:: FIN_TOPO semantic presentation
theorem Th1: :: FIN_TOPO:1
theorem Th2: :: FIN_TOPO:2
for b
1 being
set for b
2 being
FinSequence of
bool b
1 holds
( ( for b
3 being
Nat holds
( 1
<= b
3 & b
3 < len b
2 implies b
2 /. b
3 c= b
2 /. (b3 + 1) ) ) implies for b
3, b
4 being
Nat holds
( b
3 < b
4 & 1
<= b
3 & b
4 <= len b
2 & b
2 /. b
4 c= b
2 /. b
3 implies for b
5 being
Nat holds
( b
3 <= b
5 & b
5 <= b
4 implies b
2 /. b
4 = b
2 /. b
5 ) ) )
Lemma3:
for b1 being Function holds
( ( for b2 being Nat holds b1 . b2 c= b1 . (b2 + 1) ) implies for b2, b3 being Nat holds
( b2 <= b3 implies b1 . b2 c= b1 . b3 ) )
by MEASURE2:22;
:: deftheorem Def1 defines U_FT FIN_TOPO:def 1 :
:: deftheorem Def2 defines FT{0} FIN_TOPO:def 2 :
:: deftheorem Def3 defines filled FIN_TOPO:def 3 :
theorem Th3: :: FIN_TOPO:3
canceled;
theorem Th4: :: FIN_TOPO:4
canceled;
theorem Th5: :: FIN_TOPO:5
canceled;
theorem Th6: :: FIN_TOPO:6
canceled;
theorem Th7: :: FIN_TOPO:7
theorem Th8: :: FIN_TOPO:8
:: deftheorem Def4 FIN_TOPO:def 4 :
canceled;
:: deftheorem Def5 defines is_a_cover_of FIN_TOPO:def 5 :
theorem Th9: :: FIN_TOPO:9
:: deftheorem Def6 defines ^delta FIN_TOPO:def 6 :
theorem Th10: :: FIN_TOPO:10
:: deftheorem Def7 defines ^deltai FIN_TOPO:def 7 :
:: deftheorem Def8 defines ^deltao FIN_TOPO:def 8 :
theorem Th11: :: FIN_TOPO:11
:: deftheorem Def9 defines ^i FIN_TOPO:def 9 :
:: deftheorem Def10 defines ^b FIN_TOPO:def 10 :
:: deftheorem Def11 defines ^s FIN_TOPO:def 11 :
:: deftheorem Def12 defines ^n FIN_TOPO:def 12 :
:: deftheorem Def13 defines ^f FIN_TOPO:def 13 :
:: deftheorem Def14 defines symmetric FIN_TOPO:def 14 :
theorem Th12: :: FIN_TOPO:12
theorem Th13: :: FIN_TOPO:13
theorem Th14: :: FIN_TOPO:14
theorem Th15: :: FIN_TOPO:15
theorem Th16: :: FIN_TOPO:16
theorem Th17: :: FIN_TOPO:17
:: deftheorem Def15 defines open FIN_TOPO:def 15 :
:: deftheorem Def16 defines closed FIN_TOPO:def 16 :
:: deftheorem Def17 defines connected FIN_TOPO:def 17 :
:: deftheorem Def18 defines ^fb FIN_TOPO:def 18 :
:: deftheorem Def19 defines ^fi FIN_TOPO:def 19 :
theorem Th18: :: FIN_TOPO:18
theorem Th19: :: FIN_TOPO:19
theorem Th20: :: FIN_TOPO:20
theorem Th21: :: FIN_TOPO:21
theorem Th22: :: FIN_TOPO:22
theorem Th23: :: FIN_TOPO:23
theorem Th24: :: FIN_TOPO:24
theorem Th25: :: FIN_TOPO:25
theorem Th26: :: FIN_TOPO:26
theorem Th27: :: FIN_TOPO:27
theorem Th28: :: FIN_TOPO:28
theorem Th29: :: FIN_TOPO:29
for b
1 being
set for b
2, b
3 being
Subset of b
1 holds
( b
2 = b
3 iff b
2 ` = b
3 ` )
theorem Th30: :: FIN_TOPO:30
theorem Th31: :: FIN_TOPO:31