:: BINARITH semantic presentation
theorem Th1: :: BINARITH:1
canceled;
theorem Th2: :: BINARITH:2
theorem Th3: :: BINARITH:3
theorem Th4: :: BINARITH:4
canceled;
theorem Th5: :: BINARITH:5
:: deftheorem Def1 BINARITH:def 1 :
canceled;
:: deftheorem Def2 BINARITH:def 2 :
canceled;
theorem Th6: :: BINARITH:6
canceled;
theorem Th7: :: BINARITH:7
theorem Th8: :: BINARITH:8
canceled;
theorem Th9: :: BINARITH:9
theorem Th10: :: BINARITH:10
theorem Th11: :: BINARITH:11
canceled;
theorem Th12: :: BINARITH:12
theorem Th13: :: BINARITH:13
theorem Th14: :: BINARITH:14
theorem Th15: :: BINARITH:15
theorem Th16: :: BINARITH:16
theorem Th17: :: BINARITH:17
theorem Th18: :: BINARITH:18
theorem Th19: :: BINARITH:19
theorem Th20: :: BINARITH:20
theorem Th21: :: BINARITH:21
theorem Th22: :: BINARITH:22
theorem Th23: :: BINARITH:23
theorem Th24: :: BINARITH:24
theorem Th25: :: BINARITH:25
theorem Th26: :: BINARITH:26
theorem Th27: :: BINARITH:27
theorem Th28: :: BINARITH:28
canceled;
theorem Th29: :: BINARITH:29
canceled;
theorem Th30: :: BINARITH:30
canceled;
theorem Th31: :: BINARITH:31
canceled;
theorem Th32: :: BINARITH:32
canceled;
theorem Th33: :: BINARITH:33
theorem Th34: :: BINARITH:34
theorem Th35: :: BINARITH:35
theorem Th36: :: BINARITH:36
theorem Th37: :: BINARITH:37
theorem Th38: :: BINARITH:38
:: deftheorem Def3 defines -' BINARITH:def 3 :
theorem Th39: :: BINARITH:39
:: deftheorem Def4 defines 'not' BINARITH:def 4 :
:: deftheorem Def5 defines carry BINARITH:def 5 :
definition
let n be
Element of
NAT ;
let x be
Tuple of
n,
BOOLEAN ;
func Binary c2 -> Tuple of
a1,
NAT means :
Def6:
:: BINARITH:def 6
for
i being
Element of
NAT st
i in Seg n holds
it /. i = IFEQ (x /. i),
FALSE ,0,
(2 to_power (i -' 1));
existence
ex b1 being Tuple of n,NAT st
for i being Element of NAT st i in Seg n holds
b1 /. i = IFEQ (x /. i),FALSE ,0,(2 to_power (i -' 1))
uniqueness
for b1, b2 being Tuple of n,NAT st ( for i being Element of NAT st i in Seg n holds
b1 /. i = IFEQ (x /. i),FALSE ,0,(2 to_power (i -' 1)) ) & ( for i being Element of NAT st i in Seg n holds
b2 /. i = IFEQ (x /. i),FALSE ,0,(2 to_power (i -' 1)) ) holds
b1 = b2
end;
:: deftheorem Def6 defines Binary BINARITH:def 6 :
:: deftheorem Def7 defines Absval BINARITH:def 7 :
definition
let n be non
empty Element of
NAT ;
let x be
Tuple of
n,
BOOLEAN ;
let y be
Tuple of
n,
BOOLEAN ;
func c2 + c3 -> Tuple of
a1,
BOOLEAN means :
Def8:
:: BINARITH:def 8
for
i being
Element of
NAT st
i in Seg n holds
it /. i = ((x /. i) 'xor' (y /. i)) 'xor' ((carry x,y) /. i);
existence
ex b1 being Tuple of n,BOOLEAN st
for i being Element of NAT st i in Seg n holds
b1 /. i = ((x /. i) 'xor' (y /. i)) 'xor' ((carry x,y) /. i)
uniqueness
for b1, b2 being Tuple of n,BOOLEAN st ( for i being Element of NAT st i in Seg n holds
b1 /. i = ((x /. i) 'xor' (y /. i)) 'xor' ((carry x,y) /. i) ) & ( for i being Element of NAT st i in Seg n holds
b2 /. i = ((x /. i) 'xor' (y /. i)) 'xor' ((carry x,y) /. i) ) holds
b1 = b2
end;
:: deftheorem Def8 defines + BINARITH:def 8 :
:: deftheorem Def9 defines add_ovfl BINARITH:def 9 :
:: deftheorem Def10 defines are_summable BINARITH:def 10 :
theorem Th40: :: BINARITH:40
theorem Th41: :: BINARITH:41
theorem Th42: :: BINARITH:42
theorem Th43: :: BINARITH:43
theorem Th44: :: BINARITH:44
theorem Th45: :: BINARITH:45
theorem Th46: :: BINARITH:46
theorem Th47: :: BINARITH:47
theorem Th48: :: BINARITH:48
theorem Th49: :: BINARITH:49
theorem Th50: :: BINARITH:50
theorem Th51: :: BINARITH:51
theorem Th52: :: BINARITH:52
theorem Th53: :: BINARITH:53
theorem Th54: :: BINARITH:54
for
n,
i being
Nat st
n -' i = 0 holds
n <= i
theorem Th55: :: BINARITH:55
for
i,
j,
k being
Nat st
i <= j holds
(j + k) -' i = (j + k) - i
theorem Th56: :: BINARITH:56
for
i,
j,
k being
Nat st
i <= j holds
(j + k) -' i = (j -' i) + k
theorem Th57: :: BINARITH:57
theorem Th58: :: BINARITH:58
theorem Th59: :: BINARITH:59
theorem Th60: :: BINARITH:60
theorem Th61: :: BINARITH:61
theorem Th62: :: BINARITH:62
theorem Th63: :: BINARITH:63
theorem Th64: :: BINARITH:64
theorem Th65: :: BINARITH:65
theorem Th66: :: BINARITH:66
theorem Th67: :: BINARITH:67
theorem Th68: :: BINARITH:68
theorem Th69: :: BINARITH:69
theorem Th70: :: BINARITH:70
theorem Th71: :: BINARITH:71
theorem Th72: :: BINARITH:72
theorem Th73: :: BINARITH:73
theorem Th74: :: BINARITH:74
theorem Th75: :: BINARITH:75