:: While Macro Instructions of SCM+FSA
:: by Jing-Chao Chen
::
:: Received December 10, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users


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

canceled;

canceled;

canceled;

:: WHILE Statement
theorem :: SCMFSA_9:1
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:2
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:3
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:4
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:5
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:6
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:7
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:8
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:9
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:10
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:11
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:12
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:13
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:14
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:15
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:16
canceled;

:: WHILE Statement
theorem :: SCMFSA_9:17
canceled;

::$CD 3
::$CT 17
theorem Th1: :: SCMFSA_9:18
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 read-write Int-Location st s . a <> 0 holds
while=0 (a,I) is_halting_on s,P
proof end;

theorem Th2: :: SCMFSA_9:19
for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA
for k being Nat st 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))) + 3 & 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)))) + 3 & DataPart (Comput ((P +* (while=0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) )
proof end;

:: Twierdzeinie zmienia znaczenie, jezeli w oryginale mowilo
:: ze znajdziemy sie w miejscu "przed powrotem" i wykonujemy Next
:: to w tym miejscu jest teraz powrot i wykonujemy goto 0
theorem Th3: :: SCMFSA_9:20
for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA st 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)))))) + 3 holds
CurInstr ((P +* (while=0 (a,I))),(Comput ((P +* (while=0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto 0
proof end;

theorem :: SCMFSA_9:21
canceled;

::$CT
theorem Th4: :: SCMFSA_9:22
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 a being read-write Int-Location st I is_halting_on s,P & s . a = 0 holds
( IC (Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 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 MacroInstruction of SCM+FSA ;
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))),(Initialize $2),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize $2))) + 2));
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) -> sequence of (product (the_Values_of SCM+FSA)) means :Def1: :: SCMFSA_9:def 1
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (it . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (it . i)))) + 2)) ) );
existence
ex b1 being sequence of (product (the_Values_of SCM+FSA)) st
( b1 . 0 = s & ( for i being Nat holds b1 . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b1 . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b1 . i)))) + 2)) ) )
proof end;
uniqueness
for b1, b2 being sequence of (product (the_Values_of SCM+FSA)) st b1 . 0 = s & ( for i being Nat holds b1 . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b1 . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b1 . i)))) + 2)) ) & b2 . 0 = s & ( for i being Nat holds b2 . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b2 . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b2 . i)))) + 2)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines StepWhile=0 SCMFSA_9:def 1 :
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for P being Instruction-Sequence of SCM+FSA
for b5 being sequence of (product (the_Values_of SCM+FSA)) holds
( b5 = StepWhile=0 (a,I,P,s) iff ( b5 . 0 = s & ( for i being Nat holds b5 . (i + 1) = Comput ((P +* (while=0 (a,I))),(Initialize (b5 . i)),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize (b5 . i)))) + 2)) ) ) );

theorem Th5: :: SCMFSA_9:23
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 read-write Int-Location
for k being 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;

::$CT
theorem Th6: :: SCMFSA_9:25
for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile=0 (a,I,P,s)) . 1 = Comput ((P +* (while=0 (a,I))),(Initialize s),((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize s))) + 2))
proof end;

theorem Th7: :: SCMFSA_9:26
for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being 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)))) + 2))) )
proof end;

theorem Th8: :: SCMFSA_9:27
for P being Instruction-Sequence of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds 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
proof end;

theorem Th9: :: SCMFSA_9:28
for P being Instruction-Sequence of SCM+FSA
for I being really-closed parahalting MacroInstruction of SCM+FSA
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
proof end;

theorem :: SCMFSA_9:29
for I being really-closed parahalting MacroInstruction of SCM+FSA
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 :: SCMFSA_9:30
canceled;

::$CT
theorem Th11: :: 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 MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster if=0 (a,I,J) -> good ;
correctness
coherence
if=0 (a,I,J) is good ;
proof end;
end;

registration
let I be good MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster if=0 (a,I) -> good ;
correctness
coherence
if=0 (a,I) is good ;
proof end;
end;

registration
let I be good MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster while=0 (a,I) -> good ;
correctness
coherence
while=0 (a,I) is good ;
proof end;
end;

theorem :: SCMFSA_9:32
canceled;

theorem :: SCMFSA_9:33
canceled;

theorem :: SCMFSA_9:34
canceled;

theorem :: SCMFSA_9:35
canceled;

theorem :: SCMFSA_9:36
canceled;

theorem :: SCMFSA_9:37
canceled;

:: -----------------------------------------------------------
:: WHILE>0 Statement
::$CT 6
theorem Th12: :: SCMFSA_9:38
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 read-write Int-Location st s . a <= 0 holds
while>0 (a,I) is_halting_on s,P
proof end;

theorem Th13: :: SCMFSA_9:39
for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA
for k being Nat st 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))) + 3 & 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)))) + 3 & DataPart (Comput ((P +* (while>0 (a,I))),(Initialize s),((1 + k) + 1))) = DataPart (Comput ((P +* I),(Initialize s),(k + 1))) )
proof end;

theorem Th14: :: SCMFSA_9:40
for P being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being really-closed MacroInstruction of SCM+FSA
for s being State of SCM+FSA st 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)))))) + 3 holds
CurInstr ((P +* (while>0 (a,I))),(Comput ((P +* (while>0 (a,I))),(Initialize s),(1 + (LifeSpan ((P +* I),(Initialize s))))))) = goto 0
proof end;

theorem :: SCMFSA_9:41
canceled;

::$CT
theorem Th15: :: SCMFSA_9:42
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 a being read-write Int-Location st I is_halting_on s,P & s . a > 0 holds
( IC (Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = 0 & ( for k being Nat st k <= (LifeSpan ((P +* I),(Initialize s))) + 2 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 MacroInstruction of SCM+FSA ;
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))),(Initialize $2),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize $2))) + 2));
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) -> sequence of (product (the_Values_of SCM+FSA)) means :Def2: :: SCMFSA_9:def 2
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (it . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (it . i)))) + 2)) ) );
existence
ex b1 being sequence of (product (the_Values_of SCM+FSA)) st
( b1 . 0 = s & ( for i being Nat holds b1 . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b1 . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b1 . i)))) + 2)) ) )
proof end;
uniqueness
for b1, b2 being sequence of (product (the_Values_of SCM+FSA)) st b1 . 0 = s & ( for i being Nat holds b1 . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b1 . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b1 . i)))) + 2)) ) & b2 . 0 = s & ( for i being Nat holds b2 . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b2 . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b2 . i)))) + 2)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines StepWhile>0 SCMFSA_9:def 2 :
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for P being Instruction-Sequence of SCM+FSA
for b5 being sequence of (product (the_Values_of SCM+FSA)) holds
( b5 = StepWhile>0 (a,I,P,s) iff ( b5 . 0 = s & ( for i being Nat holds b5 . (i + 1) = Comput ((P +* (while>0 (a,I))),(Initialize (b5 . i)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize (b5 . i)))) + 2)) ) ) );

theorem Th16: :: SCMFSA_9:43
for P being Instruction-Sequence of SCM+FSA
for k being Nat
for s being State of SCM+FSA
for I being MacroInstruction of SCM+FSA
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 Th17: :: SCMFSA_9:44
for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile>0 (a,I,P,s)) . 1 = Comput ((P +* (while>0 (a,I))),(Initialize s),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize s))) + 2))
proof end;

theorem Th18: :: SCMFSA_9:45
for P being Instruction-Sequence of SCM+FSA
for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being 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)))) + 2))) )
proof end;

theorem Th19: :: SCMFSA_9:46
for P being Instruction-Sequence of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds 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
proof end;

theorem Th20: :: SCMFSA_9:47
for P being Instruction-Sequence of SCM+FSA
for I being really-closed parahalting MacroInstruction of SCM+FSA
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
proof end;

theorem :: SCMFSA_9:48
for I being really-closed parahalting MacroInstruction of SCM+FSA
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 MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster if>0 (a,I,J) -> good ;
coherence
if>0 (a,I,J) is good
proof end;
end;

registration
let I be good MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster if>0 (a,I) -> good ;
correctness
coherence
if>0 (a,I) is good ;
proof end;
end;

registration
let I be good MacroInstruction of SCM+FSA ;
let a be Int-Location;
cluster while>0 (a,I) -> good ;
correctness
coherence
while>0 (a,I) is good ;
proof end;
end;

theorem :: SCMFSA_9:49
for p being preProgram of SCM+FSA
for l being Nat
for ic being Instruction of SCM+FSA st ex pc being Instruction of SCM+FSA st
( pc = p . l & UsedIntLoc pc = UsedIntLoc ic ) holds
UsedILoc p = UsedILoc (p +* (l,ic))
proof end;

theorem :: SCMFSA_9:50
for p being preProgram of SCM+FSA
for l being Nat
for ic being Instruction of SCM+FSA st ex pc being Instruction of SCM+FSA st
( pc = p . l & UsedInt*Loc pc = UsedInt*Loc ic ) holds
UsedI*Loc p = UsedI*Loc (p +* (l,ic))
proof end;