:: JORDAN9 semantic presentation
Lemma1:
for b1 being Nat holds
( 1 <= b1 implies (b1 -' 1) + 2 = b1 + 1 )
theorem Th1: :: JORDAN9:1
canceled;
theorem Th2: :: JORDAN9:2
canceled;
theorem Th3: :: JORDAN9:3
theorem Th4: :: JORDAN9:4
for b
1 being non
empty set for b
2, b
3 being
FinSequence of b
1 holds
( ( for b
4 being
Nat holds b
2 | b
4 = b
3 | b
4 ) implies b
2 = b
3 )
theorem Th5: :: JORDAN9:5
theorem Th6: :: JORDAN9:6
theorem Th7: :: JORDAN9:7
theorem Th8: :: JORDAN9:8
Lemma8:
for b1 being Nat
for b2 being Go-board
for b3 being FinSequence of (TOP-REAL 2) holds
not ( b3 is_sequence_on b2 & 1 <= b1 & b1 <= len b3 & ( for b4, b5 being Nat holds
not ( [b4,b5] in Indices b2 & b3 /. b1 = b2 * b4,b5 ) ) )
theorem Th9: :: JORDAN9:9
theorem Th10: :: JORDAN9:10
theorem Th11: :: JORDAN9:11
theorem Th12: :: JORDAN9:12
theorem Th13: :: JORDAN9:13
theorem Th14: :: JORDAN9:14
theorem Th15: :: JORDAN9:15
theorem Th16: :: JORDAN9:16
theorem Th17: :: JORDAN9:17
theorem Th18: :: JORDAN9:18
theorem Th19: :: JORDAN9:19
theorem Th20: :: JORDAN9:20
theorem Th21: :: JORDAN9:21
theorem Th22: :: JORDAN9:22
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 ( b
3 * b
1,b
2 in cell b
3,b
1,b
2 & b
3 * b
1,
(b2 + 1) in cell b
3,b
1,b
2 & b
3 * (b1 + 1),
(b2 + 1) in cell b
3,b
1,b
2 & b
3 * (b1 + 1),b
2 in cell b
3,b
1,b
2 ) )
theorem Th23: :: JORDAN9:23
theorem Th24: :: JORDAN9:24
theorem Th25: :: JORDAN9:25
theorem Th26: :: JORDAN9:26
theorem Th27: :: JORDAN9:27
theorem Th28: :: JORDAN9:28
theorem Th29: :: JORDAN9:29
theorem Th30: :: JORDAN9:30
theorem Th31: :: JORDAN9:31
for b
1 being
Natfor b
2 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 + 1
<= len (Gauge b2,b1) &
N-min b
2 in cell (Gauge b2,b1),b
3,
((width (Gauge b2,b1)) -' 1) &
N-min b
2 <> (Gauge b2,b1) * b
3,
((width (Gauge b2,b1)) -' 1) & 1
<= b
4 & b
4 + 1
<= len (Gauge b2,b1) &
N-min b
2 in cell (Gauge b2,b1),b
4,
((width (Gauge b2,b1)) -' 1) &
N-min b
2 <> (Gauge b2,b1) * b
4,
((width (Gauge b2,b1)) -' 1) implies b
3 = b
4 )
theorem Th32: :: JORDAN9:32
for b
1 being
Natfor b
2 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
3 being non
constant standard special_circular_sequence holds
( b
3 is_sequence_on Gauge b
2,b
1 & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 + 1
<= len b
3 implies (
left_cell b
3,b
4,
(Gauge b2,b1) misses b
2 &
right_cell b
3,b
4,
(Gauge b2,b1) meets b
2 ) ) ) & ex b
4 being
Nat st
( 1
<= b
4 & b
4 + 1
<= len (Gauge b2,b1) & b
3 /. 1
= (Gauge b2,b1) * b
4,
(width (Gauge b2,b1)) & b
3 /. 2
= (Gauge b2,b1) * (b4 + 1),
(width (Gauge b2,b1)) &
N-min b
2 in cell (Gauge b2,b1),b
4,
((width (Gauge b2,b1)) -' 1) &
N-min b
2 <> (Gauge b2,b1) * b
4,
((width (Gauge b2,b1)) -' 1) ) implies
N-min (L~ b3) = b
3 /. 1 )
definition
let c
1 be non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2);
let c
2 be
Nat;
assume E30:
c
1 is
connected
;
func Cage c
1,c
2 -> non
constant standard clockwise_oriented special_circular_sequence means :
Def1:
:: JORDAN9:def 1
( a
3 is_sequence_on Gauge a
1,a
2 & ex b
1 being
Nat st
( 1
<= b
1 & b
1 + 1
<= len (Gauge a1,a2) & a
3 /. 1
= (Gauge a1,a2) * b
1,
(width (Gauge a1,a2)) & a
3 /. 2
= (Gauge a1,a2) * (b1 + 1),
(width (Gauge a1,a2)) &
N-min a
1 in cell (Gauge a1,a2),b
1,
((width (Gauge a1,a2)) -' 1) &
N-min a
1 <> (Gauge a1,a2) * b
1,
((width (Gauge a1,a2)) -' 1) ) & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 + 2
<= len a
3 implies ( (
front_left_cell a
3,b
1,
(Gauge a1,a2) misses a
1 &
front_right_cell a
3,b
1,
(Gauge a1,a2) misses a
1 implies a
3 turns_right b
1,
Gauge a
1,a
2 ) & (
front_left_cell a
3,b
1,
(Gauge a1,a2) misses a
1 &
front_right_cell a
3,b
1,
(Gauge a1,a2) meets a
1 implies a
3 goes_straight b
1,
Gauge a
1,a
2 ) & (
front_left_cell a
3,b
1,
(Gauge a1,a2) meets a
1 implies a
3 turns_left b
1,
Gauge a
1,a
2 ) ) ) ) );
existence
ex b1 being non constant standard clockwise_oriented special_circular_sequence st
( b1 is_sequence_on Gauge c1,c2 & ex b2 being Nat st
( 1 <= b2 & b2 + 1 <= len (Gauge c1,c2) & b1 /. 1 = (Gauge c1,c2) * b2,(width (Gauge c1,c2)) & b1 /. 2 = (Gauge c1,c2) * (b2 + 1),(width (Gauge c1,c2)) & N-min c1 in cell (Gauge c1,c2),b2,((width (Gauge c1,c2)) -' 1) & N-min c1 <> (Gauge c1,c2) * b2,((width (Gauge c1,c2)) -' 1) ) & ( for b2 being Nat holds
( 1 <= b2 & b2 + 2 <= len b1 implies ( ( front_left_cell b1,b2,(Gauge c1,c2) misses c1 & front_right_cell b1,b2,(Gauge c1,c2) misses c1 implies b1 turns_right b2, Gauge c1,c2 ) & ( front_left_cell b1,b2,(Gauge c1,c2) misses c1 & front_right_cell b1,b2,(Gauge c1,c2) meets c1 implies b1 goes_straight b2, Gauge c1,c2 ) & ( front_left_cell b1,b2,(Gauge c1,c2) meets c1 implies b1 turns_left b2, Gauge c1,c2 ) ) ) ) )
uniqueness
for b1, b2 being non constant standard clockwise_oriented special_circular_sequence holds
( b1 is_sequence_on Gauge c1,c2 & ex b3 being Nat st
( 1 <= b3 & b3 + 1 <= len (Gauge c1,c2) & b1 /. 1 = (Gauge c1,c2) * b3,(width (Gauge c1,c2)) & b1 /. 2 = (Gauge c1,c2) * (b3 + 1),(width (Gauge c1,c2)) & N-min c1 in cell (Gauge c1,c2),b3,((width (Gauge c1,c2)) -' 1) & N-min c1 <> (Gauge c1,c2) * b3,((width (Gauge c1,c2)) -' 1) ) & ( for b3 being Nat holds
( 1 <= b3 & b3 + 2 <= len b1 implies ( ( front_left_cell b1,b3,(Gauge c1,c2) misses c1 & front_right_cell b1,b3,(Gauge c1,c2) misses c1 implies b1 turns_right b3, Gauge c1,c2 ) & ( front_left_cell b1,b3,(Gauge c1,c2) misses c1 & front_right_cell b1,b3,(Gauge c1,c2) meets c1 implies b1 goes_straight b3, Gauge c1,c2 ) & ( front_left_cell b1,b3,(Gauge c1,c2) meets c1 implies b1 turns_left b3, Gauge c1,c2 ) ) ) ) & b2 is_sequence_on Gauge c1,c2 & ex b3 being Nat st
( 1 <= b3 & b3 + 1 <= len (Gauge c1,c2) & b2 /. 1 = (Gauge c1,c2) * b3,(width (Gauge c1,c2)) & b2 /. 2 = (Gauge c1,c2) * (b3 + 1),(width (Gauge c1,c2)) & N-min c1 in cell (Gauge c1,c2),b3,((width (Gauge c1,c2)) -' 1) & N-min c1 <> (Gauge c1,c2) * b3,((width (Gauge c1,c2)) -' 1) ) & ( for b3 being Nat holds
( 1 <= b3 & b3 + 2 <= len b2 implies ( ( front_left_cell b2,b3,(Gauge c1,c2) misses c1 & front_right_cell b2,b3,(Gauge c1,c2) misses c1 implies b2 turns_right b3, Gauge c1,c2 ) & ( front_left_cell b2,b3,(Gauge c1,c2) misses c1 & front_right_cell b2,b3,(Gauge c1,c2) meets c1 implies b2 goes_straight b3, Gauge c1,c2 ) & ( front_left_cell b2,b3,(Gauge c1,c2) meets c1 implies b2 turns_left b3, Gauge c1,c2 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Cage JORDAN9:def 1 :
for b
1 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
1 is
connected implies for b
3 being non
constant standard clockwise_oriented special_circular_sequence holds
( b
3 = Cage b
1,b
2 iff ( b
3 is_sequence_on Gauge b
1,b
2 & ex b
4 being
Nat st
( 1
<= b
4 & b
4 + 1
<= len (Gauge b1,b2) & b
3 /. 1
= (Gauge b1,b2) * b
4,
(width (Gauge b1,b2)) & b
3 /. 2
= (Gauge b1,b2) * (b4 + 1),
(width (Gauge b1,b2)) &
N-min b
1 in cell (Gauge b1,b2),b
4,
((width (Gauge b1,b2)) -' 1) &
N-min b
1 <> (Gauge b1,b2) * b
4,
((width (Gauge b1,b2)) -' 1) ) & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 + 2
<= len b
3 implies ( (
front_left_cell b
3,b
4,
(Gauge b1,b2) misses b
1 &
front_right_cell b
3,b
4,
(Gauge b1,b2) misses b
1 implies b
3 turns_right b
4,
Gauge b
1,b
2 ) & (
front_left_cell b
3,b
4,
(Gauge b1,b2) misses b
1 &
front_right_cell b
3,b
4,
(Gauge b1,b2) meets b
1 implies b
3 goes_straight b
4,
Gauge b
1,b
2 ) & (
front_left_cell b
3,b
4,
(Gauge b1,b2) meets b
1 implies b
3 turns_left b
4,
Gauge b
1,b
2 ) ) ) ) ) ) );
theorem Th33: :: JORDAN9:33
for b
1, b
2 being
Natfor b
3 being non
empty compact non
horizontal non
vertical Subset of
(TOP-REAL 2) holds
( b
3 is
connected & 1
<= b
1 & b
1 + 1
<= len (Cage b3,b2) implies (
left_cell (Cage b3,b2),b
1,
(Gauge b3,b2) misses b
3 &
right_cell (Cage b3,b2),b
1,
(Gauge b3,b2) meets b
3 ) )
theorem Th34: :: JORDAN9:34