:: SCMPDS_8 semantic presentation

set A = the Instruction-Locations of SCMPDS ;

set D = SCM-Data-Loc ;

theorem Th1: :: SCMPDS_8:1

for a being Int_position ex i being Nat st a = intpos i

proof end;

definition

let t be State of SCMPDS ;
func Dstate t -> State of SCMPDS means :Def1: :: SCMPDS_8:def 1
for x being set holds
( ( x in SCM-Data-Loc implies it . x = t . x ) & ( x in the Instruction-Locations of SCMPDS implies it . x = goto 0 ) & ( x = IC SCMPDS implies it . x = inspos 0 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines Dstate SCMPDS_8:def 1 :

theorem Th2: :: SCMPDS_8:2

for t1, t2 being State of SCMPDS st t1 | SCM-Data-Loc = t2 | SCM-Data-Loc holds
Dstate t1 = Dstate t2

proof end;

theorem Th3: :: SCMPDS_8:3

for t being State of SCMPDS
for i being Instruction of SCMPDS st InsCode i in {0,4,5,6} holds
Dstate t = Dstate (Exec i,t)

proof end;

theorem Th4: :: SCMPDS_8:4

for a being Int_position
for s being State of SCMPDS holds (Dstate s) . a = s . a

proof end;

theorem Th5: :: SCMPDS_8:5

for a being Int_position ex f being Function of product the Object-Kind of SCMPDS , NAT st
for s being State of SCMPDS holds
( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) )

proof end;

definition

let a be Int_position ;
let i be Integer;
let I be Program-block;
func while<0 a,i,I -> Program-block equals :: SCMPDS_8:def 2
((a,i >=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1)));
coherence ;

end;

:: deftheorem defines while<0 SCMPDS_8:def 2 :

registration

let I be shiftable Program-block;
let a be Int_position ;
let i be Integer;
cluster while<0 a,i,I -> shiftable ;
correctness

proof end;

end;

registration

let I be No-StopCode Program-block;
let a be Int_position ;
let i be Integer;
cluster while<0 a,i,I -> No-StopCode ;
correctness ;

end;

theorem Th6: :: SCMPDS_8:6

for a being Int_position
for i being Integer
for I being Program-block holds card (while<0 a,i,I) = (card I) + 2

proof end;

Lm1: for a being Int_position
for i being Integer
for I being Program-block holds card (stop (while<0 a,i,I)) = (card I) + 3

proof end;

theorem Th7: :: SCMPDS_8:7

for a being Int_position
for i being Integer
for m being Nat
for I being Program-block holds
( m < (card I) + 2 iff inspos m in dom (while<0 a,i,I) )

proof end;

theorem Th8: :: SCMPDS_8:8

for a being Int_position
for i being Integer
for I being Program-block holds
( (while<0 a,i,I) . (inspos 0) = a,i >=0_goto ((card I) + 2) & (while<0 a,i,I) . (inspos ((card I) + 1)) = goto (- ((card I) + 1)) )

proof end;

Lm2: for a being Int_position
for i being Integer
for I being Program-block holds while<0 a,i,I = (a,i >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
by SCMPDS_4:51;

theorem Th9: :: SCMPDS_8:9

for s being State of SCMPDS
for I being Program-block
for a being Int_position
for i being Integer st s . (DataLoc (s . a),i) >= 0 holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )

proof end;

theorem Th10: :: SCMPDS_8:10

for s being State of SCMPDS
for I being Program-block
for a, c being Int_position
for i being Integer st s . (DataLoc (s . a),i) >= 0 holds
IExec (while<0 a,i,I),s = s +* (Start-At (inspos ((card I) + 2)))

proof end;

theorem :: SCMPDS_8:11

for s being State of SCMPDS
for I being Program-block
for a being Int_position
for i being Integer st s . (DataLoc (s . a),i) >= 0 holds
IC (IExec (while<0 a,i,I),s) = inspos ((card I) + 2)

proof end;

theorem :: SCMPDS_8:12

for s being State of SCMPDS
for I being Program-block
for a, b being Int_position
for i being Integer st s . (DataLoc (s . a),i) >= 0 holds
(IExec (while<0 a,i,I),s) . b = s . b

proof end;

Lm3: for I being Program-block
for a being Int_position
for i being Integer holds Shift I,1 c= while<0 a,i,I

proof end;

scheme :: SCMPDS_8:sch 1
WhileLHalt{ F₁( State of SCMPDS ) -> Nat, F₂() -> State of SCMPDS , F₃() -> shiftable No-StopCode Program-block, F₄() -> Int_position , F₅() -> Integer, P₁[ State of SCMPDS ] } :

( ( F₁(F₂()) = F₁(F₂()) or P₁[F₂()] ) & while<0 F₄(),F₅(),F₃() is_closed_on F₂() & while<0 F₄(),F₅(),F₃() is_halting_on F₂() )

provided

A1: card F₃() > 0 and
A2: for t being State of SCMPDS st P₁[ Dstate t] & F₁((Dstate t)) = 0 holds
t . (DataLoc (F₂() . F₄()),F₅()) >= 0 and
A3: P₁[ Dstate F₂()] and
A4: for t being State of SCMPDS st P₁[ Dstate t] & t . F₄() = F₂() . F₄() & t . (DataLoc (F₂() . F₄()),F₅()) < 0 holds
( (IExec F₃(),t) . F₄() = t . F₄() & F₃() is_closed_on t & F₃() is_halting_on t & F₁((Dstate (IExec F₃(),t))) < F₁((Dstate t)) & P₁[ Dstate (IExec F₃(),t)] )

proof end;

scheme :: SCMPDS_8:sch 2
WhileLExec{ F₁( State of SCMPDS ) -> Nat, F₂() -> State of SCMPDS , F₃() -> shiftable No-StopCode Program-block, F₄() -> Int_position , F₅() -> Integer, P₁[ State of SCMPDS ] } :

( ( F₁(F₂()) = F₁(F₂()) or P₁[F₂()] ) & IExec (while<0 F₄(),F₅(),F₃()),F₂() = IExec (while<0 F₄(),F₅(),F₃()),(IExec F₃(),F₂()) )

provided

A1: card F₃() > 0 and
A2: F₂() . (DataLoc (F₂() . F₄()),F₅()) < 0 and
A3: for t being State of SCMPDS st P₁[ Dstate t] & F₁((Dstate t)) = 0 holds
t . (DataLoc (F₂() . F₄()),F₅()) >= 0 and
A4: P₁[ Dstate F₂()] and
A5: for t being State of SCMPDS st P₁[ Dstate t] & t . F₄() = F₂() . F₄() & t . (DataLoc (F₂() . F₄()),F₅()) < 0 holds
( (IExec F₃(),t) . F₄() = t . F₄() & F₃() is_closed_on t & F₃() is_halting_on t & F₁((Dstate (IExec F₃(),t))) < F₁((Dstate t)) & P₁[ Dstate (IExec F₃(),t)] )

proof end;

theorem :: SCMPDS_8:13

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i being Integer
for X being set
for f being Function of product the Object-Kind of SCMPDS , NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc (s . a),i) >= 0 ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) < 0 holds
( (IExec I,t) . a = t . a & f . (Dstate (IExec I,t)) < f . (Dstate t) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )

proof end;

theorem :: SCMPDS_8:14

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i being Integer
for X being set
for f being Function of product the Object-Kind of SCMPDS , NAT st card I > 0 & s . (DataLoc (s . a),i) < 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc (s . a),i) >= 0 ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) < 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & f . (Dstate (IExec I,t)) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
IExec (while<0 a,i,I),s = IExec (while<0 a,i,I),(IExec I,s)

proof end;

theorem Th15: :: SCMPDS_8:15

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i being Integer
for X being set st card I > 0 & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) < 0 holds
( (IExec I,t) . a = t . a & (IExec I,t) . (DataLoc (s . a),i) > t . (DataLoc (s . a),i) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while<0 a,i,I is_closed_on s & while<0 a,i,I is_halting_on s )

proof end;

theorem :: SCMPDS_8:16

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i being Integer
for X being set st s . (DataLoc (s . a),i) < 0 & card I > 0 & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) < 0 holds
( (IExec I,t) . a = t . a & (IExec I,t) . (DataLoc (s . a),i) > t . (DataLoc (s . a),i) & I is_closed_on t & I is_halting_on t & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
IExec (while<0 a,i,I),s = IExec (while<0 a,i,I),(IExec I,s)

proof end;

definition

let a be Int_position ;
let i be Integer;
let I be Program-block;
func while>0 a,i,I -> Program-block equals :: SCMPDS_8:def 3
((a,i <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1)));
coherence ;

end;

:: deftheorem defines while>0 SCMPDS_8:def 3 :

registration

let I be shiftable Program-block;
let a be Int_position ;
let i be Integer;
cluster while>0 a,i,I -> shiftable ;
correctness

proof end;

end;

registration

let I be No-StopCode Program-block;
let a be Int_position ;
let i be Integer;
cluster while>0 a,i,I -> No-StopCode ;
correctness ;

end;

theorem Th17: :: SCMPDS_8:17

for a being Int_position
for i being Integer
for I being Program-block holds card (while>0 a,i,I) = (card I) + 2

proof end;

Lm4: for a being Int_position
for i being Integer
for I being Program-block holds card (stop (while>0 a,i,I)) = (card I) + 3

proof end;

theorem Th18: :: SCMPDS_8:18

for a being Int_position
for i being Integer
for m being Nat
for I being Program-block holds
( m < (card I) + 2 iff inspos m in dom (while>0 a,i,I) )

proof end;

theorem Th19: :: SCMPDS_8:19

for a being Int_position
for i being Integer
for I being Program-block holds
( (while>0 a,i,I) . (inspos 0) = a,i <=0_goto ((card I) + 2) & (while>0 a,i,I) . (inspos ((card I) + 1)) = goto (- ((card I) + 1)) )

proof end;

Lm5: for a being Int_position
for i being Integer
for I being Program-block holds while>0 a,i,I = (a,i <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
by SCMPDS_4:51;

theorem Th20: :: SCMPDS_8:20

for s being State of SCMPDS
for I being Program-block
for a being Int_position
for i being Integer st s . (DataLoc (s . a),i) <= 0 holds
( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s )

proof end;

theorem Th21: :: SCMPDS_8:21

for s being State of SCMPDS
for I being Program-block
for a, c being Int_position
for i being Integer st s . (DataLoc (s . a),i) <= 0 holds
IExec (while>0 a,i,I),s = s +* (Start-At (inspos ((card I) + 2)))

proof end;

theorem :: SCMPDS_8:22

for s being State of SCMPDS
for I being Program-block
for a being Int_position
for i being Integer st s . (DataLoc (s . a),i) <= 0 holds
IC (IExec (while>0 a,i,I),s) = inspos ((card I) + 2)

proof end;

theorem :: SCMPDS_8:23

for s being State of SCMPDS
for I being Program-block
for a, b being Int_position
for i being Integer st s . (DataLoc (s . a),i) <= 0 holds
(IExec (while>0 a,i,I),s) . b = s . b

proof end;

Lm6: for I being Program-block
for a being Int_position
for i being Integer holds Shift I,1 c= while>0 a,i,I

proof end;

scheme :: SCMPDS_8:sch 3
WhileGHalt{ F₁( State of SCMPDS ) -> Nat, F₂() -> State of SCMPDS , F₃() -> shiftable No-StopCode Program-block, F₄() -> Int_position , F₅() -> Integer, P₁[ State of SCMPDS ] } :

( ( F₁(F₂()) = F₁(F₂()) or P₁[F₂()] ) & while>0 F₄(),F₅(),F₃() is_closed_on F₂() & while>0 F₄(),F₅(),F₃() is_halting_on F₂() )

provided

A1: card F₃() > 0 and
A2: for t being State of SCMPDS st P₁[ Dstate t] & F₁((Dstate t)) = 0 holds
t . (DataLoc (F₂() . F₄()),F₅()) <= 0 and
A3: P₁[ Dstate F₂()] and
A4: for t being State of SCMPDS st P₁[ Dstate t] & t . F₄() = F₂() . F₄() & t . (DataLoc (F₂() . F₄()),F₅()) > 0 holds
( (IExec F₃(),t) . F₄() = t . F₄() & F₃() is_closed_on t & F₃() is_halting_on t & F₁((Dstate (IExec F₃(),t))) < F₁((Dstate t)) & P₁[ Dstate (IExec F₃(),t)] )

proof end;

scheme :: SCMPDS_8:sch 4
WhileGExec{ F₁( State of SCMPDS ) -> Nat, F₂() -> State of SCMPDS , F₃() -> shiftable No-StopCode Program-block, F₄() -> Int_position , F₅() -> Integer, P₁[ State of SCMPDS ] } :

( ( F₁(F₂()) = F₁(F₂()) or P₁[F₂()] ) & IExec (while>0 F₄(),F₅(),F₃()),F₂() = IExec (while>0 F₄(),F₅(),F₃()),(IExec F₃(),F₂()) )

provided

A1: card F₃() > 0 and
A2: F₂() . (DataLoc (F₂() . F₄()),F₅()) > 0 and
A3: for t being State of SCMPDS st P₁[ Dstate t] & F₁((Dstate t)) = 0 holds
t . (DataLoc (F₂() . F₄()),F₅()) <= 0 and
A4: P₁[ Dstate F₂()] and
A5: for t being State of SCMPDS st P₁[ Dstate t] & t . F₄() = F₂() . F₄() & t . (DataLoc (F₂() . F₄()),F₅()) > 0 holds
( (IExec F₃(),t) . F₄() = t . F₄() & F₃() is_closed_on t & F₃() is_halting_on t & F₁((Dstate (IExec F₃(),t))) < F₁((Dstate t)) & P₁[ Dstate (IExec F₃(),t)] )

proof end;

theorem Th24: :: SCMPDS_8:24

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i, c being Integer
for X, Y being set
for f being Function of product the Object-Kind of SCMPDS , NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc (s . a),i) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc (s . a),i)) ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & f . (Dstate (IExec I,t)) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec I,t) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
(IExec I,t) . x = t . x ) ) ) holds
( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s )

proof end;

theorem Th25: :: SCMPDS_8:25

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i, c being Integer
for X, Y being set
for f being Function of product the Object-Kind of SCMPDS , NAT st s . (DataLoc (s . a),i) > 0 & card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc (s . a),i) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc (s . a),i)) ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & f . (Dstate (IExec I,t)) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec I,t) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
(IExec I,t) . x = t . x ) ) ) holds
IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s)

proof end;

theorem :: SCMPDS_8:26

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i being Integer
for X being set
for f being Function of product the Object-Kind of SCMPDS , NAT st card I > 0 & ( for t being State of SCMPDS st f . (Dstate t) = 0 holds
t . (DataLoc (s . a),i) <= 0 ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & f . (Dstate (IExec I,t)) < f . (Dstate t) & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s & ( s . (DataLoc (s . a),i) > 0 implies IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s) ) )

proof end;

theorem Th27: :: SCMPDS_8:27

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i, c being Integer
for X, Y being set st card I > 0 & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc (s . a),i)) ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & (IExec I,t) . (DataLoc (s . a),i) < t . (DataLoc (s . a),i) & ( for x being Int_position st x in X holds
(IExec I,t) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) ) & ( for x being Int_position st x in Y holds
(IExec I,t) . x = t . x ) ) ) holds
( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s & ( s . (DataLoc (s . a),i) > 0 implies IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s) ) )

proof end;

theorem :: SCMPDS_8:28

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i being Integer
for X being set st card I > 0 & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & (IExec I,t) . (DataLoc (s . a),i) < t . (DataLoc (s . a),i) & ( for x being Int_position st x in X holds
(IExec I,t) . x = t . x ) ) ) holds
( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s & ( s . (DataLoc (s . a),i) > 0 implies IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s) ) )

proof end;

theorem :: SCMPDS_8:29

for s being State of SCMPDS
for I being shiftable No-StopCode Program-block
for a being Int_position
for i, c being Integer
for X being set st card I > 0 & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc (s . a),i)) ) & ( for t being State of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (s . a),i)) ) & t . a = s . a & t . (DataLoc (s . a),i) > 0 holds
( (IExec I,t) . a = t . a & I is_closed_on t & I is_halting_on t & (IExec I,t) . (DataLoc (s . a),i) < t . (DataLoc (s . a),i) & ( for x being Int_position st x in X holds
(IExec I,t) . x >= c + ((IExec I,t) . (DataLoc (s . a),i)) ) ) ) holds
( while>0 a,i,I is_closed_on s & while>0 a,i,I is_halting_on s & ( s . (DataLoc (s . a),i) > 0 implies IExec (while>0 a,i,I),s = IExec (while>0 a,i,I),(IExec I,s) ) )

proof end;