:: Conditional branch macro instructions of SCM+FSA, Part II
:: by Noriko Asamoto
::
:: 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,())))) . 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,())))) . 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) ";" ();
coherence
((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" () 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) ";" ();
coherence
((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" () 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) ";" ();

:: 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) ";" ();

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) ";" () 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 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 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 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),())) = 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),()) = 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),(),k)) . b = (Comput ((P2 +* I),(),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(),k)) . f = (Comput ((P2 +* I),(),k)) . f ) & IC (Comput ((P1 +* I),(),k)) = IC (Comput ((P2 +* I),(),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(),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 . () = 1 holds
( I is_halting_on s,p iff I is_halting_on Initialized s,p )
proof end;