:: SCMFSA8A semantic presentation
set A = the Instruction-Locations of SCM+FSA ;
set D = Int-Locations \/ FinSeq-Locations ;
theorem Th1: :: SCMFSA8A:1
canceled;
theorem Th2: :: SCMFSA8A:2
theorem Th3: :: SCMFSA8A:3
theorem Th4: :: SCMFSA8A:4
theorem Th5: :: SCMFSA8A:5
theorem Th6: :: SCMFSA8A:6
theorem Th7: :: SCMFSA8A:7
theorem Th8: :: SCMFSA8A:8
theorem Th9: :: SCMFSA8A:9
theorem Th10: :: SCMFSA8A:10
theorem Th11: :: SCMFSA8A:11
theorem Th12: :: SCMFSA8A:12
theorem Th13: :: SCMFSA8A:13
theorem Th14: :: SCMFSA8A:14
theorem Th15: :: SCMFSA8A:15
theorem Th16: :: SCMFSA8A:16
theorem Th17: :: SCMFSA8A:17
:: deftheorem Def1 defines Directed SCMFSA8A:def 1 :
theorem Th18: :: SCMFSA8A:18
theorem Th19: :: SCMFSA8A:19
theorem Th20: :: SCMFSA8A:20
theorem Th21: :: SCMFSA8A:21
theorem Th22: :: SCMFSA8A:22
theorem Th23: :: SCMFSA8A:23
theorem Th24: :: SCMFSA8A:24
theorem Th25: :: SCMFSA8A:25
theorem Th26: :: SCMFSA8A:26
theorem Th27: :: SCMFSA8A:27
Lemma91:
for l being Instruction-Location of SCM+FSA holds
( dom ((insloc 0) .--> (goto l)) = {(insloc 0)} & insloc 0 in dom ((insloc 0) .--> (goto l)) & ((insloc 0) .--> (goto l)) . (insloc 0) = goto l & card ((insloc 0) .--> (goto l)) = 1 & not halt SCM+FSA in rng ((insloc 0) .--> (goto l)) )
:: deftheorem Def2 defines Goto SCMFSA8A:def 2 :
:: deftheorem Def3 defines is_pseudo-closed_on SCMFSA8A:def 3 :
:: deftheorem Def4 defines pseudo-paraclosed SCMFSA8A:def 4 :
:: deftheorem Def5 defines pseudo-LifeSpan SCMFSA8A:def 5 :
theorem Th28: :: SCMFSA8A:28
theorem Th29: :: SCMFSA8A:29
theorem Th30: :: SCMFSA8A:30
theorem Th31: :: SCMFSA8A:31
theorem Th32: :: SCMFSA8A:32
theorem Th33: :: SCMFSA8A:33
theorem Th34: :: SCMFSA8A:34
theorem Th35: :: SCMFSA8A:35
theorem Th36: :: SCMFSA8A:36
Lemma115:
for s being State of SCM+FSA
for I being Macro-Instruction st I is_closed_on s & I is_halting_on s holds
( Directed I is_pseudo-closed_on s & pseudo-LifeSpan s,(Directed I) = (LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 1 )
theorem Th37: :: SCMFSA8A:37
theorem Th38: :: SCMFSA8A:38
theorem Th39: :: SCMFSA8A:39
theorem Th40: :: SCMFSA8A:40
theorem Th41: :: SCMFSA8A:41
theorem Th42: :: SCMFSA8A:42
theorem Th43: :: SCMFSA8A:43
theorem Th44: :: SCMFSA8A:44
theorem Th45: :: SCMFSA8A:45
Lemma126:
for I being Macro-Instruction
for s being State of SCM+FSA st I is_closed_on s & I is_halting_on s holds
( IC ((Computation (s +* ((I ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 1)) = insloc (card I) & ((Computation (s +* (I +* (Start-At (insloc 0))))) . (LifeSpan (s +* (I +* (Start-At (insloc 0)))))) | (Int-Locations \/ FinSeq-Locations ) = ((Computation (s +* ((I ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 1)) | (Int-Locations \/ FinSeq-Locations ) & s +* ((I ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))) is halting & LifeSpan (s +* ((I ';' SCM+FSA-Stop ) +* (Start-At (insloc 0)))) = (LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 1 & I ';' SCM+FSA-Stop is_closed_on s & I ';' SCM+FSA-Stop is_halting_on s )
theorem Th46: :: SCMFSA8A:46
theorem Th47: :: SCMFSA8A:47
theorem Th48: :: SCMFSA8A:48
theorem Th49: :: SCMFSA8A:49
theorem Th50: :: SCMFSA8A:50
theorem Th51: :: SCMFSA8A:51
theorem Th52: :: SCMFSA8A:52
Lemma136:
for I being Macro-Instruction
for s being State of SCM+FSA st I is_closed_on Initialize s & I is_halting_on Initialize s holds
( IC ((Computation (s +* (Initialized (I ';' SCM+FSA-Stop )))) . ((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & ((Computation (s +* (Initialized I))) . (LifeSpan (s +* (Initialized I)))) | (Int-Locations \/ FinSeq-Locations ) = ((Computation (s +* (Initialized (I ';' SCM+FSA-Stop )))) . ((LifeSpan (s +* (Initialized I))) + 1)) | (Int-Locations \/ FinSeq-Locations ) & s +* (Initialized (I ';' SCM+FSA-Stop )) is halting & LifeSpan (s +* (Initialized (I ';' SCM+FSA-Stop ))) = (LifeSpan (s +* (Initialized I))) + 1 )
theorem Th53: :: SCMFSA8A:53
theorem Th54: :: SCMFSA8A:54
theorem Th55: :: SCMFSA8A:55
theorem Th56: :: SCMFSA8A:56
theorem Th57: :: SCMFSA8A:57
Lemma137:
for I, J being Macro-Instruction
for s being State of SCM+FSA st I is_closed_on s & I is_halting_on s holds
( IC ((Computation (s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 2)) = insloc (((card I) + (card J)) + 1) & ((Computation (s +* (I +* (Start-At (insloc 0))))) . (LifeSpan (s +* (I +* (Start-At (insloc 0)))))) | (Int-Locations \/ FinSeq-Locations ) = ((Computation (s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 2)) | (Int-Locations \/ FinSeq-Locations ) & ( for k being Element of NAT st k < (LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 2 holds
CurInstr ((Computation (s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . k) <> halt SCM+FSA ) & ( for k being Element of NAT st k <= LifeSpan (s +* (I +* (Start-At (insloc 0)))) holds
IC ((Computation (s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . k) = IC ((Computation (s +* (I +* (Start-At (insloc 0))))) . k) ) & IC ((Computation (s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 1)) = insloc (card I) & s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0))) is halting & LifeSpan (s +* ((((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ) +* (Start-At (insloc 0)))) = (LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 2 )
theorem Th58: :: SCMFSA8A:58
theorem Th59: :: SCMFSA8A:59
Lemma152:
for I, J being Macro-Instruction
for s being State of SCM+FSA st I is_closed_on Initialize s & I is_halting_on Initialize s holds
( IC ((Computation (s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop )))) . ((LifeSpan (s +* (Initialized I))) + 2)) = insloc (((card I) + (card J)) + 1) & ((Computation (s +* (Initialized I))) . (LifeSpan (s +* (Initialized I)))) | (Int-Locations \/ FinSeq-Locations ) = ((Computation (s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop )))) . ((LifeSpan (s +* (Initialized I))) + 2)) | (Int-Locations \/ FinSeq-Locations ) & ( for k being Element of NAT st k < (LifeSpan (s +* (Initialized I))) + 2 holds
CurInstr ((Computation (s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop )))) . k) <> halt SCM+FSA ) & ( for k being Element of NAT st k <= LifeSpan (s +* (Initialized I)) holds
IC ((Computation (s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop )))) . k) = IC ((Computation (s +* (Initialized I))) . k) ) & IC ((Computation (s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop )))) . ((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop )) is halting & LifeSpan (s +* (Initialized (((I ';' (Goto (insloc ((card J) + 1)))) ';' J) ';' SCM+FSA-Stop ))) = (LifeSpan (s +* (Initialized I))) + 2 )
theorem Th60: :: SCMFSA8A:60
theorem Th61: :: SCMFSA8A:61
theorem Th62: :: SCMFSA8A:62