:: SCMFSA8C semantic presentation
theorem Th1: :: SCMFSA8C:1
canceled;
theorem Th2: :: SCMFSA8C:2
theorem Th3: :: SCMFSA8C:3
canceled;
theorem Th4: :: SCMFSA8C:4
canceled;
theorem Th5: :: SCMFSA8C:5
canceled;
theorem Th6: :: SCMFSA8C:6
theorem Th7: :: SCMFSA8C:7
theorem Th8: :: SCMFSA8C:8
theorem Th9: :: SCMFSA8C:9
theorem Th10: :: SCMFSA8C:10
theorem Th11: :: SCMFSA8C:11
theorem Th12: :: SCMFSA8C:12
theorem Th13: :: SCMFSA8C:13
theorem Th14: :: SCMFSA8C:14
theorem Th15: :: SCMFSA8C:15
theorem Th16: :: SCMFSA8C:16
theorem Th17: :: SCMFSA8C:17
theorem Th18: :: SCMFSA8C:18
theorem Th19: :: SCMFSA8C:19
theorem Th20: :: SCMFSA8C:20
theorem Th21: :: SCMFSA8C:21
theorem Th22: :: SCMFSA8C:22
theorem Th23: :: SCMFSA8C:23
theorem Th24: :: SCMFSA8C:24
theorem Th25: :: SCMFSA8C:25
theorem Th26: :: SCMFSA8C:26
theorem Th27: :: SCMFSA8C:27
theorem Th28: :: SCMFSA8C:28
theorem Th29: :: SCMFSA8C:29
theorem Th30: :: SCMFSA8C:30
theorem Th31: :: SCMFSA8C:31
theorem Th32: :: SCMFSA8C:32
theorem Th33: :: SCMFSA8C:33
theorem Th34: :: SCMFSA8C:34
theorem Th35: :: SCMFSA8C:35
theorem Th36: :: SCMFSA8C:36
theorem Th37: :: SCMFSA8C:37
theorem Th38: :: SCMFSA8C:38
theorem Th39: :: SCMFSA8C:39
theorem Th40: :: SCMFSA8C:40
theorem Th41: :: SCMFSA8C:41
theorem Th42: :: SCMFSA8C:42
theorem Th43: :: SCMFSA8C:43
theorem Th44: :: SCMFSA8C:44
theorem Th45: :: SCMFSA8C:45
theorem Th46: :: SCMFSA8C:46
E116:
now
let s be
State of
SCM+FSA ;
let I be
Macro-Instruction;
E31:
ProgramPart (Initialized I) = I
by ;
hereby
assume E32:
Initialized I is_pseudo-closed_on s
;
set k =
pseudo-LifeSpan s,
(Initialized I);
(
IC ((Computation (s +* ((Initialized I) +* (Start-At (insloc 0))))) . (pseudo-LifeSpan s,(Initialized I))) = insloc (card (ProgramPart (Initialized I))) & ( for
n being
Element of
NAT st
n < pseudo-LifeSpan s,
(Initialized I) holds
IC ((Computation (s +* ((Initialized I) +* (Start-At (insloc 0))))) . n) in dom (Initialized I) ) )
by Th8, SCMFSA8A:def 5;
then
IC ((Computation ((Initialize s) +* (I +* (Start-At (insloc 0))))) . (pseudo-LifeSpan s,(Initialized I))) = insloc (card (ProgramPart (Initialized I)))
by ;
then E33:
IC ((Computation ((Initialize s) +* (I +* (Start-At (insloc 0))))) . (pseudo-LifeSpan s,(Initialized I))) = insloc (card (ProgramPart I))
by , AMI_5:72;
then E44:
for
n being
Element of
NAT st not
IC ((Computation ((Initialize s) +* (I +* (Start-At (insloc 0))))) . n) in dom I holds
pseudo-LifeSpan s,
(Initialized I) <= n
;
thus
I is_pseudo-closed_on Initialize s
by , E41, SCMFSA8A:def 3;
hence
pseudo-LifeSpan s,
(Initialized I) = pseudo-LifeSpan (Initialize s),
I
by , , SCMFSA8A:def 5;
end;
assume E45:
I is_pseudo-closed_on Initialize s
;
set k =
pseudo-LifeSpan (Initialize s),
I;
(
IC ((Computation ((Initialize s) +* (I +* (Start-At (insloc 0))))) . (pseudo-LifeSpan (Initialize s),I)) = insloc (card (ProgramPart I)) & ( for
n being
Element of
NAT st
n < pseudo-LifeSpan (Initialize s),
I holds
IC ((Computation ((Initialize s) +* (I +* (Start-At (insloc 0))))) . n) in dom I ) )
by Th8, SCMFSA8A:def 5;
then
IC ((Computation (s +* ((Initialized I) +* (Start-At (insloc 0))))) . (pseudo-LifeSpan (Initialize s),I)) = insloc (card (ProgramPart I))
by ;
then E46:
IC ((Computation (s +* ((Initialized I) +* (Start-At (insloc 0))))) . (pseudo-LifeSpan (Initialize s),I)) = insloc (card (ProgramPart (Initialized I)))
by , AMI_5:72;
then E49:
for
n being
Element of
NAT st not
IC ((Computation (s +* ((Initialized I) +* (Start-At (insloc 0))))) . n) in dom (Initialized I) holds
pseudo-LifeSpan (Initialize s),
I <= n
;
thus
Initialized I is_pseudo-closed_on s
by , , SCMFSA8A:def 3;
hence
pseudo-LifeSpan s,
(Initialized I) = pseudo-LifeSpan (Initialize s),
I
by , , SCMFSA8A:def 5;
end;
theorem Th47: :: SCMFSA8C:47
theorem Th48: :: SCMFSA8C:48
theorem Th49: :: SCMFSA8C:49
theorem Th50: :: SCMFSA8C:50
theorem Th51: :: SCMFSA8C:51
theorem Th52: :: SCMFSA8C:52
theorem Th53: :: SCMFSA8C:53
Lemma137:
for l being Instruction-Location of SCM+FSA holds goto l <> halt SCM+FSA
by SCMFSA_2:47, SCMFSA_2:124;
Lemma138:
for a being Int-Location
for l being Instruction-Location of SCM+FSA holds a =0_goto l <> halt SCM+FSA
by SCMFSA_2:48, SCMFSA_2:124;
Lemma139:
for a being Int-Location
for l being Instruction-Location of SCM+FSA holds a >0_goto l <> halt SCM+FSA
by SCMFSA_2:49, SCMFSA_2:124;
Lemma140:
for I, J being Macro-Instruction holds ProgramPart (Relocated J,(card I)) c= I ';' J
theorem Th54: :: SCMFSA8C:54
theorem Th55: :: SCMFSA8C:55
theorem Th56: :: SCMFSA8C:56
theorem Th57: :: SCMFSA8C:57
theorem Th58: :: SCMFSA8C:58
theorem Th59: :: SCMFSA8C:59
theorem Th60: :: SCMFSA8C:60
theorem Th61: :: SCMFSA8C:61
theorem Th62: :: SCMFSA8C:62
theorem Th63: :: SCMFSA8C:63
theorem Th64: :: SCMFSA8C:64
theorem Th65: :: SCMFSA8C:65
theorem Th66: :: SCMFSA8C:66
theorem Th67: :: SCMFSA8C:67
theorem Th68: :: SCMFSA8C:68
theorem Th69: :: SCMFSA8C:69
theorem Th70: :: SCMFSA8C:70
theorem Th71: :: SCMFSA8C:71
theorem Th72: :: SCMFSA8C:72
theorem Th73: :: SCMFSA8C:73
theorem Th74: :: SCMFSA8C:74
theorem Th75: :: SCMFSA8C:75
theorem Th76: :: SCMFSA8C:76
theorem Th77: :: SCMFSA8C:77
theorem Th78: :: SCMFSA8C:78
theorem Th79: :: SCMFSA8C:79
theorem Th80: :: SCMFSA8C:80
theorem Th81: :: SCMFSA8C:81
theorem Th82: :: SCMFSA8C:82
theorem Th83: :: SCMFSA8C:83
theorem Th84: :: SCMFSA8C:84
theorem Th85: :: SCMFSA8C:85
theorem Th86: :: SCMFSA8C:86
theorem Th87: :: SCMFSA8C:87
theorem Th88: :: SCMFSA8C:88
theorem Th89: :: SCMFSA8C:89
theorem Th90: :: SCMFSA8C:90
theorem Th91: :: SCMFSA8C:91
theorem Th92: :: SCMFSA8C:92
theorem Th93: :: SCMFSA8C:93
theorem Th94: :: SCMFSA8C:94
theorem Th95: :: SCMFSA8C:95
theorem Th96: :: SCMFSA8C:96
theorem Th97: :: SCMFSA8C:97
theorem Th98: :: SCMFSA8C:98
theorem Th99: :: SCMFSA8C:99
theorem Th100: :: SCMFSA8C:100
theorem Th101: :: SCMFSA8C:101
theorem Th102: :: SCMFSA8C:102
canceled;
theorem Th103: :: SCMFSA8C:103
:: deftheorem Def1 SCMFSA8C:def 1 :
canceled;
:: deftheorem Def2 SCMFSA8C:def 2 :
canceled;
:: deftheorem Def3 SCMFSA8C:def 3 :
canceled;
:: deftheorem Def4 defines loop SCMFSA8C:def 4 :
theorem Th104: :: SCMFSA8C:104
theorem Th105: :: SCMFSA8C:105
theorem Th106: :: SCMFSA8C:106
theorem Th107: :: SCMFSA8C:107
theorem Th108: :: SCMFSA8C:108
theorem Th109: :: SCMFSA8C:109
theorem Th110: :: SCMFSA8C:110
Lemma237:
for s being State of SCM+FSA
for I being Macro-Instruction st I is_closed_on s & I is_halting_on s holds
( CurInstr ((Computation (s +* ((loop I) +* (Start-At (insloc 0))))) . (LifeSpan (s +* (I +* (Start-At (insloc 0)))))) = goto (insloc 0) & ( for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0)))) holds
CurInstr ((Computation (s +* ((loop I) +* (Start-At (insloc 0))))) . m) <> halt SCM+FSA ) )
theorem Th111: :: SCMFSA8C:111
theorem Th112: :: SCMFSA8C:112
theorem Th113: :: SCMFSA8C:113
theorem Th114: :: SCMFSA8C:114
definition
let a be
Int-Location ;
let I be
Macro-Instruction;
func Times c1,
c2 -> Macro-Instruction equals :: SCMFSA8C:def 5
if>0 a,
(loop (if=0 a,(Goto (insloc 2)),(I ';' (SubFrom a,(intloc 0))))),
SCM+FSA-Stop ;
correctness
coherence
if>0 a,(loop (if=0 a,(Goto (insloc 2)),(I ';' (SubFrom a,(intloc 0))))),SCM+FSA-Stop is Macro-Instruction;
;
end;
:: deftheorem Def5 defines Times SCMFSA8C:def 5 :
theorem Th115: :: SCMFSA8C:115
theorem Th116: :: SCMFSA8C:116
theorem Th117: :: SCMFSA8C:117
theorem Th118: :: SCMFSA8C:118
theorem Th119: :: SCMFSA8C:119
theorem Th120: :: SCMFSA8C:120
theorem Th121: :: SCMFSA8C:121
theorem Th122: :: SCMFSA8C:122
for
s being
State of
SCM+FSA for
I being
parahalting good Macro-Instruction for
a being
read-write Int-Location st
I does_not_destroy a &
s . (intloc 0) = 1 &
s . a > 0 holds
ex
s2 being
State of
SCM+FSA ex
k being
Element of
NAT st
(
s2 = s +* ((loop (if=0 a,(Goto (insloc 2)),(I ';' (SubFrom a,(intloc 0))))) +* (Start-At (insloc 0))) &
k = (LifeSpan (s +* ((if=0 a,(Goto (insloc 2)),(I ';' (SubFrom a,(intloc 0)))) +* (Start-At (insloc 0))))) + 1 &
((Computation s2) . k) . a = (s . a) - 1 &
((Computation s2) . k) . (intloc 0) = 1 & ( for
b being
read-write Int-Location st
b <> a holds
((Computation s2) . k) . b = (IExec I,s) . b ) & ( for
f being
FinSeq-Location holds
((Computation s2) . k) . f = (IExec I,s) . f ) &
IC ((Computation s2) . k) = insloc 0 & ( for
n being
Element of
NAT st
n <= k holds
IC ((Computation s2) . n) in dom (loop (if=0 a,(Goto (insloc 2)),(I ';' (SubFrom a,(intloc 0))))) ) )
theorem Th123: :: SCMFSA8C:123
theorem Th124: :: SCMFSA8C:124
theorem Th125: :: SCMFSA8C:125