:: SCMFSA7B semantic presentation
set c1 = the Instruction-Locations of SCM+FSA ;
theorem Th1: :: SCMFSA7B:1
theorem Th2: :: SCMFSA7B:2
theorem Th3: :: SCMFSA7B:3
theorem Th4: :: SCMFSA7B:4
theorem Th5: :: SCMFSA7B:5
theorem Th6: :: SCMFSA7B:6
theorem Th7: :: SCMFSA7B:7
theorem Th8: :: SCMFSA7B:8
Lemma6:
for b1 being State of SCM+FSA holds
( IC b1 = insloc 0 implies for b2 being Int-Location
for b3 being Integer holds
( b2 := b3 c= b1 implies b1 is halting ) )
theorem Th9: :: SCMFSA7B:9
Lemma7:
for b1, b2, b3, b4 being FinSequence holds ((b1 ^ b2) ^ b3) ^ b4 = b1 ^ ((b2 ^ b3) ^ b4)
Lemma8:
for b1, b2, b3 being FinSequence holds
( ((len b1) + (len b2)) + (len b3) = len ((b1 ^ b2) ^ b3) & ((len b1) + (len b2)) + (len b3) = len (b1 ^ (b2 ^ b3)) & (len b1) + ((len b2) + (len b3)) = len (b1 ^ (b2 ^ b3)) & (len b1) + ((len b2) + (len b3)) = len ((b1 ^ b2) ^ b3) )
Lemma9:
for b1 being State of SCM+FSA holds
( IC b1 = insloc 0 & b1 . (intloc 0) = 1 implies for b2 being FinSeq-Location
for b3 being FinSequence of INT holds
( b2 := b3 c= b1 implies ( b1 is halting & (Result b1) . b2 = b3 & ( for b4 being Int-Location holds
( b4 <> intloc 1 & b4 <> intloc 2 implies (Result b1) . b4 = b1 . b4 ) ) & ( for b4 being FinSeq-Location holds
( b4 <> b2 implies (Result b1) . b4 = b1 . b4 ) ) ) ) )
Lemma10:
for b1 being State of SCM+FSA
for b2 being Nat holds
( IC b1 = insloc b2 implies for b3 being Int-Location
for b4 being Integer holds
( ( for b5 being Nat holds
( b5 < len (aSeq b3,b4) implies (aSeq b3,b4) . (b5 + 1) = b1 . (insloc (b2 + b5)) ) ) implies for b5 being Nat holds
( b5 <= len (aSeq b3,b4) implies IC ((Computation b1) . b5) = insloc (b2 + b5) ) ) )
Lemma11:
for b1 being State of SCM+FSA holds
( IC b1 = insloc 0 implies for b2 being Int-Location
for b3 being Integer holds
( Load (aSeq b2,b3) c= b1 implies for b4 being Nat holds
( b4 <= len (aSeq b2,b3) implies IC ((Computation b1) . b4) = insloc b4 ) ) )
Lemma12:
for b1 being State of SCM+FSA holds
( IC b1 = insloc 0 implies for b2 being FinSeq-Location
for b3 being FinSequence of INT holds
( b2 := b3 c= b1 implies b1 is halting ) )
theorem Th10: :: SCMFSA7B:10
definition
let c
2 be
Instruction of
SCM+FSA ;
let c
3 be
Int-Location ;
pred c
1 does_not_refer c
2 means :: SCMFSA7B:def 1
for b
1 being
Int-Location for b
2 being
Instruction-Location of
SCM+FSA for b
3 being
FinSeq-Location holds
( b
1 := a
2 <> a
1 &
AddTo b
1,a
2 <> a
1 &
SubFrom b
1,a
2 <> a
1 &
MultBy b
1,a
2 <> a
1 &
Divide b
1,a
2 <> a
1 &
Divide a
2,b
1 <> a
1 & a
2 =0_goto b
2 <> a
1 & a
2 >0_goto b
2 <> a
1 & b
1 := b
3,a
2 <> a
1 & b
3,b
1 := a
2 <> a
1 & b
3,a
2 := b
1 <> a
1 & b
3 :=<0,...,0> a
2 <> a
1 );
end;
:: deftheorem Def1 defines does_not_refer SCMFSA7B:def 1 :
for b
1 being
Instruction of
SCM+FSA for b
2 being
Int-Location holds
( b
1 does_not_refer b
2 iff for b
3 being
Int-Location for b
4 being
Instruction-Location of
SCM+FSA for b
5 being
FinSeq-Location holds
( b
3 := b
2 <> b
1 &
AddTo b
3,b
2 <> b
1 &
SubFrom b
3,b
2 <> b
1 &
MultBy b
3,b
2 <> b
1 &
Divide b
3,b
2 <> b
1 &
Divide b
2,b
3 <> b
1 & b
2 =0_goto b
4 <> b
1 & b
2 >0_goto b
4 <> b
1 & b
3 := b
5,b
2 <> b
1 & b
5,b
3 := b
2 <> b
1 & b
5,b
2 := b
3 <> b
1 & b
5 :=<0,...,0> b
2 <> b
1 ) );
:: deftheorem Def2 defines does_not_refer SCMFSA7B:def 2 :
:: deftheorem Def3 defines does_not_destroy SCMFSA7B:def 3 :
:: deftheorem Def4 defines does_not_destroy SCMFSA7B:def 4 :
:: deftheorem Def5 defines good SCMFSA7B:def 5 :
:: deftheorem Def6 defines halt-free SCMFSA7B:def 6 :
theorem Th11: :: SCMFSA7B:11
theorem Th12: :: SCMFSA7B:12
theorem Th13: :: SCMFSA7B:13
theorem Th14: :: SCMFSA7B:14
theorem Th15: :: SCMFSA7B:15
theorem Th16: :: SCMFSA7B:16
theorem Th17: :: SCMFSA7B:17
theorem Th18: :: SCMFSA7B:18
theorem Th19: :: SCMFSA7B:19
theorem Th20: :: SCMFSA7B:20
theorem Th21: :: SCMFSA7B:21
theorem Th22: :: SCMFSA7B:22
theorem Th23: :: SCMFSA7B:23
:: deftheorem Def7 defines is_closed_on SCMFSA7B:def 7 :
:: deftheorem Def8 defines is_halting_on SCMFSA7B:def 8 :
theorem Th24: :: SCMFSA7B:24
theorem Th25: :: SCMFSA7B:25
theorem Th26: :: SCMFSA7B:26
theorem Th27: :: SCMFSA7B:27
theorem Th28: :: SCMFSA7B:28
Lemma27:
SCM+FSA-Stop is parahalting
theorem Th29: :: SCMFSA7B:29
theorem Th30: :: SCMFSA7B:30