:: While Macro Instructions of SCM+FSA
:: by Jing-Chao Chen
::
:: Received December 10, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users

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

canceled;

canceled;

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:1

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:2

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:3

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:4

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:5

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:6

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:7

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:8

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:9

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:10

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:11

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:12

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:13

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:14

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:15

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:16

canceled;

:: WHILE Statement

theorem :: SCMFSA_9:17

canceled;

::$CD 3
::$CT 17

theorem Th1: :: SCMFSA_9:18

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a <> 0 holds
while=0 (a,I) is_halting_on s,P

proof end;

theorem Th2: :: SCMFSA_9:19

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA
for k being Nat st I is_halting_on s,P & k < LifeSpan ((P +* I),(Initialize s)) & IC (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + k))) = (IC (Comput ((P +* I),(Initialize s),k))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + k))) = DataPart (Comput ((P +* I),(Initialize s),k)) holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = (IC (Comput ((P +* I),(Initialize s),(k + 1)))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) )

proof end;

:: Twierdzeinie zmienia znaczenie, jezeli w oryginale mowilo
:: ze znajdziemy sie w miejscu "przed powrotem" i wykonujemy Next
:: to w tym miejscu jest teraz powrot i wykonujemy goto 0

theorem Th3: :: SCMFSA_9:20

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA st I is_halting_on s,P & IC (Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s)))))) = (IC (Comput ((P +* I),(Initialize s),(LifeSpan ((P +* I),(Initialize s)))))) + 3 holds
CurInstr ((P +* (while=0 (a,I))),(Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto 0

proof end;

theorem :: SCMFSA_9:21

canceled;

::$CT

theorem Th4: :: SCMFSA_9:22

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_on s,P & s . a = 0 holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while=0 (a,I))),(Initialize s),k)) in dom (while=0 (a,I)) ) )

proof end;

definition

let s be State of SCM+FSA;
let I be MacroInstruction of SCM+FSA ;
let a be read-write Int-Location;
let P be Instruction-Sequence of SCM+FSA;
deffunc H₁( Nat, State of SCM+FSA) -> set = Comput ((P +* (while=0 (a,I))),(Initialize $2),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize $2))) + 2));
deffunc H₂( Nat, State of SCM+FSA) -> Element of product (the_Values_of SCM+FSA) = down H₁($1,$2);

func StepWhile=0 (a,I,P,s) -> sequence of (product (the_Values_of SCM+FSA)) means :Def1: :: SCMFSA_9:def 1
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (it . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (it . i)))) + 2)) ) );

existence
ex b₁ being sequence of (product (the_Values_of SCM+FSA)) st
( b₁ . 0 = s & ( for i being Nat holds b₁ . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b₁ . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b₁ . i)))) + 2)) ) )

proof end;

uniqueness
for b₁, b₂ being sequence of (product (the_Values_of SCM+FSA)) st b₁ . 0 = s & ( for i being Nat holds b₁ . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b₁ . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b₁ . i)))) + 2)) ) & b₂ . 0 = s & ( for i being Nat holds b₂ . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b₂ . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b₂ . i)))) + 2)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines StepWhile=0 SCMFSA_9:def 1 :
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for P being Instruction-Sequence of SCM+FSA
for b₅ being sequence of (product (the_Values_of SCM+FSA)) holds
( b₅ = StepWhile=0 (a,I,P,s) iff ( b₅ . 0 = s & ( for i being Nat holds b₅ . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b₅ . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b₅ . i)))) + 2)) ) ) );

theorem Th5: :: SCMFSA_9:23

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for k being Nat holds (StepWhile=0 (a,I,P,s)) . (k + 1) = (StepWhile=0 (a,I,P,((StepWhile=0 (a,I,P,s)) . k))) . 1

proof end;

theorem :: SCMFSA_9:24

canceled;

::$CT

theorem Th6: :: SCMFSA_9:25

for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile=0 (a,I,P,s)) . 1 = Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize s))) + 2))

proof end;

theorem Th7: :: SCMFSA_9:26

for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Nat st IC ((StepWhile=0 (a,I,P,s)) . k) = 0 & (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) holds
( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) )

proof end;

theorem Th8: :: SCMFSA_9:27

for P being Instruction-Sequence of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds I is_halting_on (StepWhile=0 (a,I,P,s)) . k,P +* (while=0 (a,I)) ) & ex f being Function of (product (the_Values_of SCM+FSA)),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) holds
while=0 (a,I) is_halting_on s,P

proof end;

theorem Th9: :: SCMFSA_9:28

for P being Instruction-Sequence of SCM+FSA
for I being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of (product (the_Values_of SCM+FSA)),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) holds
while=0 (a,I) is_halting_on s,P

proof end;

theorem :: SCMFSA_9:29

for I being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location st ex f being Function of (product (the_Values_of SCM+FSA)),NAT st
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds
( ( f . ((StepWhile=0 (a,I,P,s)) . 1) < f . s or f . s = 0 ) & ( f . s = 0 implies s . a <> 0 ) & ( s . a <> 0 implies f . s = 0 ) ) holds
while=0 (a,I) is parahalting

proof end;

theorem :: SCMFSA_9:30

canceled;

::$CT

theorem Th11: :: SCMFSA_9:31

for i being Instruction of SCM+FSA st not i destroys intloc 0 holds
Macro i is good

proof end;

registration

let I, J be good MacroInstruction of SCM+FSA ;
let a be Int-Location;

cluster if=0 (a,I,J) -> good ;

correctness
coherence
if=0 (a,I,J) is good ;

proof end;

end;

registration

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

cluster if=0 (a,I) -> good ;

correctness
coherence
if=0 (a,I) is good ;

proof end;

end;

registration

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

cluster while=0 (a,I) -> good ;

correctness
coherence
while=0 (a,I) is good ;

proof end;

end;

theorem :: SCMFSA_9:32

canceled;

theorem :: SCMFSA_9:33

canceled;

theorem :: SCMFSA_9:34

canceled;

theorem :: SCMFSA_9:35

canceled;

theorem :: SCMFSA_9:36

canceled;

theorem :: SCMFSA_9:37

canceled;

:: -----------------------------------------------------------
:: WHILE>0 Statement
::$CT 6

theorem Th12: :: SCMFSA_9:38

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a <= 0 holds
while>0 (a,I) is_halting_on s,P

proof end;

theorem Th13: :: SCMFSA_9:39

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA
for k being Nat st I is_halting_on s,P & k < LifeSpan ((P +* I),(Initialize s)) & IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + k))) = (IC (Comput ((P +* I),(Initialize s),k))) + 3 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + k))) = DataPart (Comput ((P +* I),(Initialize s),k)) holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = (IC (Comput ((P +* I),(Initialize s),(k + 1)))) + 3 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) )

proof end;

theorem Th14: :: SCMFSA_9:40

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA st I is_halting_on s,P & IC (Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s)))))) = (IC (Comput ((P +* I),(Initialize s),(LifeSpan ((P +* I),(Initialize s)))))) + 3 holds
CurInstr ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto 0

proof end;

theorem :: SCMFSA_9:41

canceled;

::$CT

theorem Th15: :: SCMFSA_9:42

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_on s,P & s . a > 0 holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialize s),k)) in dom (while>0 (a,I)) ) )

proof end;

definition

let s be State of SCM+FSA;
let I be MacroInstruction of SCM+FSA ;
let a be read-write Int-Location;
let P be Instruction-Sequence of SCM+FSA;
deffunc H₁( Nat, State of SCM+FSA) -> set = Comput ((P +* (while>0 (a,I))),(Initialize $2),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize $2))) + 2));
deffunc H₂( Nat, State of SCM+FSA) -> Element of product (the_Values_of SCM+FSA) = down H₁($1,$2);

func StepWhile>0 (a,I,P,s) -> sequence of (product (the_Values_of SCM+FSA)) means :Def2: :: SCMFSA_9:def 2
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (it . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (it . i)))) + 2)) ) );

existence
ex b₁ being sequence of (product (the_Values_of SCM+FSA)) st
( b₁ . 0 = s & ( for i being Nat holds b₁ . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b₁ . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b₁ . i)))) + 2)) ) )

proof end;

uniqueness
for b₁, b₂ being sequence of (product (the_Values_of SCM+FSA)) st b₁ . 0 = s & ( for i being Nat holds b₁ . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b₁ . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b₁ . i)))) + 2)) ) & b₂ . 0 = s & ( for i being Nat holds b₂ . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b₂ . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b₂ . i)))) + 2)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines StepWhile>0 SCMFSA_9:def 2 :
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for P being Instruction-Sequence of SCM+FSA
for b₅ being sequence of (product (the_Values_of SCM+FSA)) holds
( b₅ = StepWhile>0 (a,I,P,s) iff ( b₅ . 0 = s & ( for i being Nat holds b₅ . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b₅ . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b₅ . i)))) + 2)) ) ) );

theorem Th16: :: SCMFSA_9:43

for P being Instruction-Sequence of SCM+FSA
for k being Nat
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location holds (StepWhile>0 (a,I,P,s)) . (k + 1) = (StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 1

proof end;

theorem Th17: :: SCMFSA_9:44

for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile>0 (a,I,P,s)) . 1 = Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize s))) + 2))

proof end;

theorem Th18: :: SCMFSA_9:45

for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Nat st IC ((StepWhile>0 (a,I,P,s)) . k) = 0 & (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(Initialize s),n) holds
( (StepWhile>0 (a,I,P,s)) . k = Initialize ((StepWhile>0 (a,I,P,s)) . k) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize ((StepWhile>0 (a,I,P,s)) . k)))) + 2))) )

proof end;

theorem Th19: :: SCMFSA_9:46

for P being Instruction-Sequence of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds I is_halting_on (StepWhile>0 (a,I,P,s)) . k,P +* (while>0 (a,I)) ) & ex f being Function of (product (the_Values_of SCM+FSA)),NAT st
for k being Nat holds
( ( f . ((StepWhile>0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile>0 (a,I,P,s)) . k) or f . ((StepWhile>0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile>0 (a,I,P,s)) . k) = 0 implies ((StepWhile>0 (a,I,P,s)) . k) . a <= 0 ) & ( ((StepWhile>0 (a,I,P,s)) . k) . a <= 0 implies f . ((StepWhile>0 (a,I,P,s)) . k) = 0 ) ) holds
while>0 (a,I) is_halting_on s,P

proof end;

theorem Th20: :: SCMFSA_9:47

for P being Instruction-Sequence of SCM+FSA
for I being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of (product (the_Values_of SCM+FSA)),NAT st
for k being Nat holds
( ( f . ((StepWhile>0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile>0 (a,I,P,s)) . k) or f . ((StepWhile>0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile>0 (a,I,P,s)) . k) = 0 implies ((StepWhile>0 (a,I,P,s)) . k) . a <= 0 ) & ( ((StepWhile>0 (a,I,P,s)) . k) . a <= 0 implies f . ((StepWhile>0 (a,I,P,s)) . k) = 0 ) ) holds
while>0 (a,I) is_halting_on s,P

proof end;

theorem :: SCMFSA_9:48

for I being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location st ex f being Function of (product (the_Values_of SCM+FSA)),NAT st
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds
( ( f . ((StepWhile>0 (a,I,P,s)) . 1) < f . s or f . s = 0 ) & ( f . s = 0 implies s . a <= 0 ) & ( s . a <= 0 implies f . s = 0 ) ) holds
while>0 (a,I) is parahalting

proof end;

registration

let I, J be good MacroInstruction of SCM+FSA ;
let a be Int-Location;

cluster if>0 (a,I,J) -> good ;

coherence
if>0 (a,I,J) is good

proof end;

end;

registration

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

cluster if>0 (a,I) -> good ;

correctness
coherence
if>0 (a,I) is good ;

proof end;

end;

registration

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

cluster while>0 (a,I) -> good ;

correctness
coherence
while>0 (a,I) is good ;

proof end;

end;

theorem :: SCMFSA_9:49

for p being preProgram of SCM+FSA
for l being Nat
for ic being Instruction of SCM+FSA st ex pc being Instruction of SCM+FSA st
( pc = p . l & UsedIntLoc pc = UsedIntLoc ic ) holds
UsedILoc p = UsedILoc (p +* (l,ic))

proof end;

theorem :: SCMFSA_9:50

for p being preProgram of SCM+FSA
for l being Nat
for ic being Instruction of SCM+FSA st ex pc being Instruction of SCM+FSA st
( pc = p . l & UsedInt*Loc pc = UsedInt*Loc ic ) holds
UsedI*Loc p = UsedI*Loc (p +* (l,ic))

proof end;