Article SCMPDS_5, MML version 4.99.1005
:: SCMPDS_5:th 2
theorem
for b1 being set
for b2 being Element of the Instructions of SCMPDS holds
b1 in proj1 Load b2
iff
b1 = inspos 0;
:: SCMPDS_5:th 3
theorem
for b1 being Instruction-Location of SCMPDS
for b2 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS
st b1 in proj1 stop b2 & (stop b2) . b1 <> halt SCMPDS
holds b1 in proj1 b2;
:: SCMPDS_5:th 4
theorem
for b1 being Element of the Instructions of SCMPDS holds
proj1 Load b1 = {inspos 0} &
(Load b1) . inspos 0 = b1;
:: SCMPDS_5:th 5
theorem
for b1 being Element of the Instructions of SCMPDS holds
inspos 0 in proj1 Load b1;
:: SCMPDS_5:th 6
theorem
for b1 being Element of the Instructions of SCMPDS holds
card Load b1 = 1;
:: SCMPDS_5:th 7
theorem
for b1 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS holds
card stop b1 = (card b1) + 1;
:: SCMPDS_5:th 8
theorem
for b1 being Element of the Instructions of SCMPDS holds
card stop Load b1 = 2;
:: SCMPDS_5:th 9
theorem
for b1 being Element of the Instructions of SCMPDS holds
inspos 0 in proj1 stop Load b1 & inspos 1 in proj1 stop Load b1;
:: SCMPDS_5:th 10
theorem
for b1 being Element of the Instructions of SCMPDS holds
(stop Load b1) . inspos 0 = b1 &
(stop Load b1) . inspos 1 = halt SCMPDS;
:: SCMPDS_5:th 11
theorem
for b1 being set
for b2 being Element of the Instructions of SCMPDS holds
b1 in proj1 stop Load b2
iff
(b1 = inspos 0 or b1 = inspos 1);
:: SCMPDS_5:th 12
theorem
for b1 being Element of the Instructions of SCMPDS holds
proj1 stop Load b1 = {inspos 0,inspos 1};
:: SCMPDS_5:th 13
theorem
for b1 being Element of the Instructions of SCMPDS holds
inspos 0 in proj1 Initialized stop Load b1 &
inspos 1 in proj1 Initialized stop Load b1 &
(Initialized stop Load b1) . inspos 0 = b1 &
(Initialized stop Load b1) . inspos 1 = halt SCMPDS;
:: SCMPDS_5:th 15
theorem
for b1, b2 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS holds
Initialized b1 c= Initialized stop (b1 ';' b2);
:: SCMPDS_5:th 16
theorem
for b1, b2 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS holds
proj1 stop b1 c= proj1 stop (b1 ';' b2);
:: SCMPDS_5:th 17
theorem
for b1, b2 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS holds
(Initialized stop b1) +* Initialized stop (b1 ';' b2) = Initialized stop (b1 ';' b2);
:: SCMPDS_5:th 18
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS
st Initialized b2 c= b1
holds IC b1 = inspos 0;
:: SCMPDS_5:th 19
theorem
for b1 being Int_position
for b2 being Element of product the Object-Kind of SCMPDS
for b3 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS holds
(b2 +* Initialized b3) . b1 = b2 . b1;
:: SCMPDS_5:th 20
theorem
for b1, b2 being Element of product the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
st Initialized stop b3 c= b1 & Initialized stop b3 c= b2 & b1,b2 equal_outside NAT
for b4 being Element of NAT holds
Computation(b1,b4),Computation(b2,b4) equal_outside NAT & CurInstr Computation(b1,b4) = CurInstr Computation(b2,b4);
:: SCMPDS_5:th 21
theorem
for b1, b2 being Element of product the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
st Initialized stop b3 c= b1 & Initialized stop b3 c= b2 & b1,b2 equal_outside NAT
holds LifeSpan b1 = LifeSpan b2 & Result b1,Result b2 equal_outside NAT;
:: SCMPDS_5:th 22
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS holds
IC IExec(b2,b1) = IC Result (b1 +* Initialized stop b2);
:: SCMPDS_5:th 23
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS
st Initialized stop b2 c= b1
for b4 being Element of NAT
st b4 <= LifeSpan b1
holds Computation(b1,b4),Computation(b1 +* (b2 ';' b3),b4) equal_outside NAT;
:: SCMPDS_5:th 24
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS
st Initialized stop b2 c= b1
for b4 being Element of NAT
st b4 <= LifeSpan b1
holds Computation(b1,b4),Computation(b1 +* Initialized stop (b2 ';' b3),b4) equal_outside NAT;
:: SCMPDS_5:attrnot 1 => SCMPDS_5:attr 1
definition
let a1 be Element of the Instructions of SCMPDS;
attr a1 is No-StopCode means
a1 <> halt SCMPDS;
end;
:: SCMPDS_5:dfs 1
definiens
let a1 be Element of the Instructions of SCMPDS;
To prove
a1 is No-StopCode
it is sufficient to prove
thus a1 <> halt SCMPDS;
:: SCMPDS_5:def 1
theorem
for b1 being Element of the Instructions of SCMPDS holds
b1 is No-StopCode
iff
b1 <> halt SCMPDS;
:: SCMPDS_5:attrnot 2 => SCMPDS_5:attr 2
definition
let a1 be Element of the Instructions of SCMPDS;
attr a1 is parahalting means
Load a1 is parahalting;
end;
:: SCMPDS_5:dfs 2
definiens
let a1 be Element of the Instructions of SCMPDS;
To prove
a1 is parahalting
it is sufficient to prove
thus Load a1 is parahalting;
:: SCMPDS_5:def 2
theorem
for b1 being Element of the Instructions of SCMPDS holds
b1 is parahalting
iff
Load b1 is parahalting;
:: SCMPDS_5:exreg 1
registration
cluster shiftable No-StopCode parahalting Element of the Instructions of SCMPDS;
end;
:: SCMPDS_5:th 25
theorem
for b1 being integer set
st b1 <> 0
holds goto b1 is No-StopCode;
:: SCMPDS_5:funcreg 1
registration
let a1 be Int_position;
cluster return a1 -> No-StopCode;
end;
:: SCMPDS_5:funcreg 2
registration
let a1 be Int_position;
let a2 be integer set;
cluster a1 := a2 -> No-StopCode;
end;
:: SCMPDS_5:funcreg 3
registration
let a1 be Int_position;
let a2 be integer set;
cluster saveIC(a1,a2) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 4
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster (a1,a2)<>0_goto a3 -> No-StopCode;
end;
:: SCMPDS_5:funcreg 5
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster (a1,a2)<=0_goto a3 -> No-StopCode;
end;
:: SCMPDS_5:funcreg 6
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster (a1,a2)>=0_goto a3 -> No-StopCode;
end;
:: SCMPDS_5:funcreg 7
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster (a1,a2):= a3 -> No-StopCode;
end;
:: SCMPDS_5:funcreg 8
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster AddTo(a1,a2,a3) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 9
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster AddTo(a1,a3,a2,a4) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 10
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster SubFrom(a1,a3,a2,a4) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 11
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster MultBy(a1,a3,a2,a4) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 12
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster Divide(a1,a3,a2,a4) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 13
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster (a1,a3):=(a2,a4) -> No-StopCode;
end;
:: SCMPDS_5:funcreg 14
registration
cluster halt SCMPDS -> parahalting;
end;
:: SCMPDS_5:funcreg 15
registration
let a1 be parahalting Element of the Instructions of SCMPDS;
cluster Load a1 -> finite programmed initial parahalting;
end;
:: SCMPDS_5:funcreg 16
registration
let a1 be Int_position;
let a2 be integer set;
cluster a1 := a2 -> parahalting;
end;
:: SCMPDS_5:funcreg 17
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster (a1,a2):= a3 -> parahalting;
end;
:: SCMPDS_5:funcreg 18
registration
let a1 be Int_position;
let a2, a3 be integer set;
cluster AddTo(a1,a2,a3) -> parahalting;
end;
:: SCMPDS_5:funcreg 19
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster AddTo(a1,a3,a2,a4) -> parahalting;
end;
:: SCMPDS_5:funcreg 20
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster SubFrom(a1,a3,a2,a4) -> parahalting;
end;
:: SCMPDS_5:funcreg 21
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster MultBy(a1,a3,a2,a4) -> parahalting;
end;
:: SCMPDS_5:funcreg 22
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster Divide(a1,a3,a2,a4) -> parahalting;
end;
:: SCMPDS_5:funcreg 23
registration
let a1, a2 be Int_position;
let a3, a4 be integer set;
cluster (a1,a3):=(a2,a4) -> parahalting;
end;
:: SCMPDS_5:th 26
theorem
for b1 being Element of the Instructions of SCMPDS
st InsCode b1 = 1
holds b1 is not parahalting;
:: SCMPDS_5:attrnot 3 => SCMPDS_5:attr 3
definition
let a1 be finite Element of sproduct the Object-Kind of SCMPDS;
attr a1 is No-StopCode means
for b1 being Instruction-Location of SCMPDS
st b1 in proj1 a1
holds a1 . b1 <> halt SCMPDS;
end;
:: SCMPDS_5:dfs 3
definiens
let a1 be finite Element of sproduct the Object-Kind of SCMPDS;
To prove
a1 is No-StopCode
it is sufficient to prove
thus for b1 being Instruction-Location of SCMPDS
st b1 in proj1 a1
holds a1 . b1 <> halt SCMPDS;
:: SCMPDS_5:def 3
theorem
for b1 being finite Element of sproduct the Object-Kind of SCMPDS holds
b1 is No-StopCode
iff
for b2 being Instruction-Location of SCMPDS
st b2 in proj1 b1
holds b1 . b2 <> halt SCMPDS;
:: SCMPDS_5:exreg 2
registration
cluster Relation-like Function-like finite programmed initial parahalting shiftable No-StopCode Element of sproduct the Object-Kind of SCMPDS;
end;
:: SCMPDS_5:funcreg 24
registration
let a1, a2 be finite programmed initial No-StopCode Element of sproduct the Object-Kind of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial No-StopCode;
end;
:: SCMPDS_5:funcreg 25
registration
let a1 be No-StopCode Element of the Instructions of SCMPDS;
cluster Load a1 -> finite programmed initial No-StopCode;
end;
:: SCMPDS_5:funcreg 26
registration
let a1 be No-StopCode Element of the Instructions of SCMPDS;
let a2 be finite programmed initial No-StopCode Element of sproduct the Object-Kind of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial No-StopCode;
end;
:: SCMPDS_5:funcreg 27
registration
let a1 be finite programmed initial No-StopCode Element of sproduct the Object-Kind of SCMPDS;
let a2 be No-StopCode Element of the Instructions of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial No-StopCode;
end;
:: SCMPDS_5:funcreg 28
registration
let a1, a2 be No-StopCode Element of the Instructions of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial No-StopCode;
end;
:: SCMPDS_5:th 27
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
st Initialized stop b2 c= b1
holds IC Computation(b1,LifeSpan (b1 +* Initialized stop b2)) = inspos card b2;
:: SCMPDS_5:th 28
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
for b3 being Element of NAT
st b3 < LifeSpan (b1 +* Initialized stop b2)
holds IC Computation(b1 +* Initialized stop b2,b3) in proj1 b2;
:: SCMPDS_5:th 29
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
for b3 being Element of NAT
st Initialized b2 c= b1 & b3 <= LifeSpan (b1 +* Initialized stop b2)
holds Computation(b1,b3),Computation(b1 +* Initialized stop b2,b3) equal_outside NAT;
:: SCMPDS_5:th 30
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
st Initialized b2 c= b1
holds IC Computation(b1,LifeSpan (b1 +* Initialized stop b2)) = inspos card b2;
:: SCMPDS_5:th 31
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
st Initialized b2 c= b1 &
CurInstr Computation(b1,LifeSpan (b1 +* Initialized stop b2)) <> halt SCMPDS
holds IC Computation(b1,LifeSpan (b1 +* Initialized stop b2)) = inspos card b2;
:: SCMPDS_5:th 32
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
for b3 being Element of NAT
st Initialized b2 c= b1 &
b3 < LifeSpan (b1 +* Initialized stop b2)
holds CurInstr Computation(b1,b3) <> halt SCMPDS;
:: SCMPDS_5:th 33
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS
for b4 being Element of NAT
st b4 <= LifeSpan (b1 +* Initialized stop b2)
holds Computation(b1 +* Initialized stop b2,b4),Computation(b1 +* ((b2 ';' b3) +* Start-At inspos 0),b4) equal_outside NAT;
:: SCMPDS_5:th 34
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial Element of sproduct the Object-Kind of SCMPDS
for b4 being Element of NAT
st b4 <= LifeSpan (b1 +* Initialized stop b2)
holds Computation(b1 +* Initialized stop b2,b4),Computation(b1 +* Initialized stop (b2 ';' b3),b4) equal_outside NAT;
:: SCMPDS_5:funcreg 29
registration
let a1 be finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS;
let a2 be finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial parahalting;
end;
:: SCMPDS_5:funcreg 30
registration
let a1 be parahalting Element of the Instructions of SCMPDS;
let a2 be finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial parahalting;
end;
:: SCMPDS_5:funcreg 31
registration
let a1 be finite programmed initial parahalting Element of sproduct the Object-Kind of SCMPDS;
let a2 be shiftable parahalting Element of the Instructions of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial parahalting;
end;
:: SCMPDS_5:funcreg 32
registration
let a1 be parahalting Element of the Instructions of SCMPDS;
let a2 be shiftable parahalting Element of the Instructions of SCMPDS;
cluster a1 ';' a2 -> finite programmed initial parahalting;
end;
:: SCMPDS_5:th 35
theorem
for b1, b2 being Element of NAT
for b3, b4 being Element of product the Object-Kind of SCMPDS
for b5 being finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS
st b3 = Computation(b4 +* Initialized stop b5,b1)
holds Exec(CurInstr b3,b3 +* Start-At ((IC b3) + b2)) = (Following b3) +* Start-At ((IC Following b3) + b2);
:: SCMPDS_5:th 36
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS
for b4 being Element of NAT
st Initialized stop (b2 ';' b3) c= b1
holds (Computation((Result (b1 +* Initialized stop b2)) +* Initialized stop b3,b4)) +* Start-At ((IC Computation((Result (b1 +* Initialized stop b2)) +* Initialized stop b3,b4)) + card b2),Computation(b1 +* Initialized stop (b2 ';' b3),(LifeSpan (b1 +* Initialized stop b2)) + b4) equal_outside NAT;
:: SCMPDS_5:th 37
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS holds
LifeSpan (b1 +* Initialized stop (b2 ';' b3)) = (LifeSpan (b1 +* Initialized stop b2)) + LifeSpan ((Result (b1 +* Initialized stop b2)) +* Initialized stop b3);
:: SCMPDS_5:th 38
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS holds
IExec(b2 ';' b3,b1) = (IExec(b3,IExec(b2,b1))) +* Start-At ((IC IExec(b3,IExec(b2,b1))) + card b2);
:: SCMPDS_5:th 39
theorem
for b1 being Int_position
for b2 being Element of product the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
for b4 being finite programmed initial parahalting shiftable Element of sproduct the Object-Kind of SCMPDS holds
(IExec(b3 ';' b4,b2)) . b1 = (IExec(b4,IExec(b3,b2))) . b1;
:: SCMPDS_5:funcnot 1 => SCMPDS_5:func 1
definition
let a1 be Element of product the Object-Kind of SCMPDS;
func Initialized A1 -> Element of product the Object-Kind of SCMPDS equals
a1 +* Start-At inspos 0;
end;
:: SCMPDS_5:def 4
theorem
for b1 being Element of product the Object-Kind of SCMPDS holds
Initialized b1 = b1 +* Start-At inspos 0;
:: SCMPDS_5:th 40
theorem
for b1 being Int_position
for b2 being Element of product the Object-Kind of SCMPDS
for b3 being Instruction-Location of SCMPDS holds
IC Initialized b2 = inspos 0 & (Initialized b2) . b1 = b2 . b1 & (Initialized b2) . b3 = b2 . b3;
:: SCMPDS_5:th 41
theorem
for b1, b2 being Element of product the Object-Kind of SCMPDS holds
b1,b2 equal_outside NAT
iff
b1 | (SCM-Data-Loc \/ {IC SCMPDS}) = b2 | (SCM-Data-Loc \/ {IC SCMPDS});
:: SCMPDS_5:th 43
theorem
for b1 being Element of the Instructions of SCMPDS
for b2, b3 being Element of product the Object-Kind of SCMPDS
st b2 | SCM-Data-Loc = b3 | SCM-Data-Loc & InsCode b1 <> 3
holds (Exec(b1,b2)) | SCM-Data-Loc = (Exec(b1,b3)) | SCM-Data-Loc;
:: SCMPDS_5:th 44
theorem
for b1, b2 being Element of product the Object-Kind of SCMPDS
for b3 being shiftable Element of the Instructions of SCMPDS
st b1 | SCM-Data-Loc = b2 | SCM-Data-Loc
holds (Exec(b3,b1)) | SCM-Data-Loc = (Exec(b3,b2)) | SCM-Data-Loc;
:: SCMPDS_5:th 45
theorem
for b1 being Element of product the Object-Kind of SCMPDS
for b2 being parahalting Element of the Instructions of SCMPDS holds
Exec(b2,Initialized b1) = IExec(Load b2,b1);
:: SCMPDS_5:th 46
theorem
for b1 being Int_position
for b2 being Element of product the Object-Kind of SCMPDS
for b3 being finite programmed initial parahalting No-StopCode Element of sproduct the Object-Kind of SCMPDS
for b4 being shiftable parahalting Element of the Instructions of SCMPDS holds
(IExec(b3 ';' b4,b2)) . b1 = (Exec(b4,IExec(b3,b2))) . b1;
:: SCMPDS_5:th 47
theorem
for b1 being Int_position
for b2 being Element of product the Object-Kind of SCMPDS
for b3 being No-StopCode parahalting Element of the Instructions of SCMPDS
for b4 being shiftable parahalting Element of the Instructions of SCMPDS holds
(IExec(b3 ';' b4,b2)) . b1 = (Exec(b4,Exec(b3,Initialized b2))) . b1;