:: Conditional branch macro instructions of SCM+FSA, Part II
:: by Noriko Asamoto
::
:: Received August 27, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users

set A = NAT ;

set D = Data-Locations ;

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

theorem :: SCMFSA8B:1

canceled;

theorem :: SCMFSA8B:2

canceled;

theorem :: SCMFSA8B:3

canceled;

theorem :: SCMFSA8B:4

canceled;

::$CT 4

theorem Th1: :: SCMFSA8B:5

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being really-closed Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_halting_on s1,P1 holds
I is_halting_on s2,P2

proof end;

theorem :: SCMFSA8B:6

canceled;

theorem :: SCMFSA8B:7

canceled;

:: theorem Th5:
:: for s being State of SCM+FSA, I,J being Program of SCM+FSA holds
:: I is_closed_on Initialized s,P iff
:: I is_closed_on s +* Initialize((intloc 0).-->1),P+*J
:: proof
:: let s be State of SCM+FSA;
:: let I,J be Program of SCM+FSA;
:: DataPart Initialized s = DataPart(s +* Initialize((intloc 0).-->1));
:: hence thesis by Th2;
:: end;
:: theorem Th6:
:: for s being State of SCM+FSA, I,J being Program of SCM+FSA, l
:: being Element of NAT holds I is_closed_on s,P iff I is_closed_on
:: s +* (Start-At(0,SCM+FSA)),P+*I
:: proof
:: let s be State of SCM+FSA;
:: let I,J be Program of SCM+FSA;
:: let l be Element of NAT;
:: DataPart s = DataPart(Initialize s) by MEMSTR_0:79;
:: hence thesis by Th2;
:: end;
::$CT 2

theorem Th2: :: SCMFSA8B:8

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1 being 0 -started State of SCM+FSA
for s2 being State of SCM+FSA
for I being really-closed Program of SCM+FSA st I c= P1 holds
for n being Nat st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )

proof end;

theorem Th3: :: SCMFSA8B:9

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

proof end;

theorem Th4: :: SCMFSA8B:10

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

proof end;

definition

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

func if=0 (a,I,J) -> Program of SCM+FSA equals :: SCMFSA8B:def 1
((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);

coherence
((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is Program of SCM+FSA ;

func if>0 (a,I,J) -> Program of SCM+FSA equals :: SCMFSA8B:def 2
((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);

coherence
((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is Program of SCM+FSA ;

end;

:: deftheorem defines if=0 SCMFSA8B:def 1 :
for a being Int-Location
for I, J being MacroInstruction of SCM+FSA holds if=0 (a,I,J) = ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);

:: deftheorem defines if>0 SCMFSA8B:def 2 :
for a being Int-Location
for I, J being MacroInstruction of SCM+FSA holds if>0 (a,I,J) = ((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);

canceled;

Lm1: for a being Int-Location
for I, J being MacroInstruction of SCM+FSA holds
( 1 in dom (if=0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) )

proof end;

Lm2: for a being Int-Location
for I, J being MacroInstruction of SCM+FSA holds
( (if=0 (a,I,J)) . 0 = a =0_goto ((card J) + 3) & (if=0 (a,I,J)) . 1 = goto 2 & (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )

proof end;

theorem :: SCMFSA8B:11

for I, J being MacroInstruction of SCM+FSA
for a being Int-Location holds card (if=0 (a,I,J)) = ((card I) + (card J)) + 4

proof end;

theorem :: SCMFSA8B:12

for I, J being MacroInstruction of SCM+FSA
for a being Int-Location holds card (if>0 (a,I,J)) = ((card I) + (card J)) + 4

proof end;

theorem Th7: :: SCMFSA8B:13

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 J being MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_halting_on s,P holds
if=0 (a,I,J) is_halting_on s,P

proof end;

theorem Th8: :: SCMFSA8B:14

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 J being MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

Lm3: for I, J being really-closed MacroInstruction of SCM+FSA holds ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is really-closed

proof end;

registration

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

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

coherence
if=0 (a,I,J) is really-closed

proof end;

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

coherence
if>0 (a,I,J) is really-closed

proof end;

end;

theorem Th9: :: SCMFSA8B:15

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

proof end;

theorem Th10: :: SCMFSA8B:16

for P being Instruction-Sequence of SCM+FSA
for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <> 0 & J is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th11: :: SCMFSA8B:17

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location holds
( if=0 (a,I,J) is parahalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )

proof end;

theorem Th12: :: SCMFSA8B:18

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )

proof end;

theorem Th13: :: SCMFSA8B:19

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_halting_on s,P holds
if>0 (a,I,J) is_halting_on s,P

proof end;

theorem Th14: :: SCMFSA8B:20

for P being Instruction-Sequence of SCM+FSA
for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a > 0 & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th15: :: SCMFSA8B:21

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

proof end;

theorem Th16: :: SCMFSA8B:22

for P being Instruction-Sequence of SCM+FSA
for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <= 0 & J is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th17: :: SCMFSA8B:23

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )

proof end;

theorem Th18: :: SCMFSA8B:24

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )

proof end;

theorem :: SCMFSA8B:25

canceled;

theorem :: SCMFSA8B:26

canceled;

theorem :: SCMFSA8B:27

canceled;

theorem :: SCMFSA8B:28

canceled;

theorem :: SCMFSA8B:29

canceled;

theorem :: SCMFSA8B:30

canceled;

theorem :: SCMFSA8B:31

canceled;

::$CT 7

registration

let I, J be really-closed parahalting MacroInstruction of SCM+FSA ;
let a be read-write Int-Location;

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

correctness
coherence
if=0 (a,I,J) is parahalting ;
by Th11;

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

correctness
coherence
if>0 (a,I,J) is parahalting ;
by Th17;

end;

definition

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

func if=0 (a,b,I,J) -> Program of SCM+FSA equals :: SCMFSA8B:def 3
(SubFrom (a,b)) ";" (if=0 (a,I,J));

coherence
(SubFrom (a,b)) ";" (if=0 (a,I,J)) is Program of SCM+FSA ;

func if>0 (a,b,I,J) -> Program of SCM+FSA equals :: SCMFSA8B:def 4
(SubFrom (a,b)) ";" (if>0 (a,I,J));

coherence
(SubFrom (a,b)) ";" (if>0 (a,I,J)) is Program of SCM+FSA ;

end;

:: deftheorem defines if=0 SCMFSA8B:def 3 :
for a, b being Int-Location
for I, J being MacroInstruction of SCM+FSA holds if=0 (a,b,I,J) = (SubFrom (a,b)) ";" (if=0 (a,I,J));

:: deftheorem defines if>0 SCMFSA8B:def 4 :
for a, b being Int-Location
for I, J being MacroInstruction of SCM+FSA holds if>0 (a,b,I,J) = (SubFrom (a,b)) ";" (if>0 (a,I,J));

registration

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

cluster if=0 (a,I,J) -> halt-ending unique-halt ;

coherence
( if=0 (a,I,J) is halt-ending & if=0 (a,I,J) is unique-halt ) ;

cluster if>0 (a,I,J) -> halt-ending unique-halt ;

coherence
( if>0 (a,I,J) is halt-ending & if>0 (a,I,J) is unique-halt ) ;

end;

registration

let a, b be Int-Location;
let I, J be really-closed MacroInstruction of SCM+FSA ;

cluster if=0 (a,b,I,J) -> really-closed ;

coherence
if=0 (a,b,I,J) is really-closed ;

cluster if>0 (a,b,I,J) -> really-closed ;

coherence
if>0 (a,b,I,J) is really-closed ;

end;

registration

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

cluster if=0 (a,b,I,J) -> halt-ending unique-halt ;

coherence
( if=0 (a,b,I,J) is halt-ending & if=0 (a,b,I,J) is unique-halt ) ;

cluster if>0 (a,b,I,J) -> halt-ending unique-halt ;

coherence
( if>0 (a,b,I,J) is halt-ending & if>0 (a,b,I,J) is unique-halt ) ;

end;

registration

let I, J be really-closed parahalting MacroInstruction of SCM+FSA ;
let a, b be read-write Int-Location;

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

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

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

correctness
coherence
if>0 (a,b,I,J) is parahalting ;
;

end;

registration

let I, J be really-closed MacroInstruction of SCM+FSA ;
let a, b be read-write Int-Location;

cluster if=0 (a,b,I,J) -> really-closed ;

correctness
coherence
if=0 (a,b,I,J) is really-closed ;
;

cluster if>0 (a,b,I,J) -> really-closed ;

correctness
coherence
if>0 (a,b,I,J) is really-closed ;
;

end;

theorem :: SCMFSA8B:32

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA holds DataPart (Result ((P +* I),(Initialized s))) = DataPart (IExec (I,P,s)) by SCMFSA6B:def 1;

theorem Th20: :: SCMFSA8B:33

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA holds Result ((P +* I),(Initialized s)) = IExec (I,P,s) by SCMFSA6B:def 1;

theorem Th21: :: SCMFSA8B:34

for s1, s2 being State of SCM+FSA
for i being Instruction of SCM+FSA
for a being Int-Location st ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not i refers a & IC s1 = IC s2 holds
( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) )

proof end;

theorem Th22: :: SCMFSA8B:35

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being really-closed Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) holds
for k being Nat holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )

proof end;

theorem :: SCMFSA8B:36

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being really-closed Program of SCM+FSA
for l being Nat holds
( I is_halting_on s,P iff I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I )

proof end;

theorem Th24: :: SCMFSA8B:37

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being really-closed Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_halting_on s1,P1 holds
I is_halting_on s2,P2

proof end;

theorem Th25: :: SCMFSA8B:38

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being really-closed Program of SCM+FSA
for a being Int-Location st ( for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not I refers a & I is_halting_on Initialized s1,P1 holds
( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )

proof end;

theorem :: SCMFSA8B:39

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )

proof end;

theorem :: SCMFSA8B:40

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being really-closed parahalting MacroInstruction of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )

proof end;

theorem :: SCMFSA8B:41

canceled;

:: theorem
:: s.intloc 0 = 1 implies (I is_closed_on s,p iff I is_closed_on
:: Initialized s,p)
:: proof
:: assume s.intloc 0 = 1;
:: then DataPart Initialized s = DataPart s by SCMFSA_M:19;
:: hence thesis by Th2;
:: end;
::$CT

theorem :: SCMFSA8B:42

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for I being really-closed Program of SCM+FSA st s . (intloc 0) = 1 holds
( I is_halting_on s,p iff I is_halting_on Initialized s,p )

proof end;