:: SCMFSA_9 semantic presentation

for I being Macro-Instruction
for a being Int-Location holds card (if=0 a,(I ';' (Goto (insloc 0))),SCM+FSA-Stop ) = (card I) + 6

proof end;

theorem Th2: :: SCMFSA_9:2

for I being Macro-Instruction
for a being Int-Location holds card (if>0 a,(I ';' (Goto (insloc 0))),SCM+FSA-Stop ) = (card I) + 6

proof end;

definition

let a be Int-Location ;
let I be Macro-Instruction;
func while=0 a,I -> Macro-Instruction equals :: SCMFSA_9:def 1
(if=0 a,(I ';' (Goto (insloc 0))),SCM+FSA-Stop ) +* ((insloc ((card I) + 4)) .--> (goto (insloc 0)));
correctness

proof end;

func while>0 a,I -> Macro-Instruction equals :: SCMFSA_9:def 2
(if>0 a,(I ';' (Goto (insloc 0))),SCM+FSA-Stop ) +* ((insloc ((card I) + 4)) .--> (goto (insloc 0)));
correctness

proof end;

end;

:: deftheorem defines while=0 SCMFSA_9:def 1 :

:: deftheorem defines while>0 SCMFSA_9:def 2 :

theorem Th3: :: SCMFSA_9:3

for I being Macro-Instruction
for a being Int-Location holds card (if=0 a,SCM+FSA-Stop ,(if>0 a,SCM+FSA-Stop ,(I ';' (Goto (insloc 0))))) = (card I) + 11

proof end;

definition

let a be Int-Location ;
let I be Macro-Instruction;
func while<0 a,I -> Macro-Instruction equals :: SCMFSA_9:def 3
(if=0 a,SCM+FSA-Stop ,(if>0 a,SCM+FSA-Stop ,(I ';' (Goto (insloc 0))))) +* ((insloc ((card I) + 4)) .--> (goto (insloc 0)));
correctness

proof end;

end;

:: deftheorem defines while<0 SCMFSA_9:def 3 :

theorem Th4: :: SCMFSA_9:4

for I being Macro-Instruction
for a being Int-Location holds card (while=0 a,I) = (card I) + 6

proof end;

theorem Th5: :: SCMFSA_9:5

for I being Macro-Instruction
for a being Int-Location holds card (while>0 a,I) = (card I) + 6

proof end;

theorem :: SCMFSA_9:6

for I being Macro-Instruction
for a being Int-Location holds card (while<0 a,I) = (card I) + 11

proof end;

theorem Th7: :: SCMFSA_9:7

for a being Int-Location
for l being Instruction-Location of SCM+FSA holds a =0_goto l <> halt SCM+FSA by SCMFSA_2:48, SCMFSA_2:124;

theorem Th8: :: SCMFSA_9:8

for a being Int-Location
for l being Instruction-Location of SCM+FSA holds a >0_goto l <> halt SCM+FSA by SCMFSA_2:49, SCMFSA_2:124;

theorem Th9: :: SCMFSA_9:9

for l being Instruction-Location of SCM+FSA holds goto l <> halt SCM+FSA by SCMFSA_2:47, SCMFSA_2:124;

theorem Th10: :: SCMFSA_9:10

for a being Int-Location
for I being Macro-Instruction holds
( insloc 0 in dom (while=0 a,I) & insloc 1 in dom (while=0 a,I) & insloc 0 in dom (while>0 a,I) & insloc 1 in dom (while>0 a,I) )

proof end;

theorem Th11: :: SCMFSA_9:11

for a being Int-Location
for I being Macro-Instruction holds
( (while=0 a,I) . (insloc 0) = a =0_goto (insloc 4) & (while=0 a,I) . (insloc 1) = goto (insloc 2) & (while>0 a,I) . (insloc 0) = a >0_goto (insloc 4) & (while>0 a,I) . (insloc 1) = goto (insloc 2) )

proof end;

theorem Th12: :: SCMFSA_9:12

for a being Int-Location
for I being Macro-Instruction
for k being Nat st k < 6 holds
insloc k in dom (while=0 a,I)

proof end;

theorem Th13: :: SCMFSA_9:13

for a being Int-Location
for I being Macro-Instruction
for k being Nat st k < 6 holds
insloc ((card I) + k) in dom (while=0 a,I)

proof end;

theorem Th14: :: SCMFSA_9:14

for a being Int-Location
for I being Macro-Instruction holds (while=0 a,I) . (insloc ((card I) + 5)) = halt SCM+FSA

proof end;

theorem Th15: :: SCMFSA_9:15

for a being Int-Location
for I being Macro-Instruction holds (while=0 a,I) . (insloc 3) = goto (insloc ((card I) + 5))

proof end;

theorem Th16: :: SCMFSA_9:16

for a being Int-Location
for I being Macro-Instruction holds (while=0 a,I) . (insloc 2) = goto (insloc 3)

proof end;

theorem :: SCMFSA_9:17

for a being Int-Location
for I being Macro-Instruction
for k being Nat st k < (card I) + 6 holds
insloc k in dom (while=0 a,I)

proof end;

theorem Th18: :: SCMFSA_9:18

for s being State of SCM+FSA
for I being Macro-Instruction
for a being read-write Int-Location st s . a <> 0 holds
( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s )

proof end;

theorem Th19: :: SCMFSA_9:19

proof end;

theorem Th20: :: SCMFSA_9:20

for a being Int-Location
for I being Macro-Instruction
for s being State of SCM+FSA st I is_closed_on s & I is_halting_on s & IC ((Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . (1 + (LifeSpan (s +* (I +* (Start-At (insloc 0))))))) = (IC ((Computation (s +* (I +* (Start-At (insloc 0))))) . (LifeSpan (s +* (I +* (Start-At (insloc 0))))))) + 4 holds
CurInstr ((Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . (1 + (LifeSpan (s +* (I +* (Start-At (insloc 0))))))) = goto (insloc ((card I) + 4))

proof end;

theorem Th21: :: SCMFSA_9:21

for a being Int-Location
for I being Macro-Instruction holds (while=0 a,I) . (insloc ((card I) + 4)) = goto (insloc 0)

proof end;

theorem Th22: :: SCMFSA_9:22

for s being State of SCM+FSA
for I being Macro-Instruction
for a being read-write Int-Location st I is_closed_on s & I is_halting_on s & s . a = 0 holds
( IC ((Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 3)) = insloc 0 & ( for k being Nat st k <= (LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 3 holds
IC ((Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . k) in dom (while=0 a,I) ) )

proof end;

set sl0 = Start-At (insloc 0);

definition

let s be State of SCM+FSA ;
let I be Macro-Instruction;
let a be read-write Int-Location ;
func StepWhile=0 a,I,s -> Function of NAT , product the Object-Kind of SCM+FSA means :Def4: :: SCMFSA_9:def 4
( it . 0 = s & ( for i being Nat holds it . (i + 1) = (Computation ((it . i) +* ((while=0 a,I) +* (Start-At (insloc 0))))) . ((LifeSpan ((it . i) +* (I +* (Start-At (insloc 0))))) + 3) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines StepWhile=0 SCMFSA_9:def 4 :

theorem :: SCMFSA_9:23

canceled;

theorem :: SCMFSA_9:24

canceled;

theorem Th25: :: SCMFSA_9:25

for s being State of SCM+FSA
for I being Macro-Instruction
for a being read-write Int-Location
for k being Nat holds (StepWhile=0 a,I,s) . (k + 1) = (StepWhile=0 a,I,((StepWhile=0 a,I,s) . k)) . 1

proof end;

scheme :: SCMFSA_9:sch 1
MinIndex{ F₁( Nat) -> Nat, F₂() -> Nat } :

ex k being Nat st
( F₁(k) = 0 & ( for n being Nat st F₁(n) = 0 holds
k <= n ) )

provided

A1: F₁(0) = F₂() and
A2: for k being Nat holds
( F₁((k + 1)) < F₁(k) or F₁(k) = 0 )

proof end;

theorem :: SCMFSA_9:26

for f, g being Function holds (f +* g) +* g = f +* g

proof end;

theorem Th27: :: SCMFSA_9:27

for f, g, h being Function
for D being set st (f +* g) | D = h | D holds
(h +* g) | D = (f +* g) | D

proof end;

theorem :: SCMFSA_9:28

for f, g, h being Function
for D being set st f | D = h | D holds
(h +* g) | D = (f +* g) | D

proof end;

theorem Th29: :: SCMFSA_9:29

for s1, s2 being State of SCM+FSA st IC s1 = IC s2 & s1 | (Int-Locations \/ FinSeq-Locations ) = s2 | (Int-Locations \/ FinSeq-Locations ) & s1 | the Instruction-Locations of SCM+FSA = s2 | the Instruction-Locations of SCM+FSA holds
s1 = s2

proof end;

theorem Th30: :: SCMFSA_9:30

for I being Macro-Instruction
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile=0 a,I,s) . (0 + 1) = (Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 3)

proof end;

theorem Th31: :: SCMFSA_9:31

for I being Macro-Instruction
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,s) . k) = insloc 0 & (StepWhile=0 a,I,s) . k = (Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . n holds
( (StepWhile=0 a,I,s) . k = ((StepWhile=0 a,I,s) . k) +* ((while=0 a,I) +* (Start-At (insloc 0))) & (StepWhile=0 a,I,s) . (k + 1) = (Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0))))) . (n + ((LifeSpan (((StepWhile=0 a,I,s) . k) +* (I +* (Start-At (insloc 0))))) + 3)) )

proof end;

theorem Th32: :: SCMFSA_9:32

for I being Macro-Instruction
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds
( I is_closed_on (StepWhile=0 a,I,s) . k & I is_halting_on (StepWhile=0 a,I,s) . k ) ) & ex f being Function of product the Object-Kind of SCM+FSA , NAT st
for k being Nat holds
( ( f . ((StepWhile=0 a,I,s) . (k + 1)) < f . ((StepWhile=0 a,I,s) . k) or f . ((StepWhile=0 a,I,s) . k) = 0 ) & ( f . ((StepWhile=0 a,I,s) . k) = 0 implies ((StepWhile=0 a,I,s) . k) . a <> 0 ) & ( ((StepWhile=0 a,I,s) . k) . a <> 0 implies f . ((StepWhile=0 a,I,s) . k) = 0 ) ) holds
( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s )

proof end;

theorem Th33: :: SCMFSA_9:33

for I being parahalting Macro-Instruction
for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of product the Object-Kind of SCM+FSA , NAT st
for k being Nat holds
( ( f . ((StepWhile=0 a,I,s) . (k + 1)) < f . ((StepWhile=0 a,I,s) . k) or f . ((StepWhile=0 a,I,s) . k) = 0 ) & ( f . ((StepWhile=0 a,I,s) . k) = 0 implies ((StepWhile=0 a,I,s) . k) . a <> 0 ) & ( ((StepWhile=0 a,I,s) . k) . a <> 0 implies f . ((StepWhile=0 a,I,s) . k) = 0 ) ) holds
( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s )

proof end;

theorem :: SCMFSA_9:34

for I being parahalting Macro-Instruction
for a being read-write Int-Location st ex f being Function of product the Object-Kind of SCM+FSA , NAT st
for s being State of SCM+FSA holds
( ( f . ((StepWhile=0 a,I,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 Th35: :: SCMFSA_9:35

for l1, l2 being Instruction-Location of SCM+FSA
for a being Int-Location holds l1 .--> (goto l2) does_not_destroy a

proof end;

theorem Th36: :: SCMFSA_9:36

for i being Instruction of SCM+FSA st i does_not_destroy intloc 0 holds
Macro i is good

proof end;

registration

let I, J be good Macro-Instruction;
let a be Int-Location ;
cluster if=0 a,I,J -> good ;
correctness

proof end;

end;

registration

let I be good Macro-Instruction;
let a be Int-Location ;
cluster while=0 a,I -> good ;
correctness

proof end;

end;

theorem Th37: :: SCMFSA_9:37

for a being Int-Location
for I being Macro-Instruction
for k being Nat st k < 6 holds
insloc k in dom (while>0 a,I)

proof end;

theorem Th38: :: SCMFSA_9:38

for a being Int-Location
for I being Macro-Instruction
for k being Nat st k < 6 holds
insloc ((card I) + k) in dom (while>0 a,I)

proof end;

theorem Th39: :: SCMFSA_9:39

for a being Int-Location
for I being Macro-Instruction holds (while>0 a,I) . (insloc ((card I) + 5)) = halt SCM+FSA

proof end;

theorem Th40: :: SCMFSA_9:40

for a being Int-Location
for I being Macro-Instruction holds (while>0 a,I) . (insloc 3) = goto (insloc ((card I) + 5))

proof end;

theorem Th41: :: SCMFSA_9:41

for a being Int-Location
for I being Macro-Instruction holds (while>0 a,I) . (insloc 2) = goto (insloc 3)

proof end;

theorem :: SCMFSA_9:42

for a being Int-Location
for I being Macro-Instruction
for k being Nat st k < (card I) + 6 holds
insloc k in dom (while>0 a,I)

proof end;

theorem Th43: :: SCMFSA_9:43

for s being State of SCM+FSA
for I being Macro-Instruction
for a being read-write Int-Location st s . a <= 0 holds
( while>0 a,I is_halting_on s & while>0 a,I is_closed_on s )

proof end;

theorem Th44: :: SCMFSA_9:44

proof end;

theorem Th45: :: SCMFSA_9:45

for a being Int-Location
for I being Macro-Instruction
for s being State of SCM+FSA st I is_closed_on s & I is_halting_on s & IC ((Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . (1 + (LifeSpan (s +* (I +* (Start-At (insloc 0))))))) = (IC ((Computation (s +* (I +* (Start-At (insloc 0))))) . (LifeSpan (s +* (I +* (Start-At (insloc 0))))))) + 4 holds
CurInstr ((Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . (1 + (LifeSpan (s +* (I +* (Start-At (insloc 0))))))) = goto (insloc ((card I) + 4))

proof end;

theorem Th46: :: SCMFSA_9:46

for a being Int-Location
for I being Macro-Instruction holds (while>0 a,I) . (insloc ((card I) + 4)) = goto (insloc 0)

proof end;

theorem Th47: :: SCMFSA_9:47

for s being State of SCM+FSA
for I being Macro-Instruction
for a being read-write Int-Location st I is_closed_on s & I is_halting_on s & s . a > 0 holds
( IC ((Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 3)) = insloc 0 & ( for k being Nat st k <= (LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 3 holds
IC ((Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . k) in dom (while>0 a,I) ) )

proof end;

set sl0 = Start-At (insloc 0);

definition

let s be State of SCM+FSA ;
let I be Macro-Instruction;
let a be read-write Int-Location ;
func StepWhile>0 a,I,s -> Function of NAT , product the Object-Kind of SCM+FSA means :Def5: :: SCMFSA_9:def 5
( it . 0 = s & ( for i being Nat holds it . (i + 1) = (Computation ((it . i) +* ((while>0 a,I) +* (Start-At (insloc 0))))) . ((LifeSpan ((it . i) +* (I +* (Start-At (insloc 0))))) + 3) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines StepWhile>0 SCMFSA_9:def 5 :

theorem :: SCMFSA_9:48

canceled;

theorem :: SCMFSA_9:49

canceled;

theorem Th50: :: SCMFSA_9:50

for k being Nat
for s being State of SCM+FSA
for I being Macro-Instruction
for a being read-write Int-Location holds (StepWhile>0 a,I,s) . (k + 1) = (StepWhile>0 a,I,((StepWhile>0 a,I,s) . k)) . 1

proof end;

theorem Th51: :: SCMFSA_9:51

for I being Macro-Instruction
for a being read-write Int-Location
for s being State of SCM+FSA holds (StepWhile>0 a,I,s) . (0 + 1) = (Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . ((LifeSpan (s +* (I +* (Start-At (insloc 0))))) + 3)

proof end;

theorem Th52: :: SCMFSA_9:52

for I being Macro-Instruction
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,s) . k) = insloc 0 & (StepWhile>0 a,I,s) . k = (Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . n holds
( (StepWhile>0 a,I,s) . k = ((StepWhile>0 a,I,s) . k) +* ((while>0 a,I) +* (Start-At (insloc 0))) & (StepWhile>0 a,I,s) . (k + 1) = (Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0))))) . (n + ((LifeSpan (((StepWhile>0 a,I,s) . k) +* (I +* (Start-At (insloc 0))))) + 3)) )

proof end;

theorem Th53: :: SCMFSA_9:53

for I being Macro-Instruction
for a being read-write Int-Location
for s being State of SCM+FSA st ( for k being Nat holds
( I is_closed_on (StepWhile>0 a,I,s) . k & I is_halting_on (StepWhile>0 a,I,s) . k ) ) & ex f being Function of product the Object-Kind of SCM+FSA , NAT st
for k being Nat holds
( ( f . ((StepWhile>0 a,I,s) . (k + 1)) < f . ((StepWhile>0 a,I,s) . k) or f . ((StepWhile>0 a,I,s) . k) = 0 ) & ( f . ((StepWhile>0 a,I,s) . k) = 0 implies ((StepWhile>0 a,I,s) . k) . a <= 0 ) & ( ((StepWhile>0 a,I,s) . k) . a <= 0 implies f . ((StepWhile>0 a,I,s) . k) = 0 ) ) holds
( while>0 a,I is_halting_on s & while>0 a,I is_closed_on s )

proof end;

theorem Th54: :: SCMFSA_9:54

for I being parahalting Macro-Instruction
for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of product the Object-Kind of SCM+FSA , NAT st
for k being Nat holds
( ( f . ((StepWhile>0 a,I,s) . (k + 1)) < f . ((StepWhile>0 a,I,s) . k) or f . ((StepWhile>0 a,I,s) . k) = 0 ) & ( f . ((StepWhile>0 a,I,s) . k) = 0 implies ((StepWhile>0 a,I,s) . k) . a <= 0 ) & ( ((StepWhile>0 a,I,s) . k) . a <= 0 implies f . ((StepWhile>0 a,I,s) . k) = 0 ) ) holds
( while>0 a,I is_halting_on s & while>0 a,I is_closed_on s )

proof end;

theorem :: SCMFSA_9:55

for I being parahalting Macro-Instruction
for a being read-write Int-Location st ex f being Function of product the Object-Kind of SCM+FSA , NAT st
for s being State of SCM+FSA holds
( ( f . ((StepWhile>0 a,I,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 Macro-Instruction;
let a be Int-Location ;
cluster if>0 a,I,J -> good ;
coherence

proof end;

end;

registration

let I be good Macro-Instruction;
let a be Int-Location ;
cluster while>0 a,I -> good ;
correctness

proof end;

end;