:: AMISTD_2 semantic presentation
Lemma36:
for R being Relation st dom R <> {} holds
R <> {}
by RELAT_1:60;
theorem Th1: :: AMISTD_2:1
theorem Th2: :: AMISTD_2:2
theorem Th3: :: AMISTD_2:3
theorem Th4: :: AMISTD_2:4
:: deftheorem Def1 defines PA AMISTD_2:def 1 :
theorem Th5: :: AMISTD_2:5
theorem Th6: :: AMISTD_2:6
:: deftheorem Def2 defines product-like AMISTD_2:def 2 :
theorem Th7: :: AMISTD_2:7
theorem Th8: :: AMISTD_2:8
theorem Th9: :: AMISTD_2:9
theorem Th10: :: AMISTD_2:10
Lemma84:
for k being natural number holds - 1 < k
Lemma85:
for k being natural number
for a, b, c being Element of NAT st 1 <= a & 2 <= b & not k < a - 1 & not ( a <= k & k <= (a + b) - 3 ) & not k = (a + b) - 2 & not (a + b) - 2 < k holds
k = a - 1
theorem Th11: :: AMISTD_2:11
theorem Th12: :: AMISTD_2:12
theorem Th13: :: AMISTD_2:13
theorem Th14: :: AMISTD_2:14
theorem Th15: :: AMISTD_2:15
:: deftheorem Def3 defines AddressPart AMISTD_2:def 3 :
theorem Th16: :: AMISTD_2:16
:: deftheorem Def4 defines homogeneous AMISTD_2:def 4 :
theorem Th17: :: AMISTD_2:17
:: deftheorem Def5 defines AddressParts AMISTD_2:def 5 :
:: deftheorem Def6 defines with_explicit_jumps AMISTD_2:def 6 :
:: deftheorem Def7 defines without_implicit_jumps AMISTD_2:def 7 :
:: deftheorem Def8 defines with_explicit_jumps AMISTD_2:def 8 :
:: deftheorem Def9 defines without_implicit_jumps AMISTD_2:def 9 :
:: deftheorem Def10 defines with-non-trivial-Instruction-Locations AMISTD_2:def 10 :
theorem Th18: :: AMISTD_2:18
:: deftheorem Def11 defines regular AMISTD_2:def 11 :
theorem Th19: :: AMISTD_2:19
theorem Th20: :: AMISTD_2:20
theorem Th21: :: AMISTD_2:21
:: deftheorem Def12 defines ins-loc-free AMISTD_2:def 12 :
theorem Th22: :: AMISTD_2:22
theorem Th23: :: AMISTD_2:23
theorem Th24: :: AMISTD_2:24
:: deftheorem Def13 defines Stop AMISTD_2:def 13 :
Lemma124:
for N being with_non-empty_elements set
for S being non empty non void halting IC-Ins-separated definite standard AMI-Struct of N holds (Stop S) . (il. S,0) = halt S
by CQC_LANG:6;
theorem Th25: :: AMISTD_2:25
theorem Th26: :: AMISTD_2:26
Lemma127:
for N being with_non-empty_elements set
for S being non empty non void halting IC-Ins-separated definite standard AMI-Struct of N holds (card (Stop S)) -' 1 = 0
theorem Th27: :: AMISTD_2:27
:: deftheorem Def14 defines IncAddr AMISTD_2:def 14 :
theorem Th28: :: AMISTD_2:28
theorem Th29: :: AMISTD_2:29
theorem Th30: :: AMISTD_2:30
theorem Th31: :: AMISTD_2:31
theorem Th32: :: AMISTD_2:32
theorem Th33: :: AMISTD_2:33
theorem Th34: :: AMISTD_2:34
theorem Th35: :: AMISTD_2:35
theorem Th36: :: AMISTD_2:36
theorem Th37: :: AMISTD_2:37
definition
let N be
with_non-empty_elements set ;
let S be non
empty non
void IC-Ins-separated definite standard regular AMI-Struct of
N;
let p be
programmed FinPartState of
S;
let k be
natural number ;
E44:
dom p c= the
Instruction-Locations of
S
by AMI_3:def 13;
func IncAddr c3,
c4 -> FinPartState of
a2 means :
Def15:
:: AMISTD_2:def 15
(
dom it = dom p & ( for
m being
natural number st
il. S,
m in dom p holds
it . (il. S,m) = IncAddr (pi p,(il. S,m)),
k ) );
existence
ex b1 being FinPartState of S st
( dom b1 = dom p & ( for m being natural number st il. S,m in dom p holds
b1 . (il. S,m) = IncAddr (pi p,(il. S,m)),k ) )
uniqueness
for b1, b2 being FinPartState of S st dom b1 = dom p & ( for m being natural number st il. S,m in dom p holds
b1 . (il. S,m) = IncAddr (pi p,(il. S,m)),k ) & dom b2 = dom p & ( for m being natural number st il. S,m in dom p holds
b2 . (il. S,m) = IncAddr (pi p,(il. S,m)),k ) holds
b1 = b2
end;
:: deftheorem Def15 defines IncAddr AMISTD_2:def 15 :
theorem Th38: :: AMISTD_2:38
theorem Th39: :: AMISTD_2:39
definition
let N be
with_non-empty_elements set ;
let S be non
empty non
void IC-Ins-separated definite standard AMI-Struct of
N;
let p be
FinPartState of
S;
let k be
natural number ;
func Shift c3,
c4 -> FinPartState of
a2 means :
Def16:
:: AMISTD_2:def 16
(
dom it = { (il. S,(m + k)) where m is Element of NAT : il. S,m in dom p } & ( for
m being
Element of
NAT st
il. S,
m in dom p holds
it . (il. S,(m + k)) = p . (il. S,m) ) );
existence
ex b1 being FinPartState of S st
( dom b1 = { (il. S,(m + k)) where m is Element of NAT : il. S,m in dom p } & ( for m being Element of NAT st il. S,m in dom p holds
b1 . (il. S,(m + k)) = p . (il. S,m) ) )
uniqueness
for b1, b2 being FinPartState of S st dom b1 = { (il. S,(m + k)) where m is Element of NAT : il. S,m in dom p } & ( for m being Element of NAT st il. S,m in dom p holds
b1 . (il. S,(m + k)) = p . (il. S,m) ) & dom b2 = { (il. S,(m + k)) where m is Element of NAT : il. S,m in dom p } & ( for m being Element of NAT st il. S,m in dom p holds
b2 . (il. S,(m + k)) = p . (il. S,m) ) holds
b1 = b2
end;
:: deftheorem Def16 defines Shift AMISTD_2:def 16 :
theorem Th40: :: AMISTD_2:40
theorem Th41: :: AMISTD_2:41
theorem Th42: :: AMISTD_2:42
theorem Th43: :: AMISTD_2:43
:: deftheorem Def17 defines IC-good AMISTD_2:def 17 :
:: deftheorem Def18 defines IC-good AMISTD_2:def 18 :
:: deftheorem Def19 defines Exec-preserving AMISTD_2:def 19 :
:: deftheorem Def20 defines Exec-preserving AMISTD_2:def 20 :
theorem Th44: :: AMISTD_2:44
theorem Th45: :: AMISTD_2:45
theorem Th46: :: AMISTD_2:46
:: deftheorem Def21 defines CutLastLoc AMISTD_2:def 21 :
theorem Th47: :: AMISTD_2:47
theorem Th48: :: AMISTD_2:48
theorem Th49: :: AMISTD_2:49
theorem Th50: :: AMISTD_2:50
theorem Th51: :: AMISTD_2:51
:: deftheorem Def22 defines ';' AMISTD_2:def 22 :
Lemma177:
for N being with_non-empty_elements set
for S being non empty non void IC-Ins-separated definite standard regular AMI-Struct of N
for F, G being non empty programmed FinPartState of S holds dom (F ';' G) = (dom (CutLastLoc F)) \/ (dom (Shift (IncAddr G,((card F) -' 1)),((card F) -' 1)))
by FUNCT_4:def 1;
theorem Th52: :: AMISTD_2:52
theorem Th53: :: AMISTD_2:53
theorem Th54: :: AMISTD_2:54
theorem Th55: :: AMISTD_2:55
theorem Th56: :: AMISTD_2:56
theorem Th57: :: AMISTD_2:57
theorem Th58: :: AMISTD_2:58
theorem Th59: :: AMISTD_2:59
theorem Th60: :: AMISTD_2:60
theorem Th61: :: AMISTD_2:61