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

begin

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

proof end;

cluster AddTo a,b -> good ;
coherence
AddTo a,b is good

proof end;

cluster SubFrom a,b -> good ;
coherence
SubFrom a,b is good

proof end;

cluster MultBy a,b -> good ;
coherence
MultBy a,b is good

proof end;

end;

registration

cluster parahalting good Element of the Instructions of SCM+FSA ;
existence
ex b₁ being Instruction of SCM+FSA st
( b₁ is good & b₁ is parahalting )

proof end;

end;

registration

let a, b be read-write Int-Location ;
cluster Divide a,b -> good ;
coherence
Divide a,b is good

proof end;

end;

registration

let l be Element of NAT ;
cluster goto l -> good ;
coherence
goto l is good

proof end;

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

proof end;

cluster a >0_goto l -> good ;
coherence
a >0_goto l is good

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 -> good ;
coherence
b := f,a is good

proof end;

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

proof end;

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

proof end;

let b be Int-Location ;
cluster f,a := b -> good ;
coherence
f,a := b is good

proof end;

end;

registration

cluster good Element of the Instructions of SCM+FSA ;
existence
ex b₁ being Instruction of SCM+FSA st b₁ is good

proof end;

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 Program of SCM+FSA ;
let a be read-write Int-Location ;
cluster Times a,I -> good ;
coherence
Times a,I is good

proof end;

end;

theorem Th1: :: SFMASTR1:1

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

proof end;

begin

set D = Int-Locations \/ FinSeq-Locations ;

theorem Th2: :: SFMASTR1:2

for I being Program of SCM+FSA holds DataPart (I +* (Start-At 0 ,SCM+FSA )) = {}

proof end;

theorem Th3: :: SFMASTR1:3

for S being State of SCM+FSA
for I, J being Program of SCM+FSA st I is_halting_on Initialize S & I is_closed_on Initialize S & J is_closed_on IExec I,S holds
I ';' J is_closed_on Initialize S

proof end;

theorem Th4: :: SFMASTR1:4

for S being State of SCM+FSA
for I, J being Program of SCM+FSA st I is_halting_on Initialize S & J is_halting_on IExec I,S & I is_closed_on Initialize S & J is_closed_on IExec I,S holds
I ';' J is_halting_on Initialize S

proof end;

theorem Th5: :: SFMASTR1:5

for s being State of SCM+FSA
for I, J being Program of SCM+FSA st I is_closed_on s & I +* (Start-At 0 ,SCM+FSA ) c= s & ProgramPart s halts_on s holds
for m being Element of NAT st m <= LifeSpan s holds
Comput (ProgramPart s),s,m, Comput (ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m equal_outside NAT

proof end;

Lm1: for I being good Program of SCM+FSA
for J being Program of SCM+FSA
for s being State of SCM+FSA st s . (intloc 0 ) = 1 & I is_halting_on s & J is_halting_on IExec I,s & I is_closed_on s & J is_closed_on IExec I,s & Initialized (I ';' J) c= s holds
( IC (Comput (ProgramPart s),s,((LifeSpan (s +* I)) + 1)) = card I & DataPart (Comput (ProgramPart s),s,((LifeSpan (s +* I)) + 1)) = DataPart ((Comput (ProgramPart (s +* I)),(s +* I),(LifeSpan (s +* I))) +* (Initialized J)) & ProgramPart (Relocated J,(card I)) c= Comput (ProgramPart s),s,((LifeSpan (s +* I)) + 1) & (Comput (ProgramPart s),s,((LifeSpan (s +* I)) + 1)) . (intloc 0 ) = 1 & ProgramPart s halts_on s & LifeSpan s = ((LifeSpan (s +* I)) + 1) + (LifeSpan ((Result (s +* I)) +* (Initialized J))) & ( J is good implies (Result s) . (intloc 0 ) = 1 ) )

proof end;

theorem Th6: :: SFMASTR1:6

for s being State of SCM+FSA
for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA st Ig is_halting_on Initialize s & J is_halting_on IExec Ig,s & Ig is_closed_on Initialize s & J is_closed_on IExec Ig,s holds
LifeSpan (s +* (Initialized (Ig ';' J))) = ((LifeSpan (s +* (Initialized Ig))) + 1) + (LifeSpan ((Result (s +* (Initialized Ig))) +* (Initialized J)))

proof end;

theorem Th7: :: SFMASTR1:7

for s being State of SCM+FSA
for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA st Ig is_halting_on Initialize s & J is_halting_on IExec Ig,s & Ig is_closed_on Initialize s & J is_closed_on IExec Ig,s holds
IExec (Ig ';' J),s = (IExec J,(IExec Ig,s)) +* (Start-At ((IC (IExec J,(IExec Ig,s))) + (card Ig)),SCM+FSA )

proof end;

theorem Th8: :: SFMASTR1:8

for s being State of SCM+FSA
for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA
for a being Int-Location st ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) & ( J is parahalting or ( J is_halting_on IExec Ig,s & J is_closed_on IExec Ig,s ) ) holds
(IExec (Ig ';' J),s) . a = (IExec J,(IExec Ig,s)) . a

proof end;

theorem Th9: :: SFMASTR1:9

for s being State of SCM+FSA
for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA
for f being FinSeq-Location st ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) & ( J is parahalting or ( J is_halting_on IExec Ig,s & J is_closed_on IExec Ig,s ) ) holds
(IExec (Ig ';' J),s) . f = (IExec J,(IExec Ig,s)) . f

proof end;

theorem :: SFMASTR1:10

for s being State of SCM+FSA
for J being Program of SCM+FSA
for Ig being good Program of SCM+FSA st ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) & ( J is parahalting or ( J is_halting_on IExec Ig,s & J is_closed_on IExec Ig,s ) ) holds
DataPart (IExec (Ig ';' J),s) = DataPart (IExec J,(IExec Ig,s))

proof end;

theorem Th11: :: SFMASTR1:11

for s being State of SCM+FSA
for Ig being good Program of SCM+FSA st ( Ig is parahalting or ( Ig is_closed_on Initialize s & Ig is_halting_on Initialize s ) ) holds
DataPart (Initialize (IExec Ig,s)) = DataPart (IExec Ig,s)

proof end;

theorem Th12: :: SFMASTR1:12

for s being State of SCM+FSA
for Ig being good Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA
for a being Int-Location st ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) holds
(IExec (Ig ';' j),s) . a = (Exec j,(IExec Ig,s)) . a

proof end;

theorem Th13: :: SFMASTR1:13

for s being State of SCM+FSA
for Ig being good Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA
for f being FinSeq-Location st ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) holds
(IExec (Ig ';' j),s) . f = (Exec j,(IExec Ig,s)) . f

proof end;

theorem :: SFMASTR1:14

for s being State of SCM+FSA
for Ig being good Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA st ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) holds
DataPart (IExec (Ig ';' j),s) = DataPart (Exec j,(IExec Ig,s))

proof end;

theorem Th15: :: SFMASTR1:15

for s being State of SCM+FSA
for J being Program of SCM+FSA
for i being parahalting good Instruction of SCM+FSA
for a being Int-Location st ( J is parahalting or ( J is_halting_on Exec i,(Initialize s) & J is_closed_on Exec i,(Initialize s) ) ) holds
(IExec (i ';' J),s) . a = (IExec J,(Exec i,(Initialize s))) . a

proof end;

theorem Th16: :: SFMASTR1:16

for s being State of SCM+FSA
for J being Program of SCM+FSA
for i being parahalting good Instruction of SCM+FSA
for f being FinSeq-Location st ( J is parahalting or ( J is_halting_on Exec i,(Initialize s) & J is_closed_on Exec i,(Initialize s) ) ) holds
(IExec (i ';' J),s) . f = (IExec J,(Exec i,(Initialize s))) . f

proof end;

theorem :: SFMASTR1:17

for s being State of SCM+FSA
for J being Program of SCM+FSA
for i being parahalting good Instruction of SCM+FSA st ( J is parahalting or ( J is_halting_on Exec i,(Initialize s) & J is_closed_on Exec i,(Initialize s) ) ) holds
DataPart (IExec (i ';' J),s) = DataPart (IExec J,(Exec i,(Initialize s)))

proof end;

begin

definition

let d be Int-Location ;
:: original: {
redefine func {d} -> Subset of Int-Locations ;
coherence
{d} is Subset of Int-Locations

proof end;

let e be Int-Location ;
:: original: {
redefine func {d,e} -> Subset of Int-Locations ;
coherence
{d,e} is Subset of Int-Locations

proof end;

let f be Int-Location ;
:: original: {
redefine func {d,e,f} -> Subset of Int-Locations ;
coherence
{d,e,f} is Subset of Int-Locations

proof end;

let g be Int-Location ;
:: original: {
redefine func {d,e,f,g} -> Subset of Int-Locations ;
coherence
{d,e,f,g} is Subset of Int-Locations

proof end;

end;

definition

let L be finite Subset of Int-Locations ;
func RWNotIn-seq L -> Function of NAT ,(bool NAT ) means :Def2: :: SFMASTR1:def 2
( it . 0 = { k where k is Element of NAT : ( not intloc k in L & k <> 0 ) } & ( for i being Element of NAT
for sn being non empty Subset of NAT st it . i = sn holds
it . (i + 1) = sn \ {(min sn)} ) & ( for i being Element of NAT holds not it . i is finite ) );
existence
ex b₁ being Function of NAT ,(bool NAT ) st
( b₁ . 0 = { k where k is Element of NAT : ( not intloc k in L & k <> 0 ) } & ( for i being Element of NAT
for sn being non empty Subset of NAT st b₁ . i = sn holds
b₁ . (i + 1) = sn \ {(min sn)} ) & ( for i being Element of NAT holds not b₁ . i is finite ) )

proof end;

uniqueness
for b₁, b₂ being Function of NAT ,(bool NAT ) st b₁ . 0 = { k where k is Element of NAT : ( not intloc k in L & k <> 0 ) } & ( for i being Element of NAT
for sn being non empty Subset of NAT st b₁ . i = sn holds
b₁ . (i + 1) = sn \ {(min sn)} ) & ( for i being Element of NAT holds not b₁ . i is finite ) & b₂ . 0 = { k where k is Element of NAT : ( not intloc k in L & k <> 0 ) } & ( for i being Element of NAT
for sn being non empty Subset of NAT st b₂ . i = sn holds
b₂ . (i + 1) = sn \ {(min sn)} ) & ( for i being Element of NAT holds not b₂ . i is finite ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines RWNotIn-seq SFMASTR1:def 2 :

for L being finite Subset of Int-Locations
for b₂ being Function of NAT ,(bool NAT ) holds
( b₂ = RWNotIn-seq L iff ( b₂ . 0 = { k where k is Element of NAT : ( not intloc k in L & k <> 0 ) } & ( for i being Element of NAT
for sn being non empty Subset of NAT st b₂ . i = sn holds
b₂ . (i + 1) = sn \ {(min sn)} ) & ( for i being Element of NAT holds not b₂ . i is finite ) ) );

registration

let L be finite Subset of Int-Locations ;
let n be Element of NAT ;
cluster (RWNotIn-seq L) . n -> non empty ;
coherence
not (RWNotIn-seq L) . n is empty by Def2;

end;

theorem Th18: :: SFMASTR1:18

for L being finite Subset of Int-Locations
for n being Element of NAT holds
( not 0 in (RWNotIn-seq L) . n & ( for m being Element of NAT st m in (RWNotIn-seq L) . n holds
not intloc m in L ) )

proof end;

theorem Th19: :: SFMASTR1:19

for L being finite Subset of Int-Locations
for n being Element of NAT holds min ((RWNotIn-seq L) . n) < min ((RWNotIn-seq L) . (n + 1))

proof end;

theorem Th20: :: SFMASTR1:20

for L being finite Subset of Int-Locations
for n, m being Element of NAT st n < m holds
min ((RWNotIn-seq L) . n) < min ((RWNotIn-seq L) . m)

proof end;

definition

let n be Element of NAT ;
let L be finite Subset of Int-Locations ;
func n -thRWNotIn L -> Int-Location equals :: SFMASTR1:def 3
intloc (min ((RWNotIn-seq L) . n));
correctness
coherence
intloc (min ((RWNotIn-seq L) . n)) is Int-Location ;
;

end;

:: deftheorem defines -thRWNotIn SFMASTR1:def 3 :

for n being Element of NAT
for L being finite Subset of Int-Locations holds n -thRWNotIn L = intloc (min ((RWNotIn-seq L) . n));

notation

let n be Element of NAT ;
let L be finite Subset of Int-Locations ;
synonym n -stRWNotIn L for n -thRWNotIn L;
synonym n -ndRWNotIn L for n -thRWNotIn L;
synonym n -rdRWNotIn L for n -thRWNotIn L;

end;

registration

let n be Element of NAT ;
let L be finite Subset of Int-Locations ;
cluster n -thRWNotIn L -> read-write ;
coherence
not n -thRWNotIn L is read-only

proof end;

end;

theorem Th21: :: SFMASTR1:21

for L being finite Subset of Int-Locations
for n being Element of NAT holds not n -thRWNotIn L in L

proof end;

theorem Th22: :: SFMASTR1:22

for L being finite Subset of Int-Locations
for n, m being Element of NAT st n <> m holds
n -thRWNotIn L <> m -thRWNotIn L

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

end;

:: 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 (UsedIntLoc 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;

begin

theorem Th23: :: SFMASTR1:23

for a, b being Int-Location holds
( a in UsedIntLoc (swap a,b) & b in UsedIntLoc (swap a,b) )

proof end;

definition

let N, result be Int-Location ;
func Fib_macro N,result -> Program of SCM+FSA equals :: SFMASTR1:def 5
((((((2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))) := N) ';' (SubFrom result,result)) ';' ((1 -stRWNotIn {N,result}) := (intloc 0 ))) ';' ((1 -stRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))) := (2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))))) ';' (Times (1 -stRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))),((AddTo result,(1 -stRWNotIn {N,result})) ';' (swap result,(1 -stRWNotIn {N,result}))))) ';' (N := (2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))));
correctness
coherence
((((((2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))) := N) ';' (SubFrom result,result)) ';' ((1 -stRWNotIn {N,result}) := (intloc 0 ))) ';' ((1 -stRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))) := (2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))))) ';' (Times (1 -stRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))),((AddTo result,(1 -stRWNotIn {N,result})) ';' (swap result,(1 -stRWNotIn {N,result}))))) ';' (N := (2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result}))))) is Program of SCM+FSA ;
;

end;

:: deftheorem defines Fib_macro SFMASTR1:def 5 :

for N, result being Int-Location holds Fib_macro N,result = ((((((2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))) := N) ';' (SubFrom result,result)) ';' ((1 -stRWNotIn {N,result}) := (intloc 0 ))) ';' ((1 -stRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))) := (2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))))) ';' (Times (1 -stRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))),((AddTo result,(1 -stRWNotIn {N,result})) ';' (swap result,(1 -stRWNotIn {N,result}))))) ';' (N := (2 -ndRWNotIn (UsedIntLoc (swap result,(1 -stRWNotIn {N,result})))));

theorem :: SFMASTR1:24

for s being State of SCM+FSA
for N, result being read-write Int-Location st N <> result holds
for n being Element of NAT st n = s . N holds
( (IExec (Fib_macro N,result),s) . result = Fib n & (IExec (Fib_macro N,result),s) . N = s . N )

proof end;