:: JORDAN11 semantic presentation
Lemma1:
for b1, b2, b3 being Nat
for b4 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
( b1 > 0 & 1 <= b2 & b2 <= width (Gauge b4,b1) & b3 <= b2 & 1 <= b3 & b3 <= width (Gauge b4,b1) & (LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),b2),((Gauge b4,b1) * (Center (Gauge b4,b1)),b3)) /\ (Upper_Arc (L~ (Cage b4,b1))) = {((Gauge b4,b1) * (Center (Gauge b4,b1)),b2)} & (LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),b2),((Gauge b4,b1) * (Center (Gauge b4,b1)),b3)) /\ (Lower_Arc (L~ (Cage b4,b1))) = {((Gauge b4,b1) * (Center (Gauge b4,b1)),b3)} implies LSeg ((Gauge b4,b1) * (Center (Gauge b4,b1)),b2),((Gauge b4,b1) * (Center (Gauge b4,b1)),b3) c= Cl (RightComp (Cage b4,b1)) )
Lemma2:
for b1 being being_simple_closed_curve Subset of (TOP-REAL 2) holds
ex b2 being Nat st b2 is_sufficiently_large_for b1
:: deftheorem Def1 defines ApproxIndex JORDAN11:def 1 :
theorem Th1: :: JORDAN11:1
definition
let c
1 be
being_simple_closed_curve Subset of
(TOP-REAL 2);
func Y-InitStart c
1 -> Nat means :
Def2:
:: JORDAN11:def 2
( a
2 < width (Gauge a1,(ApproxIndex a1)) &
cell (Gauge a1,(ApproxIndex a1)),
((X-SpanStart a1,(ApproxIndex a1)) -' 1),a
2 c= BDD a
1 & ( for b
1 being
Nat holds
( b
1 < width (Gauge a1,(ApproxIndex a1)) &
cell (Gauge a1,(ApproxIndex a1)),
((X-SpanStart a1,(ApproxIndex a1)) -' 1),b
1 c= BDD a
1 implies b
1 >= a
2 ) ) );
existence
ex b1 being Nat st
( b1 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b1 c= BDD c1 & ( for b2 being Nat holds
( b2 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b2 c= BDD c1 implies b2 >= b1 ) ) )
uniqueness
for b1, b2 being Nat holds
( b1 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b1 c= BDD c1 & ( for b3 being Nat holds
( b3 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b3 c= BDD c1 implies b3 >= b1 ) ) & b2 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b2 c= BDD c1 & ( for b3 being Nat holds
( b3 < width (Gauge c1,(ApproxIndex c1)) & cell (Gauge c1,(ApproxIndex c1)),((X-SpanStart c1,(ApproxIndex c1)) -' 1),b3 c= BDD c1 implies b3 >= b2 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Y-InitStart JORDAN11:def 2 :
for b
1 being
being_simple_closed_curve Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
2 = Y-InitStart b
1 iff ( b
2 < width (Gauge b1,(ApproxIndex b1)) &
cell (Gauge b1,(ApproxIndex b1)),
((X-SpanStart b1,(ApproxIndex b1)) -' 1),b
2 c= BDD b
1 & ( for b
3 being
Nat holds
( b
3 < width (Gauge b1,(ApproxIndex b1)) &
cell (Gauge b1,(ApproxIndex b1)),
((X-SpanStart b1,(ApproxIndex b1)) -' 1),b
3 c= BDD b
1 implies b
3 >= b
2 ) ) ) );
theorem Th2: :: JORDAN11:2
theorem Th3: :: JORDAN11:3
definition
let c
1 be
being_simple_closed_curve Subset of
(TOP-REAL 2);
let c
2 be
Nat;
assume E8:
c
2 is_sufficiently_large_for c
1
;
E9:
c
2 >= ApproxIndex c
1
by E8, Def1;
set c
3 =
X-SpanStart c
1,c
2;
func Y-SpanStart c
1,c
2 -> Nat means :
Def3:
:: JORDAN11:def 3
( a
3 <= width (Gauge a1,a2) & ( for b
1 being
Nat holds
( a
3 <= b
1 & b
1 <= ((2 |^ (a2 -' (ApproxIndex a1))) * ((Y-InitStart a1) -' 2)) + 2 implies
cell (Gauge a1,a2),
((X-SpanStart a1,a2) -' 1),b
1 c= BDD a
1 ) ) & ( for b
1 being
Nat holds
( b
1 <= width (Gauge a1,a2) & ( for b
2 being
Nat holds
( b
1 <= b
2 & b
2 <= ((2 |^ (a2 -' (ApproxIndex a1))) * ((Y-InitStart a1) -' 2)) + 2 implies
cell (Gauge a1,a2),
((X-SpanStart a1,a2) -' 1),b
2 c= BDD a
1 ) ) implies b
1 >= a
3 ) ) );
existence
ex b1 being Nat st
( b1 <= width (Gauge c1,c2) & ( for b2 being Nat holds
( b1 <= b2 & b2 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b2 c= BDD c1 ) ) & ( for b2 being Nat holds
( b2 <= width (Gauge c1,c2) & ( for b3 being Nat holds
( b2 <= b3 & b3 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b3 c= BDD c1 ) ) implies b2 >= b1 ) ) )
uniqueness
for b1, b2 being Nat holds
( b1 <= width (Gauge c1,c2) & ( for b3 being Nat holds
( b1 <= b3 & b3 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b3 c= BDD c1 ) ) & ( for b3 being Nat holds
( b3 <= width (Gauge c1,c2) & ( for b4 being Nat holds
( b3 <= b4 & b4 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b4 c= BDD c1 ) ) implies b3 >= b1 ) ) & b2 <= width (Gauge c1,c2) & ( for b3 being Nat holds
( b2 <= b3 & b3 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b3 c= BDD c1 ) ) & ( for b3 being Nat holds
( b3 <= width (Gauge c1,c2) & ( for b4 being Nat holds
( b3 <= b4 & b4 <= ((2 |^ (c2 -' (ApproxIndex c1))) * ((Y-InitStart c1) -' 2)) + 2 implies cell (Gauge c1,c2),((X-SpanStart c1,c2) -' 1),b4 c= BDD c1 ) ) implies b3 >= b2 ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines Y-SpanStart JORDAN11:def 3 :
for b
1 being
being_simple_closed_curve Subset of
(TOP-REAL 2)for b
2 being
Nat holds
( b
2 is_sufficiently_large_for b
1 implies for b
3 being
Nat holds
( b
3 = Y-SpanStart b
1,b
2 iff ( b
3 <= width (Gauge b1,b2) & ( for b
4 being
Nat holds
( b
3 <= b
4 & b
4 <= ((2 |^ (b2 -' (ApproxIndex b1))) * ((Y-InitStart b1) -' 2)) + 2 implies
cell (Gauge b1,b2),
((X-SpanStart b1,b2) -' 1),b
4 c= BDD b
1 ) ) & ( for b
4 being
Nat holds
( b
4 <= width (Gauge b1,b2) & ( for b
5 being
Nat holds
( b
4 <= b
5 & b
5 <= ((2 |^ (b2 -' (ApproxIndex b1))) * ((Y-InitStart b1) -' 2)) + 2 implies
cell (Gauge b1,b2),
((X-SpanStart b1,b2) -' 1),b
5 c= BDD b
1 ) ) implies b
4 >= b
3 ) ) ) ) );
theorem Th4: :: JORDAN11:4
theorem Th5: :: JORDAN11:5
theorem Th6: :: JORDAN11:6
theorem Th7: :: JORDAN11:7
theorem Th8: :: JORDAN11:8
theorem Th9: :: JORDAN11:9
theorem Th10: :: JORDAN11:10
theorem Th11: :: JORDAN11:11
theorem Th12: :: JORDAN11:12
theorem Th13: :: JORDAN11:13
theorem Th14: :: JORDAN11:14
theorem Th15: :: JORDAN11:15
theorem Th16: :: JORDAN11:16
theorem Th17: :: JORDAN11:17