:: BINARI_3 semantic presentation
theorem Th1: :: BINARI_3:1
theorem Th2: :: BINARI_3:2
theorem Th3: :: BINARI_3:3
theorem Th4: :: BINARI_3:4
theorem Th5: :: BINARI_3:5
theorem Th6: :: BINARI_3:6
theorem Th7: :: BINARI_3:7
theorem Th8: :: BINARI_3:8
theorem Th9: :: BINARI_3:9
theorem Th10: :: BINARI_3:10
theorem Th11: :: BINARI_3:11
theorem Th12: :: BINARI_3:12
theorem Th13: :: BINARI_3:13
theorem Th14: :: BINARI_3:14
theorem Th15: :: BINARI_3:15
theorem Th16: :: BINARI_3:16
theorem Th17: :: BINARI_3:17
theorem Th18: :: BINARI_3:18
theorem Th19: :: BINARI_3:19
theorem Th20: :: BINARI_3:20
theorem Th21: :: BINARI_3:21
theorem Th22: :: BINARI_3:22
theorem Th23: :: BINARI_3:23
theorem Th24: :: BINARI_3:24
theorem Th25: :: BINARI_3:25
definition
let c
1, c
2 be
Nat;
func c
1 -BinarySequence c
2 -> Tuple of a
1,
BOOLEAN means :
Def1:
:: BINARI_3:def 1
for b
1 being
Nat holds
( b
1 in Seg a
1 implies a
3 /. b
1 = IFEQ ((a2 div (2 to_power (b1 -' 1))) mod 2),0,
FALSE ,
TRUE );
existence
ex b1 being Tuple of c1,BOOLEAN st
for b2 being Nat holds
( b2 in Seg c1 implies b1 /. b2 = IFEQ ((c2 div (2 to_power (b2 -' 1))) mod 2),0,FALSE ,TRUE )
uniqueness
for b1, b2 being Tuple of c1,BOOLEAN holds
( ( for b3 being Nat holds
( b3 in Seg c1 implies b1 /. b3 = IFEQ ((c2 div (2 to_power (b3 -' 1))) mod 2),0,FALSE ,TRUE ) ) & ( for b3 being Nat holds
( b3 in Seg c1 implies b2 /. b3 = IFEQ ((c2 div (2 to_power (b3 -' 1))) mod 2),0,FALSE ,TRUE ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines -BinarySequence BINARI_3:def 1 :
theorem Th26: :: BINARI_3:26
theorem Th27: :: BINARI_3:27
theorem Th28: :: BINARI_3:28
Lemma23:
for b1 being non empty Nat holds (b1 + 1) -BinarySequence (2 to_power b1) = (0* b1) ^ <*TRUE *>
Lemma24:
for b1 being non empty Nat
for b2 being Nat holds
( 2 to_power b1 <= b2 & b2 < 2 to_power (b1 + 1) implies ((b1 + 1) -BinarySequence b2) . (b1 + 1) = TRUE )
Lemma25:
for b1 being non empty Nat
for b2 being Nat holds
( 2 to_power b1 <= b2 & b2 < 2 to_power (b1 + 1) implies (b1 + 1) -BinarySequence b2 = (b1 -BinarySequence (b2 -' (2 to_power b1))) ^ <*TRUE *> )
Lemma26:
for b1 being non empty Nat
for b2 being Nat holds
( b2 < 2 to_power b1 implies for b3 being Tuple of b1,BOOLEAN holds
( b3 = 0* b1 implies ( b1 -BinarySequence b2 = 'not' b3 iff b2 = (2 to_power b1) - 1 ) ) )
theorem Th29: :: BINARI_3:29
theorem Th30: :: BINARI_3:30
theorem Th31: :: BINARI_3:31
theorem Th32: :: BINARI_3:32
theorem Th33: :: BINARI_3:33
theorem Th34: :: BINARI_3:34
theorem Th35: :: BINARI_3:35
theorem Th36: :: BINARI_3:36
theorem Th37: :: BINARI_3:37