & b3 = b5 & b1 = cell c2,(b4 -' 1),b5 ) & not ( b2 = b4 & b3 = b5 + 1 & b1 = cell c2,(b4 -' 1),(b5 -' 1) ) )
uniqueness
for b1, b2 being Subset of (TOP-REAL 2) holds
( ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b1 = cell c2,b5,b6 ) & not ( b3 + 1 = b5 & b4 = b6 & b1 = cell c2,b5,(b6 -' 1) ) & not ( b3 = b5 + 1 & b4 = b6 & b1 = cell c2,(b5 -' 1),b6 ) & not ( b3 = b5 & b4 = b6 + 1 & b1 = cell c2,(b5 -' 1),(b6 -' 1) ) ) ) & ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b2 = cell c2,b5,b6 ) & not ( b3 + 1 = b5 & b4 = b6 & b2 = cell c2,b5,(b6 -' 1) ) & not ( b3 = b5 + 1 & b4 = b6 & b2 = cell c2,(b5 -' 1),b6 ) & not ( b3 = b5 & b4 = b6 + 1 & b2 = cell c2,(b5 -' 1),(b6 -' 1) ) ) ) implies b1 = b2 )
func front_left_cell c
1,c
3,c
2 -> Subset of
(TOP-REAL 2) means :
Def5:
:: GOBRD13:def 5
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
2 &
[b3,b4] in Indices a
2 & a
1 /. a
3 = a
2 * b
1,b
2 & a
1 /. (a3 + 1) = a
2 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 & a
4 = cell a
2,
(b3 -' 1),b
4 ) & not ( b
1 + 1
= b
3 & b
2 = b
4 & a
4 = cell a
2,b
3,b
4 ) & not ( b
1 = b
3 + 1 & b
2 = b
4 & a
4 = cell a
2,
(b3 -' 1),
(b4 -' 1) ) & not ( b
1 = b
3 & b
2 = b
4 + 1 & a
4 = cell a
2,b
3,
(b4 -' 1) ) );
existence
ex b1 being Subset of (TOP-REAL 2) st
for b2, b3, b4, b5 being Nat holds
not ( [b2,b3] in Indices c2 & [b4,b5] in Indices c2 & c1 /. c3 = c2 * b2,b3 & c1 /. (c3 + 1) = c2 * b4,b5 & not ( b2 = b4 & b3 + 1 = b5 & b1 = cell c2,(b4 -' 1),b5 ) & not ( b2 + 1 = b4 & b3 = b5 & b1 = cell c2,b4,b5 ) & not ( b2 = b4 + 1 & b3 = b5 & b1 = cell c2,(b4 -' 1),(b5 -' 1) ) & not ( b2 = b4 & b3 = b5 + 1 & b1 = cell c2,b4,(b5 -' 1) ) )
uniqueness
for b1, b2 being Subset of (TOP-REAL 2) holds
( ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b1 = cell c2,(b5 -' 1),b6 ) & not ( b3 + 1 = b5 & b4 = b6 & b1 = cell c2,b5,b6 ) & not ( b3 = b5 + 1 & b4 = b6 & b1 = cell c2,(b5 -' 1),(b6 -' 1) ) & not ( b3 = b5 & b4 = b6 + 1 & b1 = cell c2,b5,(b6 -' 1) ) ) ) & ( for b3, b4, b5, b6 being Nat holds
not ( [b3,b4] in Indices c2 & [b5,b6] in Indices c2 & c1 /. c3 = c2 * b3,b4 & c1 /. (c3 + 1) = c2 * b5,b6 & not ( b3 = b5 & b4 + 1 = b6 & b2 = cell c2,(b5 -' 1),b6 ) & not ( b3 + 1 = b5 & b4 = b6 & b2 = cell c2,b5,b6 ) & not ( b3 = b5 + 1 & b4 = b6 & b2 = cell c2,(b5 -' 1),(b6 -' 1) ) & not ( b3 = b5 & b4 = b6 + 1 & b2 = cell c2,b5,(b6 -' 1) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines front_right_cell GOBRD13:def 4 :
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Go-boardfor b
3 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len b
1 & b
1 is_sequence_on b
2 implies for b
4 being
Subset of
(TOP-REAL 2) holds
( b
4 = front_right_cell b
1,b
3,b
2 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
2 &
[b7,b8] in Indices b
2 & b
1 /. b
3 = b
2 * b
5,b
6 & b
1 /. (b3 + 1) = b
2 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 & b
4 = cell b
2,b
7,b
8 ) & not ( b
5 + 1
= b
7 & b
6 = b
8 & b
4 = cell b
2,b
7,
(b8 -' 1) ) & not ( b
5 = b
7 + 1 & b
6 = b
8 & b
4 = cell b
2,
(b7 -' 1),b
8 ) & not ( b
5 = b
7 & b
6 = b
8 + 1 & b
4 = cell b
2,
(b7 -' 1),
(b8 -' 1) ) ) ) );
:: deftheorem Def5 defines front_left_cell GOBRD13:def 5 :
for b
1 being
FinSequence of
(TOP-REAL 2)for b
2 being
Go-boardfor b
3 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len b
1 & b
1 is_sequence_on b
2 implies for b
4 being
Subset of
(TOP-REAL 2) holds
( b
4 = front_left_cell b
1,b
3,b
2 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
2 &
[b7,b8] in Indices b
2 & b
1 /. b
3 = b
2 * b
5,b
6 & b
1 /. (b3 + 1) = b
2 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 & b
4 = cell b
2,
(b7 -' 1),b
8 ) & not ( b
5 + 1
= b
7 & b
6 = b
8 & b
4 = cell b
2,b
7,b
8 ) & not ( b
5 = b
7 + 1 & b
6 = b
8 & b
4 = cell b
2,
(b7 -' 1),
(b8 -' 1) ) & not ( b
5 = b
7 & b
6 = b
8 + 1 & b
4 = cell b
2,b
7,
(b8 -' 1) ) ) ) );
theorem Th35: :: GOBRD13:35
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[b2,(b3 + 1)] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * b
2,
(b3 + 1) implies
front_left_cell b
4,b
1,b
5 = cell b
5,
(b2 -' 1),
(b3 + 1) )
theorem Th36: :: GOBRD13:36
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[b2,(b3 + 1)] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * b
2,
(b3 + 1) implies
front_right_cell b
4,b
1,b
5 = cell b
5,b
2,
(b3 + 1) )
theorem Th37: :: GOBRD13:37
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * (b2 + 1),b
3 implies
front_left_cell b
4,b
1,b
5 = cell b
5,
(b2 + 1),b
3 )
theorem Th38: :: GOBRD13:38
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,b
3 & b
4 /. (b1 + 1) = b
5 * (b2 + 1),b
3 implies
front_right_cell b
4,b
1,b
5 = cell b
5,
(b2 + 1),
(b3 -' 1) )
theorem Th39: :: GOBRD13:39
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * (b2 + 1),b
3 & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
front_left_cell b
4,b
1,b
5 = cell b
5,
(b2 -' 1),
(b3 -' 1) )
theorem Th40: :: GOBRD13:40
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,b3] in Indices b
5 &
[(b2 + 1),b3] in Indices b
5 & b
4 /. b
1 = b
5 * (b2 + 1),b
3 & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
front_right_cell b
4,b
1,b
5 = cell b
5,
(b2 -' 1),b
3 )
theorem Th41: :: GOBRD13:41
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,(b3 + 1)] in Indices b
5 &
[b2,b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,
(b3 + 1) & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
front_left_cell b
4,b
1,b
5 = cell b
5,b
2,
(b3 -' 1) )
theorem Th42: :: GOBRD13:42
for b
1, b
2, b
3 being
Natfor b
4 being
FinSequence of
(TOP-REAL 2)for b
5 being
Go-board holds
( 1
<= b
1 & b
1 + 1
<= len b
4 & b
4 is_sequence_on b
5 &
[b2,(b3 + 1)] in Indices b
5 &
[b2,b3] in Indices b
5 & b
4 /. b
1 = b
5 * b
2,
(b3 + 1) & b
4 /. (b1 + 1) = b
5 * b
2,b
3 implies
front_right_cell b
4,b
1,b
5 = cell b
5,
(b2 -' 1),
(b3 -' 1) )
theorem Th43: :: GOBRD13:43
definition
let c
1 be
set ;
let c
2 be
FinSequence of c
1;
let c
3 be
Matrix of c
1;
let c
4 be
Nat;
pred c
2 turns_right c
4,c
3 means :
Def6:
:: GOBRD13:def 6
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
3 &
[b3,b4] in Indices a
3 & a
2 /. a
4 = a
3 * b
1,b
2 & a
2 /. (a4 + 1) = a
3 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 &
[(b3 + 1),b4] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * (b3 + 1),b
4 ) & not ( b
1 + 1
= b
3 & b
2 = b
4 &
[b3,(b4 -' 1)] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * b
3,
(b4 -' 1) ) & not ( b
1 = b
3 + 1 & b
2 = b
4 &
[b3,(b4 + 1)] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * b
3,
(b4 + 1) ) & not ( b
1 = b
3 & b
2 = b
4 + 1 &
[(b3 -' 1),b4] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * (b3 -' 1),b
4 ) );
pred c
2 turns_left c
4,c
3 means :
Def7:
:: GOBRD13:def 7
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
3 &
[b3,b4] in Indices a
3 & a
2 /. a
4 = a
3 * b
1,b
2 & a
2 /. (a4 + 1) = a
3 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 &
[(b3 -' 1),b4] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * (b3 -' 1),b
4 ) & not ( b
1 + 1
= b
3 & b
2 = b
4 &
[b3,(b4 + 1)] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * b
3,
(b4 + 1) ) & not ( b
1 = b
3 + 1 & b
2 = b
4 &
[b3,(b4 -' 1)] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * b
3,
(b4 -' 1) ) & not ( b
1 = b
3 & b
2 = b
4 + 1 &
[(b3 + 1),b4] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * (b3 + 1),b
4 ) );
pred c
2 goes_straight c
4,c
3 means :
Def8:
:: GOBRD13:def 8
for b
1, b
2, b
3, b
4 being
Nat holds
not (
[b1,b2] in Indices a
3 &
[b3,b4] in Indices a
3 & a
2 /. a
4 = a
3 * b
1,b
2 & a
2 /. (a4 + 1) = a
3 * b
3,b
4 & not ( b
1 = b
3 & b
2 + 1
= b
4 &
[b3,(b4 + 1)] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * b
3,
(b4 + 1) ) & not ( b
1 + 1
= b
3 & b
2 = b
4 &
[(b3 + 1),b4] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * (b3 + 1),b
4 ) & not ( b
1 = b
3 + 1 & b
2 = b
4 &
[(b3 -' 1),b4] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * (b3 -' 1),b
4 ) & not ( b
1 = b
3 & b
2 = b
4 + 1 &
[b3,(b4 -' 1)] in Indices a
3 & a
2 /. (a4 + 2) = a
3 * b
3,
(b4 -' 1) ) );
end;
:: deftheorem Def6 defines turns_right GOBRD13:def 6 :
for b
1 being
set for b
2 being
FinSequence of b
1for b
3 being
Matrix of b
1for b
4 being
Nat holds
( b
2 turns_right b
4,b
3 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
3 &
[b7,b8] in Indices b
3 & b
2 /. b
4 = b
3 * b
5,b
6 & b
2 /. (b4 + 1) = b
3 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 &
[(b7 + 1),b8] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * (b7 + 1),b
8 ) & not ( b
5 + 1
= b
7 & b
6 = b
8 &
[b7,(b8 -' 1)] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * b
7,
(b8 -' 1) ) & not ( b
5 = b
7 + 1 & b
6 = b
8 &
[b7,(b8 + 1)] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * b
7,
(b8 + 1) ) & not ( b
5 = b
7 & b
6 = b
8 + 1 &
[(b7 -' 1),b8] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * (b7 -' 1),b
8 ) ) );
:: deftheorem Def7 defines turns_left GOBRD13:def 7 :
for b
1 being
set for b
2 being
FinSequence of b
1for b
3 being
Matrix of b
1for b
4 being
Nat holds
( b
2 turns_left b
4,b
3 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
3 &
[b7,b8] in Indices b
3 & b
2 /. b
4 = b
3 * b
5,b
6 & b
2 /. (b4 + 1) = b
3 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 &
[(b7 -' 1),b8] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * (b7 -' 1),b
8 ) & not ( b
5 + 1
= b
7 & b
6 = b
8 &
[b7,(b8 + 1)] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * b
7,
(b8 + 1) ) & not ( b
5 = b
7 + 1 & b
6 = b
8 &
[b7,(b8 -' 1)] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * b
7,
(b8 -' 1) ) & not ( b
5 = b
7 & b
6 = b
8 + 1 &
[(b7 + 1),b8] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * (b7 + 1),b
8 ) ) );
:: deftheorem Def8 defines goes_straight GOBRD13:def 8 :
for b
1 being
set for b
2 being
FinSequence of b
1for b
3 being
Matrix of b
1for b
4 being
Nat holds
( b
2 goes_straight b
4,b
3 iff for b
5, b
6, b
7, b
8 being
Nat holds
not (
[b5,b6] in Indices b
3 &
[b7,b8] in Indices b
3 & b
2 /. b
4 = b
3 * b
5,b
6 & b
2 /. (b4 + 1) = b
3 * b
7,b
8 & not ( b
5 = b
7 & b
6 + 1
= b
8 &
[b7,(b8 + 1)] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * b
7,
(b8 + 1) ) & not ( b
5 + 1
= b
7 & b
6 = b
8 &
[(b7 + 1),b8] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * (b7 + 1),b
8 ) & not ( b
5 = b
7 + 1 & b
6 = b
8 &
[(b7 -' 1),b8] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * (b7 -' 1),b
8 ) & not ( b
5 = b
7 & b
6 = b
8 + 1 &
[b7,(b8 -' 1)] in Indices b
3 & b
2 /. (b4 + 2) = b
3 * b
7,
(b8 -' 1) ) ) );
theorem Th44: :: GOBRD13:44
theorem Th45: :: GOBRD13:45
theorem Th46: :: GOBRD13:46
theorem Th47: :: GOBRD13:47
theorem Th48: :: GOBRD13:48
theorem Th49: :: GOBRD13:49
theorem Th50: :: GOBRD13:50