:: JORDAN8 semantic presentation
theorem Th1: :: JORDAN8:1
theorem Th2: :: JORDAN8:2
theorem Th3: :: JORDAN8:3
theorem Th4: :: JORDAN8:4
canceled;
theorem Th5: :: JORDAN8:5
theorem Th6: :: JORDAN8:6
theorem Th7: :: JORDAN8:7
theorem Th8: :: JORDAN8:8
theorem Th9: :: JORDAN8:9
for b
1 being
Go-boardfor b
2 being
Point of
(TOP-REAL 2)for b
3 being non
empty FinSequence of
(TOP-REAL 2) holds
( b
3 is_sequence_on b
1 & ex b
4, b
5 being
Nat st
(
[b4,b5] in Indices b
1 & b
2 = b
1 * b
4,b
5 ) & ( for b
4, b
5, b
6, b
7 being
Nat holds
(
[b4,b5] in Indices b
1 &
[b6,b7] in Indices b
1 & b
3 /. (len b3) = b
1 * b
4,b
5 & b
2 = b
1 * b
6,b
7 implies
(abs (b6 - b4)) + (abs (b7 - b5)) = 1 ) ) implies b
3 ^ <*b2*> is_sequence_on b
1 )
theorem Th10: :: JORDAN8:10
theorem Th11: :: JORDAN8:11
theorem Th12: :: JORDAN8:12
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2 be
Nat;
deffunc H
1(
Nat,
Nat)
-> Element of the
carrier of
(TOP-REAL 2) =
|[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (a1 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (a2 - 2)))]|;
E4:
(2 |^ c2) + 3
> 0
by NAT_1:19;
func Gauge c
1,c
2 -> Matrix of
(TOP-REAL 2) means :
Def1:
:: JORDAN8:def 1
(
len a
3 = (2 |^ a2) + 3 &
len a
3 = width a
3 & ( for b
1, b
2 being
Nat holds
(
[b1,b2] in Indices a
3 implies a
3 * b
1,b
2 = |[((W-bound a1) + ((((E-bound a1) - (W-bound a1)) / (2 |^ a2)) * (b1 - 2))),((S-bound a1) + ((((N-bound a1) - (S-bound a1)) / (2 |^ a2)) * (b2 - 2)))]| ) ) );
existence
ex b1 being Matrix of (TOP-REAL 2) st
( len b1 = (2 |^ c2) + 3 & len b1 = width b1 & ( for b2, b3 being Nat holds
( [b2,b3] in Indices b1 implies b1 * b2,b3 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (b2 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (b3 - 2)))]| ) ) )
uniqueness
for b1, b2 being Matrix of (TOP-REAL 2) holds
( len b1 = (2 |^ c2) + 3 & len b1 = width b1 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices b1 implies b1 * b3,b4 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (b3 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (b4 - 2)))]| ) ) & len b2 = (2 |^ c2) + 3 & len b2 = width b2 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices b2 implies b2 * b3,b4 = |[((W-bound c1) + ((((E-bound c1) - (W-bound c1)) / (2 |^ c2)) * (b3 - 2))),((S-bound c1) + ((((N-bound c1) - (S-bound c1)) / (2 |^ c2)) * (b4 - 2)))]| ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Gauge JORDAN8:def 1 :
theorem Th13: :: JORDAN8:13
theorem Th14: :: JORDAN8:14
theorem Th15: :: JORDAN8:15
theorem Th16: :: JORDAN8:16
theorem Th17: :: JORDAN8:17
theorem Th18: :: JORDAN8:18
theorem Th19: :: JORDAN8:19
theorem Th20: :: JORDAN8:20
theorem Th21: :: JORDAN8:21