:: BINARI_4 semantic presentation
theorem Th1: :: BINARI_4:1
for b
1 being
Nat holds
( b
1 > 0 implies b
1 * 2
>= b
1 + 1 )
theorem Th2: :: BINARI_4:2
theorem Th3: :: BINARI_4:3
theorem Th4: :: BINARI_4:4
for b
1, b
2, b
3 being
Nat holds
not ( b
3 <= b
1 & b
1 <= b
2 & not b
3 = b
1 & not ( b
3 + 1
<= b
1 & b
1 <= b
2 ) )
theorem Th5: :: BINARI_4:5
theorem Th6: :: BINARI_4:6
theorem Th7: :: BINARI_4:7
theorem Th8: :: BINARI_4:8
for b
1, b
2, b
3 being
Nat holds
( b
1 + b
2 <= b
3 - 1 implies ( b
1 < b
3 & b
2 < b
3 ) )
theorem Th9: :: BINARI_4:9
for b
1, b
2, b
3 being
Integer holds
( b
1 <= b
2 + b
3 & b
2 < 0 & b
3 < 0 implies ( b
1 < b
2 & b
1 < b
3 ) )
theorem Th10: :: BINARI_4:10
theorem Th11: :: BINARI_4:11
theorem Th12: :: BINARI_4:12
theorem Th13: :: BINARI_4:13
theorem Th14: :: BINARI_4:14
theorem Th15: :: BINARI_4:15
theorem Th16: :: BINARI_4:16
theorem Th17: :: BINARI_4:17
theorem Th18: :: BINARI_4:18
theorem Th19: :: BINARI_4:19
theorem Th20: :: BINARI_4:20
:: deftheorem Def1 defines MajP BINARI_4:def 1 :
theorem Th21: :: BINARI_4:21
for b
1, b
2, b
3 being
Nat holds
( b
1 >= b
2 implies
MajP b
3,b
1 >= MajP b
3,b
2 )
theorem Th22: :: BINARI_4:22
for b
1, b
2, b
3 being
Nat holds
( b
1 >= b
2 implies
MajP b
1,b
3 >= MajP b
2,b
3 )
theorem Th23: :: BINARI_4:23
for b
1 being
Nat holds
( b
1 >= 1 implies
MajP b
1,1
= b
1 )
theorem Th24: :: BINARI_4:24
theorem Th25: :: BINARI_4:25
:: deftheorem Def2 defines 2sComplement BINARI_4:def 2 :
theorem Th26: :: BINARI_4:26
theorem Th27: :: BINARI_4:27
Lemma19:
for b1 being non empty Nat
for b2, b3 being Nat holds
not ( b2 mod b1 = b3 mod b1 & b2 > b3 & ( for b4 being Integer holds
not b2 = b3 + (b4 * b1) ) )
Lemma20:
for b1 being non empty Nat
for b2, b3 being Nat holds
not ( b2 mod b1 = b3 mod b1 & ( for b4 being Integer holds
not b2 = b3 + (b4 * b1) ) )
Lemma21:
for b1 being non empty Nat
for b2, b3, b4 being Nat holds
( b4 < b1 & b2 mod (2 to_power b1) = b3 mod (2 to_power b1) implies (b2 div (2 to_power b4)) mod 2 = (b3 div (2 to_power b4)) mod 2 )
Lemma22:
for b1 being non empty Nat
for b2, b3 being Integer holds
( b2 mod (2 to_power b1) = b3 mod (2 to_power b1) implies ((2 to_power (MajP b1,(abs b2))) + b2) mod (2 to_power b1) = ((2 to_power (MajP b1,(abs b3))) + b3) mod (2 to_power b1) )
Lemma23:
for b1 being non empty Nat
for b2, b3 being Integer holds
( b2 >= 0 & b3 >= 0 & b2 mod (2 to_power b1) = b3 mod (2 to_power b1) implies 2sComplement b1,b2 = 2sComplement b1,b3 )
Lemma24:
for b1 being non empty Nat
for b2, b3 being Integer holds
( b2 < 0 & b3 < 0 & b2 mod (2 to_power b1) = b3 mod (2 to_power b1) implies 2sComplement b1,b2 = 2sComplement b1,b3 )
theorem Th28: :: BINARI_4:28
theorem Th29: :: BINARI_4:29
theorem Th30: :: BINARI_4:30
theorem Th31: :: BINARI_4:31
theorem Th32: :: BINARI_4:32
theorem Th33: :: BINARI_4:33
theorem Th34: :: BINARI_4:34
theorem Th35: :: BINARI_4:35
theorem Th36: :: BINARI_4:36
theorem Th37: :: BINARI_4:37
theorem Th38: :: BINARI_4:38