SCMFSA_9: While Macro Instructions of SCM+FSA

:: While Macro Instructions of SCM+FSA
:: by Jing-Chao Chen
::
:: Received December 10, 1997
:: Copyright (c) 1997-2012 Association of Mizar Users

begin

Lm1: card (Stop SCM+FSA) = 1
by AFINSQ_1:33;

Lm2: (Stop SCM+FSA) . 0 = halt SCM+FSA
by AFINSQ_1:34;

Lm3: 0 in dom (Stop SCM+FSA)
by COMPOS_1:3;

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

theorem Th1: :: SCMFSA_9:1

for I being Program of
for a being Int-Location holds card (if=0 (a,(I ";" (Goto 0)),(Stop SCM+FSA))) = (card I) + 6

proof end;

theorem Th2: :: SCMFSA_9:2

for I being Program of
for a being Int-Location holds card (if>0 (a,(I ";" (Goto 0)),(Stop SCM+FSA))) = (card I) + 6

proof end;

definition

let a be Int-Location;
let I be Program of ;

func while=0 (a,I) -> Program of equals :: SCMFSA_9:def 1
(if=0 (a,(I ";" (Goto 0)),(Stop SCM+FSA))) +* (((card I) + 4) .--> (goto 0));

correctness

proof end;

func while>0 (a,I) -> Program of equals :: SCMFSA_9:def 2
(if>0 (a,(I ";" (Goto 0)),(Stop SCM+FSA))) +* (((card I) + 4) .--> (goto 0));

correctness

proof end;

end;

:: deftheorem defines while=0 SCMFSA_9:def 1 :

:: deftheorem defines while>0 SCMFSA_9:def 2 :

theorem Th3: :: SCMFSA_9:3

for I being Program of
for a being Int-Location holds card (if=0 (a,(Stop SCM+FSA),(if>0 (a,(Stop SCM+FSA),(I ";" (Goto 0)))))) = (card I) + 11

proof end;

definition

let a be Int-Location;
let I be Program of ;

func while<0 (a,I) -> Program of equals :: SCMFSA_9:def 3
(if=0 (a,(Stop SCM+FSA),(if>0 (a,(Stop SCM+FSA),(I ";" (Goto 0)))))) +* (((card I) + 4) .--> (goto 0));

correctness

proof end;

end;

:: deftheorem defines while<0 SCMFSA_9:def 3 :

theorem Th4: :: SCMFSA_9:4

for I being Program of
for a being Int-Location holds card (while=0 (a,I)) = (card I) + 6

proof end;

theorem Th5: :: SCMFSA_9:5

for I being Program of
for a being Int-Location holds card (while>0 (a,I)) = (card I) + 6

proof end;

theorem :: SCMFSA_9:6

for I being Program of
for a being Int-Location holds card (while<0 (a,I)) = (card I) + 11

proof end;

theorem :: SCMFSA_9:7

canceled;

theorem :: SCMFSA_9:8

canceled;

theorem :: SCMFSA_9:9

canceled;

theorem Th10: :: SCMFSA_9:10

for a being Int-Location
for I being Program of holds
( 0 in dom (while=0 (a,I)) & 1 in dom (while=0 (a,I)) & 0 in dom (while>0 (a,I)) & 1 in dom (while>0 (a,I)) )

proof end;

theorem Th11: :: SCMFSA_9:11

for a being Int-Location
for I being Program of holds
( (while=0 (a,I)) . 0 = a =0_goto 4 & (while=0 (a,I)) . 1 = goto 2 & (while>0 (a,I)) . 0 = a >0_goto 4 & (while>0 (a,I)) . 1 = goto 2 )

proof end;

theorem Th12: :: SCMFSA_9:12

for a being Int-Location
for I being Program of
for k being Element of NAT st k < 6 holds
k in dom (while=0 (a,I))

proof end;

theorem Th13: :: SCMFSA_9:13

for a being Int-Location
for I being Program of
for k being Element of NAT st k < 6 holds
(card I) + k in dom (while=0 (a,I))

proof end;

theorem Th14: :: SCMFSA_9:14

for a being Int-Location
for I being Program of holds (while=0 (a,I)) . ((card I) + 5) = halt SCM+FSA

proof end;

theorem Th15: :: SCMFSA_9:15

for a being Int-Location
for I being Program of holds (while=0 (a,I)) . 3 = goto ((card I) + 5)

proof end;

theorem Th16: :: SCMFSA_9:16

for a being Int-Location
for I being Program of holds (while=0 (a,I)) . 2 = goto 3

proof end;

theorem :: SCMFSA_9:17

for a being Int-Location
for I being Program of
for k being Element of NAT st k < (card I) + 6 holds
k in dom (while=0 (a,I))

proof end;

theorem Th18: :: SCMFSA_9:18

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

proof end;

theorem Th19: :: SCMFSA_9:19

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being Program of
for s being State of SCM+FSA
for k being Element of NAT st I is_closed_on s,P & 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))) + 4 & 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)))) + 4 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) )

proof end;

theorem Th20: :: SCMFSA_9:20

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being Program of
for s being State of SCM+FSA st I is_closed_on s,P & 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)))))) + 4 holds
CurInstr ((P +* (while=0 (a,I))),(Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto ((card I) + 4)

proof end;

theorem Th21: :: SCMFSA_9:21

for a being Int-Location
for I being Program of holds (while=0 (a,I)) . ((card I) + 4) = goto 0

proof end;

theorem Th22: :: SCMFSA_9:22

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of
for a being read-write Int-Location st I is_closed_on s,P & I is_halting_on s,P & s . a = 0 holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 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 Program of ;
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))),($2 +* (Start-At (0,SCM+FSA))),((LifeSpan (((P +* (while=0 (a,I))) +* I),($2 +* (Start-At (0,SCM+FSA))))) + 3));
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) -> Function of NAT,(product (the_Values_of SCM+FSA)) means :Def4: :: SCMFSA_9:def 4
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Comput ((P +* (while=0 (a,I))),((it . i) +* (Start-At (0,SCM+FSA))),((LifeSpan (((P +* (while=0 (a,I))) +* I),((it . i) +* (Start-At (0,SCM+FSA))))) + 3)) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines StepWhile=0 SCMFSA_9:def 4 :

theorem Th23: :: SCMFSA_9:23

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of
for a being read-write Int-Location
for k being Element of 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;

theorem Th25: :: SCMFSA_9:25

for P being Instruction-Sequence of SCM+FSA
for I being Program of
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile=0 (a,I,P,s)) . (0 + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize s))) + 3))

proof end;

theorem Th26: :: SCMFSA_9:26

for P being Instruction-Sequence of SCM+FSA
for I being Program of
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Element of 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)))) + 3))) )

proof end;

theorem Th27: :: SCMFSA_9:27

for P being Instruction-Sequence of SCM+FSA
for I being Program of
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds
( I is_closed_on (StepWhile=0 (a,I,P,s)) . k,P +* (while=0 (a,I)) & 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 & while=0 (a,I) is_closed_on s,P )

proof end;

theorem Th28: :: SCMFSA_9:28

for P being Instruction-Sequence of SCM+FSA
for I being parahalting Program of
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 & while=0 (a,I) is_closed_on s,P )

proof end;

theorem :: SCMFSA_9:29

for I being parahalting Program of
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 Th30: :: SCMFSA_9:30

for l1, l2 being Element of NAT
for a being Int-Location holds not l1 .--> (goto l2) destroys a

proof end;

theorem Th31: :: 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 Program of ;
let a be Int-Location;

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

correctness

proof end;

end;

registration

let I be good Program of ;
let a be Int-Location;

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

correctness

proof end;

end;

:: -----------------------------------------------------------
:: WHILE>0 Statement

theorem Th32: :: SCMFSA_9:32

for a being Int-Location
for I being Program of
for k being Element of NAT st k < 6 holds
k in dom (while>0 (a,I))

proof end;

theorem Th33: :: SCMFSA_9:33

for a being Int-Location
for I being Program of
for k being Element of NAT st k < 6 holds
(card I) + k in dom (while>0 (a,I))

proof end;

theorem Th34: :: SCMFSA_9:34

for a being Int-Location
for I being Program of holds (while>0 (a,I)) . ((card I) + 5) = halt SCM+FSA

proof end;

theorem Th35: :: SCMFSA_9:35

for a being Int-Location
for I being Program of holds (while>0 (a,I)) . 3 = goto ((card I) + 5)

proof end;

theorem Th36: :: SCMFSA_9:36

for a being Int-Location
for I being Program of holds (while>0 (a,I)) . 2 = goto 3

proof end;

theorem :: SCMFSA_9:37

for a being Int-Location
for I being Program of
for k being Element of NAT st k < (card I) + 6 holds
k in dom (while>0 (a,I))

proof end;

theorem Th38: :: SCMFSA_9:38

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

proof end;

theorem Th39: :: SCMFSA_9:39

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being Program of
for s being State of SCM+FSA
for k being Element of NAT st I is_closed_on s,P & 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))) + 4 & 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)))) + 4 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),(k + 1))) )

proof end;

theorem Th40: :: SCMFSA_9:40

for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being Program of
for s being State of SCM+FSA st I is_closed_on s,P & 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)))))) + 4 holds
CurInstr ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto ((card I) + 4)

proof end;

theorem Th41: :: SCMFSA_9:41

for a being Int-Location
for I being Program of holds (while>0 (a,I)) . ((card I) + 4) = goto 0

proof end;

theorem Th42: :: SCMFSA_9:42

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of
for a being read-write Int-Location st I is_closed_on s,P & I is_halting_on s,P & s . a > 0 holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 3))) = 0 & ( for k being Element of NAT st k <= (LifeSpan ((P +* I),(Initialize s))) + 3 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 Program of ;
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))),($2 +* (Start-At (0,SCM+FSA))),((LifeSpan (((P +* (while>0 (a,I))) +* I),($2 +* (Start-At (0,SCM+FSA))))) + 3));
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) -> Function of NAT,(product (the_Values_of SCM+FSA)) means :Def5: :: SCMFSA_9:def 5
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Comput ((P +* (while>0 (a,I))),((it . i) +* (Start-At (0,SCM+FSA))),((LifeSpan (((P +* (while>0 (a,I))) +* I),((it . i) +* (Start-At (0,SCM+FSA))))) + 3)) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines StepWhile>0 SCMFSA_9:def 5 :

theorem Th43: :: SCMFSA_9:43

for P being Instruction-Sequence of SCM+FSA
for k being Element of NAT
for s being State of SCM+FSA
for I being Program of
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 Th44: :: SCMFSA_9:44

for P being Instruction-Sequence of SCM+FSA
for I being Program of
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile>0 (a,I,P,s)) . (0 + 1) = Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize s))) + 3))

proof end;

theorem Th45: :: SCMFSA_9:45

for P being Instruction-Sequence of SCM+FSA
for I being Program of
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Element of 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)))) + 3))) )

proof end;

theorem Th46: :: SCMFSA_9:46

for P being Instruction-Sequence of SCM+FSA
for I being Program of
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds
( I is_closed_on (StepWhile>0 (a,I,P,s)) . k,P +* (while>0 (a,I)) & 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 & while>0 (a,I) is_closed_on s,P )

proof end;

theorem Th47: :: SCMFSA_9:47

for P being Instruction-Sequence of SCM+FSA
for I being parahalting Program of
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 & while>0 (a,I) is_closed_on s,P )

proof end;

theorem :: SCMFSA_9:48

for I being parahalting Program of
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 Program of ;
let a be Int-Location;

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

coherence

proof end;

end;

registration

let I be good Program of ;
let a be Int-Location;

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

correctness

proof end;

end;