SCMFSA6A: On the compositions of macro instructions

:: On the compositions of macro instructions
:: by Andrzej Trybulec , Yatsuka Nakamura and Noriko Asamoto
::
:: Received June 20, 1996
:: Copyright (c) 1996-2012 Association of Mizar Users

begin

definition

let P be NAT -defined the InstructionsF of SCM+FSA -valued finite Function;
let l be Element of NAT ;

func Directed (P,l) -> preProgram of SCM+FSA equals :: SCMFSA6A:def 1
P +~ ((halt SCM+FSA),(goto l));

end;

:: deftheorem defines Directed SCMFSA6A:def 1 :

definition

let P be NAT -defined the InstructionsF of SCM+FSA -valued finite Function;

func Directed P -> preProgram of SCM+FSA equals :: SCMFSA6A:def 2
Directed (P,(card P));

end;

:: deftheorem defines Directed SCMFSA6A:def 2 :

registration

let I be Program of ;

cluster Directed I -> non empty initial ;

proof end;

end;

theorem :: SCMFSA6A:1

for I being Program of holds not halt SCM+FSA in rng (Directed I) by FUNCT_4:100;

theorem :: SCMFSA6A:2

for m being Element of NAT
for I being Program of holds Reloc ((Directed I),m) = ((id the InstructionsF of SCM+FSA) +* ((halt SCM+FSA) .--> (goto (m + (card I))))) * (Reloc (I,m))

proof end;

set q = (intloc 0) .--> 1;

set f = the_Values_of SCM+FSA;

theorem Th23: :: SCMFSA6A:3

for i being Instruction of SCM+FSA
for s being State of SCM+FSA holds
( InsCode i in {0,6,7,8} or (Exec (i,s)) . (IC ) = succ (IC s) )

proof end;

theorem :: SCMFSA6A:4

canceled;

theorem :: SCMFSA6A:5

canceled;

theorem :: SCMFSA6A:6

canceled;

theorem :: SCMFSA6A:7

canceled;

theorem :: SCMFSA6A:8

for s1, s2 being State of SCM+FSA
for n being Element of NAT
for i being Instruction of SCM+FSA st (IC s1) + n = IC s2 & DataPart s1 = DataPart s2 holds
( (IC (Exec (i,s1))) + n = IC (Exec ((IncAddr (i,n)),s2)) & DataPart (Exec (i,s1)) = DataPart (Exec ((IncAddr (i,n)),s2)) )

proof end;

theorem :: SCMFSA6A:9

canceled;

theorem :: SCMFSA6A:10

canceled;

theorem :: SCMFSA6A:11

canceled;

theorem :: SCMFSA6A:12

canceled;

theorem :: SCMFSA6A:13

canceled;

theorem :: SCMFSA6A:14

canceled;

begin

definition

canceled;
let I, J be Program of ;

func I ";" J -> Program of equals :: SCMFSA6A:def 4
(Directed (CutLastLoc (stop I))) +* (Reloc (J,(card I)));

proof end;

end;

:: deftheorem SCMFSA6A:def 3 :

:: deftheorem defines ";" SCMFSA6A:def 4 :

registration

let I be Program of ;
let J be non halt-free Program of ;

cluster I ";" J -> non halt-free ;

end;

theorem :: SCMFSA6A:15

for I, J being Program of
for l being Element of NAT st l in dom I & I . l <> halt SCM+FSA holds
(I ";" J) . l = I . l

proof end;

theorem :: SCMFSA6A:16

for I, J being Program of holds Directed I c= I ";" J

proof end;

theorem Th56: :: SCMFSA6A:17

for I, J being Program of holds dom I c= dom (I ";" J)

proof end;

theorem :: SCMFSA6A:18

for I, J being Program of holds I +* (I ";" J) = I ";" J

proof end;

begin

definition

let i be Instruction of SCM+FSA;
let J be Program of ;

func i ";" J -> Program of equals :: SCMFSA6A:def 5
(Macro i) ";" J;

end;

:: deftheorem defines ";" SCMFSA6A:def 5 :

definition

let I be Program of ;
let j be Instruction of SCM+FSA;

func I ";" j -> Program of equals :: SCMFSA6A:def 6
I ";" (Macro j);

end;

:: deftheorem defines ";" SCMFSA6A:def 6 :

definition

let i, j be Instruction of SCM+FSA;

func i ";" j -> Program of equals :: SCMFSA6A:def 7
(Macro i) ";" (Macro j);

end;

:: deftheorem defines ";" SCMFSA6A:def 7 :

theorem :: SCMFSA6A:19

for i, j being Instruction of SCM+FSA holds i ";" j = (Macro i) ";" j ;

theorem :: SCMFSA6A:20

for i, j being Instruction of SCM+FSA holds i ";" j = i ";" (Macro j) ;

theorem Th61: :: SCMFSA6A:21

for I, J being Program of holds card (I ";" J) = (card I) + (card J)

proof end;

registration

let P be preProgram of SCM+FSA;
let l be Element of NAT ;

cluster Directed (P,l) -> halt-free ;

proof end;

end;

registration

let P be preProgram of SCM+FSA;

cluster Directed P -> halt-free ;

end;

theorem Th63: :: SCMFSA6A:22

for I being preProgram of SCM+FSA
for l being Element of NAT st I is halt-free holds
Directed (I,l) = I

proof end;

theorem Th65: :: SCMFSA6A:23

for I being preProgram of SCM+FSA
for k being Element of NAT holds Reloc ((Directed I),k) = Directed ((Reloc (I,k)),((card I) + k))

proof end;

theorem Th66: :: SCMFSA6A:24

for I, J being Program of holds Directed (I ";" J) = I ";" (Directed J)

proof end;

theorem Th67: :: SCMFSA6A:25

for I, J, K being Program of holds (I ";" J) ";" K = I ";" (J ";" K)

proof end;

theorem :: SCMFSA6A:26

for k being Instruction of SCM+FSA
for I, J being Program of holds (I ";" J) ";" k = I ";" (J ";" k) by Th67;

theorem :: SCMFSA6A:27

for j being Instruction of SCM+FSA
for I, K being Program of holds (I ";" j) ";" K = I ";" (j ";" K) by Th67;

theorem :: SCMFSA6A:28

for j, k being Instruction of SCM+FSA
for I being Program of holds (I ";" j) ";" k = I ";" (j ";" k) by Th67;

theorem :: SCMFSA6A:29

for i being Instruction of SCM+FSA
for J, K being Program of holds (i ";" J) ";" K = i ";" (J ";" K) by Th67;

theorem :: SCMFSA6A:30

for i, k being Instruction of SCM+FSA
for J being Program of holds (i ";" J) ";" k = i ";" (J ";" k) by Th67;

theorem :: SCMFSA6A:31

for i, j being Instruction of SCM+FSA
for K being Program of holds (i ";" j) ";" K = i ";" (j ";" K) by Th67;

theorem :: SCMFSA6A:32

for i, j, k being Instruction of SCM+FSA holds (i ";" j) ";" k = i ";" (j ";" k) by Th67;

theorem :: SCMFSA6A:33

for i being Instruction of SCM+FSA
for J being Program of holds card (i ";" J) = (card J) + 2

proof end;

theorem :: SCMFSA6A:34

for j being Instruction of SCM+FSA
for I being Program of holds card (I ";" j) = (card I) + 2

proof end;

theorem :: SCMFSA6A:35

for i, j being Instruction of SCM+FSA holds card (i ";" j) = 4

proof end;