:: JORDAN_A semantic presentation
theorem Th1: :: JORDAN_A:1
theorem Th2: :: JORDAN_A:2
theorem Th3: :: JORDAN_A:3
theorem Th4: :: JORDAN_A:4
theorem Th5: :: JORDAN_A:5
theorem Th6: :: JORDAN_A:6
theorem Th7: :: JORDAN_A:7
definition
let c
1 be
Nat;
E6:
[:the carrier of (TOP-REAL c1),the carrier of (TOP-REAL c1):] = the
carrier of
[:(TOP-REAL c1),(TOP-REAL c1):]
by BORSUK_1:def 5;
func Eucl_dist c
1 -> RealMap of
[:(TOP-REAL a1),(TOP-REAL a1):] means :
Def1:
:: JORDAN_A:def 1
for b
1, b
2 being
Point of
(TOP-REAL a1) holds a
2 . b
1,b
2 = |.(b1 - b2).|;
existence
ex b1 being RealMap of [:(TOP-REAL c1),(TOP-REAL c1):] st
for b2, b3 being Point of (TOP-REAL c1) holds b1 . b2,b3 = |.(b2 - b3).|
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL c1),(TOP-REAL c1):] holds
( ( for b3, b4 being Point of (TOP-REAL c1) holds b1 . b3,b4 = |.(b3 - b4).| ) & ( for b3, b4 being Point of (TOP-REAL c1) holds b2 . b3,b4 = |.(b3 - b4).| ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Eucl_dist JORDAN_A:def 1 :
:: deftheorem Def2 defines continuous JORDAN_A:def 2 :
theorem Th8: :: JORDAN_A:8
theorem Th9: :: JORDAN_A:9
for b
1 being
Simple_closed_curvefor b
2, b
3 being
Point of
(TOP-REAL 2) holds
(
LE b
2,b
3,b
1 &
LE b
3,
E-max b
1,b
1 & b
2 <> b
3 implies
Segment b
2,b
3,b
1 = Segment (Upper_Arc b1),
(W-min b1),
(E-max b1),b
2,b
3 )
theorem Th10: :: JORDAN_A:10
theorem Th11: :: JORDAN_A:11
theorem Th12: :: JORDAN_A:12
for b
1 being
Simple_closed_curvefor b
2, b
3 being
Point of
(TOP-REAL 2) holds
(
LE b
2,b
3,b
1 &
LE E-max b
1,b
2,b
1 implies
Segment b
2,b
3,b
1 = Segment (Lower_Arc b1),
(E-max b1),
(W-min b1),b
2,b
3 )
theorem Th13: :: JORDAN_A:13
for b
1 being
Simple_closed_curvefor b
2, b
3 being
Point of
(TOP-REAL 2) holds
(
LE b
2,
E-max b
1,b
1 &
LE E-max b
1,b
3,b
1 implies
Segment b
2,b
3,b
1 = (R_Segment (Upper_Arc b1),(W-min b1),(E-max b1),b2) \/ (L_Segment (Lower_Arc b1),(E-max b1),(W-min b1),b3) )
theorem Th14: :: JORDAN_A:14
for b
1 being
Simple_closed_curvefor b
2 being
Point of
(TOP-REAL 2) holds
(
LE b
2,
E-max b
1,b
1 implies
Segment b
2,
(W-min b1),b
1 = (R_Segment (Upper_Arc b1),(W-min b1),(E-max b1),b2) \/ (L_Segment (Lower_Arc b1),(E-max b1),(W-min b1),(W-min b1)) )
theorem Th15: :: JORDAN_A:15
theorem Th16: :: JORDAN_A:16
theorem Th17: :: JORDAN_A:17
theorem Th18: :: JORDAN_A:18
theorem Th19: :: JORDAN_A:19
theorem Th20: :: JORDAN_A:20
theorem Th21: :: JORDAN_A:21
definition
let c
1 be
Simple_closed_curve;
mode Segmentation of c
1 -> FinSequence of
(TOP-REAL 2) means :
Def3:
:: JORDAN_A:def 3
( a
2 /. 1
= W-min a
1 & a
2 is
one-to-one & 8
<= len a
2 &
rng a
2 c= a
1 & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 < len a
2 implies
LE a
2 /. b
1,a
2 /. (b1 + 1),a
1 ) ) & ( for b
1 being
Nat holds
( 1
<= b
1 & b
1 + 1
< len a
2 implies
(Segment (a2 /. b1),(a2 /. (b1 + 1)),a1) /\ (Segment (a2 /. (b1 + 1)),(a2 /. (b1 + 2)),a1) = {(a2 /. (b1 + 1))} ) ) &
(Segment (a2 /. (len a2)),(a2 /. 1),a1) /\ (Segment (a2 /. 1),(a2 /. 2),a1) = {(a2 /. 1)} &
(Segment (a2 /. ((len a2) -' 1)),(a2 /. (len a2)),a1) /\ (Segment (a2 /. (len a2)),(a2 /. 1),a1) = {(a2 /. (len a2))} &
Segment (a2 /. ((len a2) -' 1)),
(a2 /. (len a2)),a
1 misses Segment (a2 /. 1),
(a2 /. 2),a
1 & ( for b
1, b
2 being
Nat holds
( 1
<= b
1 & b
1 < b
2 & b
2 < len a
2 & not b
1,b
2 are_adjacent1 implies
Segment (a2 /. b1),
(a2 /. (b1 + 1)),a
1 misses Segment (a2 /. b2),
(a2 /. (b2 + 1)),a
1 ) ) & ( for b
1 being
Nat holds
( 1
< b
1 & b
1 + 1
< len a
2 implies
Segment (a2 /. (len a2)),
(a2 /. 1),a
1 misses Segment (a2 /. b1),
(a2 /. (b1 + 1)),a
1 ) ) );
existence
ex b1 being FinSequence of (TOP-REAL 2) st
( b1 /. 1 = W-min c1 & b1 is one-to-one & 8 <= len b1 & rng b1 c= c1 & ( for b2 being Nat holds
( 1 <= b2 & b2 < len b1 implies LE b1 /. b2,b1 /. (b2 + 1),c1 ) ) & ( for b2 being Nat holds
( 1 <= b2 & b2 + 1 < len b1 implies (Segment (b1 /. b2),(b1 /. (b2 + 1)),c1) /\ (Segment (b1 /. (b2 + 1)),(b1 /. (b2 + 2)),c1) = {(b1 /. (b2 + 1))} ) ) & (Segment (b1 /. (len b1)),(b1 /. 1),c1) /\ (Segment (b1 /. 1),(b1 /. 2),c1) = {(b1 /. 1)} & (Segment (b1 /. ((len b1) -' 1)),(b1 /. (len b1)),c1) /\ (Segment (b1 /. (len b1)),(b1 /. 1),c1) = {(b1 /. (len b1))} & Segment (b1 /. ((len b1) -' 1)),(b1 /. (len b1)),c1 misses Segment (b1 /. 1),(b1 /. 2),c1 & ( for b2, b3 being Nat holds
( 1 <= b2 & b2 < b3 & b3 < len b1 & not b2,b3 are_adjacent1 implies Segment (b1 /. b2),(b1 /. (b2 + 1)),c1 misses Segment (b1 /. b3),(b1 /. (b3 + 1)),c1 ) ) & ( for b2 being Nat holds
( 1 < b2 & b2 + 1 < len b1 implies Segment (b1 /. (len b1)),(b1 /. 1),c1 misses Segment (b1 /. b2),(b1 /. (b2 + 1)),c1 ) ) )
end;
:: deftheorem Def3 defines Segmentation JORDAN_A:def 3 :
for b
1 being
Simple_closed_curvefor b
2 being
FinSequence of
(TOP-REAL 2) holds
( b
2 is
Segmentation of b
1 iff ( b
2 /. 1
= W-min b
1 & b
2 is
one-to-one & 8
<= len b
2 &
rng b
2 c= b
1 & ( for b
3 being
Nat holds
( 1
<= b
3 & b
3 < len b
2 implies
LE b
2 /. b
3,b
2 /. (b3 + 1),b
1 ) ) & ( for b
3 being
Nat holds
( 1
<= b
3 & b
3 + 1
< len b
2 implies
(Segment (b2 /. b3),(b2 /. (b3 + 1)),b1) /\ (Segment (b2 /. (b3 + 1)),(b2 /. (b3 + 2)),b1) = {(b2 /. (b3 + 1))} ) ) &
(Segment (b2 /. (len b2)),(b2 /. 1),b1) /\ (Segment (b2 /. 1),(b2 /. 2),b1) = {(b2 /. 1)} &
(Segment (b2 /. ((len b2) -' 1)),(b2 /. (len b2)),b1) /\ (Segment (b2 /. (len b2)),(b2 /. 1),b1) = {(b2 /. (len b2))} &
Segment (b2 /. ((len b2) -' 1)),
(b2 /. (len b2)),b
1 misses Segment (b2 /. 1),
(b2 /. 2),b
1 & ( for b
3, b
4 being
Nat holds
( 1
<= b
3 & b
3 < b
4 & b
4 < len b
2 & not b
3,b
4 are_adjacent1 implies
Segment (b2 /. b3),
(b2 /. (b3 + 1)),b
1 misses Segment (b2 /. b4),
(b2 /. (b4 + 1)),b
1 ) ) & ( for b
3 being
Nat holds
( 1
< b
3 & b
3 + 1
< len b
2 implies
Segment (b2 /. (len b2)),
(b2 /. 1),b
1 misses Segment (b2 /. b3),
(b2 /. (b3 + 1)),b
1 ) ) ) );
theorem Th22: :: JORDAN_A:22
:: deftheorem Def4 defines Segm JORDAN_A:def 4 :
theorem Th23: :: JORDAN_A:23
theorem Th24: :: JORDAN_A:24
theorem Th25: :: JORDAN_A:25
theorem Th26: :: JORDAN_A:26
theorem Th27: :: JORDAN_A:27
theorem Th28: :: JORDAN_A:28
theorem Th29: :: JORDAN_A:29
theorem Th30: :: JORDAN_A:30
theorem Th31: :: JORDAN_A:31
theorem Th32: :: JORDAN_A:32
:: deftheorem Def5 defines diameter JORDAN_A:def 5 :
:: deftheorem Def6 defines diameter JORDAN_A:def 6 :
theorem Th33: :: JORDAN_A:33
theorem Th34: :: JORDAN_A:34
theorem Th35: :: JORDAN_A:35
definition
let c
1 be
Simple_closed_curve;
let c
2 be
Segmentation of c
1;
func S-Gap c
2 -> Real means :
Def7:
:: JORDAN_A:def 7
ex b
1, b
2 being non
empty finite Subset of
REAL st
( b
1 = { (dist_min (Segm a2,b3),(Segm a2,b4)) where B is Nat, B is Nat : ( 1 <= b3 & b3 < b4 & b4 < len a2 & not b3,b4 are_adjacent1 ) } & b
2 = { (dist_min (Segm a2,(len a2)),(Segm a2,b3)) where B is Nat : ( 1 < b3 & b3 < (len a2) -' 1 ) } & a
3 = min (min b1),
(min b2) );
existence
ex b1 being Realex b2, b3 being non empty finite Subset of REAL st
( b2 = { (dist_min (Segm c2,b4),(Segm c2,b5)) where B is Nat, B is Nat : ( 1 <= b4 & b4 < b5 & b5 < len c2 & not b4,b5 are_adjacent1 ) } & b3 = { (dist_min (Segm c2,(len c2)),(Segm c2,b4)) where B is Nat : ( 1 < b4 & b4 < (len c2) -' 1 ) } & b1 = min (min b2),(min b3) )
correctness
uniqueness
for b1, b2 being Real holds
( ex b3, b4 being non empty finite Subset of REAL st
( b3 = { (dist_min (Segm c2,b5),(Segm c2,b6)) where B is Nat, B is Nat : ( 1 <= b5 & b5 < b6 & b6 < len c2 & not b5,b6 are_adjacent1 ) } & b4 = { (dist_min (Segm c2,(len c2)),(Segm c2,b5)) where B is Nat : ( 1 < b5 & b5 < (len c2) -' 1 ) } & b1 = min (min b3),(min b4) ) & ex b3, b4 being non empty finite Subset of REAL st
( b3 = { (dist_min (Segm c2,b5),(Segm c2,b6)) where B is Nat, B is Nat : ( 1 <= b5 & b5 < b6 & b6 < len c2 & not b5,b6 are_adjacent1 ) } & b4 = { (dist_min (Segm c2,(len c2)),(Segm c2,b5)) where B is Nat : ( 1 < b5 & b5 < (len c2) -' 1 ) } & b2 = min (min b3),(min b4) ) implies b1 = b2 );
;
end;
:: deftheorem Def7 defines S-Gap JORDAN_A:def 7 :
for b
1 being
Simple_closed_curvefor b
2 being
Segmentation of b
1for b
3 being
Real holds
( b
3 = S-Gap b
2 iff ex b
4, b
5 being non
empty finite Subset of
REAL st
( b
4 = { (dist_min (Segm b2,b6),(Segm b2,b7)) where B is Nat, B is Nat : ( 1 <= b6 & b6 < b7 & b7 < len b2 & not b6,b7 are_adjacent1 ) } & b
5 = { (dist_min (Segm b2,(len b2)),(Segm b2,b6)) where B is Nat : ( 1 < b6 & b6 < (len b2) -' 1 ) } & b
3 = min (min b4),
(min b5) ) );
theorem Th36: :: JORDAN_A:36
theorem Th37: :: JORDAN_A:37