:: On the compositions of macro instructions, Part III
:: by Noriko Asamoto , Yatsuka Nakamura , Piotr Rudnicki and Andrzej Trybulec
::
:: Received July 22, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users

set SA0 = Start-At (0,SCM+FSA);

theorem :: SCMFSA6C:1

for a being Int-Location
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being keeping_0 really-closed parahalting Program of SCM+FSA
for J being really-closed parahalting Program of SCM+FSA holds (IExec ((I ";" J),P,s)) . a = (IExec (J,P,(IExec (I,P,s)))) . a

proof end;

theorem :: SCMFSA6C:2

for f being FinSeq-Location
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being keeping_0 really-closed parahalting Program of SCM+FSA
for J being really-closed parahalting Program of SCM+FSA holds (IExec ((I ";" J),P,s)) . f = (IExec (J,P,(IExec (I,P,s)))) . f

proof end;

canceled;

::$CD

definition

let i be Instruction of SCM+FSA;

:: attr i is parahalting means
:: :Def1:
:: Macro i is parahalting;

attr i is keeping_0 means :Def1: :: SCMFSA6C:def 1
Macro i is keeping_0 ;

end;

:: deftheorem Def1 defines keeping_0 SCMFSA6C:def 1 :
for i being Instruction of SCM+FSA holds
( i is keeping_0 iff Macro i is keeping_0 );

Lm1: Macro (halt SCM+FSA) is keeping_0

proof end;

registration

cluster halt SCM+FSA -> keeping_0 ;

coherence
halt SCM+FSA is keeping_0 by Lm1;

end;

registration

let i be sequential Instruction of SCM+FSA;

cluster Macro i -> parahalting ;

coherence
Macro i is parahalting

proof end;

end;

registration

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

cluster (f,a) := b -> keeping_0 ;

coherence
(f,a) := b is keeping_0

proof end;

end;

registration

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

cluster f :=<0,...,0> a -> keeping_0 ;

coherence
f :=<0,...,0> a is keeping_0

proof end;

end;

registration

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

cluster a := b -> keeping_0 ;

coherence
a := b is keeping_0

proof end;

cluster AddTo (a,b) -> keeping_0 ;

coherence
AddTo (a,b) is keeping_0

proof end;

cluster SubFrom (a,b) -> keeping_0 ;

coherence
SubFrom (a,b) is keeping_0

proof end;

cluster MultBy (a,b) -> keeping_0 ;

coherence
MultBy (a,b) is keeping_0

proof end;

end;

registration

cluster sequential V155( Segm 3, SCM+FSA ) V157( Segm 3, SCM+FSA ) keeping_0 for M3( the InstructionsF of SCM+FSA);

existence
ex b₁ being Instruction of SCM+FSA st
( b₁ is keeping_0 & b₁ is sequential )

proof end;

end;

registration

let i be keeping_0 Instruction of SCM+FSA;

cluster Macro i -> keeping_0 ;

coherence
Macro i is keeping_0 by Def1;

end;

registration

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

cluster Divide (a,b) -> keeping_0 ;

coherence
Divide (a,b) is keeping_0

proof end;

end;

registration

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

cluster b := (f,a) -> keeping_0 ;

coherence
b := (f,a) is keeping_0

proof end;

end;

registration

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

cluster b :=len f -> keeping_0 ;

coherence
b :=len f is keeping_0

proof end;

end;

registration

let i be sequential Instruction of SCM+FSA;

cluster Macro i -> really-closed ;

coherence
Macro i is really-closed

proof end;

end;

:: registration
:: let i be keeping_0 Instruction of SCM+FSA;
:: cluster Macro i -> really-closed;
:: coherence;
:: end;

registration

let i be sequential Instruction of SCM+FSA;
let J be really-closed parahalting Program of SCM+FSA;

cluster i ";" J -> really-closed parahalting ;

coherence
( i ";" J is parahalting & i ";" J is really-closed ) ;

end;

registration

let I be really-closed parahalting Program of SCM+FSA;
let j be sequential Instruction of SCM+FSA;

cluster I ";" j -> really-closed parahalting ;

coherence
( I ";" j is parahalting & I ";" j is really-closed ) ;

end;

registration

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

cluster i ";" j -> really-closed parahalting ;

coherence
( i ";" j is parahalting & i ";" j is really-closed ) ;

end;

registration

let i be sequential keeping_0 Instruction of SCM+FSA;
let J be keeping_0 really-closed Program of SCM+FSA;

cluster i ";" J -> keeping_0 ;

coherence
i ";" J is keeping_0 ;

end;

registration

let I be keeping_0 really-closed Program of SCM+FSA;
let j be sequential keeping_0 Instruction of SCM+FSA;

cluster I ";" j -> keeping_0 ;

coherence
I ";" j is keeping_0 ;

end;

registration

let i, j be sequential keeping_0 Instruction of SCM+FSA;

cluster i ";" j -> keeping_0 ;

coherence
i ";" j is keeping_0 ;

end;

theorem :: SCMFSA6C:3

canceled;

::$CT

theorem Th3: :: SCMFSA6C:4

for i being Instruction of SCM+FSA
for s1, s2 being State of SCM+FSA st DataPart s1 = DataPart s2 holds
DataPart (Exec (i,s1)) = DataPart (Exec (i,s2))

proof end;

Lm2: now :: thesis: for I being keeping_0 parahalting Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds DataPart (Initialized (IExec (I,P,s))) = DataPart (IExec (I,P,s))

set IF = Data-Locations ;
let I be keeping_0 parahalting Program of SCM+FSA; :: thesis: for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds DataPart (Initialized (IExec (I,P,s))) = DataPart (IExec (I,P,s))
let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA holds DataPart (Initialized (IExec (I,P,s))) = DataPart (IExec (I,P,s))
let P be Instruction-Sequence of SCM+FSA; :: thesis: DataPart (Initialized (IExec (I,P,s))) = DataPart (IExec (I,P,s))
set IE = IExec (I,P,s);

now :: thesis: ( dom (DataPart (Initialized (IExec (I,P,s)))) = (dom (IExec (I,P,s))) /\ (Data-Locations ) & ( for x being object st x in dom (DataPart (Initialized (IExec (I,P,s)))) holds
(DataPart (Initialized (IExec (I,P,s)))) . x = (IExec (I,P,s)) . x ) )

A1: dom (Initialized (IExec (I,P,s))) = the carrier of SCM+FSA by PARTFUN1:def 2;
A2: the carrier of SCM+FSA = {(IC )} \/ (Data-Locations ) by STRUCT_0:4;
A3: dom (IExec (I,P,s)) = the carrier of SCM+FSA by PARTFUN1:def 2;
hence dom (DataPart (Initialized (IExec (I,P,s)))) = (dom (IExec (I,P,s))) /\ (Data-Locations ) by A1, RELAT_1:61; :: thesis: for x being object st x in dom (DataPart (Initialized (IExec (I,P,s)))) holds
(DataPart (Initialized (IExec (I,P,s)))) . b₂ = (IExec (I,P,s)) . b₂
then A4: dom (DataPart (Initialized (IExec (I,P,s)))) = Data-Locations by A3, A2, XBOOLE_1:21;
let x be object ; :: thesis: ( x in dom (DataPart (Initialized (IExec (I,P,s)))) implies (DataPart (Initialized (IExec (I,P,s)))) . b₁ = (IExec (I,P,s)) . b₁ )
assume A5: x in dom (DataPart (Initialized (IExec (I,P,s)))) ; :: thesis: (DataPart (Initialized (IExec (I,P,s)))) . b₁ = (IExec (I,P,s)) . b₁

per cases ( x in Int-Locations or x in FinSeq-Locations ) by A5, A4, SCMFSA_2:100, XBOOLE_0:def 3;

suppose x in Int-Locations ; :: thesis: (DataPart (Initialized (IExec (I,P,s)))) . b₁ = (IExec (I,P,s)) . b₁

then reconsider x9 = x as Int-Location by AMI_2:def 16;

per cases ( x9 is read-write or x9 is read-only ) ;

suppose A6: x9 is read-write ; :: thesis: (DataPart (Initialized (IExec (I,P,s)))) . b₁ = (IExec (I,P,s)) . b₁

thus (DataPart (Initialized (IExec (I,P,s)))) . x = (Initialized (IExec (I,P,s))) . x by A5, A4, FUNCT_1:49
.= (IExec (I,P,s)) . x by A6, SCMFSA_M:37 ; :: thesis: verum

end;

suppose x9 is read-only ; :: thesis: (DataPart (Initialized (IExec (I,P,s)))) . b₁ = (IExec (I,P,s)) . b₁

then A7: x9 = intloc 0 ;
thus (DataPart (Initialized (IExec (I,P,s)))) . x = (Initialized (IExec (I,P,s))) . x9 by A5, A4, FUNCT_1:49
.= 1 by A7, SCMFSA_M:9
.= (IExec (I,P,s)) . x by A7, SCMFSA6B:11 ; :: thesis: verum

end;

end;

end;

suppose x in FinSeq-Locations ; :: thesis: (DataPart (Initialized (IExec (I,P,s)))) . b₁ = (IExec (I,P,s)) . b₁

then reconsider x9 = x as FinSeq-Location by SCMFSA_2:def 5;
thus (DataPart (Initialized (IExec (I,P,s)))) . x = (Initialized (IExec (I,P,s))) . x9 by A5, A4, FUNCT_1:49
.= (IExec (I,P,s)) . x by SCMFSA_M:37 ; :: thesis: verum

end;

end;

end;

hence DataPart (Initialized (IExec (I,P,s))) = DataPart (IExec (I,P,s)) by FUNCT_1:46; :: thesis: verum

end;

theorem Th4: :: SCMFSA6C:5

for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for i being sequential Instruction of SCM+FSA holds Exec (i,(Initialized s)) = IExec ((Macro i),P,s)

proof end;

theorem Th5: :: SCMFSA6C:6

for a being Int-Location
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being keeping_0 really-closed parahalting Program of SCM+FSA
for j being sequential Instruction of SCM+FSA holds (IExec ((I ";" j),P,s)) . a = (Exec (j,(IExec (I,P,s)))) . a

proof end;

theorem Th6: :: SCMFSA6C:7

for f being FinSeq-Location
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being keeping_0 really-closed parahalting Program of SCM+FSA
for j being sequential Instruction of SCM+FSA holds (IExec ((I ";" j),P,s)) . f = (Exec (j,(IExec (I,P,s)))) . f

proof end;

theorem Th7: :: SCMFSA6C:8

for a being Int-Location
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for i being sequential keeping_0 Instruction of SCM+FSA
for j being sequential Instruction of SCM+FSA holds (IExec ((i ";" j),P,s)) . a = (Exec (j,(Exec (i,(Initialized s))))) . a

proof end;

theorem :: SCMFSA6C:9

for f being FinSeq-Location
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for i being sequential keeping_0 Instruction of SCM+FSA
for j being sequential Instruction of SCM+FSA holds (IExec ((i ";" j),P,s)) . f = (Exec (j,(Exec (i,(Initialized s))))) . f

proof end;

definition

let a, b be Int-Location;

func swap (a,b) -> Program of SCM+FSA equals :: SCMFSA6C:def 2
(((FirstNotUsed (Macro (a := b))) := a) ";" (a := b)) ";" (b := (FirstNotUsed (Macro (a := b))));

correctness
coherence
(((FirstNotUsed (Macro (a := b))) := a) ";" (a := b)) ";" (b := (FirstNotUsed (Macro (a := b)))) is Program of SCM+FSA;
;

end;

:: deftheorem defines swap SCMFSA6C:def 2 :
for a, b being Int-Location holds swap (a,b) = (((FirstNotUsed (Macro (a := b))) := a) ";" (a := b)) ";" (b := (FirstNotUsed (Macro (a := b))));

registration

let a, b be Int-Location;

cluster swap (a,b) -> really-closed parahalting ;

coherence
( swap (a,b) is parahalting & swap (a,b) is really-closed ) ;

end;

registration

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

cluster swap (a,b) -> keeping_0 ;

coherence
swap (a,b) is keeping_0 ;

end;

theorem :: SCMFSA6C:10

for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for a, b being read-write Int-Location holds
( (IExec ((swap (a,b)),P,s)) . a = s . b & (IExec ((swap (a,b)),P,s)) . b = s . a )

proof end;

theorem :: SCMFSA6C:11

for a, b being Int-Location holds UsedI*Loc (swap (a,b)) = {}

proof end;

registration

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

cluster i ';' j -> really-closed ;

coherence
i ';' j is really-closed ;

end;

registration

let I be really-closed MacroInstruction of SCM+FSA ;
let j be sequential Instruction of SCM+FSA;

cluster I ';' j -> really-closed ;

coherence
I ';' j is really-closed ;

end;

registration

let i be sequential Instruction of SCM+FSA;
let J be really-closed MacroInstruction of SCM+FSA ;

cluster i ';' J -> really-closed ;

coherence
i ';' J is really-closed ;

end;