:: Another { \bf times } Macro Instruction
:: by Piotr Rudnicki
::
:: Copyright (c) 1998-2021 Association of Mizar Users

set D = Data-Locations ;

theorem Th1: :: SFMASTR2:1
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for b being Int-Location
for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds
(IExec (I,p,s)) . b = () . b
proof end;

theorem :: SFMASTR2:2
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for f being FinSeq-Location
for I being really-closed Program of st I is_halting_on Initialized s,p & not f in UsedI*Loc I holds
(IExec (I,p,s)) . f = () . f
proof end;

theorem :: SFMASTR2:3
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed Program of st ( I is_halting_on Initialized s,p or I is parahalting ) & ( s . () = 1 or a is read-write ) & not a in UsedILoc I holds
(IExec (I,p,s)) . a = s . a
proof end;

theorem :: SFMASTR2:4
canceled;

theorem :: SFMASTR2:5
canceled;

definition
let a be Int-Location;
let I be MacroInstruction of SCM+FSA ;
func times* (a,I) -> MacroInstruction of SCM+FSA equals :: SFMASTR2:def 1
while>0 ((1 -stRWNotIn ({a} \/ ())),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ ())),()))));
correctness
coherence
while>0 ((1 -stRWNotIn ({a} \/ ())),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ ())),())))) is MacroInstruction of SCM+FSA
;
;
end;

:: deftheorem defines times* SFMASTR2:def 1 :
for a being Int-Location
for I being MacroInstruction of SCM+FSA holds times* (a,I) = while>0 ((1 -stRWNotIn ({a} \/ ())),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ ())),()))));

definition
let a be Int-Location;
let I be MacroInstruction of SCM+FSA ;
func times (a,I) -> MacroInstruction of SCM+FSA equals :: SFMASTR2:def 2
((1 -stRWNotIn ({a} \/ ())) := a) ";" (times* (a,I));
correctness
coherence
((1 -stRWNotIn ({a} \/ ())) := a) ";" (times* (a,I)) is MacroInstruction of SCM+FSA
;
;
end;

:: deftheorem defines times SFMASTR2:def 2 :
for a being Int-Location
for I being MacroInstruction of SCM+FSA holds times (a,I) = ((1 -stRWNotIn ({a} \/ ())) := a) ";" (times* (a,I));

registration
let a be Int-Location;
let I be really-closed MacroInstruction of SCM+FSA ;
cluster times* (a,I) -> really-closed ;
coherence
times* (a,I) is really-closed
;
end;

registration
let a be Int-Location;
let I be really-closed MacroInstruction of SCM+FSA ;
cluster times (a,I) -> really-closed ;
coherence
times (a,I) is really-closed
;
end;

theorem :: SFMASTR2:6
canceled;

theorem :: SFMASTR2:7
canceled;

::\$CT 2
theorem :: SFMASTR2:8
for b being Int-Location
for I being MacroInstruction of SCM+FSA holds {b} \/ () c= UsedILoc (times (b,I))
proof end;

theorem :: SFMASTR2:9
for b being Int-Location
for I being MacroInstruction of SCM+FSA holds UsedI*Loc (times (b,I)) = UsedI*Loc I
proof end;

registration
let I be good MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster times (a,I) -> good ;
coherence
times (a,I) is good
;
end;

definition
let p be Instruction-Sequence of SCM+FSA;
let s be State of SCM+FSA;
let I be MacroInstruction of SCM+FSA ;
let a be Int-Location;
func StepTimes (a,I,p,s) -> sequence of equals :: SFMASTR2:def 3
StepWhile>0 ((1 -stRWNotIn ({a} \/ ())),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ ())),()))),p,(Exec (((1 -stRWNotIn ({a} \/ ())) := a),())));
correctness
coherence
StepWhile>0 ((1 -stRWNotIn ({a} \/ ())),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ ())),()))),p,(Exec (((1 -stRWNotIn ({a} \/ ())) := a),()))) is sequence of
;
;
end;

:: deftheorem defines StepTimes SFMASTR2:def 3 :
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 Int-Location holds StepTimes (a,I,p,s) = StepWhile>0 ((1 -stRWNotIn ({a} \/ ())),(I ";" (SubFrom ((1 -stRWNotIn ({a} \/ ())),()))),p,(Exec (((1 -stRWNotIn ({a} \/ ())) := a),())));

theorem Th6: :: SFMASTR2:10
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being good MacroInstruction of SCM+FSA holds ((StepTimes (a,J,p,s)) . 0) . () = 1
proof end;

theorem Th7: :: SFMASTR2:11
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being good MacroInstruction of SCM+FSA st ( s . () = 1 or a is read-write ) holds
((StepTimes (a,J,p,s)) . 0) . (1 -stRWNotIn ({a} \/ ())) = s . a
proof end;

theorem Th8: :: SFMASTR2:12
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st ((StepTimes (a,J,p,s)) . k) . () = 1 & J is_halting_on (StepTimes (a,J,p,s)) . k,p +* (times* (a,J)) holds
( ((StepTimes (a,J,p,s)) . (k + 1)) . () = 1 & ( ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ ())) > 0 implies ((StepTimes (a,J,p,s)) . (k + 1)) . (1 -stRWNotIn ({a} \/ ())) = (((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ ()))) - 1 ) )
proof end;

theorem Th9: :: SFMASTR2:13
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being MacroInstruction of SCM+FSA st ( s . () = 1 or a is read-write ) holds
((StepTimes (a,I,p,s)) . 0) . a = s . a
proof end;

theorem :: SFMASTR2:14
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for f being FinSeq-Location
for I being MacroInstruction of SCM+FSA holds ((StepTimes (a,I,p,s)) . 0) . f = s . f
proof end;

definition
let p be Instruction-Sequence of SCM+FSA;
let s be State of SCM+FSA;
let a be Int-Location;
let I be MacroInstruction of SCM+FSA ;
pred ProperTimesBody a,I,s,p means :: SFMASTR2:def 4
for k being Nat st k < s . a holds
I is_halting_on (StepTimes (a,I,p,s)) . k,p +* (times* (a,I));
end;

:: deftheorem defines ProperTimesBody SFMASTR2:def 4 :
for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for a being Int-Location
for I being MacroInstruction of SCM+FSA holds
( ProperTimesBody a,I,s,p iff for k being Nat st k < s . a holds
I is_halting_on (StepTimes (a,I,p,s)) . k,p +* (times* (a,I)) );

theorem Th11: :: SFMASTR2:15
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being MacroInstruction of SCM+FSA st I is parahalting holds
ProperTimesBody a,I,s,p by SCMFSA7B:19;

theorem Th12: :: SFMASTR2:16
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being really-closed good MacroInstruction of SCM+FSA st ProperTimesBody a,J,s,p holds
for k being Nat st k <= s . a holds
((StepTimes (a,J,p,s)) . k) . () = 1
proof end;

theorem Th13: :: SFMASTR2:17
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being really-closed good MacroInstruction of SCM+FSA st ( s . () = 1 or a is read-write ) & ProperTimesBody a,J,s,p holds
for k being Nat st k <= s . a holds
(((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ ()))) + k = s . a
proof end;

theorem Th14: :: SFMASTR2:18
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being really-closed good MacroInstruction of SCM+FSA st ProperTimesBody a,J,s,p & 0 <= s . a & ( s . () = 1 or a is read-write ) holds
for k being Nat st k >= s . a holds
( ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ ())) = 0 & ((StepTimes (a,J,p,s)) . k) . () = 1 )
proof end;

theorem :: SFMASTR2:19
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being MacroInstruction of SCM+FSA st s . () = 1 holds
((StepTimes (a,I,p,s)) . 0) | () = s | ()
proof end;

theorem Th16: :: SFMASTR2:20
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st ((StepTimes (a,J,p,s)) . k) . () = 1 & J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) & ((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ ())) > 0 holds
((StepTimes (a,J,p,s)) . (k + 1)) | () = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ()
proof end;

theorem :: SFMASTR2:21
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st ( ProperTimesBody a,J,s,p or J is parahalting ) & k < s . a & ( s . () = 1 or a is read-write ) holds
((StepTimes (a,J,p,s)) . (k + 1)) | () = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ()
proof end;

theorem :: SFMASTR2:22
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA st s . a <= 0 & s . () = 1 holds
(IExec ((times (a,I)),p,s)) | () = s | ()
proof end;

theorem Th19: :: SFMASTR2:23
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for k being Nat
for J being really-closed good MacroInstruction of SCM+FSA st s . a = k & ( ProperTimesBody a,J,s,p or J is parahalting ) & ( s . () = 1 or a is read-write ) holds
DataPart (IExec ((times (a,J)),p,s)) = DataPart ((StepTimes (a,J,p,s)) . k)
proof end;

theorem :: SFMASTR2:24
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being really-closed good MacroInstruction of SCM+FSA st s . () = 1 & ( ProperTimesBody a,J,s,p or J is parahalting ) holds
times (a,J) is_halting_on s,p
proof end;

definition
func triv-times d -> MacroInstruction of SCM+FSA equals :: SFMASTR2:def 5
times (d,((while=0 (d,(Macro (d := d)))) ";" (SubFrom (d,()))));
correctness
coherence
times (d,((while=0 (d,(Macro (d := d)))) ";" (SubFrom (d,())))) is MacroInstruction of SCM+FSA
;
;
end;

:: deftheorem defines triv-times SFMASTR2:def 5 :
for d being read-write Int-Location holds triv-times d = times (d,((while=0 (d,(Macro (d := d)))) ";" (SubFrom (d,()))));

registration
coherence ;
end;

theorem :: SFMASTR2:25
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for d being read-write Int-Location st s . d <= 0 holds
(IExec ((),p,s)) . d = s . d
proof end;

theorem :: SFMASTR2:26
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for d being read-write Int-Location st 0 <= s . d holds
(IExec ((),p,s)) . d = 0
proof end;