:: On the Composition of non-parahalting Macro Instructions
:: by Piotr Rudnicki
::
:: Received June 3, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users

definition

let i be Instruction of SCM+FSA;

attr i is good means :Def1: :: SFMASTR1:def 1
Macro i is good ;

end;

:: deftheorem Def1 defines good SFMASTR1:def 1 :
for i being Instruction of SCM+FSA holds
( i is good iff Macro i is good );

registration

let a be read-write Int-Location;
let b be Int-Location;

cluster a := b -> good ;

coherence
a := b is good by SCMFSA7B:6, SCMFSA8C:70;

cluster AddTo (a,b) -> good ;

coherence
AddTo (a,b) is good by SCMFSA7B:7, SCMFSA8C:70;

cluster SubFrom (a,b) -> good ;

coherence
SubFrom (a,b) is good by SCMFSA7B:8, SCMFSA8C:70;

cluster MultBy (a,b) -> good ;

coherence
MultBy (a,b) is good by SCMFSA7B:9, SCMFSA8C:70;

end;

registration

cluster with_explicit_jumps IC-relocable sequential good for Element of the InstructionsF of SCM+FSA;

existence
ex b₁ being Instruction of SCM+FSA st
( b₁ is good & b₁ is sequential )

proof end;

end;

registration

let a, b be read-write Int-Location;

cluster Divide (a,b) -> good ;

coherence
Divide (a,b) is good by SCMFSA7B:10, SCMFSA8C:70;

end;

registration

let l be Element of NAT ;

cluster goto l -> good ;

coherence
goto l is good by SCMFSA7B:11, SCMFSA8C:70;

end;

registration

let a be Int-Location;
let l be Element of NAT ;

cluster a =0_goto l -> good ;

coherence
a =0_goto l is good by SCMFSA7B:12, SCMFSA8C:70;

cluster a >0_goto l -> good ;

coherence
a >0_goto l is good by SCMFSA7B:13, SCMFSA8C:70;

end;

registration

let a be Int-Location;
let f be FinSeq-Location ;
let b be read-write Int-Location;

cluster b := (f,a) -> good ;

coherence
b := (f,a) is good by SCMFSA7B:14, SCMFSA8C:70;

end;

registration

let f be FinSeq-Location ;
let b be read-write Int-Location;

cluster b :=len f -> good ;

coherence
b :=len f is good by SCMFSA7B:16, SCMFSA8C:70;

end;

registration

let f be FinSeq-Location ;
let a be Int-Location;

cluster f :=<0,...,0> a -> good ;

coherence
f :=<0,...,0> a is good by SCMFSA7B:17, SCMFSA8C:70;
let b be Int-Location;

cluster (f,a) := b -> good ;

coherence
(f,a) := b is good by SCMFSA7B:15, SCMFSA8C:70;

end;

registration

let i be good Instruction of SCM+FSA;

cluster Macro i -> good ;

coherence
Macro i is good by Def1;

end;

registration

let i, j be good Instruction of SCM+FSA;

cluster i ";" j -> good ;

coherence
i ";" j is good ;

end;

registration

let i be good Instruction of SCM+FSA;
let I be good Program of SCM+FSA;

cluster i ";" I -> good ;

coherence
i ";" I is good ;

cluster I ";" i -> good ;

coherence
I ";" i is good ;

end;

registration

let a, b be read-write Int-Location;

cluster swap (a,b) -> good ;

coherence
swap (a,b) is good

proof end;

end;

registration

let I be good MacroInstruction of SCM+FSA ;
let a be read-write Int-Location;

cluster Times (a,I) -> good ;

coherence
Times (a,I) is good ;

end;

theorem :: SFMASTR1:1

for a being Int-Location
for I being Program of SCM+FSA st not a in UsedILoc I holds
not I destroys a

proof end;

set D = Data-Locations ;

set SAt = Start-At (0,SCM+FSA);

theorem :: SFMASTR1:2

canceled;

::$CT

theorem Th2: :: SFMASTR1:3

for P being Instruction-Sequence of SCM+FSA
for S being State of SCM+FSA
for I, J being really-closed Program of SCM+FSA st I is_halting_on Initialized S,P & J is_halting_on IExec (I,P,S),P holds
I ";" J is_halting_on Initialized S,P

proof end;

theorem Th3: :: SFMASTR1:4

for p being Instruction-Sequence of SCM+FSA
for J being Program of SCM+FSA
for I being really-closed Program of SCM+FSA
for s being 0 -started State of SCM+FSA st I c= p & p halts_on s holds
for m being Nat st m <= LifeSpan (p,s) holds
Comput (p,s,m) = Comput ((p +* (I ";" J)),s,m)

proof end;

Lm1: for p being Instruction-Sequence of SCM+FSA
for I being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA
for s being State of SCM+FSA st s . (intloc 0) = 1 & I is_halting_on s,p & J is_halting_on IExec (I,p,s),p & Initialize ((intloc 0) .--> 1) c= s & I ";" J c= p holds
( IC (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = card I & DataPart (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = DataPart (Initialized (Comput ((p +* I),s,(LifeSpan ((p +* I),s))))) & Reloc (J,(card I)) c= p & (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) . (intloc 0) = 1 & p halts_on s & LifeSpan (p,s) = ((LifeSpan ((p +* I),s)) + 1) + (LifeSpan (((p +* I) +* J),(Initialized (Result ((p +* I),s))))) & ( J is good implies (Result (p,s)) . (intloc 0) = 1 ) )

proof end;

theorem Th4: :: SFMASTR1:5

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st Ig is_halting_on Initialized s,p & J is_halting_on IExec (Ig,p,s),p holds
LifeSpan ((p +* (Ig ";" J)),(Initialized s)) = ((LifeSpan ((p +* Ig),(Initialized s))) + 1) + (LifeSpan (((p +* Ig) +* J),(Initialized (Result ((p +* Ig),(Initialized s))))))

proof end;

theorem Th5: :: SFMASTR1:6

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st Ig is_halting_on Initialized s,p & J is_halting_on IExec (Ig,p,s),p holds
IExec ((Ig ";" J),p,s) = (IExec (J,p,(IExec (Ig,p,s)))) +* (Start-At (((IC (IExec (J,p,(IExec (Ig,p,s))))) + (card Ig)),SCM+FSA))

proof end;

theorem Th6: :: SFMASTR1:7

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for a being Int-Location
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_on IExec (Ig,p,s),p ) holds
(IExec ((Ig ";" J),p,s)) . a = (IExec (J,p,(IExec (Ig,p,s)))) . a

proof end;

theorem Th7: :: SFMASTR1:8

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for f being FinSeq-Location
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_on IExec (Ig,p,s),p ) holds
(IExec ((Ig ";" J),p,s)) . f = (IExec (J,p,(IExec (Ig,p,s)))) . f

proof end;

theorem :: SFMASTR1:9

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for Ig being good really-closed Program of SCM+FSA
for J being really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) & ( J is parahalting or J is_halting_on IExec (Ig,p,s),p ) holds
DataPart (IExec ((Ig ";" J),p,s)) = DataPart (IExec (J,p,(IExec (Ig,p,s))))

proof end;

theorem Th9: :: SFMASTR1:10

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
DataPart (Initialized (IExec (Ig,p,s))) = DataPart (IExec (Ig,p,s))

proof end;

theorem Th10: :: SFMASTR1:11

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for j being sequential Instruction of SCM+FSA
for a being Int-Location
for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
(IExec ((Ig ";" j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a

proof end;

theorem Th11: :: SFMASTR1:12

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for j being sequential Instruction of SCM+FSA
for f being FinSeq-Location
for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
(IExec ((Ig ";" j),p,s)) . f = (Exec (j,(IExec (Ig,p,s)))) . f

proof end;

theorem :: SFMASTR1:13

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for j being sequential Instruction of SCM+FSA
for Ig being good really-closed Program of SCM+FSA st ( Ig is parahalting or Ig is_halting_on Initialized s,p ) holds
DataPart (IExec ((Ig ";" j),p,s)) = DataPart (Exec (j,(IExec (Ig,p,s))))

proof end;

theorem Th13: :: SFMASTR1:14

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for i being sequential good Instruction of SCM+FSA
for a being Int-Location
for J being really-closed Program of SCM+FSA st ( J is parahalting or J is_halting_on Exec (i,(Initialized s)),p ) holds
(IExec ((i ";" J),p,s)) . a = (IExec (J,p,(Exec (i,(Initialized s))))) . a

proof end;

theorem Th14: :: SFMASTR1:15

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for i being sequential good Instruction of SCM+FSA
for f being FinSeq-Location
for J being really-closed Program of SCM+FSA st ( J is parahalting or J is_halting_on Exec (i,(Initialized s)),p ) holds
(IExec ((i ";" J),p,s)) . f = (IExec (J,p,(Exec (i,(Initialized s))))) . f

proof end;

theorem :: SFMASTR1:16

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for i being sequential good Instruction of SCM+FSA
for J being really-closed Program of SCM+FSA st ( J is parahalting or J is_halting_on Exec (i,(Initialized s)),p ) holds
DataPart (IExec ((i ";" J),p,s)) = DataPart (IExec (J,p,(Exec (i,(Initialized s)))))

proof end;

definition

let n be Element of NAT ;
let p be preProgram of SCM+FSA;

func n -thNotUsed p -> Int-Location equals :: SFMASTR1:def 4
n -thRWNotIn (UsedILoc p);

correctness
coherence
n -thRWNotIn (UsedILoc p) is Int-Location;
;

end;

:: deftheorem SFMASTR1:def 2 :
canceled;

:: deftheorem SFMASTR1:def 3 :
canceled;

:: deftheorem defines -thNotUsed SFMASTR1:def 4 :
for n being Element of NAT
for p being preProgram of SCM+FSA holds n -thNotUsed p = n -thRWNotIn (UsedILoc p);

notation

let n be Element of NAT ;
let p be preProgram of SCM+FSA;
synonym n -stNotUsed p for n -thNotUsed p;
synonym n -ndNotUsed p for n -thNotUsed p;
synonym n -rdNotUsed p for n -thNotUsed p;

end;

registration

let n be Element of NAT ;
let p be preProgram of SCM+FSA;

cluster n -thNotUsed p -> read-write ;

coherence
not n -thNotUsed p is read-only ;

end;