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Cahier des charges
Spécification
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Machine abstraite
Raffine
preuves
preuves de raffinement
Raffinement
Raffine
preuves
preuves de raffinement
Implémentation
preuves
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Code
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Sequence
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BN<>VY?78E8Þ
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isEmpty()
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notEmpty()
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LM;^O6 X£6?B
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sum()
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<>Bc6= LN6 ¾ L$O6 ;=<H8$öO6 6?BLN6 ¾ RLN6?BNF6 868$‹ TW6=BB6qV ¾ =O6 LMRBN<>VY?
select(...)
68NB\=O6 J˜RKÔHIW=O6 6iR£
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collect(uneExpression)
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O6 X£6?BN8D4|-0I7?\RKIBLN6
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includes(unObjet)
unObjet
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forAll(uneExpression)
uneExpression
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exists(uneExpression)
uneExpression
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excluding(unObjet)
Ô- VF 6=B
unObjet
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including(unObjet)
unObjet
ß LM6=BVYILM?6lÔHR JRKÔ>6IL¬JLMRK<H6x8N<
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excludes(unObjet)
unObjet
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union(set)
let variable : type = expression "%&"!
in expr
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VVV
OPERATIONS
oo creerClasse(valAttrib2) PRE
not(CLASSE-classe = B ) valAttrib2 @ typeAttrib2
THEN
ANY new WHERE
new @ CLASSE-classe
THEN
classe := classe new attrib1 := attrib1 new attrib2 := attrib2 new oo := new
END
END ;
val1 valAttrib2 VVV
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RoleClasse1
MultClasse1
assos
RoleClasse2
MultClasse2
Classe2
Ž V \ .=Š%( ! (! VVV
VARIABLES
INVARIANT
INITIALISATION
VVV
assos, classe1, classe2
assos @ classe1 classe2
assos := B
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assos @ classe1 classe2
assos @ classe1 classe2
assos @ classe1 classe2
assos @ classe1 classe2 dom(assos) = classe1
assos @ classe1 classe2 x.(x @ classe1 card(assos[x]) = n)
assos @ classe1 classe2 x.(x @ classe1 card(assos[x]) n)
assos @ classe1 classe2 x.(x @ classe1 card(assos[x]) n) dom(assos) = classe1
assos @ classe1 classe2 x.(x @ classe1 card(assos[x]) n)
assos @ classe1 classe2 x.(x @ classe1 n card(assos[x]) m)
VV
VV +
ran(assos) = classe2
assos @ classe2 classe1 x.(x @ classe2 card(assos[x]) = n)
assos @ classe2 classe1 x.(x @ classe2 card(assos[x]) n)
assos @ classe2 classe1 x.(x @ classe2 card(assos[x]) n) ran(assos) = classe2
assos @ classe2 classe1 x.(x @ classe2 card(assos[x]) n)
assos @ classe2 classe1 x.(x @ classe2 n card(assos[x]) m)
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SEES
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€
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SuperClasse
sousclasse
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VVV
SETS
VARIABLES
INVARIANT
VVV
S= S1,...,Sn s
s@ S
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VQV}V
SETS
VARIABLES
INVARIANT
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VVV
CLASS STATE = S1, . . ., Sn class state
class state @ Class CLASS STATE
class state := B
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VVV
INITIALISATION
" V O ? 2
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OPERATIONS
oo creerClasse PRE
not(CLASSE - classe = B )
THEN
ANY new WHERE
new @ CLASSE - classe
THEN
VVV
END
VVV
classe state(new) := S1
oo := new
END
" V O 1
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E
SB
...
E
SC
SD
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VVV
OPERATIONS
E IF s=SA THEN s :=SB END ... ELSIF s=SC THEN s :=SD
SELECT s=SA THEN s :=SB END ... WHEN s=SC THEN s :=SD
CASE s OF EITHER SA THEN s :=SB END ... OR SC THEN s :=SD
END
VVV
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VVV
OPERATIONS
TransitionAB ( oo ) PRE
oo @ class class state ( oo ) = SA
THEN
class state ( oo ) := SB
END ;
VVV
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S2
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VVV
OPERATIONS
E(para) IF SELECT s = S1 cond
THEN s := S1’ act END
VVV
VVV
END
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Lift
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-curDestFloor: FLOOR
-doorStatus: STATUS
+goDown()
+goUp()
+setCurDestFloor(fl:FLOOR)
+getCurDestFloor(): FLOOR
+closeDoor()
+openDoor()
liftButton
2..*
Button
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setCurDestFloor(fl);goUp()
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movement
close
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SEES
Types
VVV
VARIABLES
lift, lift dir, lift curDestFloor, lift doorStatus, lift state, button ...
INVARIANT
lift A LIFT lift dir @ lift DIRECTION ...
OPERATIONS
lift close ( ll ) PRE
ll @ lift THEN
SELECT lift State(ll) = lift visit THEN
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H
ELSE
END
END ;
VVV
END
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Types
SETS
OBJECTS ;
DIRECTION = direction up,direction down ;
STATUS = status opened,status closed ;
BUTTON STATE = button on,button off ;
LIFT STATE = lift ready,lift movement,lift visit CONSTANTS
floor ground, floor top, LIFT, BUTTON
PROPERTIES
floor ground @ NAT floor top @ NAT floor ground floor top LIFT A OBJECTS BUTTON A OBJECTS END
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SEES
Types
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INVARIANT
lift A LIFT
VVV
DEFINITIONS
FLOOR == (floor ground .. floor top)
OPERATIONS
lift transVisitReady(ll) PRE
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THEN
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END ;
lift closeDoor(ll) PRE
ll @ lift
THEN
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THEN
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VVV
END
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lift close(ll) VAR bb IN
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lift transVisitReady(ll) ;
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MACHINE
GestionDesInscriptions
SETS
ETUDIANTS ; DIPLOMES
VARIABLES
Inscrits, InscritsAuxExamens
INVARIANT
Inscrits @ ETUDIANTS DIPLOMES InscritsAuxExamens Inscrits
INITIALISATION
Inscrits := B InscritsAuxExamens := B
OPERATIONS
inscrire(etudiant, diplome) PRE
etudiant @ ETUDIANTS diplome @ DIPLOMES etudiant @ dom(Inscrits)
THEN
Inscrits := Inscrits etudiant diplome END ;
....
END
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context GestionDesInscriptions inv :
Inscrits includesAll(InscritsAuxExamens)
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ControleAccesBatiments
SETS
BATS ; PERS
VARIABLES
aut, sit
INVARIANT
aut @ PERS BATS sit @ PERS BATS sit A aut ...
END
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" MACHINE
Scheduler
SETS
PID
CONCRETE VARIABLES
active, ready, waiting
INVARIANT
active 3 PID ready 3 PID waiting 3 PID ready waiting = 5
(active = 5 or (active 2 ready
card (active) 1 active = 5 ready = 5
INITIALISATION
...
END
waiting
ready
active
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Scheduler
1
waiting
*
1
Scheduler
1
1
1
Scheduler
PID
Scheduler
*
active
E_PID
Scheduler
*
ready
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6 - Scheduler
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context Scheduler inv :
PID includesAll (active)
and PID includesAll(ready)
and PID includesAll(waiting)
and ready intersection(waiting) isEmpty()
and active isEmpty() or not (ready union(waiting) includesAll(active))
and active size() 1
and active isEmpty() implies ready isEmpty()
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6 - Scheduler
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(xx, yy).((xx yy) @ ContenuEvacuation
yy @ AutorisationEvacuation [ xx ])
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Evacuation forAll(ce : E Evacuation
AutorisationEvacuation select(
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collect(E AutorisationEvacuation C)
includes(ce.E Evacuation C))
6
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INITIALIZATION E528 DF4 > n`69DF2=L <?>R< DF2•LDF2 > TV0M{4 G _ @ 2 >R> <0y\K8J8I4 > TV0M4RE < Yy032 > 8 @QP >
context nomClasse : :nomOpération(ListeDeparamètres : Types) : typeResultat
pre : pré-condition
post : post-condition
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,?
MACHINE
Scheduler
...
INITIALISATION
active, ready, waiting := 5 , 5 , 5
OPERATIONS
creer(pp) PRE
pp @ PID pp @ 2 active pp @ 2 (ready waiting)
THEN
waiting := waiting pp END ;
activer(qq) PRE
qq @ waiting
THEN
waiting := waiting - qq +
/
5 THEN
IF
active := qq ELSE
ready := ready qq END
END ;
...
END
q036 >R< Y38
tW
6 - 0
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Scheduler
" 7 context Scheduler : :INITIALIZATION()
post :active isEmpty() and ready isEmpty() and waiting isEmpty()
context Scheduler : :creer(pp : Elt PID)
pre : active excludes (pp)
and ready union(waiting) excludes(pp)
post : waiting = waiting@pre including(pp)
context Scheduler : :activer(qq : Elt PID)
pre : waiting includes(qq)
post : waiting = waiting@pre excluding(qq)
and if active@pre isEmpty() then
active = Set qq else ready = ready@pre including(qq)
endif
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8
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5@ L + waiting := waiting - qq
6 - IF (active = 5 ) THEN active := qq ELSE ready := ready qq /
END
5@ LL ++ waiting := waiting 5@ IF (active = 5 )
1
qq THEN active := qq ELSE ready := ready
/
END
/
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/
4
4
4
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PRE @ < >
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SELECT @ < >
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ANY
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SETS
VARIABLES
INVARIANT
@ INITIALISATION OPERATIONS END
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REFINMENT
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MACHINE M1
VARIABLES x,y
...
END
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SETS X= vx ,...,vx VARIABLES x
INVARIANT x @ X
...
END
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état
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Nom de l'opération
ou de l'événement
état
source
état
cible
Evénement et
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Initialisation
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OP (
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6
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ŠKW 74A8ITVDF2C ˜eŠD†W
HEkr4 >R<?> E >R< DF12 0 HEk=4 >R<?> E >R< DF2 ! " IF P THEN S END
76
OP
SELECT P THEN S
4
WHEN P THEN Skip END
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IF P THEN x := vx END
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(
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7 OP[¬P]
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OP[P2]
OP[P2]
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6
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7 MACHINE M1
SETS X= VX1, ..., VXN ;Y= VY1, ..., VYN
VARIABLES x,y
INVARIANT x X y Y
INITIALISATION x := VX1 y := VY1
OPERATIONS
OP IF x = VX1 y = VY1 THEN x := VX2 y := VY2 END
END
6
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y
x
VX1
6
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OP(Param)
OP[IN y = VY1]
VY1
VX2
OP[IN x = VX1]
VY2
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SUB SIMPLE OP
7 7%
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SUB PRINCIPALE B CondP THEN
SUB SECONDAIRE B CondS THEN SUB SIMPLE
SUB SECONDAIRE B CondS THEN SUB SIMPLE END
‘0
END
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MACHINE M1
SETS ETAT= ouvert,ferme
VARIABLES v1,v2
INVARIANT
v1 ETAT
v2 ETAT
v1 = ouvert
v2 = ferme
v2 = ouvert
v1 = ferme
INITIALISATION
v1 := ferme v2 := ferme
OPERATIONS
ouvrir v1
SELECT v2 = ferme THEN v1 := ouvert END ;
ouvrir v2
SELECT v1 = ferme THEN v2 := ouvert END ;
fermer
BEGIN v1 := ferme v2 := ferme END
END
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MACHINE Robot0
SETS ETAT= vid,occ
VARIABLES dt0
INVARIANT
dt0
vid, occ
INITIALISATION
dt0 := vid
OPERATIONS
Chgt SELECT dt0 := vid THEN dt0 :=occ END
Dchgt SELECT dt0 := occ THEN dt0 :=vid END
arr p skip
evac skip
END
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T 8I4 8a0€
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6
Robot0
Dt0
vid
Chgt
Dchgt
occ
6
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Robot0
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) < 8Iƒ 698Ž4RE @ TV8L < 4)™Da1
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 8 >•8UG Y 8IG 28 P 8I2 > 8I4 > L 8IG 6 @-<?> L5032=4sTV0:6UTi03E54A8 OPERATIONS )=0 @ arr p Wr| cjD ) 8 G @ 0 >R< DF2 Chgt 89\ ) @Q<VP 8
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+
0 ƒ /-W
vid occ
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T?8IE ) D ) 8[4 4 8ƒ 8:LCc 8UYf036UE50 DF2 8I4 )™8I6 Nq4 8 4R036Q;5032 >7^ E8 AUCUN 69D @Q@ 8I4)™DF25L€03E
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)™8 @QP 8 >A>A@ 0xL8‚69DF25250 >A@ 8XT?8 >`_ )Œ81L8) < 8Iƒ 698
@
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|’8„L < 0 @ 0 P:P 8„LCc 8 G > 0 > 4An >A@ 03254 <?>R< DF254L58„TV0SYf0 @Q< 0fkrT?8 Type )ŒDFE @ TdcjD ) 8 G @ 0 >R< DF2 Chgt 0ZYy032 > 4 <VP n
)=T < w=6U0 >R< DF2S8I4 > 698ITVE < L58TV0•w FE @ 8ŠKWj˜ „DlEu
ƒ TV8 ) @H8IG L < 6U0 > P 8I4 > NO03E\SL=03254xT?8I4 8 G > 0 > 4 T1 8 > T2 W’|C8I4
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D >K8 G IN= occDa1
<
>
@
Dt1 = vid Type = AUCUN)
P (Da1 = occ Dt1 = vid)
8I2 >A@ 8b6 @ DH6Q;8 > 4a0Sƒ TV0S4RE <V> 8ŽL=E+2D P L8…TdcjD ) 8 G @ 0 >R< DF2lWh| 0 >A@ 03254 <?>R< DF2 @K8 G 589\ < Y38
= occ
Dt1 = vid
2 
REFINEMENT Robot1 REFINES Robot0
SETS TYPE = AUCUN, T1, T2
VARIABLES Dt1, Da1, De1, De2, Type
INVARIANT
Dt1 = Dt0 Da1
vid, occ
De1
vid, occ
De2
vid, occ
Type TYPE
Type = AUCUN Dt1 = vid
Type 2 AUCUN Dt1 = occ
INITIALISATION
Dt1, Da1, De1, De2, Type := vid, vid, vid, vid, AUCUN
OPERATIONS
Chgt
ANY pp WHERE pp
T1, T2
Da1 = occ Type = AUCUN
THEN Type := pp Da1,Dt1 : = vid, occ END
Dchgt SELECT De1 = vid Type = T1
THEN Dt1, De1, Type := vid, occ, AUCUN
WHEN De2 = vid Type = T2
THEN Dt1, De2, Type := vid, occ, AUCUN END
arr p
SELECT Da1 = vid THEN Da1 := occ END
evac
SELECT De1 = occ THEN De1 := vid
WHEN De2 = occ THEN De2 := vid END
END
ŠKW—3‰g @ 8 P:< 8 @p@ 028 P 8I2 > L5E @ D3k™D >
03E > DFE @ L8[Tdc 8 G > 0 > AUCUN L < 4 )=0 @ 0 > 6U0 @ 8ITVT?8[28 P DHL < w=87Tdc 8 G > 0 > LCcm03E56UE528JYy0 @Q< 0fk=T?83W 12bD3k >R< 8I2 > T?8
L < 0 @ 0 P:P 87L87TV0w FE @ 8xŠKWj˜
6
Robot1
Type
Chgt[P]
Chgt[P]
Chgt[P]
Chgt[P]
T1
Chgt[P]
T2
Chgt[P]
AUCUN
Chgt[P]
Chgt[P]
Chgt[P]
6
ŠKWj˜ / < 0 @ 0 P:P x8 Llc 8 G > 0 > 4 n >A@ 03254 <?>R< DF254XL8xTV0Yf0 @Q< 0fk=TV8
Type
)™DFE
@ Chgt
3‘
7 Robot1
Type
T1
Chgt[IN Da1 = occ et Dt1 = vid]
T2
Chgt[IN Da1 = occ et Dt1 = vid]
AUCUN
KŠ Wj˜ / < 0 @ 0 P:P 87Llc 8 G > 0 > 4 n >A@ 032=4 <?>R< DF254XL87TV0Yf0 @Q< 0fk=TV8 Type )ŒDFE @ Chgt 0 ) @ 8Iƒ 4s4 <VP )=T < w=6U0 >R< DF2
/‚8hTV0 8
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@Q< 0fkrT?8 Type +Ow ŠKWj˜e‘'/-W
6
Robot1
Type
Dchgt[IN De1 = vid et Dt1 = occ]
T1
Chgt[IN Da1 = occ et Dt1 = vid]
Chgt[IN Da1 = occ et Dt1 = vid]
T2
AUCUN
Dchgt[IN De2 = vid et Dt1 = occ]
ŠKWj˜e‘/ < 0 @ 0 P:P 8xLlc 8 G > 0 > 4 n >A@ 03254 <?>R< DF2=4s69D P )=TV8 > L87TV0Yf0 @Q< 0fk=TV8 Type
HE @ T?8 P 8
P 8 ) @Q< 256 < )™8 L8pL 8IG L5E=6 >R< DF2DF2L 8UG w=2 <?> T?8I4~L < 0 @ 0 P:P 8I4}Llc 8 G > 0 > 4 n >A@ 03254 <?>R< DF254~L8I4~03E >A@ 8I4
Yf0 @Q< 0fk=T?8I4pL587TV0M4) 8IG 6 < w=6U0 >R< DF2 +Ow ŠKWj˜e'/
6
Dt1
De1
De2
Da1
Robot1
Dt1
Da1
vid
vid
Chgt [IN
Da1 = occ]
Dchgt[IN De1 = vid
et Type = T1]
Dchgt[IN De2 = vid
et Type = T2]
De2
De1
vid
vid
occ
6
Chgt[IN Dt1 = vid
et Type = AUCUN]
occ
occ
Dchgt[IN Dt1 = occ
et Type = T1]
arr_p
evac
Dchgt[IN Dt1 = occ
et Type = T2]
evac
occ
ŠKWj˜e/ < 0 @ 0 P:P x8 LCc 8 G > 0 > 4An >A@ 03254 <?>R< DF254XL58I4pYy0 @-< 0fk=T?8I4 Dt1, De1, De2 8 >
Da1
3 
g DFE D3k 8I2 T?8XL 0 0 8X69D )=T?8 L8XTV0J4) 8I6 w=6U0 DF2
L5E D3kŒD T54RE L8‚6U03TV6UE5T?8
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0< fk=T?83W 12Ž69@ D P:P > 8I< 25698,)=0 > @ T?8) @ DHL5E <?@>
4 _ 256Q; @ DF2 < 4 8G 8I2 >A@ 8pT?8I4 L < 0 @ 0 P:P 8I4 @ 8I4)™8I6 >R< Nq4’L8 Dt1 8 > Type / / / / +†p; > B
{; /-B +q/‚6-; Br/16-; /
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2 Robot1
Dt1,Type
S1
Chgt[IN Da1 = occ]
Chgt[IN Da1 = occ]
Dchgt[IN
De1 = vid]
S2
Dchgt[IN
De2 = vid]
S3
ŠKWj˜3˜ / < 0 @ 0 PP x8 Llc 8 G > 0 > 4 n >A@ 032=4 <?>R< DF254
6
S1
S2
S3
6 ŠKW—
Dt1
vid
occ
occ
Type
AUCUN
T1
T2
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/ / /
+q/16Q; B™/16-; /-B™{; B=0
)CB8UYf036(
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12 )Œ8IE
0 2 8I25032 Y 8 w58 T?8I4-) D ) 8 8I4 8ITVT?8I4 E8 28)™8IE 6Q;50 38 E5258) 8Iƒ 698 E8
4 < p8I4 > T < >vk @ P 8 + < vid> /8 >> ^ EG 8 @Q< *Ž@ 8I4 > DK@ 6U6UE+@Q) <o8UGG >K8 G + occ> /HW ^ HE @ T?8… L < 0 @ 0 P:P > 8…LCc 8 G @ > 0 > @ 4An >A@ 03254 <?< >R< DF254^ T?8
6Q;=0 @ 38 P 8I2 > L8) < 8Iƒ 698+OD ) 8 G @ 0 >R< DF2 Chgt/p4Ucj8 8I6 > E8L8*)=E < 4pTdc 8 G > 0 > S4 8 > T?8I4pYf03T?8IE @ 4 @ 8I4)™8I6 >R< Y38I4sL8
8 4ADF2 >‚@ 8I4)™8I6 >R< Y38 P 8I2 > vid 8 > occW™|C84 _ 4 > 8ƒ P 8 @ 8I4)™8I6 > 8[698 >A> 8) @ D ) @Q<o8 G >t83G W 12 )™DFE @Q@ 0 <?>
Dt1 > Da1
03E54R4 < T?8Y 8 G @Q< w58 @ )™DFE @ T?8I4v03E >A@ 8I4 ) @ D ) @Q<-8 G >t8IG 4UW
y˜
7 Robot1
Dt1,Type,Da1,De1,De2
evac
S8
evac
S1
S10
arr_p
Dchgt
Dchgt
S4
Chgt
S2
Chgt
evac
S3
evac
arr_p
arr_p
S6
S5
Dchgt
Dchgt
S7
6
S9
ŠKWj˜0/ < 0 @ 0 P:P J8 Llc 8 G > 0 > 4 n >A@ 03254 <V>R< DF254 / > _ )™8 /10 /‚8 ‚/ 8Z‘
vid
occ
occ
vid
occ
occ
vid
vid
vid
vid
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
-,{“
XD
6 ŠKWj˜
AUCUN
T1
T2
AUCUN
T2
T1
AUCUN
AUCUN
T2
T2
vid
vid
vid
occ
occ
occ
occ
vid
occ
vid
vid
vid
vid
vid
vid
vid
occ
occ
vid
vid
03TV8IE @ 4vL8I4vYy0 @Q< 0fk=T?8I4 )™DFE @ ?T 8I4 8 G > 0 > 41L8
vid
vid
vid
vid
vid
vid
vid
vid
occ
occ
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A
>
@
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0
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Yf0 @Q< 0fk=T?8 feu W
!
MACHINE Parking
CONSTANTS NbMax
&
PROPERTIES NbMax 0
VARIABLES NbVoit, cc
INVARIANT
NbVoit 0..NbMax
cc -1..1
(cc = -1
NbVoit NbMax)
&
(cc = 1
NbVoit 0)
INITIALISATION
NbVoit := 0 cc := 0
OPERATIONS
entrer SELECT NbVoit NbMax cc = 0 THEN NbVoit := NbVoit + 1 cc := 1 END
controler entree SELECT cc = 1 THEN cc := 0 END
&
sortir SELECT NbVoit 0 cc = 0 THEN NbVoit := NbVoit - 1 cc := -1 END
controler sortie SELECT cc = -1 THEN cc := 0 END
END
6
ŠKWj˜e‹| 0 P 30 6Q; < 28
Parking
QInit
[][]Init
[NbVoit<NbMax][]entrer
cc=1
[][]controler_entree
6
[][]controler_sortie
cc=0
cc=-1
ŠKWj˜ - _ 4 > 8ƒ P J8 L58 A> @ 03254 <?>R< DF254 8 G >R<V^ E58 >t8UG 8I4 @ I8 Ti0 R> < N{0:ƒ Ti0 P 30 6Q; < 28
[NbVoit>0][]sortir
Parking
3‹
7 REFINEMENT Parking r1
SETS Couleur feu = vert, rouge
VARIABLES feu
INVARIANT
feu Couleur feu
((cc=0 feu=vert)
NbVoit NbMax)
((cc=0 feu=rouge)
NbVoit = NbMax)
(cc = 1
feu = vert)
((cc=-1 feu=rouge)
NbVoit = NbMax - 1)
((cc=-1 feu=vert)
NbVoit NbMax - 1)
INITIALISATION
NbVoit := 0 cc := 0 feu := vert
OPERATIONS
entrer SELECT cc = 0 feu = vert THEN NbVoit := NbVoit + 1 cc := 1 END
controler entree SELECT cc = 1 THEN
IF NbVoit = NbMax THEN feu := rouge END
cc := 0 END
&
sortir SELECT NbVoit 0 cc = 0 THEN NbVoit := NbVoit - 1 cc := -1 END
controler sortie SELECT cc = -1 THEN
IF feu = rouge THEN feu := vert END
cc := 0 END
END
6
ŠKWj˜eŠŽ|C8 @ 0 28 P I8 2 > L87TV0 P 30 6-; < 28
Parking
cc=1 & feu=vert
[][NbVoit != NbMax]
controler_entree
[][]entrer
[][NbVoit = NbMax]
controler_entree
[][]Init
cc=0
feu=
rouge
QInit
feu=
vert
[NbVoit>0][]sortir
[NbVoit>0][]sortir
[][]controler_sortie
cc=-1
ŠKWj˜e‰ _ 4 > 8ƒ P 8JL8 >A@ 03254 <?>R< DF2=4 8 G >R<i^ E8 >K8UG 8I4 @ 8ITV0 >R< N 03E @ 028 P 8I2 > Parking r1
 8 034)™8I6 L8 032tE=TV0 8X2Ccm0)=034 8 8 8 8I2KEL=03254h2DF4 0IYf03EH\6U0 2D 8‚L 8 0 6Q;58‚69DF254 4 8
0€ƒ L 8UG w=2 > <?@ E52uL > < 0 @ 0 @ P:P 8Ll@Q<?c >K8 G >G 0 > 4 n >A@ 03254 G <?>K>R<G DF@ 254[> Nq0 < 4R032 > 0 ) )r0 @ >A0 @ >A@ 8E528 @)r0 @R>R>A< @8L58IG 4xP Yf0 @ @Q< 0fk=TV8I4)™<DFE > @
P) 8 D >A) >A@ 88 8I8I2 4[@ 698ID T < 8U)=N T?E58925\H8I8 4J)=L0 8@R>RTd< 8:c 2tL5YyE]0 6903D 2P )™B D 6f@Rcj8I> 48 P no0yƒ 8In`2 L > L58•E E5425_ 8 4 > 8ƒ 8ƒ P FT?8 8Ž@ L8I82TV0 3>R8I< N74 0…ƒ DFE52 2 L=03E]4 )Œ6U8I036 ; > 8 )=0 L@R>R8I< 4[6UE56QT ;5< 8 0 @ 3L8I8I4U4 W
|’@ 0 > 0 @Q<-< TVG TV>t8G L8I4P L < 0 @ 0 PP 8I4J<LCc 8 G > 0 @->< 4An >A> @ 03254 <?>R> < DF254[<?89@ \ >A@ 0 <V> 4[@ LCcmE5284 ) 8IG 6 < >Rw=< 6U0 >R< DF2 –8I< 4 > @ E528N DF2=6 @ >R< DF2
89\ )ŒDF258I2 >R< 8ITiT?8vL5E25D P k @ 8sL8sYf0 @Q< 0fkrT?8I4~8 > L5E:2D P k @ 8‚L8vYf03T?8IE @^ Elcj8ITVTV8I4 )™8IEY38I2 > ) @ 8I25L @ 83WtˆbD < 254
6
'
DF2 @ 8*) @H8IG 4R8I2 > 8:L8[Yf0 @Q< 0fk=T?8I4)=TVE=4XT?8L < 0 @ 0 P:P 88I4 > T < 4 < k=T?88 > 89\ )=T?D <?> 0fk=TV83Wr/103254E528[D ) >R<V^ E8L8
69DF256 < 4 < DF2 L5E*L < 0 @ 0 P:P 8 < Tp8I4 > LDF256…2 8IG 698I4Q4R0 <?@ 8€L8 @ 8I4 >A@ 8 < 25L @ 8…03E P 0y\ <VP E P Tdcj8I254R8 P k=TV8…L8I4
Yf0 @Q< 0fk=T?8I4 @ 8*) @K8IG 4A8I2 >t8UG 8I4IW
!
Parking
cc
entrer
[NbVoit<NbMax]
controler_sortie
1
0
-1
controler_entree
sortir
[NbVoit>0]
KŠ W 0 / < 0 @ 0 P:P 8xLlc 8 G > 0 > 4 n >A@ 03254 <?>R< DF2=4 @ 8ITV0 >R< N{0:ƒ TV0 P 036Q; < 25870fk=4 >A@ 0 <?> 8 Parking
‡‚2803E 8 )=0 6UE5Ti0 8JL8[698 8ML 8 0 6Q;58 8I4 L8L503254‚TV04R6 4R4 DF2„L8I4X69DF2=L DF254XL8 032Hn
4 <?>R< DF2…8I2€L8I>A@EH\ @ @RDF>RE+< )™8I4 @Q<?>t69DFG 25L <V>R< DF>A25> 4sL58JG P L 8IG 6U@ T?8I256-;@H8 G P < 8I2 > B
) @ 8 P:< 8ƒ @ 8[< 69DF< 25L <?>R< DF2…8I2 >A@ <?8[>R< 6 @ DK6-;8 > >A41@ L8I4
8
/8 69DF25L <?>R< DF254:Llcm0 >A> 8 < F2=0fk < n
>AT @ 038325B 4 L<?>R8I< EDF\254:8ƒ L58„8•TV690]DF25w L FE @ DF8„2 ŠK8IWj˜ 2 - 8…+ NbVoit
NbMax > NbVoit
0 >
8
/-Wh 8 >A> 8
<?8I>K6QG ;=2 E8b< 4UP cm0 ) )rT E8]<?>R< 0 ƒ E528b>A0@ )) 6 @ DHDK6Q6-;;8 8 > 8U4Y ŠK8I2Wj˜e8 ‰ +8INbVoit
!= NbMax > NbVoit = NbMax
2
I
8
V
T
?
T
€
8
L
b
8
V
T
0
8
5
;
K
D
L
8
(
l
D
E
ƒ
V
T
0
F
0
L
…
8
>69DF25L <V^ DF2 2 8I698I4R4R0 <V^ 8 )™DFE E8Tdc @8UY 8I28 G 8I2 G 4AP 8L 8I>R6U< T?8I256Q;583W | 0 P F0 G > L8L58TV0…4RE5k=4 E @ DF.2 8I)4 > 25E5628 n
<V>R< G <V@ @7^ G G P > G
@
>R<?> >R< @Q< <
)=03T?8L8Tdc 8UG Y 8IG 258 P 8I2 > 8I4 > LDF256TV0…69DF25L <V>R< DF2 L8L 8IG 6UTV8I256Q;50fk < T <?>K8G L58Tdc 8UG Y 8IG 28 P 8I2 > W T~8I4 > )ŒDF4Q4 < k=T?8
LlcmE >R< T < 4A8 @ E=2803E >A@ 84RE5k=4 >R<?> E >R< DF2]69DF25L <V>R< DF2528ITVTV8„0€ƒ T†c < 2 >K8 G @Q< 8IE @ L8ŽTV0€4REk=4 >R<?> E >R< DF2 ) @Q< 256 < )r03T?8M8 >
TV0a69DF25L <V>R< DF2 L8…698 >A> 8bL8IEH\ < 8ƒ P 8…4RE5k=4 >R<?> E >R< DF2]8I4 > Ti0S69DF25L <?>R< DF2 Llcm0 >A> 8 < F250fk < T <V>t8•G L8…T†c 8 G > 0 > 6 < k=T?8
L8ŽTV0 >A@ 03254 <V>R< DF2lW '2 Llcm03E >A@ 8I4 > 8 @QP 8I4UB~TV0„69DF2=L <?>R< DF2 L8ŽL 8IG 6UT?8I256-;50fk < T <V>t8•G 69D @R@ 8I4 )™DF25Lu03E @ 0 289n
PL 8I2 8 > 8IL258J698xTV.0 LCFc 8 0 @ 0 L8[6 Lk=8xT?8Tdc 8ULG Y 878IG Ti250 8 P 8I032 25> 4 L8JDF2TV0 TVD P 4036-;E8< 2L8x8I0f4vkr4A4 DF>AE5@ 0 4An<?>8 870 8 > 41TV0 0:) 69)rDF0 25L0 <?4R>R4A< DF8I2…2 LlL5cm0 03>A25> 4v8 < E=F2•250f4Rk +E < )™T <V8 >t8ŽG n 8 0:ƒ 0 TV0
>A@ <?>R< @ ^
G> > @ < >
@ G> >
L87< sGTdcm@0fkr4 >A@ 036 >RG< DF> 2l> W <
6
&
Parking
cc
entrer
[feu = vert]
1
controler_sortie
0
-1
controler_entree
sortir
[NbVoit>0]
feu
controler_entree
[IN cc = 1 et NbVoit = NbMax]
controler_entree
[IN cc = 1 et NbVoit != NbMax]
vert
rouge
controler_sortie
[IN cc = -1]
controler_entree
[IN cc = 1 et NbVoit = NbMax]
ŠKW 0 / < 0 @ 0 P:P 8xLCc 8 G > 0 > 4An >A@ 03254 <?>R< DF254XL58I4pYy0 @-< 0fk=T?8I4pL5E @ 028 P 8I2 > Parking r1
T’89\ < 4 > 8E528[L < sG8 @ 8I25698[8I2 >A@ 8T?84 _ 4 > 8 ƒ P 8ML8 >A@ 032=4 <?>R< DF254 8 G >R<V^ E8 >t8UG 8I4D3k > 8I2KE+0
ƒ Tdcm0 < L58JL8[Tdcm0 )n
) @ DH6Q;58ŽL58 8 @Q> B g D > 8 > 8 > > DFE5TV4 +Ow ]ŠKWj˜e‰'/8 > T?8•L < 0 @ 0 PP 8•Llc 8 G > 0 > 4 n >A@ 03254 <V>R< DF254:69D P )rT?8 > +Ow ŠKW 03‘'/D3k > 8I2KE0ZY38I6a2D >A@ 8a0 ) ) @ DH6Q;83B < TVTVE=4 >A@K8UG 8 )=0 @ T?8I4w FE @ 8I4ŽŠKW 0 HBvŠKW 0 BpŠKW 0]8 > ŠKW 03‘KB )™DFE @ TV0
4) 8IG 6 < wr6U0 >R< DF2•L5E @ 0258 P 8I2 > L8Tdcj89\H8 P )=TV8‚L5E )=0 @ < 2 B Parking r1 W 'h2Ž8 8 > Tdc 8 G > 0 > cc = -1 L8TV0
6
3Š
7 Parking
cc, feu
sortir[NbVoit>0]
S3
S1
controler_entree
[IN NbVoit != NbMax]
controler_sortie
entrer
controler_sortie
S2
controler_entree
[IN NbVoit = NbMax]
6
S5
S4
ŠKW 03‘/ < 0 @ 0 P:P x8 Llc 8 G > 0 > 4 n >A@ 03254 <?>R< DF2=4
S1
S2
S3
S4
S5
76 ŠKW 0
cc
0
1
-1
-1
0
sortir[NbVoit>0]
feu
vert
vert
vert
rouge
rouge
L=E @ 0 28 P I8 2 > Parking r1
03TV8IE @ 4vL8I4vYy0 @Q< 0fk=T?8I4 )™DFE @ T?8I4 8 G > 0 > 4‚L8 w FE @ 8pŠKWj˜e‰169D @Q@ 8I4)™DF25L…0ƒ L8IEH\ 8 G > 0 > 4IB S3 8 > S4 L8pTV0‚w FE @ 8pŠKW 03‘KWK 8I4~L < Y38 @ 38I25698I4 ) @ DZY < 8I2528I2 > L58
TV0 032 8ƒ 8vL8v69DF254 E 8pT?8 0 )=; E583Wf|’8I4 4 4 8ƒ 8I4 L8 03254 DF254 8 E58 8U8I4{4A8ITVDF2 8 @Q> Beg D > 8 >
8 > P > DFE5< Ti4@ 4ADF2 >a8IG Ti0fkŒ>AD@ @H8I<?G @4Ž0 ƒ )=0 @ @R>R<?@ Ll<V^ cmE=28ML < 4 ADF_ 256> >R<P DF2aL8:Td>Ac <@ 2eYf0 @Q<?>R< 03< 2 > L=G 03>R25<V^ 47TV0Ž>tG 6UTV03E54R8 ASSERTION
03T?D @ 4 ^ E81T?8I4{L < 0 @ 0 P:P 8I4 LCc 8 G > 0 > 4An >A@ 03254 <?>R< DF254UB4A8IT?DF2Ž2D >A@ 8L 8 G P 0 @ 6Q;83B54ADF2 > 69DF254 >A@ E <?> 4L58*)=E < 4T?8I4
L < 0 @ 0 P:P 8I4 @ 8I4)™8I6 >R< Nq4L8I4Yf0 @Q< 0fk=T?8I4ML58ŽTi0€4 ) 8IG 6 < w=6U0 >R< DF2lW g}0 @ 89\H8 P )=TV8 )™DFE @ TdcjDFE >R< T ‚8I28 _ 4 >
jy˜ †B T8I4 2 8I698I4R4R0 8‚LCcm0 DFE > 8 @ TV0JT < F28
L503254TV0x4 ) 8IG 6 < w=6U0 >R< DF2
ASSERTION cc=-1 or cc=0 or cc=1
L87TV0< w FE > @ 8xGŠKWj˜e‹KW <V@
! " #$ &%
'#(%*) +% -,
. /1032)547036 89036 :03;=<>2?8A@5B )DCE<F4>032G2IHJB ))DKECELNM5BOCP03;=<RBSQ 0UTN<F4-BOCP4>0R:032WVNM032YX[Z\/]:0*)DM5CE2^KE032W2)_036 8ICP`D8IBa<>CEbc;52
BadD2F<F4-BOCP<F032:eHJMD; )54>bgfF0I< Yhi/j032Clk )DKm036 kS03;=<>Ba<>CPbc;52n:jHJM5; )54>bgfF0I< o2Fbc;=<]036 8U4ACP<F03203; 7h_p0q2>bcM52sr
03;52F03ktdDKP0\:5MuKEBO;5vcBavO0 w032><BO2>2F0I.x )54>b78A@0y:032KlBO;vcBavO032z:0 )54>bOvO4-BOknknBa<>CPbc;CEk )_0I6 4ABa<>CE{|2<F4ABO:5C}r
<>CPbc;5;503KE2IhD~Ba4?8Ubc;52=036 L=M503;€<KEBn8Ubck )54N036 @03;52>CEbc;\:032[CEk )9K-036 kS03;€<>Ba<>CEbc;52W032F<[d‚03BOM58Ubc+M ).)DKEM52WBOCl2=0I6 0ƒLNM0
8U03KEKP0?:032_2 )_036 8ICP`98IBa<>CPbc;52^BadD2F<F4-BOCP<F032Ih„…036 BO;5knbcCE;52_CEK†2F0I4ABOCP<^CE;=<=0I6 4>032>2>BO;=<^:eH‡0UTN<F4-BOCP4>0:032_2 )_036 8ICP`D8IBa<>CEbc;52
8Ubc;58U430IQ <F032ƒ:032y036 Km036 kS03;=<>$2 )‚0I4Akn0I<F<>BO;€<RMD;0SkS03CEKEKP03M54>0ƒ8Ubck )54N036 @03;52>CEbc;ˆvcKPbOdDBOKP0n:0SKP03M54{‰bc;58U<>CEbc;5;0Ur
kS03;=<Ih=Š?BO;52i8U0I<F<F0Gb )5<>CELNM0G;5bcM52*B3VObc;D2i8-@bcCE2>C5:jH‡03;S0UTN<F4-BOCP4>0G:03MTƒ<Œ‹ )‚032:50G:5CEBavO4ABOkkS032*LNM5C2Fbc;=<GŽ
 KEKE03032W2W:5:5CECEBaBavOvO4A4ABOBOkkkSkS03032W2W::e0HJBO2N8U03<>6 LNCPV7MCP03<N03;56 2I8Uh 0O‘

F’ “ '% #$ ” ; )†03M5<•<F4-BO:5M5CP4>0–M5;50ˆ2 )036 8ICP`D8IBa<>CPbc; 03;—M5;:5CEBavO4-BOknkS0˜:0˜2N036 LNM03;58U0™2ACKP032…:5C}šG0I6 4>03;=<F032
knBO8-@5CE;03,2 )‚03MVO03;=<8UbcknktMD;5CELNM0I43h‚/j0n;5CPVO03BOMˆbjM–Q 8U0I<F<F08UbcknktMD;5CE8IBa<>CPbc;ˆ032F<KEB )DKlM52 )54N036 2F03;=<F0S032F<
K›HJCEk )DK-036 kS03;=<>Ba<>CPbc;eh 'KlKP0?032F<W47036 BOKECE2N0I6 0 )DBa4GKP032^B ))†03Kl2^2>M58I8U032>2>CE{|2W:5032^b )0I6 4ABa<>CPbc;523h
‰œ ž ŸG D¡\¢j£¥¤g ¦c¡
” ;™:i0I6 `D;5CP<RKE032bOd7fF0I<>2 )DBa4t4AB ) )‚bO4><BOM7T–8IKEBOM52F032
0I< hjŠ?BO;52R<FbcM52RKE032R8IBO2I‘1KE032
knBO8-@5CE;032CEk )‚bO4><N0I6 032 )DBa4+KEB[8IKEBOM52>0 IMPORTS :0IV7CP03;5;IMPORTS
03;=<:032¥bOd7fs0I<>2+SEES
:DMƒ:DCEBavO4ABOknkS0_:0_2=036 L=M503;58U0Oh
/j032RknBO8A@5Cl;032?VNM503,2 )DBa4RKEBz8IKEBOM52>0 :50IVNCP03;D;03;€<:032?bOdfs0I<>2MD;5CELNM03kS03;=<?:DBO;52KP0S8IBO2bjM˜Q 2F032
b )_0I6 4ABa<>CPbc;D22Fbc;=<M<>CEKlCE2=0I6 032Ihip038IC_032FSEES
< )†bc2A2>CPdDKP0zM5;5ClL=M03kn03;€<S:5BO;52SKE0q8IBO2nbjM§Q KP032nb )0I6 4ABa<>CPbc;52:0qKEB
knBO8-@5CE;0…;0…kSb7:5CP`503;=< )DBO2SKP032¨VaBa4ACEBadDKP032¨:0…KEByknBO8-@5CE;0Oh*ŠƒHJBOM<F4>0 )9Ba4><S8A@5BOLNM0qCE;52><>BO;58U0…:eHJM5;0
k…03© kS0–knBO8-@5CE;0ˆ:bcCP<y0I© <F4>0ˆ4>*0 )547036 2F03;=<=0I6 0™:5BO;52•KP0ˆ:5CEBavO4ABOkkS0y:50y2N036 LNM03;58U0OhYª7C[M5;0yknBO8A@DCE;0y032F<
CEk )‚bO4><N0I6 0 )DKEMD2>CP03M4A2j{‰bcCE2¥0I<4>03;bckk«0I6 0¬|`Dv­Nh ¯® 03KlKP0B ))DBa4AB¯©°P<F4AB[BOM<>BO;=<+:0_{‰bcCE2¥LNMeH‡03KEKP0032F<+CEk )‚bO4><N0I6 0
:5BO;52GKP0:5CEBavO4ABOkkS0:02N036 LNM03;58U032Ih
p_@DBOL=M0SbOdfs0I<t032F<4>*0 )547036 2>03;€<N0 6 )DBa4tM5;ˆ4A038U<>BO;vcKP00I<tM5;0KECEvc;0S:0nV7CP0n:5BO;52KE0:5CEBavO4ABOkkS0n:0
2N036 LNM03;58U0•¬|`Dv­Nh²± ® h9/1BKECPvc;0?VO0I4><>CE8IBOKE0R8UbO4>4>032 )‚bc;5:™BQ K›HŒ036 8UbcM5KP03kS03;=<Y:5Mz<F03k )D2Ih
3³ ­
IMPLEMENTATION MAIN imp
IMPORTS M1.M,M2.M
...
e­Nh  [X ;0CEk)DKm036 kn03;€<>Ba<>CPbc;
"!$#
MAIN_imp
M1.M
M2.M
e­Nh²±  / B¨<F4ABO:5M58U<>CEbc;q:eHJM5;0Clk)DKm036 kS03;=<>Ba<>CPbc;…03;qX?Z\/
‰œ‰œ ŸG D¡ 9¡g¡ D¡
p_@DBOL=M0tB ) )‚03Kj:eH‡b )_0I6 4ABa<>CPbc; )‚0I4AkS0I<GKEB8UbcknktMD;5CE8IBa<>CPbc; 03;€<F4>0tKP032G:5CPšG0I6 4>03;=<>2Y8Ubck )†bc2ABO;€<>2Y:eHJM5;
)54>bgfF0I< [h [Mn;5CPVO03BOM:0WK›H IMPLEMENTATION ‘aKP032+0UT036 8IM<>CPbc;52i:032:5CPšG0I6 4>03;=<F032CE;D2F<F4AM58U<>CPbc;D2¥2>bc;€<
2N036 LNM03;€<>CE03KEKP032Ih ” ;¨03;=<>BOkS0WK›H‡0UT036 8IM<>CPbc;n:0^K›HJCl;52F<F4AM58U<>CEbc; M5;50{|bcCE2LNM0^K›H‡0UT036 8IM<>CEbc;n:0^K›HJCE;52><F4AM58U<>CPbc;
+[032F<^<F0I4AknCl;i0I6 0Oh5Š[BO;D2WKEBƒ`5vcM4>0­Nh ³ ‘5K›H‡0UT036 8IM<>CPbc;q:DM•2>‹N2F<I03Q kS0t:i0I6 dDM<F0 )DBa4WK›HJB ))†03K†:0K›H‡b )_0I6 4-Ba<>CPbc;
:0 M1 hD~Ba4GKlBS2>M5CP<F0K›H‡b )_0I6 4-Ba<>CPbc; op2 :0 M2 032F<W0UT036 8IM5<=0I6 0Oh
op1
%!&#
('
*)
IMPLEMENTATION MAIN imp
IMPORTS M1.M,M2.M
...
OPERATIONS
main BEGIN M1.op1 ;M2.op2 END
END
e­Nh ³  X[;0b )_0I6 4ABa<>CEbc;q03; [Mq;5CPVO03BOM :5M…:5CEBavO4ABOkkS0:0R2N036 LNM03;58U0z¬|`5v­Nh =® ‘9K›HJB ))†03Ke:5032^b )0I6 4ABa<>CPbc;52 M1.op1 0I<[Zy±Nh‡b )9±
032F<S4>0*)D4N036 2F03;€<N0 6 )DBa4nM5; kS032A2>BavO0 )DBa4><>BO;=<:0qKlByKlCPvc;0…:5M <F03k )D2S:50qK›H‡bOd7fF0I< MAIN ¬|bOd7fF0I<:bc;=<
K›H‡b )_0I6 4ABa<>CEbc;032F<iB ) )‚03KEBO;€<F0 ® VO0I4A2K›H‡bOdfs0I<:5bc;€<K›H‡b )0I6 4ABa<>CPbc;032F<iB ) )‚03Km0I6 0 M1.M )‚bcM4 M1.op1
0I< M2.M )‚bcM4 M2.op2 h1/jmain
0•kS032>2>BavO0032F<tKECEd†03KlK-0 6 )9Ba4RKP0;bck :0K›H‡b )_0I6 4-Ba<>CPbc;«B ) )‚03Km0I6 0Oh¥/j0n<F03k )92
:eH‡0UT036 8IM<>CPbc;•:032k]0I6 <>@b7:032_:0GK H‡bOd7fs0I<i2F0YknBa<N0I6 4AClBOKECE2F0 )9Ba4iKP0324>038U<>BO;vcKP0322>M4KlBKECPvc;0W:jHŒ036 8UbcMDKP03kS03;=<
:5M?<F03k)92Ihgp032e4A038U<>BO;vcKP03212>‹Nktd‚bcKECE2F03;=<†KE0+<F03k )D2j:eH‡0UT036 8IM<>CPbc;ƒ:0*K›H‡b )_0I6 4ABa<>CEbc;eh¯/j0+<F03k )D2j:eH‡0UT036 8IM<>CPbc;
:0K›H‡b )_0I6 4ABa<>CEbc;qB ) )‚03KEBO;=<F0 main 03;vcKPbOd‚0KP032_<F03k )92W:032_b )_0I6 4ABa<>CPbc;D2WB ) )‚03K-0I6 032 M1.op1 0I< M2.op2 h
/j0G4>0I<FbcM4i:50GK›H‡0UT036 8IM5<>CPbc;2>M54K›H‡b )_0I6 4ABa<>CPbc; main 032F<iCEk )DKECE8ICP<F0OhaŠƒHJBOM<F4>0 )DBa4><K›HJB ))†03K:50GK›H‡b )_0I6 4ABa<>CPbc;
2IH‡0Uš‚038U<>M0M5;50ƒ{|bcCE2LNM0SKP0S<F03k )D2:jH‡0UT036 8IM<>CPbc;™:0nK›HJB ) )‚03K¥:0 M1.op1 032F<y036 8UbcM5Km06 8IBa4bc;
M2.op2
;0 )‚03M< )DBO2W0UT036 8IM<F0I4?:03M7Tzb )0I6 4ABa<>CPbc;52Y2>CEkƒM5KP<>BO;036 kS03;=<W03; 7h
%!&#
+
+
MAIN_imp
M1.M
M2.M
op1
op2
e­Nh  / B¨<F4ABO:5M58U<>CPbc; :0K›HJB ) )‚03Ke:eHJM5;50[b )_0I6 4-Ba<>CPbc;
‰œ ŸG D¡
€¡ * c ¦O¢ * 9¡q¢ †¦ ¢ i¡
¦ c D¡… ¦
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:0K›H‡0UT703k)DKE02>M5CPVaBO;€<_bjM\Q K›H‡b )0I6 4ABa<>CPbc; M1.op1 4>0I<FbcM4-;0:03M7TzVgBOKE03M4A2¬|`5v­Nh ® h
%!&#
$'
'
' '
IMPLEMENTATION MAIN imp
IMPORTS M1.M
...
OPERATIONS
main VAR aa,bb IN aa,bb ! M1.op1 END
END
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Š?BO;52WKE0:5CEBavO4ABOknkn0:02=036 L=M503;58U0R8UbO4>4>032 )†bc;D:5BO;€<R¬|`Dv­Nh ® bc;qB 8-@00UT )DKECE8ICE<F03kS03;€<^KP04A0I<FbcM4
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"
MAIN_imp
$#
M1.M
op1
aa,bb
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s;52F<F4AM58U<>CEbc;
IF
IMPLEMENTATION MAIN IMP
IMPORTS M1.MX
...
OPERATIONS
main BEGIN
...
IF v=Value1 THEN M1.op1
ELSIF v=Value2 THEN M1.op2
ELSE M1.op3
END
END
;52><F4AM58U<>CPbc;
CASE
IMPLEMENTATION MAIN IMP
IMPORTS M1.MX
...
OPERATIONS
main BEGIN
...
CASE v OF
EITHER Value1 THEN M1.op1
OR Value2 THEN M1.op2
ELSE M1.op3
END
END
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p032CE;D2F<F4AM58U<>CPbc;DIMPLEMENTATION
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ELSE
Š?BO;52nKE0z:DCEBavO4ABOknkS0…:0…2N036 LNM03;58U0 8UbO4A4>032)‚bc;5:5BO;=<SBOM7T]:03MT«2)_036 8ICP`D8IBa<>CEbc;52n:0I6 `D;5CE032S:5BO;52¨KP0
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bc;…bOd5<>CE03;€<WKP0k…03© kn0R:5CEBavO4ABOkkS0:02N036 LNM03;58U0OhD/1032GkS032>2ABavO032G8Ubc;5:5CE<>CPbc;5;03KE2ibc;=<GKEBSk…03© kS0bO4-CPvcCE;0
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:0?V7CP0:0K›H‡bOd7fF0I<W:032F<>CE;DBa<>BOCP4>0:5MzkS032>2>BavO0t8IBa4GKP032_<F4>bcCE2 )‚bc2>2ACPdDCEKECE<=036 2 )‚03MVO03;=<WB ) )DBa4AB¯©°P<F4A0?2>03KPbc;…KEB
VaBOKP03M4*:0WKEB?VaBa4ACEBad9KP0 v BQ KEBk…03© kS0G:DBa<F0Oh€Š?BO;52KEB2 )036 8ICP`D8IBa<>CPbc;nBOMD8IM5;0^VaBOKP03M4*;eH‡032F<4>0I<FbcM4A;0I6 0 )9Ba4
KP032[b )_0I6 4ABa<>CPbc;D2[:5bc;58Rbc;ˆ;0ƒ4>*0 )547036 2>03;€<F0 )DBO2Y0UT )DKECE8ICP<F03kn03;€<?KP0R4A0I<FbcM4B•Q K H‡0UT036 8IM<>CPbc;–:50ƒK›H‡b )_0I6 4-Ba<>CPbc;
B ) )‚03KEBO;=<F0Oh
MAIN_IMP
M1.MX
[v!=Value1
and v!=Value2]
op3
[v=Value2]op2
[v=Value1]op1
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8U0I<F<F0tCl;52F<F4AM58U<>CEbc; )DBa4><>CE8IWHILE
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KEBd†bcMD8IKP0zB•Q 8€bO© <=0n6 :0ƒKlBKECPvc;50t:0ƒVNCE0t:032YbOd7fF0I<>2Ih;yCE;€<F4AbN:5MDCE2>BO;=<W:5M )92F03M5:barž8Ub7:0ƒbc; )‚03M<[:bc;58
:036 KECEkCP<F0I4WKEBƒd‚bcM58IKP0 5CEKe;04>032><F0 )DKEM52_LNMeH‰BSQ CE;52N0I6 4>0I4WKE032WkS032>2>BavO0R03;€VOb ‹*036 2nBSQ K›HJCE;=<=0I6 4ACP03M4W:5Mz8UbO4 )92W:0
'
'
'
€³
IMPLEMENTATION MAIN imp
IMPORTS M1.MX
...
OPERATIONS
main VAR fin IN
fin := 25
&
WHILE fin 0 DO
M1.op1 ;
fin := fin - 1 ;
INVARIANT fin NAT
VARIANT fin END
END
END
END
‚­Nh²­  X[;0d‚bcM58IKP0?03; KEB•d‚bcM58IKP0S:5BO;52?KP0S:5ClBavO4ABOknkS0n:0S2N036 LNM03;58U0n03;=<F4>0nKP032:036 KlCEknCP<F03M4-2[:50¨:0I6 d9M<0I<:0¨`D;–:0¨d‚bcM58IKP0
¬|`5v­Nh ® h
%!&#
+
+
MAIN_IMP
M1.MX
while fin > 0
do
op1
end while
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+
$#
'
'
'
'
2
IMPLEMENTATION MAIN imp
IMPORTS aa.M1,bb.M1
SEES M2
...
END
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KEB•{|bO4AkS0n:0S<F4AB¯V*0I6 032ƒ8UbcknkS0KE032CE;52F<>BO;58U032:50 M1 Ž aa 0I< bb :5BO;D2KEBz`5vcM4>0n­Nh ±Nh¥/j032knBO8-@5CE;032
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%!&#
2
%
MAIN
bb.M1
aa.M1
M2
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%!&#
' %
('
'
'
'
'
IMPLEMENTATION MAIN imp
...
INITIALISATION ...
OPERATIONS
OP1 ...
END
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"!$#
MAIN_imp
...
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b )_0I6 4ABa<>CPbc;D2Ih
%!&#
' &%
+
IMPLEMENTATION MAIN IMP
REFINES MAIN
INITIALISATION Inst ini 1 ;... ;Inst ini n
OPERATIONS
main BEGIN Inst1 ;... ;Instn END
END
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<>CPbc;52 Inst ini 1 B Q Inst ini n :50_KlB[8IKlBOM52F0 INITIALISATIONh /1B:03M7TC›03Q kS0^2N036 LNM03;58U0W4>*0 )D4N036 2F03;€<F0
KP032WCl;52F<F4AM58U<>CEbc;52 Inst1 B Q Instn :0RK›H‡b )_0I6 4ABa<>CEbc; OP1 h
"!$#
"
E1
/Inst_ini_1
/Inst1
I1
E2
...
...
In-1
En
/Inst_ini_n
/Instn
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"!$#
"
%
"
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8UbckkS0R:5BO;52^K H‡0UTN03k )9KP08IC}rž:032>2>bcM52R¬|`5v­Nh ® h
ELSE ...
IF ... THEN ...
IF aa MAX THEN Inst1
ELSE Inst2 END
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%!&#
E1
[aa<MAX]/Inst1
[else]/Inst2
E2
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var=valuex
x
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THEN aa := TRUE ELSE aa := FALSE END
BOOL
IF Cond1 THEN Inst1
ELSEIF Cond2 THEN Inst2
...
ELSE Instn END
CASE var OF
EITHER value1 THEN Inst1
OR value2 THEN Inst2
...
ELSE Instn END
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E1
[Cond1 ou var=value1]/Inst1
[!Cond1 and !Cond2 and ...
ou var!=value1 and var!=
value2 and ...]
/Instn
[Cond2 ou var=value2]
/Inst2
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aa := BOOL(bb = xx)
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INVARIANT VARIANT
03; [hD~Ba4Y8Ubc;D2=036 L=M03;=<G03KlKP032^;eHJB ) )DBa4-B¯©°P<F4>bc;=< )DBO2G:5BO;D2^KP0:5CEBavO4ABOkkS0:eHJBO8U<>CPV7CP<=0O6 h9/1032WCE;52F<F4-M58U<>CPbc;52
LNM5Ce2F0I4>bc;=<W<F4ABO:5MDCP<F032W2Fbc;=< Inst 1, ..., Inst n, Inst n+1ch
Š?BO;52˜KE0 :DCEBavO4ABOknkS0 :eH 0I6 <>Ba<>2sr <F4-BO;52>CP<>CPbc;D2˜8UbO4>4>032 )‚bc;5:DBO;€<uBQ KEB<F4ABO:5MD8U<>CPbc; :0 K›HJCE;52><F4AM58U<>CPbc;
"!$#
+
INITIALISATION Inst 0
OPERATIONS
main VAR ii IN
ii := 20 ;
WHILE ii /= 0 DO
Inst 1 ;
...
Inst n
INVARIANT ii : NAT
VARIANT ii
END ;
Inst n+1
END
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WHILE
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KEBVgBa4AClBadDKP0 )‚0I4Akn0I<Y:eH‡0UT036 8IM<F0I4KP032[CE;52F<F4AMD8U<>CPbc;52WCE;=<F0I4A;032¨BQ KlBSd‚bcM58IKP0RbcMy:eH‡03;\2FbO4><>CP4?8U0ƒLNM5Ce032><
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"!$#
+
En
/Inst_n-1
/Inst_n
En-1
E1
...
/Inst_2
[ii /= 0]/Inst_1
E2
[else]/Inst_n+1
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'
$'
'
' ­
IMPLEMENTATION MAIN IMP
...
IMPORTS MX.M
...
OPERATIONS
main VAR zz IN
zz ! MX.OP1(25) ;
END
END
IMPLEMENTATION M IMP
OPERATIONS
res ! OP1(arg1) 4 BEGIN ... END ;
...
END ;
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"!$#
MAIN
MX.M
main
E1
/^OP1(25)
E2
E3
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#
#
#
O­
#include stdlib.h #include "BWallet.h"
T BWallet *function new T BWallet( void)
T BWallet * this = (T BWallet *)malloc( sizeof(T BWallet))
memset( this, 0, sizeof(T BWallet) )
return this ;
//-------------------// INITIALISATION
//-------------------//IMPORTSs imported instances
void function T BWallet IMPORTS(T BWallet * this)
this- initialisation = FALSE ;
// end T BWallet : :IMPORTS
void function T BWallet INITIALISATION(T BWallet * this)
if ( this->initialisation == FALSE)
this->initialisation = TRUE ;
this->balance = 0 ;
//end T BWallet : :INITIALISATION;
//access methods
T int function T BWallet get balance(T BWallet * this)
return this->balance;
;
//----------------//Clause OPERATIONS
//----------------void function T BWallet setBalance(T BWallet * this,T int balanceInit)
this->balance = balanceInit ;
void function T BWallet debit(T BWallet * this,T int debitAmount)
this->balance = this->balance - debitAmount ;
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+ ] B+ 3 587 3u;0hZ + =(F 7 102C Hq O 7,9 1 5N0 / 5I0p/ 38021<;<O<02CYC8O C + Z¤b
ZD7 02G1 Kv5 / = 9 1<=E5832+60]585 7M902C GW + _3 K`K K 0 93 9 =E5¥02C8/ C8O<7:C¤5 ;0¥+2@ 3 + 1<02ZD9 021 5 g‡A - 2k W K nm zx=EKM02+ CZ-[+6+ - =(F 7102+ CDC 9 1 K¦5p/ +ž WG0* Kv04/ 02 CDž ;02C
@ Z 02O 5 02C8ZDO 021 C + ZpZD1<02C 1 ;5 3 029 Cf=(F<C 0d1 ;0202C =E3KM+ 1 02-[C +6- 0202C =h7 1<O<j 1e9 3(C Z O<+6C~587M02q 1<O<CQ0202C Zd^c 9 O0r3 ;<q O O0 1 K`+p5 3 +0 ;<OB= 587:9 1 0 @ 0 043vCu=E02O 0 T bˆ= 7 02CQ1 0
C 9[7:5 j + = 7MKM7M5I04/ 0 W
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K= M+ C8C80 1 W $VC(=(+ F 1<02CfZ =E0r=;+ 0C2t= K 021 C @U043(+ C 3 7`+ 1 c K 021 C#= O<9 381<0 =E340]= 5 021<C~C ;C 0£0.=E0 ] 5Q5 0r3 Z ;<+O =v38F 0p7 1=(0£F C 9 1 O5'0 5 3 Z + ;<O 027M5Zp02Ca021 021 @6+ 3 1 7M+ c K 02Ca;¬3¤{ 7O<1<1C0 b
58+ 1<=E0¤;0 KM+ = KM+ C8CQ0d0 5YK 02C 9*#04/ 3 +6587:9 1<Cu021PZ 0 / 5 F 9 ;02C WG* 0d= 9 1<C 5 3(O<= 5 02O3 B+ 3Y; 04/ j + O 5 ;0 KM+ = KM+ C8CQ0¥g 9 O
K 0£Z 02/ = + 1 7 C(ZD0 043(ZD0 5Q58+ 1 5~K { 7 1 7M587M+LKM7 C +6587M9 1 + OeZ 9 ZD021 5 ;0 K { 7 1BC 58+ 1<= 7M+6587:9 1 m 043(ZD0 5 ;0 5 3 + ;BO 7 380 KM+
= KM+ O<7C80 587 1+ 7:587M+LKM7 C +6587:9 1 + W š 7 0>C8=(F KM02/+ Z + 9 ;0BK 5 3 T+ 7:;<5I/O<= 587:9 1H5Q+.]5 K { - +n@U+ 581 7:9 58+6- 0J/ 58+ ;<0 5 3 /9*Kv^/ - 9 CQ04/ 3pO<+ 10 KM+ - 02-[C 58+67:- 9 1–+sC 7 9 Z B¡ K 05
;9 02C 1<C 9 1<=E+ 02C£54;W 0¤Z =(F 102C4t = Z 0 0D;0D=E0 0 02C 1 0 1 ; 0 0 O 040 O 1 0N] c 0
= 3838*,02C+ 9 1B; @ 1 7:9 @
KM+ -[+6- +N9 ¡ 5 5 / / 7
/ + 9 @ 7:9 / 7v/
+ O T Z + = =(F 1 7 104023(C C q O 1 7 1040D3(ChC 9 O<1 1 <5 B+ 1 C 7 1<C 0›58+ ]1B= 7vc04/ 02Cu0 ;<0¤02ZhO O K:c 58Œ702B1 K 040rABj = 9[704C 3¤W ;¬K { O<j +1ŸO C85£=vFq O<02Z 0¤; + ;1<0sCV= =E0¤1 ;<04043(3(1C 7 041”3V= ; + 02C ; K 00
KM+ 1 -[+6- 0¤C8O 9 3 5 0 K { O 587MKM7 C +6587:9 1e;0hZ 0 / 5 F 9 ;<02CY0 5 ;0 @6+ 3 7M+ c K 02C~;0d= KM+ C8CQ02C W$V+ 1<Cu=E0d;043(1 7 043~= + C4t KM+
KM+ 7 7:587M+LK`7 +6587:9 5 5 5 + 7:5 B+
/ 5 9 '/ 7`+LK @6+ 5 5 + K¦/ G+ KM+
5= 3 + O<;<C8OB0 = 587:1 9 1>;0 C K { 7 Z B1 K¦02/ ZD02O 021 580 +658387:0 9 1 3 7 Z ;<^O 9 3 0 58+ 1 5 3d=EO<0 5Q15 0p0dZ Z 0 + =(F F 7 1<;00W C 02= 0;0 1 0 380 0 040 3
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=« * 9 Z *9 B3 0258/ F7 021<7 C +67 c 587MK 9 0V0 5 3 02/ O 58+ 7MK`7 C + 7:c @ K 0 + B+ 3 K 0£BC #K`+ 02/ = 7 A<02O3 5 ; 0 / @ 0 K:9*9 5802+ O3 5 W K`7:5Q5I/ +LK B+ K @U+LK
W:W:W02m C C 9 1 5YZ KM+L7 C C8C 04/ 02C¤1<+pC] ;KM+ 0~=(c F + C_3 1- 0y;<0 O>OP= g‡9 38Z 02Z B7MKM+65 =E0202OZD3\02;B1 O KM;+ 021 C#-[+6= - 1<0dC = 7 c 1 K 0 02W C 043 02C 3 02C 02O3(C4t
« * 02C 9*<587 Z 7 C +6587:9 1<C ^9 O3#C +2@L9[7 3\C4{ 7MK j + O 5
043(ZD0 5Q5 380 9 Os1 9 1 KM+ =E3 0 / +6587:9 1e;|{ 7 1<C 58+ 1<=E02C£+] G+ 3 587 3
KM+ 9 5~-,/ / B+ K 5 + 5 @ K KM+ -[+6- 7 K W
« ;¬* { 02O<CY1Z 0r02=/ = + C81 C87 0rC8ZDC 021 CuC8O 043 ^04902CC 02/ C~A 3 + c 0 K 02CY3 ;;<02O<C = K`+ 021O<-[3~+6;- 0 02C¤+] 9 04c3(C ¡ 0 5 0 CYC 9 1 1 5 O 0y587M=KM7 C c 02/ C 0 9 O<3Y; 7 C 9 C8043\;|{ O<1
^9 3 5 ;0r=E3 0 / +6587:9 1P;|{ 7 1<C 58+ 1<=E02C'0 5 ;|{ O<1¥02C B+ =E0d;0r1 9 ZpZ +6- 0 W
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0 5 + 380£K = 9 W 1j‡3 9 1 5l0/ + ] ;02Ca= 9 1 5 3 +L7 1 5 02C ¡ OBC 587 A + 1 5aKM+ 3 0 / +LKM7 C +6587M9 1;0 5 3 + ;<O<= 5 02O<3(Ca= 9 Z BK 0 T 02C_0 5 ; 9 1<=u1 9 1
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=
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-,04/ 38Z 043 =vK 02F Cf1Z 0£9 C4; { 7 AB1|= +6{R 5 02O3(C'O¬;|{ O<{ +1=40V=6CQ02] 02C¤O g 0 1<C 1<=E0£;0£=E0 0Ebˆ= ; 1<C 0 5 0 3 0 m ^9 OAB3 1eK 02;C'0£j 9 11<0 = 587:9 C 1BC'0 5\3hK 02]C
@6+ 3 7M+ c K 02CY;02C~= K`+ C8CQ02C4tO<1 B+ = *U+6- 0DCQ043 @U+ 1 5 ;|{02C B+ =E0h;<0d1 9 ZpCYCQ043 + ; 04/ AB1 7 0 5 =(F + =4O<10d;<02CY= KM+ C(CQ02C
-,02/ 1 04/ 3 04/ 02C + B+ 3 5 021 + 1 5\+ O 3 9U¡ 0 l5 CQ043 + ; 04/ AB1 7 0£; + 1BC#=E0 G+ = *6+6- 0 WG* 02Cf;<3 9[7:5 C';|{ + =4=U02] C + O T 021 587M5I02/ C
CQ043 9 1 5 =E02O T ;<O G+ = *6+6- 0 W
%'& &ƒ & l "&# …B† ˆ… Ž a˜ˆ™’/Š Š +Š Ž >‘_Š„6?Ž^‘a— l[˜•Œ,™ }Ž ,^Ž
? =(F +Lq O0’Z + =(F 7 1<0’;|{ 7 Z BK¦02/ Zp021 58+6587:9 1 CQ043 +f+ C8C 9 = 7¦04/ 0aOB10’= KM+ C8CQ0 W š 0 5Q5 0a= KM+ C8C80 02O 5 380 3 02/ CQ021 5 043
KM+ Z + =vF 7 10rC2{ 7MK 1|{R +pq O¬{ O<10£CQ02O K 0 7 1<C 58+ 1<=E0r;0r=E0 K`K 0Ebˆ= 7 ; + 1<C K 0 3 9U¡ 0 5 uW
* { O 587MKM7 C +6587:9 1;|{ OB10 + O 5 3(0VZ + =(F 7 1<0uCQ0uj +L7:†5 G+ 3 K`+ = KM+ OBCQ0 IMPORTS W $ 0Vj + = ) 9 1 -,02/ 1 04/ 3 +LK 0 K 0 KM7 021
;| { 7 Z 5 9 7 3 58+67M58K`+L7:9 7 1 + 043(ZDK 0 57 ;<0Y=EKM3 + 04/ 043f-[+6;- 02C 7 +h1<C 9 58+ ¡1<=E5 02CaW;* 0Y7 Z +58+=(F 7 1<02C 7 Z ^9 3 5l+ 04/ 02C 7Wl* 0u= 9 9 1<=E0 <5 K ;|{ 7 1<9C 58+ 1B/ =E0\021 5
K 02C 02*9C '04C / 3 Z +6587:9 1<38C~0d; ] 04/ AG=E01 7 O 02C\;; 02+ C 1BCf1=E0 5Q5 0h02CyZ ]+ =vF c 7 10 0£C 0 5 CQ{02Cf1BC 3 + 1<=E1<0V02ZD;|{ 02O<1 15 0VC W Z =(F 10 C(CE02] ;0 02C#; 1<1 0402C'0
* 02C B+ 3 587 =4O KM+ 3 7:5l02/ Cf;0 C 9 1 5~K 02C\C8O 7:@6+ 1 5 02C£z
« K 02C 5 9 C(C 7 c K 0D021 ;0s1<0 B+ C¤; 9 1B1043h;<0s1 9 Z + ] O<10 7 1<C 58+ 1<=E0LtxC4{ 7MK 1|{R +Pq O¬{ O<10¥CQ02O K 0
7 1<C 58+ 1<=E0s;0=E0 5 R 0 W}wx+ 3h0 T 02Z GK 0Lt,C 7 O<10 7 Z BKM+ 1 58+6587:9 1 M1 7 1<C 58+ 1<= 7 0OB10p0 5 O<10C802O K 0
jK 9[7 7 C BKM+ KM+ Z 58+ +6=v58F 7:97 10 M2 t?0 5 C 7BK`+ l5Z B+ + =(F 7 + 10 M27:- ^W 9 C8CE02] ;0VO<10 *9 '04/ 3 +6587M9 1 f t +LK:9 3(C K { O 587MKM7 C +6587:9 1D; + 1<C
« {K Z 02C 5 1 102/ =E02C8C +L17 3(0£M1021 ; 0 ;f 0£1¬;{029 C 1<1<043'C OB1Zd1 c 9 Z O 0 + O T7 1<C 58+ 1B=E02CfC4{ 7MK R + 021 BK O<C#;|{ O<10 W 1s0 € 0 5 C 7
7 58+ 7 T 9[7 KM+ Z + =(F 7 1<0 M2 t K { O 587MKM7 C +6587:9 1¥;0 f 02C 5~+ Zdc 7:- O 0pzx+D] q O0 K`K 0 7 1<C 58+ 1<=E0
M1 1<C 1<= 0V;<02O j C
+B+ 3 587 021 5YK`+ j 9 1<= 587:9 1 f * + C 9[K O 587:9 1N02C 5 ;0d1 9 ZZD043 K 02C 7 1BC 58+ 1<=E02C4t B+ 3Y0 T 02Z BK 0
;<0 M2
5
5 / T K + 0 K ;0 K`+ j 9 1<= 587M9 1 B+ 3 K 0r1 9 Z;0 K { 7 1<C 58+ 1<=E0 q O 7 = 9 1 587 021 5 f B+ 3
i1 0 i2 tB0 ;0 3 04A 043 {
T BK W
* 0Y0 C8=v02F Z 02/ Z +y0D-}z 02/i1.f1 04/ 3 +LK 02C 5 ; 9 1<=~;|{ + C8C 9 = 7 043r+ ] =(F +Lq O0 7 1<C 58+ 1B=E0 O<10 7 1<C 58+ 1<=E0Y;|{ O<10~= K`+ C8CQ0 ?+2@6+SW
9 O 5 04j 9[7 C; + 1BCO<1 cGO 5 ;|{ *9 B587 Z 7 C +6587M9 1|t’O<10NZ + =(F 7 1<0 q O 7 10JCQ043 +L7:5s7 1<C 58+ 1<= 7¦04/ 0 q O¬{ O<10>j 9[7 C
CQK 047 3 + 9 380 +63 0258/ 7:CQ9 021 5I04/ 9 0 G+ 3dO<+ 15Y0pK = K`+ C89 CQ0p0 5 ;02C 7 @6+ 583 + 7M+ c K 02Cy;0@ = + KM+ C85 C802C W w}/ 5 9 O3y73 0 / +L/ K`7 CQB04+ 3y=EK 0 5Q5 5 0 + 9*<587 5Z 7 C +6W 587:9 1|t
{ 1<j $ 3(Z T 1¥9 = K 1<=E043(1 1 ^9 0r1 5 Zyc<3(0r;|{ 1<C 1B=E02Cf;0 3 0 380d; 0 043(Z 1 040 3 0 3 ;<O<= 02O3
« 02*,O + Z 3 + =(c F 7 02] 1ZD0 027 C\Z CQ^0 9 3 5l04/ CQ0£021 02C 5 z 1 9 ZpZ 04/ 0yZ +L7 C 7MK 1|{R +q O|{ OB10 7 1<C 58+ 1<=E0Dz<; + 1<C KM+¤-,02/ 1 04/ 3 +6587:9 1P;0
9
9 7 B+ + 5
^9 KM+L7 q K 9 KM+ KM+ ?+2@6+ 9 ^9 + 5 W
« =*,+ ;0yZ =E+ 0d=(F 1 7 1Z 0 02; C 5NC 7 Z 3 ^ 9 3 5I04/ 0 O3\B1K O<0 C 7 02OC(3(CQC043 j 9[7 O<Ce0 Z 0r+L7 1 CeO<Z 10›;<0 ;02C = 7 1<CC858CQ+0 1<=E02Ce1¬= {0238C 3(>5 02CB+ Ce1<1; 9 Z1 Z 0 04/ 0"z
; + 1<+–C + KM+¤B-}+ 02/ 1 + 04/ 35 +6587:9 K 1P;0y9 = 9 ;0 7MKMK + j + O<K`;+ 3 + =vF 5J9[7 CK 7 3~O<+L1e7:5>1 q 9 Z 7MK ^9 O+63\-[7M=E5 0 5Q5 0hZ + =(7 F 7 1580+ W<* 0d+1 9 9Z =(F 9[7 C 7
j‡043 3  380 0›1 Z ;0 = C8- CQ0 2k 0 W 0 5 2j k W m W O|{ C4{ ;|{ O<10 1<C 1<=E0 1 1IRSZD0–z
š F anonymous
+Lq O0”= KM+ C8C80 ?NomDeClasse
+2@6+ = 9 Z 9 3 5 043 + O<g‡A 10„j 9 1<= 580 7M9 1 INITIALISATION W š 0 5Q5 0 j 9 1<= 587:9 1 CQ043 +
3 02/ = 02/ ; 04/ 0 ;BOZ 9 ; 7 AG= +65 02O3
+ K +
K`+ q 7 +>¡Q+ +L7 7 58+ 7¦/ W
š 0 5Q5 0rj 9 1<= 587:9 1 043vZD0 5Q5 3 + z static ; 1<C 0J= Ce;¬{ O<10J= C8CQ0 O 10.CQ043 Z C 1<C 1<= 040
« K`+ =E3 0 / +6587:9 1¥;02C 7 1<C 58+ 1<=E02Ca;0~Z + =vF 7 10\021 [+n@U+d@l7`+rK { *9 '04/ 3 +65 02O3 new zI=E0 5_*9 '04/ 3 +65 02O3 + 0 KMK 0
K 0r= 9 1<C 5 3(O<= 5 02O<3~;0 KM+ = KM+ C(CQ0 q O 7|+ 0 KMK 0 KM+ j 9 1<= 587:9 1 INITIALISATION ;0 K { 7 1<C 58+ 1<=E0Lt
« K { 7 1 7:587M+LK`7 C +6587:9 1;02C @6+ 3 7M+ c K 02Cf;<0 K { 7 1<C 58+ 1<=E0 9 O>;0 KM+ = K`+ C8CQ0 W
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m t K`+ Z 0 / 5 F 9 ;0 INITIALISATION 02C 5¤+ 0 Kv04/ 01 9 1 B+ C B+ 3hC 9 1 3 9* 380
= 9 1<- C 25 3vk O<W ® = 5 202k OW 3­[m q WO $V7 +1|{0 T7 C 5 0 + B+ C_9 Z +L7 C 5 B+ 3 K`q + Z 0 /KM+5 F 9 K`;<+ 0 INITIALISATION
;<K`0 + KM+ = / KM5 + C89 C80 q O 7<K { O 587MK`7 CQ0
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LISATION ;
C 58+6587Mq O0g 7MK 1|{R + O<3 + B+ C\;0r= 9 1BC 5 3(O<= 5 02O3 m W
…B† † xŠ ‹#“•.Š
* 02CV0 T 02Z GK 02C q O 7 C8O 7M@ 021 5u9 1 <5 ^9 O3ucBO 5 ;|{ 7MKMK O<C 5 38043 K 02Cu; 7 €~04/ 38021 5 C£= + C£;0hA - O380 q O 7( 02O @ 021 5
CQ0 3 02/ CQ021 5 043¤g‡A - 2k W t 2k W t 2k W ® 0 5 2k W ­[m W
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IMPLEMENTATION
M1 imp /* implementation de la machine abstraite M1 */
...
END
IMPLEMENTATION
M2 imp
IMPORTS
i.M1, M1
...
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« 02K C {021<R CQ0202ZdC\c ;K 0£0£c;02CfC80y021C 5871 043(C~z 380 KM+6587 jiCVt
« K {021<CQ02Zdc K 0£;02Cfc 9l9[Kv04/ 021<C BOOL = true, false t
« K {021<CQ02Zdc K 0£;02C\=vF + M102CY;0r= + 3 + = 5 04] 3802C STRINGt
« K 02Ca021BCQ02Zyc K 02C 02/ 1lO<Z 04/ 3 02/ C~0 5'+ cGC 5 3 +L7:5 C4t; 04/ AB1 7 Ca; + 1<C KM+ = K`+ O<CQ0 SETS g @L9[7 3\C802= 587:9 1eC8O 7:@6+ 1 5 0 m W
w}9 ^9 @L9[7 7 GKv/ 5 K 9 / + KM+ -[+6- 7 9 +6587Mq 58+L7 9 5 +L7 5
9C 1 5 1 02O/ =E3 02C8C +LO 7 3802Cr3 z Z 02ZD021 043 02C}; 1<1 0402C’; 1<CxO<1 1 0 1j 3vZ O0LtU=E043 102C = 1 3 1 02C
« * 02CD021 587 043(C= 9 1B=E380 5 C; 9[7M@ 021 5+ B+ 3 5 021 7 3J+Ÿ] K { 7 1 5 043 @U+LK`K 0 INT W INT 02C 5 O<1 C 9 O<Cb•021<CQ02Zdc K 0
W\* + @6+LK 02O3P;0Ÿ=E02C
;<9 0 K {5802+ 1<CQ5 02Zdc K 0 5†BV+ t_+ c 9 3(/ 1 5 0 / + GK + 3 K 02K:9CJ= KM9 + 1<C 58++ 1 5 027 C MININT
0 5 KMK MAXINT
7
K
5
9
5
q
= 1BC / 1 5802+ C'02K C 3 KMZ+ 0 3+ c 7 0£CQ0 5#1 q 7MK Z =(F +L7:1†5 0YB= + c 0Lt?;0 / 09 0uC 3 0 5 OK:9 0 K 02C#; 9 1<+L1 K 04/ 02C'K C W 9[š 7 021 5 7
3(0 9[7:53 02CQ021 +L7 c 02+ C'C8O37:@ + Z =vF KM1+ 0Y-[0 +6- O| { ^1|9 {R q KMCa+ ;< 0u; 04@ c 3(;0258Zp7 021 C 9 3(C#B;502C#W = =4O C 02=
« ;* 02C'C CQ9 0£O<jCb•02381<0 C802ZyONc 1 K 02Ca0 ;|Oe{021 ;<58O 7 043(Ca1 0 5'K 020 C_021BCQ02ZyOc3 K 02O<C 0 + cGC 5 3 +L38027:5 O C#; 0£9[027:1@ 021 5021<0 15 3(0r0u= ;02Z C 7 1 0 5 043 @6+LKMK 02Ca;|{021Sb
587 043(C\= 9 1<=E380 5 C W
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2­ œ
class M1 ...
void INITIALISATION () /* initialisation des variables de M1 */
M1() INITIALISATION();
class M2 /* On suppose M2 instancié plusieurs fois */
...
M1 i, anonymousM1 ;
void INITIALISATION () /* Création des instances */
i = new M1() ;
anonymousM1 = new M1() ;
/* initialisation des variables de M2 */
...
M2() INITIALISATION () ;
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# 1
M1
K { O<10r1|{ 0 / 58+ 1 5;B+ C\1 9 ZpZ 04/ 0
* -[K 5 B+6- 9 5r+ / K:9[7:- /
KMK
KM+ -[+6- 9L- + +6587:9
K= M+ C8C 7M02q C£O<0340]W<* 0202CrC\;; 07:€~04/ R 38021<=E020CY;1 0 9L5Q9[7 38C 02C 1 + O>C81 C807:@ 0 +0 O>;<021 Cf040202Cy1<CQ;<020pZd=Ec 0 K 02C\02C£C 9 ;|1 { O<5~1 K 02CfC81 O 7:@6+ 0p1 5 ;020 Crz 3 3 ZpZ 1
« K { 7 Z ^9 3 58+6587:9 1¥;0yZ + =(F 7 1<02C +2@ 02=h;02C B+ 3 + Ze0] 5 3(02C~j 9 3(ZD0 K Cf;<0 5 R 0y021<CQ02Zdc K 0Lt
« K`+¤@U+LK O +6587:9 1>;|{021<C802Zyc K 02Cf;0rZ + =(F 7 10+p] K { 7 1 5l04/ 3 7 02O<3\;0 K { 7 Z GKv02/ ZD021 58+6587M9 1 W
$V+ 1<C K 02C KM+ 1 -[+6- 02C~;0 3 9L- 3 + ZpZ +6587M9 1 7 Z #04/ 3 +6587 jiC'= KM+ C8C 7Mq O02C4tSO<10 @6+ 3 7M+ c K 0u02C 5'5 R '04/ 0r0 5 C 9 1 5 R 0
1/0r/ @U=v+LF K + 1 / - +0 GK O<9 C W š 0r1|{02C 5 K B+ C K 0r+ = + C~;<O KM+ 5 1 9 -[+6- 0 u9 W ^1>9 0 €7M580 7:9 5~K 0 5 R 0r+ ;|{ O<7 1>= 9 1<C K`58+ 7:5 O + 1 5l 02O 5 0 5 38m 0 W
3* 0 + 0 + O ¡Q09 7:5IOp/ = O<3(C’7;<OD=ERSK:=5l/ 0~;0f3 5 B+61- 02ZD021 Op587M;5I0\/ = Z +C 7 1D;02C’/ Z 7 =vF + 102C\gi= O<CQ+0 IMPORTS
- 5
5 R Z0 q O0 3 K { 0u9 1 ;<02+ Ca; 0 KMK 04=43 O + 3(0202C_= ;<9 0O @ R3(02ZD020£1 5 ;;02Ca+ 1<021C KM+ 02C(C'O ;7:5 020rC#;<Z Oe;=vF 9 =41O<02ZDCa023 102C54W ;0u; 1<Ca=E02C#=vF 1 02ZD021 C';0
W 0 10hZ + =(F 7 10 + cBC 5 3 +L7:5 0 02O 5 0  5 3(0 B+ 3 + Z 0 / 5 3 04/ 0>+] K { +L7 ;0d;|{ O<1e021<C802Zyc K 0sg‡A - 2k W m W $V+ 1<Cu=E0
=+ + C K 0 5 R 7 09 ;0p+6=E580 7:5Q9 5 0s021 587:5 5I0/ B02+6C - 5d-,02/ 1 04/ 3 0Ÿ/ 5Q+ ] 5 G+ 3 58587 7:3y5I/ ;<W O š 1 9 58Z +L7 ;0 K {021B58CQ7:025IZy/ c K 0p= KM+ + 3d1 9 + OBCr7 1|{ +2@L99 1BC5
5 R OB'=404O</ 021CD0 +¤] 1K { j +L7 3v;Z 0£;0£=E1–0 5 ;020 1<C802R Zyc K 00 W C8?;O 33202=E] 0CfOB0.10 021@’04/ 3 7 0 AG= +658047:9 3 1 K:91=02+LCK 0021 +¤] KM+ 02CsZ ;<+ 0=(F 7 10rZ ;02=(C F 5 R 1 0N02CC4t 1 KM+
58+ c K 0y;02C 5 R 02C\;<02Cf021 587M5I02/ Cu;0 KM+ Z + =(F 7 1<0 A 02C 5 ; 04/ AB1 7 0r; + 1<C K 0 58+ c K 0 + O 2k W W
1<7 C8O 7M5 0y;<O- 3 2+ k 1 W 5Y[m K W{ + š 1 +L5QK 5RSCQ0d;<+ O 7 3 9U¡ 7 0 5  9 t 5KM+pKM@’+ 04/ 3 7 AB+ = +6587 7M9 1P;<W 0 *¬5 9 R 02CVCQK 0473 + ^0 9€ 0258= +65 58O 7:904/ 0pC8K O<3 KM+ Z + K b
=vF 10 B g‡A
0 0'Z =(F 1<0 Z 3 0 Z =(F 10 A 3vC};<0 { Z 3 1 {021<CQ02Zdc 0
!
!
ž
IMPLEMENTATION
M1 imp /* implementation de la machine abstraite M1 */
OPERATIONS
...
op1 ...
...
END
IMPLEMENTATION
M2 imp /* implementation de la machine abstraite M2 */
...
OPERATIONS
...
op2 ...
...
END
IMPLEMENTATION
M3 imp
IMPORTS
i.M1, M2
...
OPERATIONS
...
op ...i.op1 ;...op2 ;...END
...
END
§#¨‡©aª 2k W ®h« 0~587MK`7 C +6587:9 1¥;0yZ + =(F 7 1<02C 7 Z ^9 3 5I04/ 02C~O<1<0rCQ02O K 0Vj 9[7 C
R 20 C
1 587:5I02/ C
!
ENSEMBLE1
aa
dd
bb
$
ª 2k W V« R 0r;<02Cf021 587M5I02/ Cu;0 KM+ Z + =(F 7 1<0 A + 3402] C @’04/ 3 7 AB= +6587:9 1 K:9 = +LK 0r;02C 5 R 02C
5h/ @U+LK O 0 / G+ 3 K { 7 1 5 043 @U+LK`K 0 0..255 ;0 5 R 0 g m W $ O3 + 1 5 KM+V@’04/ 3 7 AB= +6587:9 1 KM9 = +LK 0
ENSEMBLE1 02C 0
5
;<K`+e02C 58+ R K 02C C(O3 5 B t K 0 5 R 0f;0 587:ENSEMBLE1
t7
tLK`K ; 9[7:5 0 5 5838+ 0f38K 02+Z BK`2+ k = W 0 k / W B+š 3 7 g m 5 t
I
5
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M
K
+
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7
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KM9 = +LK c 0 501;<002C ^9 R CQ0 02CdB+ ;<C¤02Cr;0 021 3 9 02c CdK 02] ;ZD0 0 9 3 Z K { + =vF 10 0 K ;<A0 ;K 0{ 9*'0204/ 13 +658=E7:9 0 1 0op1;<O ;0 c KM+ 0 Z O + =(F 7 10 A02= ; + 021BC C
K { *9 #04/ 3 +6587:9 1 op2 ;0 KM+ Z + =vF 7 10 B 02C 5 3 0 / +LK`7 C 04/ 0.+J] K { +L7 ;0p;<OŸ1 9 ZycB380 5 W š 0 5y+ 0 K 02C 5yK¦0 / -[+LK
= + 3 5 02C 5 ;0 5 R 0 g m t K 0uZe02 ZD0 q O0V=E0 K O 7 ;0 ENSEMBLE1 Z +L7 C KM+y58+ c K 0u;02C 5 R 02C';0 KM+
2
3
$
class M1 ...
static void INITIALISATION () ... /* Bloc exécuté pour l’initialisation de la classe */
static void op1() ... ...
class M2 ...
static void INITIALISATION () ... /* Bloc exécuté pour l’initialisation de la classe */
static void op2() ... ...
class M3 /* On suppose M3 instancié une fois */
void INITIALISATION () /* initialisation des variables de M3 */
M1.INITIALISATION();
M2.INITIALISATION();
...
M3() INITIALISATION () ;
... op () ... M1.op1() ; ... ; M2.op2() ; ... §#¨i©_ª 2k W ­h« 0~587MK`7 C +6587:9 1e;0r;<02O T Z + =vF 7 102Cdg M1 0 5 M2m 1 9 1 7 1<C 58+ 1<= 7¦04/ 02CY; + 1<C M3
Z+”+ / 5I=(F/ 7 1<0 + A-,02/ CW 5l^9 O3 58+ 1 5 021>; 02/ C + =4= 9 3(; +2@ 02=d=E0 3 7 1<= 7Œ 0u= + 3YCQ02O K|K 0 5 R 0r;0 ENSEMBLE1
0 0h=(F 1 0
1 587:5I02/ C
R 02m C
g
ENSEMBLE1
!
$
aa
dd
bb
243
B
ª 2k W kD« R 0y;<02Cf021 587M5I02/ CV;0 KM+ Z + =vF 7 10 A + 3402] C @’04/ 3 7 AB= +6587:9 1 K:9 = +LK 0 B+ 3 587 0 KMK 0£;02C 5 R 02CY;0
K 02C 5 ; 9 1B=u1 02/ =E02C(C +L7 380y;<0VZ 9 ; 7 A<043 K 0 5 R 0r;<0 5Q9 O 5 02C K 02C'021 587:5I02/ C~;0 KM+ Z + =(F 7 1<0 A 5 R '04/ 02CD+ ]
K { +L7 ;0d;<0 ENSEMBLE1 9 O<3 9 c 5 021 7 3YO<10 58+ c K 0h;02C 5 R 02Cp+r] ¡9 O3u= 9 ZpZD0¤; + 1<C K 0 58+ c K 0 + O
$
k
MACHINE A(ENSEMBLE1)
VARIABLES aa
INVARIANT aa ENSEMBLE1
OPERATIONS
bb op1(dd) PRE
dd ENSEMBLE1
THEN
bb : ENSEMBLE1
END
END
§#¨i©_ª 2k W « T 20 Z BK r0 ;0rZ + =(F 7 1<0 2+ @ 20 =h;02C B+ 3 + Ze0] 5 3(02C\021<CQ02Zdc KM7 C 5 02C
!
IMPLEMENTATION B
IMPORTS
A(0..255)
...
OPERATIONS
cc op2 BEGIN
cc op1(5)
END
END
§#¨‡©aª 2k W h« ’T 20 Z GK r0 ;|{ 7 Z ^9 3 58+6587M9 1s;<0rZ + =(F 7 102C +2@ 02=h;02C G+ 3 + Z¥0] 5 3802CY021<CQ02Zdc KM7 C 5 02C
2k W lW
R 02m C
1 587:5l02/ C
g
ENSEMBLE1
!
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$
aa
dd
bb
ª 2k W « R 0£;02C'021 587:5I02/ C~;0 KM+ Z + =vF 7 10 A + 3402] C @’04/ 3 7 AB= +6587:9 1 K:9 = +LK 0£= 9 Z BK 0] 5 0d;02C 5 R 02Cf;0
B
k W~w}9 O3 KM+£@6+LK O +6587:9 1;|{021<CQ02Zdc K 02C K 0 3 9 c K 02] ZD0\02C 5 C 7 Z 7MK`+L7 380Lt 9 1 B+ 3 5 ;0 KM+ Z¥02 ZD0Yj + = ) 9 1s;|{ O<10
Z$V+ + =(F 7 1<0 A+ g‡A - 2k W œ m 5VW 9 7 BK¦/ 58+6587:9 - 2k W 4ž?m K 587:5I/
K
5 R 1<0 CyO<g 1P m 3 W?* 0 15 02R ZD 020r1 ;02C O @6+ O<3 17M+0 c K Z 02C aa,02Zpbb021 0 5 dd1–; g‡9[A 7M5#+ O<C(C 7 0 5 3({0y021 Z 9 ;0 7 AENSEMBLE2
/0VC 7 1 9 1s=E02= 7i 3 389 02@L1<9l; q O00
OB10J043(3802O3e= + 3 K 02+C¥;02m O W¬T"* 5 R 02C¥10K C 9 1 55DB@U+ +LC K 7 ;/ 02q 1 587Mq O<02CPg K 0 K 5 R 9[70 ;0 9 aa,K bb 0 9 5 ddK B+ C8CQ05
;<0 9 9[7 ] 9 5 +L7:- {02+1<CQ025 Zdc9 0> 1¬{02C 5D7 ^O 9 0 5 O|{ OB10sC802+ O 0¥7 j CDG;K 1<7 = 0 9[3 7 c +202] @ ZD0¥02C
c^0 + ENSEMBLE2
O<= O Z 1<C¤= 1 3 1 1 g 1 02O Z 3 043¤O<10eZ =vF 10 O<C 02O<3(Crj C 02=N;02C
243
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MACHINE A
SETS ENSEMBLE2
CONCRETE VARIABLES
aa
INVARIANT
aa ENSEMBLE2
...
OPERATIONS
bb op1(dd) PRE
dd ENSEMBLE2
THEN
bb : ENSEMBLE2
END
END
§#¨i©_ª 2k W œ « T 02Z B K 0r;0rZ + =(F 7 1<0 +2@ 02=yOB1e021<CQ02Zdc K 0 + cBC 5 3 +L7:5
!
MACHINE A imp
REFINES A
VALUES
ENSEMBLE2 = 10..100
...
OPERATIONS
...
END
#§ ¨i©_ª 2k W 4žD« T 02Z BK 02Cf;|{ 7 Z BK¦02/ Zp021 58+6587:9 1 +2@ 02= 5 R G+6- 0d;|{ O<1¥021BCQ02Zyc K 0 + cBC 5 3 +L7M5
@6+LK O +6587M9 1<C’;|{021<C802Zyc K 0~; 7 €~04/ 38021 5 02C#Z +L7 C_; + 1<C K 0Y=ERl= K 0Y;0\3 + 1<02ZD021 5a9 110 02O 5_@U+LK O<043_O<1
021BCQ02Zyc K 0 q O¬{ O<10rCQ02O K 0uj 9[7 C m W
0 10 3802Z 7 04] 380\C 9[K O 587M9 1DC 7 =E02C 5 R 02C’C 9 1 5 ;02C’= K`+ C8CQ02C 02C 5 ;|{ O 587MKM7 CQ043 K`+ = KM+ C8CQ0 Object ;0 $?(!‚?
m W<w}+ 3V= 9 1<C 0 / q O021 5
;7MK 9 1 + 5Y5Q5 9 OBC587MKMK 702C 5 R 02CYF + 04/ 3 5 7:5 021 5 9[ZK +L587:7 9 Cu=E02q C 5 7VR + 02CV+LC 7:9 5 1 5 C 7 Z 5 BK 02Cyg int/ 0 5 + boolean
8
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;|{ 1BC 1<=E02C\;0rZ =(F 1<0 02C 1<C 1<=E02C\;<0rZ =(F 102C CQ043 021 ; 0438021<= 0402C 3 0 R 0rO C 0
^9 O3 @U+LK O043 K {021<CQ02Zdc K 0 W6* 0 3 9 c K 02] ZD0 7 1<F 04/ 38021 5 + ] =E0 5Q5 0YC 5 3 +65I0 / -[7 0\02C 5’K 0a38021 9 ZpZ +6- 0\;0 KM+ Z + =(F 7 10
;<7 O3 + 581 7 5y+ K`+e5 5 3 B+ K;<O<= 7 587M9 1 W KM+ 1Ÿ0 € 0 ?5¤+2@6K 02+7Cd^Z 9 + =(F 7 102C q O K7 O 587`KM+ 7 CQ0217 5 =E0 5Qm 5 0 587 1<7MKMC7 58+ 1<=E09 1<0K C 7 + O3 589 + 1 57MB+658+ 7:C 9 q O0 K7
;KMK 021 A 5 1 / g O<+LC7 02Om 3vC\5 = ^9 C8CQ02C / 5 9O3uO<1-W 0d0 C802O 0dZ 7 =(F 1<0 9[K 587M9O +CQ04/ 3 5I/ / KMO<+ 3 9 { 1B/ C W a1<KM= K 1„giC 5
0 0D02C / 1 02=E02C8C K`+ 3(0 0 +n@ OK 3r5 CQ02CdZ 0 58F 7 ;<02C G+ 10T ;04B3(1 K 04] 380pC m O 5 1 5 0+ 0 0 9 c 3 04K0 7 0 587 043(+ ZD05
;09I9[=EK¦3 / 04043¤O<10p= C8CQ0 02= 0 R 0p58027:@ 1 043g 5 3y0 02Z 0 5 intK:9Sq 0 ;K 0 583 7MK`7 1<+6CQj587:9 3(ZD043 025 C ;021 A 1 C
c^9 0402K 1B7 C 58FALSE,TRUE
3802C 02= 02Zp021 021 .+ B0 + c O049 3 K { O C 5 1>B+6;B- O R K 0 STRING
O587MK`7 3 / { 1<5~C K + 1<5 = 7M+6 587M9 1 W 587 $ 0W =E0 5Q5 0¥Z + 1 7 04] 380 7`K 1|{R + 0,1
O3 Cy;0 3 c 02] ZD0;0 R 0Pz 0pC802O K’5 R 0
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jBshort 0 jBbyte = ZpZp0r;02C 1 043
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int
9
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jBshort
short g
= 0 t<b
jBbyte
byte g
STRING
String
ª 2k W « w 3 *9 ^9 C 7:587:9 1p;¬{ 0 / q O 7:@6+LK 021<=E0V1 9 1380 5 021lO0V021 5 380 K 02C 5 R 02C#;<0~c + CQ0 0 5'K 02C 5 R 02C ?+2@6+
9[7 7 t7 G+ T 9 K 9 9 5 9 7:5 +p+ + 9
5Q5 / 5 9
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« 02* 1 02Cf04Z 3(C\+ =(c F 7 10204C'021<;Cf0V0 c + 02CQ01IO<5 Z0 KMK04023 C 02C q O0 BASIC IO O 587MKM7 CQ021 5#K 0 5 R 0 INT W š 0 5 R 0£= 9 383(02C ^9 1<;
9 O jBbyte W KB+ O3 +L7:5 j +LKMK Op=E3 04/ 043fO<10~Z + =vF 7 10 3 02/ = 9 ; 04/ 0
C80 K:9 1 K 0u= + C + O Tp5 R 02C 54jBshort
G+ 3 5 R 0p= 7M5I0 / 3 02/ = 02/ ;02ZpjBint,
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W S+ =(F + 1 5'q O<0 K 0YcBO 5 ;0u=E0 5 3 + ;<O<= 5 02O3#02C 5 ;0Y=E3 04/ 043f;02C + BKM7 = +6587:9 1<C ^9 O3 [+2@6+ š + 3(; K 02C
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5 R 02CrO 587MK`7 C 02/ CV; 9[7:@ 021 5 0 5 3(0 9*<587 Z 7 C 02/ C W K 02C 5 ; 9 1<=¤1 02/ =E02C8C +L7 380;0 9 O @L9[7 3£=(F 9[7 C 7 3 K 0 5 R 0pO 587MKM7 C 0 /
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@6+LK 02O3vC\CQRSZyc 9[KM7Mq O02C W
k W 0 1 021<CQ02Zdc K 0 + cBC 5 3 +L7:5 02C 5 O 587MK`7 C 0 / ^9 O3d; 02/ C 7:- 1043d;<02C 9 c ¡ 0 5 Ch; 9 1 5d9 1 1<0 @ 02O 5B+ Cy; 04/ AG1 7 3
K`+ C 5 3(O<= 5 O3(0 + ON1 7:@ 0 + O>;0 K { + cGC 5 3 + = 587:9 1 W 9 O<C K 02C'021<CQ02Zdc K 02C + cBC 5 3 +L7:5 CfC 9 1 5 3 + ZD021 02/ C¤+ ] O<1
7 1 5 043 @6+LKMK 0f021 587 043aAB1 7 1 9 1 @l7 ;<0Lt + O1 7:@ 0 + O;0 K { 7 Z BKM+ 1 58+6587:9 1>gi= KM+ O<C80 VALUESm W K R + ;02O T
O« 587MKM7 / C +6587M7 9 1<C';0r9 =E02@ C + 7 1 5 Te043 5 @6+L KMK 02CVz 9 + K B+ +6- + 2k W ® W « ;; 0404/ ABAB11 7 3\3 K;020rC\1 c 9 O 3v1002C O 7 1j 04/ R 3 7 0202OC\3802= C'Zp0 5 ZDC8O0h'; 04/ 3 1<7 02C O<380 02Cf;|3 { O<1 3 58+ c FK 0 0 + O W t
$V+ 5Q9 K + K 7 / 5 K`+L7 K 9 B7MKM+65 [+2@6+ / +LK`7 K *9 <587 7 +6587:9 WoY9
+Z LK 02Zp1<021C 57`O<K C ;0 02@ C~3 +L= 7:5 C4t ^9 { O ; @L049[0£7 3D02C 3802Z;0 GKM+ =EC804CQ3 043 K 02Cp0r== 9 Z1BC 58+ 1 5 0202CsO3 0 5K 02CD30 0 fT 38CQ0204C83C 7:9 021BC CD= 9 1<Z C 58+ C 1 5 02C 1<BC + 3 K 023QCb
@6+LK 02O3(Cf1lO<Z 04/ 3 7Mq O02C\= 9 3(3802C ^9 1<; + 1 5 02C W
$
## *
K + 5 +L7:5 + 5 + 7 5 @U+LKMK - 2k W nm q 7 9 5 587`KM7 / ^9
/; 04AB1 7023_CD;02021<C CQ@U02+Zd3 c7`+ c02C K 02C cG02C1<C83 02Zyc CpK`7 C8C 0>5 023 C 9 ZeOs02] ;10202C’1 021B] CQ02;Zy02cC K 021 C_;|043 { 7 1<; 7 02=EC>02C’g‡;<A 0 58+ c K 0 + O W O K C’CQC 043 1 9 1 5O 3(0 C 3 0202/C CQ021 5IO02/ C3
021 '[/ +2@67 + G+ W<3Y* O<10d7 = 5 KM+ C8@6CQ0h+LKM;K + 1<C K`+L9 q O5 0 K`K 0r CQ/043 + 5I; / 04/ ABG1 +7 0r02KM+1P= KM9 + 1<C 58+ 1 5 0 KM+ c 9 q3(10 7¬+ 7 1j 04/ + 3 7 02O380£T 0 5V+KM+ c 9 3(10
02O380 + 02C K¦/ 1 043 0258C\7:@ CQ043 1 5 380 3 02CQ021 02C 3 = C(CQ5 0 BRange O O3 ;02OWI* =(F 9 Z 5 C~;0
5C8RO 0043 int
0 + @L9[7 - 2k W 2k[m W 0¤= 1<C 3(O<=¦b
3802C 02= 02ZD021 5Q5 borneInferieure
5 02O3~;<0 KM+ = KM+ C80 CQ0 02CYBRange
043vZD0 3 + ;|{ 7 1 7:587M+LKM7 C8043a=E02C~;<02O T =vborneSuperieure
F Z Cdg 3~A
SETS
VITESSE
VALUES
VITESSE = 0..100 ;
#§ ¨i©_ª 2k W £« 0 1>021<CQ02Zdc K 0 + cBC 5 3 +L7:5 021 * 02C 5 R 02C BOOL, NATURAL 0 5 INT K:9 3vC q O¬{ 7MK CrCQ043 9 1 5 O 587MK`7 C 02/ Cr021 1 9 1 ^9 O3 5 R 043DO<1<0
@6+ 3 7M+ c K 0YZ +L7 C#= 9 ZpZD0 7 1 5 043 @U+LKMK 0~CQ043 9 1 5a5 3 + ;<O 7:5 C_021 ?+2@6+G+ 3#O<1<0 7 1<C 58+ 1<=E0Y;0 KM+ = K`+ C8CQ0 BRange W
oY9 O<C +2@L9 1BCy=(F 9[7 C 7 K 0 5 R 0 int O 587MKM7 C 0D/ ; + 1<C K`+¥5 3 + ;<O<= 587:9 1Ÿ;0 K {021<CQ02Zdc K 0 VITESSE g‡A - 2k W 2k[m
+L7 K 5 + O3 +L7M5 O K O 7¬+ O<C8C 7 0 5 380¤=(F 9[7 C 7 = + 3 7MK ; 0 / 021<;J;<O 5 R 0h;0yc + C80d=(F 9[7 C 7x^9 O3
KMZ +¤5 3 C + ;B0 O<= R 587:9 0 1 short
W
## * KM+ -[+6- [+n@U+ ^9 B+
/ + 7 / / 7 ^9 / 7
/ / +6587M9 W
* { 7 ; 04/ 0¤0 = 9 1 1BC 7 C 5 0 0 7 = 7 + ] ; 1040/ AG1 7 32C8t^C402]= ;9 0ZpZD0DCd; ;+ 01<ZC K0202= Cu1A - C(OZD3(0202CC 2k 3 02W ; 04AB0 15 C2k W Ot 3 OB021=E0h3 38= 0sK`+ O<C8CQ10 0 B02+ 1l3VO<02Z 1<04CQ3 02Zdc K 1 0 W
š 0 5Q5 0y= K`+ C8CQ0y10r= 9 1 587 021 5Yq O0r;02C\= 9 1BC 58+ 1 5 02C Oc KM7Mq O02C_0 5 10rCQ043 +G+ C 7 1<C 58+ 1B= 7v04/ 0 W
&
(
­
class BRange final int borneInferieure;
final int borneSuperieure;
BRange (int bi, int bs) borneInferieure = bi ;
borneSuperieure = bs ;
...
BRange VITESSE = new BRange (0, 100) ;
...
§#¨i©_ª 2k W 2k¤« 3 + <; O<= 587:9 1>;¬{ O<1¥021<CQ02Zdc K 0 + cBC 5 3 +L7:5 021 ? +2@6+
$
SETS
ETAT = haut, milieu, bas ;
§#¨‡©aª 2k W « 0 e1 021BCQ02Zyc K 0 02/ I1 O<Z 04/ 3 0h/ 021
&
class ETAT static final short haut = 0 ;
static final short milieu = 1 ;
static final short bas = 2 ;
static final short borneInferieure = haut ;
static final short borneSuperieure = bas ;
static Brange getBRange() ... /* Fonction pour utiliser les ensembles énumérés */
/* comme arguments de machine */
§#¨‡©aª 2k W « 3 + ;BO<= 587:9 1e;|{ O<1>021<CQ02Zdc K 0 02/ 1IOBZ 04/ 3 0d/ 021 ?+2@6+
…B† )( ‰\Š. SŒ ™ LŽ|™,6/Š SŒ ™SŠLŠ.
* 02C 7 ;021 587 A + 1 5 C¬;02C|= 9 1<C 58+ 1 5 02C}= 9 1<=E320] 5 02CxC 9 1 5u02/ 1IO<Z 04/ 3 02/ Cx; + 1BC|O<10’= KM+ O<C80 CONCRETE CON9 5u5 R #02/ C£; + 1<CVO<10h= KM+ O<CQ0 PROPERTIES 0 5 C 9 1 5u@6+LK O 02/ CV; + 1<CuO<10h= KM+ O<CQ0 VALUES
STANTS tGC 1
g‡A - * 2k W 2®[9 m 021 58+ u5W 9 5 9 ^9 5 +KM+ / 7:587:9 9 58+ 5
5+ 7
9; 1<1 0402/ 0›CJ=; 9 1<1 C 5eKM+„1 02@6C +LK 02= O3e1<=E13200] 02@UCŸ+ 3 = 7 0 38G3802+ C C Wf* 1<+;021 1 9L587:] 9 1 0 / q ; O 047:AG@61 +LK 021 5 1 0 ;0 0T7 C = 5 0 1<C ; + 1B1 C 02KC402t£C =6K`{+ 021C -[b¦+6] - bˆ; 02C 380 7 1O<j 9 13Q0 b
Z +658« 7Mq OK 02C W<*,7 €~+D/ 5 3 + 5;<OB= / Kv58/7:9 1¥025C 5 3809 K`+6587:@ 02ZD+ 025u1 5~K +L7 C 049 / 0sg‡58A + - 5 2k W 2­[9 m 5 z TS5 +L7:5 7:€~/ 5 KM+
« K`02+ C\= ; 9 1<C 045838+ 021 1 5 0yC 020 C 5 02ZD; 0402/ AG1 1 Cu7 0r= ; 1<+ =E1<04C 3(1KM+ 1= KM+ C802CQCf0£= 0 5 1<1 C 9 1>1 ; 02+ CY1<CC K 1 02C 07 1<C 583 + 1B=ECY02C\;02;C\0r; Z + 04=v38F 027 1 10 02W C~= OBCQ02C4t
(
$
#
CONCRETE CONSTANTS
constante
PROPERTIES
constante NAT
VALUES
constante = 10 ;
...
§#¨‡©aª 2k W 2®h« 0 1 0r= 9 1BC 58+ 1 5 0d021
static final int constante = 10 ;
§#¨‡©aª 2k W 2­D« 3 + ;BO<= 587:9 1>;|{ OB10£= 9 1<C 58+ 1 5 0y021 ?+2@6+
0
9 58+ 5 5 5
587
9I9[K¦/ / Kv/ 5
K / //9
+ cBC 5 3 +L17:0r5 m = tBO<1<1 C 7 1 1 5 040 3 @6+LKMK 0V02ABO 1 7 0 0 385 0y1 OB9 1¥1 02@S1 7 ;0 043n9 tBO>O<O<1¥1 c 58+ c 04K 002+1¬O t<O<W 1 0 02ZD021 ;|{ O<1¥021<C802Zyc 0g 021lO<Z 043 0 O
+
+q
« 02*,Z + 3 = KM+ OO<02C8C£0 z CONCRETE CONSTANTS 02O 5 0 5 380J; 04/ AG1 7 0 7 1B; 7 €~04/ 3(02ZpZD021 5 ; + 1<C K 02CpZ + b
=« v* F 7 102@6C +LK + cBC 5 3 +L7:9 5 02C2t</ ; + 1<+ C TK 02C\9 3 + 58+102ZD5 021 5 C+ 9 OJK ; + 1<CKM+ K 02C 7 Z BKM+ 1 58+6587:9 1< C4t @ 5 +L7 7 5 @ 7
02C 02O3(CV; 1<1 0402C O = 1BC 1 02Cd; 1<C 02Cy= O<CQ02C 5 + 7 02O W 021 j 380 1 043 021 3
;<02Cf021 587M5I02/ C¤g‡021<CQ02Zdc K 02C2t<= 9 1<C 58+ 1 5 02C m ; 04/ AG1 7 02Cf; + 1<C~;|{ + O VALUES
3802C\Z =(F 1<02C
…B† ‰\.Š ,Ž^?˜ˆŽ a“•/Š SŒ ™ S Š<6/Š
š 02C @U+ 3 7M+ c K 02C’= 9 3(3802C ^9 1<;021 5 + ] O<1<0~; 04/ AB1 7:587M9 1p;0~; 9 1B1 04/ 02C_Z 9 ; 7 A + c K 02C W _KMK 02C’C 9 1 5 ; 04/ AB1 7 02C 021
g‡A - 2k W m ; + 1<C KM+ = K`+ O<CQ0
5 '/ + KM+ KM+
CONCRETE VARIABLES t R 0402C¤; 1<C
50 D@6+LK O 04/ 02CD; + 1<C KM+ = KM+ O<C80 INITIALISATION
W-0 1<0 @6+ 3 7M+ c K 0= 9 1<=E320] 5 0 C8= 043 +POBCQ5 0 3 + INVARIANT
7:5 0¥021 O<1<0
<
;
O
@6+ 3 7M+ c K 0h; + 1<CVO<1 KM+ 1 -[+6- 0 7 1<j 9 3(Z +6587Mq O<0 .W 0 1<0 @6+ 3 7M+ c K 0d= 9 1<=E320] 5 0 02O 5 0 5 380DO<1.C8= +LKM+L7 380 9 O.O<1
58+ c K 0 + O W
*,+ 9L587:9 @6+ 7M+ K T7 5 + K KM+ -[+6- 7 9 +6587Mq W?* +h5 + 587:9 - 2k W [m 02C 5
K380 M+6587M@ 021 ZD021 5Y1+L;<7 C 0 04/ 0z 3 c 0~0 C 0£; 1<C 02C 1 02C 1<j 3(Z O<02C 3 ;BO<= 1.g‡A
« K 02C\; 7 €~04/ 38021 5 C 0 / Kv02/ ZD021 5 Cu= 9 1<=E043(1 + 1 5uK 02C @U+ 3 7M+ c K 02CfC 9 1 5 0 TS5 3 +L7:5 CY;02C\; 7 €~04/ 38021 5 02C~= KM+ O<C802C4t
« C2{ 7MK 1|{0 T7 C 5 0 q O|{ O<10 7 1<C 58+ 1<=E0>;0 KM+ Z + =vF 7 10Lt K 02C @U+ 3 7M+ c K 02CDC 9 1 57 Z BKv02/ ZD021 5I04/ 02C;0ej + = ) 9 1
C 97 Z 5~7MKM+L7 7 380BK¦+ / O T = 5I9 / 1<C 58+ 91 5 02Ch; + 1<C KM@6+ + = 7MK`+ + C8CQK 0P+e] K {0 T KM+ =E0 <587:9 1Ÿ;<O›Z 9L5 = K¦0 / final z K 02C @6+ 3 7M+ c K 02C
« CC2{ 7M1K 0 TZ7 C 5 0 02BZDK O<02C 1 7 0204O023(CYC = 7 1<ZpC 58+Zp1<0y=E02;C#02;C 0 KM+ 3 Z c + 02=(FCf7 ;1<0r0Lt = K 02C C8CQ@602+ C23 t 7M+ c K 02CaC 9 1 5'7 Z BK¦02/ ZD021 5I04/ 02Cf= 9 ZpZD0V;02C
@6+ 3 7M+ c K 02Cf;|{ 7 1<C 58+ 1<=E0£0 5~K 0rZ 9L5 = Kv0 / static ; 7 C G+ 3 +  54W
+
+ q 9 K 9 58+ 5 9 5 K @6+ 7M+ K 9 5 @ 5 5 / 7
+; 1<C 02K 02Z C 5 3 3 9[O7 C 05 z^R = 02ZpCY;ZD0r0 = 9 02Z CY^= 9 1BC +C 1 5 1 C W 02C£= 1B=E340] 02C4t 02C 3 c 02C~= 1<=E320] 02C 02O 021 0 3(0¤; 04AG1 02C
…B† ‰\.Š J˜ˆ™,LŠ< ,^Ž|“ˆ“•/Š
*,+y5 3 + ;BO<= 587:9 1s;02C 7 1 5 043 @6+LKMK 02C_02C 5'7 ;021 587Mq O<0h+ ] =E0 K`K 0Y;02Ca021<CQ02Zdc K 02C + cBC 5 3 +L7:5 C~z 7MK C2{ +6-[7 3 + ;|{ O<10
7 1<C 58+ 1<=E0r;0 KM+ = KM+ C(CQ0 BRangeW
$
!
%
CONCRETE VARIABLES
monEntier,
monBooleen
INVARIANT
monEntier INT
monBooleen BOOL
INITIALISATION
monEntier := 10 ; monBooleen := TRUE
§#¨i©_ª 2k W « $ 20 C 6@ + 3 7M+ c K 02C'021
int monEntier ;
int monBooleen ;
void INITIALISATION(...) monEntier = 10 ;
monBooleen = true ;
§#¨i©_ª 2k W D« 3 + <; O<= 587:9 1e;02C U@ + 3 7`+ c K 02C'021 ? +2@6+
$
…B† ###
‰\Š.>LŽ #“•Š<Ž|—
# * 02C 58+ c K 0 + O T C 9 1 5 ;02C'= 9 1<C 58+ 1 5 02C~= 9 1B=E340] 5 02C 9 Oe;02C @U+ 3 7M+ c K 02C#= 9 1<=E320] 5 02C~Z +L7 C q O 7o 3 02/ CQ021 5 021 5
C588+O K C ++ ZpZD021 5 5 ;5 0 B+ 3 5877 =4O 5 KM+ 3@U7:+L5lK`02/K C ^9 O/ 3~0 7f5 38/ 0@ 5 3 5+L7:5I02/ KMC'K ;0~j + 5b= ) 9B1+ C 0 / G+ 3 04/ 0 W<* {02K1<CQ02+ Zdc 5 K 0u+L7M;5 02m C 7 1<; 7 =E02C_;|{ O<K 1
043 0dW gi; 04AB1 0 021 O<0 02ZD021 3#OB1p021<CQ02Zdc 0 cBC 3 tIO<1021<CQ02Zdc 0
/021Ic OBZ 0 04/ O 3 0 / 9 02O O K {0 021<38CQ0£02ZdO<1 c K 01 BOOL
* { 7 1 7:587M+LK`7 C +6587:9 1›g KM+@6+LK O +6587:9 1 m ;|{ O<1 58+ c K 0 + O.CQ0hj +L7:5 C8O3 5Q9 O<C K 02C 7 1<; 7 =E02C B+ 3 9 3(;380h=E3 9[7 C8C + 1 5
;K 0 KM@6+ +LKc 9 3(10 / 7 1j 04/ 3 7/ 02O/ 380s+s] KM+ c 9 3v1 0y7 C8O '04/ 3 7 02O38^0 9 ^9 OK 3 K 02@6C +LK 7 1 5 043 @6+L9lKMK 9[02K¦C2/ t B+ 3 9 W 3(;380h;|{ +B+ 3 7M587:9 1 ^9 O3
02C 02O3vC 021IO<Z 043 0402C2t FALSE O C TRUE O3 02C 02O3vC'c 04021<102C
+ 02Z + 3 q O0Dz K 02C 58+ c K 0 + O T 02O @ 021 5 CQ043 @l7 3¤+D] 7 Z BK¦02/ ZD021 5 043~;02C\021<CQ02Zdc K 02C W
### (
3 3
* 58+ K + T 9 58+ 1 5 C£+ ] O<1<0~; 7 ZD021BC 7:9 1¤C 9 1 5 ; 04/ AB1 7 C 021 g‡A - 2k W œ m = 9 ZpZp0 K 02C’= 9 1BC 58+ 1 5 02C
5
Wx* +›5 3 + ;<O<= 587:9 1
›+ ] K { +L7 02;C 0e;c 02Cp0 = OKM+ O<C8= 02C 1BC CONCRETE
CONSTANTS t PROPERTIES 0 VALUES
1+ 0¤5lj +L/ 7:57 G+ C + +K B+ 3 B+ +  5 3(0¤q ;<02C 7MK 58+ c587`K KM0 7 + O T 9[= 7:9 5r1BK C 587 + 1 55 Cy; +L+ KM1<KM9 C K 0D/ +2= @6+ + C 9 5 O ] 7MK K02C +U5r€ 7 1 7:5 587`+LKMG7 C + 0 / +n@ 02=p@6O<+L1 K 3 9 ;<O 7:- 5
=2k 3 W k602ž?C m W 021„= 3 {02C =E0 O|{ O CQ0D; O 0 380 O 0 1 ;0 { 02= 043 3d;02C 02O3vCg‡A
%
#
6œ
CONCRETE CONSTANTS
tableau1, tableau2, tableau3, tableau4
PROPERTIES
tableau1 (0..2)
INT /* Un tableau commençant à l’indice 0 */
tableau2 (-1..1)
NAT /* Un tableau commençant à l’indice -1 */
tableau3 (0..6)
INT /* Tableau initialisé avec une valeur */
tableau4 (2..8)
INT
VALUES
tableau1 = 0 2, 1 4, 2 -6 ;
tableau2 = -1 3, 0 5, 1 7 ;
tableau3 = (0..6)* 6 ;
tableau4 = (2..8)* 9 ;
§#¨‡©aª 2k W œ « š 9 <1 C 58+ 1 5 02C 85 + c K 0 + O T + ] O<10r; 7 Zp021<C 7:9 1s021
static
static
static
static
final
final
final
final
int[]
int[]
int[]
int[]
tableau1
tableau2
tableau3
tableau4
=
=
=
=
2,
3,
new
new
4, -6 ;
5, 7 ;
int [6-(0)+1] ;
int [8-(2)+1] ;
void INITIALISATION() Arrays.fill(tableau3, 6) ;
Arrays.fill(tableau4, 9) ;
§#¨i©_ª 2k W k6žD« 3 + <; O<= 587:9 1N;0r= 9 1<C 58+ 1 5 02CY;<0 5 R 0 85 + c K 0 + O T 021 [ +n@U+
$
CONCRETE VARIABLES
tableau1, tableau2, tableau3, tableau4
INVARIANT
tableau1 (0..2)
INT /* Un tableau commençant en 0 */
tableau2 (-1..1)
NAT /* Un tableau commençant en -1 */
INT /* Tableau initialisé avec une valeur */
tableau3 (0..6)
tableau4 (2..8)
NAT
INITIALISATION
tableau1 := 0 2, 1 4, 2 -6 ;
tableau2 := -1 3, 0 5, 1 7 ;
tableau3 := (0..6)* 6 ;
tableau4 := (2..8)* 9 ;
§#¨i©_ª 2k W k r« !’+ 3 7M+ c K 02C\;0 5 R 0 85 + c K 0 + O T 021
6ž
int[]
int[]
int[]
int[]
tableau1
tableau2
tableau3
tableau4
2,
3,
new
new
=
=
=
=
4, -6 ;
5, 7 ;
int[6-(0)+1] ;
int[8-(2)+1] ;
void INITIALISATION() Arrays.fill(tableau3, 6 ) ;
Arrays.fill(tableau4, 9) ;
§#¨i©_ª 2k W kLk¤« 3 + ;<O<= 587:9 1N;02C @6+ 3 7M+ c K 02Cf;<0 5 R 0 58+ c K 0 + O T 021 [ +n@U+
$
###
(
###
(
3
*3
3 * 0Z 9L5 = Kv0/ C 58+6587 =sCQ043 +L7M5 O 587MKM7 C 0p/ C4{ 7MK 1¬{R +n@U+L7M5dq O|{ O<10 7 1<C 58+ 1<=E0;<0 KM+ Z + =(F 7 1<0 W-0 1 0 / Kv02/ ZD021 5
;0 - 58+ 2c k K W 0 k + nO>m W 020C 5u+ =45 =E02K>C8C / 7Kvc / K 0y; + 5 1<C K 0258C + 9*'K 04/ + 3 +6587M9 1<C+ 5 +B+ 3Y7:O<5 1<0yCQ¡8RS+21 @658+ +UT B0d+ ;0 KM+ j 9 3(ZD0 identifiant(index)
g‡A
1 0 0 02W4ZDwx02+ 1 T ;0 BK c 0 O CQ043 3 ;<O @6+ 021 5 + KM3+ identifiant[index
- O380 2k W kLk 0 5 CQ043 +V5 3 + ;<O 7:-5
_
3
0
2
0
Z
0
;
B
1
C
A
borneInferieure]
tableau4(3) O
021 ?+2@6+ B+ 3 tableau4[3-2] W
3
*,+r5 3 + ;<OB= 587:9 1;02C + c K 0 + O T ; 9 1 5aK 02C 7 1<; 7 =E02C C 9 1 5 ;02C @U+LK 02O<3(C 02/ 1lO<Z 04/ 3 04/ 02C 9 Opc 9l9[Kv04/ 021<102CVg‡A + 7 G+ 587 KM7 W K 5 / +L7 587MKM7
587 9
2k W k m 9 5\5 + 7M5
7 1<; 7 =E02C4Ct B1 + 3 = 39 1<;<C O0 / q OC\02;|1 5{ O<1 19 0dOBZC 5 3 1 + ;<04] O 380 7 C 9 1<C3 K 0 =4O5 R 04] 0 3(0 02/ 1lO<Z 02C04/ 3 0 / 1 9 02Or=E02c C8C9l9[K¦3804/0d021 ;¬{ BO + 3,;C80204C¬3'02;<1<02C8C\02Zy021 c K 0204C|3(C~;|{= 021 Z587 ZD043(0C
g‡A - 2k W k m W
$
&
(
SETS
Enumere = e1, e2, e3, e4 CONCRETE CONSTANTS
tableau1, tableau2
PROPERTIES
tableau1 Enumere
INT
tableau2 BOOL INT
VALUES
tableau1 = e1 0, e2 1, e3 2, e4 3 ;
tableau2 = FALSE 0, TRUE 1 ;
§#¨‡©aª 2k W k « + c K 0 + O T ; 9 1 5~K 02C 7 1<; 7 =E02C#C 9 1 5 ;<02C 02/ I1 OBZ 04/ 3 02/ C 9 OJ;02Cfc 9I9[K¦04/ 021BCf021
&
###
(
$
3
( * 02C 58+ c K 0 + O T 20 O @ 021 5 C8043 l@ 7 J3 + ] 7 Z BK¦20 / ZD021 5 043p;02C¤021<C802Zyc K 02CD= 9 ZpZD0N; + 1BC K { 0 T 02Z BK e0 ;0 KM+
A - O3(0 2k W kL® W
#
static final int[] tableau1 = 0, 1, 2, 3 ;
static final int[] tableau2 = 0, 1 ;
#§ ¨‡©aª 2k W k « 3 + ;BO<= 587:9 1.;02C 85 + c K 0 + O T ; 9 1 5VK 02C 7 1<; 7 =E02CuC 9 1 5 ;<02C 02/ l1 O<Z 04/ 3 02/ C 9 O ;02Cuc 9l9[K¦04/ 021<C
021 ?+2@6+
(
$
MACHINE
M1
...
SETS
VITESSE
ABSTRACT VARIABLES
Vitesse VITESSE
...
END
IMPLEMENTATION
...
VALUES
VITESSE = 0..100 ;
CONCRETE VARIABLES
VitesseImp
INVARIANT
VitesseImp VITESSE
...
END
BOOL
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* 58+ K + T 58+ K + T - 2 k W kL­[m C 9 1 5d5 3 + ;<O 7:5 C£021 [ +n@U+.B+ 3d;<02C 58+ c K 0 + O T +>] BK OBC 7 02O3(C
7; Zp021<02CC 7:9 1<c C£0g‡A -O 2k ;W 0k m W c 0 O g‡A
…B† ##
‰\Š.N‹_Ž^[Ž ŠL[Š.>‘_Š.
Ža˜ˆ™_Š.
* B+ + 5 9 K - 2k W k[m ;02C~Z + =(F 7 102C 02O @ 021 5 0 5 83 0pz
« 02;<C 02Cf023 1<CQZe02Zd0] c3(02K 02C\C j + 3(cGZDC 5 0 3 +LCy7:5 g‡C4A t
« ;<02C 0 / Kv02/ ZD021 5 C 5 R '02/ Cu; + 1<C KM+ = K`+ O<CQ0 CONSTRAINTS t
« ;<02C 7 1<C 58+ 1B=E02C\;0£c 9I9[K¦04/ 021BC4tB;|{021 587 043(C4t
« ;<02C 0 / Kv02/ ZD021 5 C +G+ 3 5 021 + 1 5 + ] O<1e021<C802Zyc K 0 + cBC 5 3 +L7:54W
Lk
CONCRETE CONSTANTS
tableau1, tableau2
PROPERTIES
tableau1 (0..2)
((0..1)
INT)
tableau2 (0..1)*(0..3)
INT
VALUES
tableau1 = 0 0 -12, 1 0 ,
1 0 -13, 1 1 ,
2 0 -14, 1 2 ;
tableau2 = (0..1)*(0..3)*5 ;
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$
static final int[][] tableau1 = -12,0 , -13,1 , -14,2 ;
static final int[][] tableau2 = new int [1-(0)+1][3-(0)+1];
static for (int i = 0 ; i = 1 ; i += 1) Arrays.fill(tableau2[i], 5) ;
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## ##
* 02C B+ 3 + Ze0] 5 3802Cy0 € 02= 587 j‡Cg‡A - 2k W k[m q O 7 C8O 7:@ 021 5rK 0D1 9 Z ;0 K`+ Z + =(F 7 1<0p; + 1<C KM+ = KM+ O<C80 IM @ 5 5
PORTS 02O 021 0  3(0z
$
« 02CYC8= +LKM+L7 3802Crz<;02C\021 587 043(C4t<;<02Cfc 9I9[K¦04/ 021<C2t<;02C @6+LK 02O3(Cf;¬{ O<1¥021<CQ02Zdc K 0 02/ 1lO<Z 04/ 3 0 /
« $ 02C~021<CQ02Zdc K 02C£z 02/ 1lO<Z 04/ 3 02/ C4t 7 1 5 043 @U+LK`K 02CfC8O3 K 02C\021 587 043vC4t 3 02/ ; 04/ AB1 7 Cyg BOOL t NAT 0 5 INTm
š 02Cy; 7 €~04/ 3(021 5 C B+ 3 + Ze0] 5 3802C +B+ 3 +L7 C(CQ021 5 ; + 1<C K { 9 3(;3(0¤; + 1<C KM+ j 9 1<= 587:9 1 INITIALISATION
- 2k W k œ m W š 5Q5 9 587:9 5a+ K¦/ 9[7:5†B+ K 9 5 5 KM+ K`+ q 7 ) 9[7:5aK 7
Kg‡02A C B+ 3 + Ze0] 5 38020 C#0 0~5 jCQ0Y1<= = 9 1 5 1021 025 C 0r;0 K 020 C_043 0u0 / C 043(=4O 5 043 3f; 0u+ = 1<C 1<K C 0Y3(= OB+ = C'02;<OO3#Zd;<O0 KM587 b =7 1<C C858CQ+ 0 1B=E0LO t?C 3(9[0 7:= 5 ; 7 3802O = 5 bˆ02Z¥ZD02 02ZD1 05
; + 1<C K 0r= + C~;|{ OB10 7 1<C 58+ 1<=E0rOB1 7Mq O0 W
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*,+ = KM+ OBCQ0 OPERATIONS g‡A - 2k W ž?m 043(ZD0 5 ;0P; 04/ AB1 7 3 K 02C *9 #04/ 3 +6587:9 1<Ce;|{ OB1"= 9 Z ^9 C + 1 54W
* 02C *9 '04/ 3 +6587M9 1<C£; 04/ AB1 7 C(CQ021 5VKM9+ B+ 3 587 0¤;RS1 + Z 7Mq O0h;<O K`+ 1 -[+6- 0 YW aKMK 02C 02O @ 021 5 j +L7 380 0 / @L9[K O043 K 02C
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IMPLEMENTATION
M1 imp (capacite, MONENSEMBLE)
/* implementation de la machine abstraite M1 */
...
END
IMPLEMENTATION
M2 imp
IMPORTS
i.M1 (10, 2..8), M1(30, 6..9)
...
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class M1 ...
void INITIALISATION (int capacite, BRange MONENSEMBLE) ...
M1(int capacite, BRange MONENSEMBLE) /* initialisation des variables de M1 */
...
INITIALISATION(capacite, MONENSEMBLE) ;
class M2 ...
M1 i, anonymousM1 ;
void INITIALISATION () /* initialisation des variables de M2 */
...
/* Création des instances */
i = new M1(10, new BRange(2,8)) ;
anonymousM1 = new M1(30, new BRange(6,9));
M2() INITIALISATION () ;
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$
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(
@6+ 3 7M+ c K 02Cy;<O = 9 Z ^9 C + 1 54W 0 10 9*'04/ 3 +6587:9 1 02O 5r+2@L9[7 3h;02C B+ 3 + Ze0] 5 3802Ch;|{021 5 3 04/ 00 5 ;0pC 9 3 587 0Pg K 02C
3 02/ C8O KM58+65 C m W
0 10 *9 '04/ 3 +6587M9 1>02C 5 = 9 Z 9 C 04/ 0h;0r;02O TB+ 3 587 02CVz
« K {021Sb 5 0 5 0V; 04/ AG1 7:5_K 0u1 9 Z ;<0 KM+ j 9 1<= 587:9 1 +L7 1<C 7<q O<0 K 02C B+ 3 + Ze0] 5 3802C#j 9 3(ZD0 K Ca;|{021 5 3 04/ 0u0 5 9 O¥;0
9 587 / @ 5 K
« CK 0'3 = 9 30 0 C ;0201 K O{ *9 0 'C404/ t 3 +6587:9 1p02C 5 O<10'C8OcBC 587:5 O 587:9 1¥g 02= 587:9 1 lW ­[m q O 7 = 9 383802C 9 1B;e+£] K { 7 Z BKM+ 1 58+6587:9 1
5Q5 *9 '/ +6587:9 W
oY9 ;<O<0rCe=E1 0 9 O<0 C 7 1 045I3 04/ 3802C8C 1 9 1BC 7 = 7 OB1 7Mq O02ZD021 5 +”] KM+„5 3 + ;<O<= 587M9 1;<0 K {021b 5 0 5 0 ;0 K { 9*#04/ 3 +6587:9 1 W0 10
9*'04/ 3 +6587:9 1 CQ0 5 3 + ;<O 7:5 1 +65 O380 K`K 02ZD021 5B+ 3'O<10uj 9 1B= 587:9 1 ?+2@6+SWw,9 O<3 K 02C B+ 3 + Z¥0] 5 3802C\;|{021 5 3 04/ 0Lt K 02C
; 7 €~04«/ 3802K 1 5 C~5 = + C~C 9 1 5~B+ K 02Cf+ C8O 7:5 @6+ 1 5 Cr5 z 5 + +LKM9 K 7 7 5d5 + 7M5
0?+2@6+R ^09 ;<O / + 3 5YZeK 0] 380s02C 9OB1 R 5 09 ;05~K c 5 CQ0 5~3(KM+¤Cd5=E0 + O bˆ= 587:029 C 3 5;<O 021ŸO<+1<0 @Uu+ 3W 7M+ c K 0
« 7|K 0 5 R C80rC 02;¬; { O<1 1 B0r+ 3 Ze+ Ze02 Zp0] 5 0y380 1 Z 02C 0 5 O<; 1 1 7 1 5 040 3 @UR +LK`K 0£0Lt 02K C 0 5 R 0r3 =(;<F OB9[7= C 7 021>1 ;<[O+2@6+ R 02C 0y5Y;K 00£5c R CQ0r0 ;<0£c + CQ0
0« 21 7}-[K K:9 cB++ 1 + 54W 5 5 @6+LK
0 3 Z¥0] K:9380p02K C 5 O< 1<0 02O+ 3£;|{ O<9[1P7 02i7 1<BC802+ ZycK K 0 587M02KM/ 71lO<+6Z5 04/ 3 0L/ W t K 0 5 R 0p=(F 9[7 C 7 021 [+n@U+ 02C 5 int
9 O short
CQ0 1 0 R 0y;<0£c CQ0r=(F C 3 { O C 02O3
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*,+›5 3 + ;<OB= 587:9 1";|{ OB10 9*'04/ 3 +6587:9 1021 [+2@6+ 9 C80JO<10J; 7 =4O KM5I0J/ Z + ¡ 02O<380Lta; + 1BC K 0N= + C 9 O ] K 02C
3[02+n/ C8@UO + KM58+65 C- C 9 1 5B++ OK 1 9 ZycB5Q389 0N;0N;<02K:O 58T7tGK 9 O WB0 K OBC W *9 + 58; 7:7 9 =4?O +2K:5l@60e/+ 3 02/ C 7 ;5 0N9[; 7:+ 5 1<C 5QK 0e9 j +L7:5q O0 K 0 K`+ B1 + -[+6- 0
10 04] 3(0 C 02Cr380 O3(C£ZhO 02C 10¤j 1<= 1
02O C 380 O3v1043 5 BK g 7 Cr;<0
3809[5QK 9 O587:3 9 m C 9[7:5 O<1 5 R 0£;0uc + CQ0 9 OeO<1 9 c ¡ 0 54W 1B= 9 380£O<10uj 9[7 C2tl1 9 OBC +n@L9 1<C K 0£=vF 9[7 T 02void
1 380 OBC 02O3(C
C O « & 1<CV5Qz5 K B+ + 5
0+ 380 027MCK + 3 5yZeq 0] 3802K Ch;B0p+ C +9 3 587 0p5 ; + 1<C 9[7 K 02C 5 B+ 3 + Ze9 0] 5¡ 38025 Cd;|+ {021 [5 3 +n04/@U0 +JW}5w}9 + O3 q O5y0pK =E0 5Q5 + 0- C 9[K O 587:9 5 1
Z 3(=(F0Lt j O O0 02C 3 Z¥0] 3802C¤C 021 ;02C c 0 Cd= 3
3 1<¡9 C8Zp0 W 02C 3 O<ZD021 C
G+ 3 @U+LK 02O3f0 5~K 02C @6+LK 02O3vC\;0rC 9 3 587 0r10rCQ043 +L7 021 5 ; 9 1B= ¡Q+ Z +L7 C\Z 7 CQ02Cy+V
« + 0 5Q9 O3(1043,O<1 58+ c K 0 + O q O 7 = 9 1 587 021<;3 +L7:5¬K 02C¬; 7:€~04/ 38021 5 02C @6+LK 02O3(C|;<0 380 5Q]9 O3 WO0 3 1 + O 5 380 3 9 c K 02] ZD0
C80 7 ^q9 CQ0~z ^9 O9 3 = ¡ 9 1 5 5 021 7 3 5 7 1<; 7:€~04/ 3802ZpZp021 W 5 š 1|{ 7 Z 7’^7 9 3B5 K`0 7Mq q O0 K?5 R 0L9tUO<@1 5 0 K?7M9 58+ c K 0 + Od5 1 0 02O@ 5 = 9 1K 5 0Eb
1 K`+ 3 O<0p;@ 02C KM*9 c 0 'W Cy? ;K 0 7 R@ 0 Object9 @ 02= 7:9 Z KM+ O<0dKM+O<10p= 1 @ 04K:*9 3(C 1Ÿ@ ;02C K R 5 02C 043(C 02+ O3
= 9 C8CQ0V^0291 0 + 5 0 5 / { 1 04+L3(7 CQ0LtS+ O< 10£= K 1 043(5QC9 1eW ;0 = C8C80u021 0 0 043(C 0 R 0£;0Vc CQ0
« =+ 023(C 385 0238C 0 7 1B;1<38; 0V1 021 02C K 0r1 021 =E9 02ZdC8C c<38380r0 ;<0£3320202]/ C8C O K:0£58+6385 0 C O9 3 C(C 7 c K 02Cr+ ] + O>Z +UT7 ZdOBZ W
« 3 +n@U+L7MK`K 043 +2@ 02=h;02C @6+ 3 7M+ c K 02C -[K:9 c +LK 02C W
*,+ C 9[K O 587M9 1e=(F 9[7 C 7 0£02C 5\KM+ C8O 7:@U+ 1 5 0g‡A - 2k W nm z
« * 0 3802Z 7 043'3 02/ C8O K:58+6#5 02C 5 38021 @L9 R 0¤/ = 9 ZpZD0 K 0£3 02/ C8O K:58+65 1 9 3(Z +LK ;<0 KM+ j 9 1B= 587:9 1 [+n@U+
« * 02C + O 5 3802C_3 02/ C8O K:58+65 C ;0 @l7 021<1021 5 ;02C B+ 3 + Ze0] 5 3(02C_;|{021 5 3 04/ 0Y;0 KM+ j 9 1B= 587:9 1 [+n@U+ z K 02C 5 R 02C
C(= +LKM+L7 3802C 0 / 58+ 1 5 5 021<= + C(O + Kv02/ C£; + 1<C K 02O<3 5 R m W 0D021 @ 0 K:9* 0>g INT ; + 1<C BInteger t NAT ; + 1<C
* { +2BInteger
@6+ 1 58+6- 0y;<0V0 =E0 5QBOOL
5 0rC 9[K O ; 587:9 1<1C 02BBoolean
5 5Q5 5 + + K + @l7 9
5
9= 1 5 3 +L7 1 5 CD= 9 ZpZD0>O<10¥= + 3 5 0 "+ ] O<C =E0 ;0 g K 0e04C 3v'ZD02/ = 0 7 AB02380VO3 ;|{ 0 38; 0£9[07M5 C4= { + =EcG0rC 5 ; 021 1<7 3¤C ; 0V+ 1<= C¤Cf=E;|0>{02= 1 + ;383 0e1<;|1{ 02O ZD587MKM027 1 CQ04C3
BK OBC_;|{ O<1s3 02/ C8O K:58+6b
5 B+ 3 *9 #04/ 3 +6587:9 1 m 0 5 ;0u3802C 5 043 3 9 =(F<0V;¬{ O<10 3 9L- 3 + ZpZ +6587M9 1s= K`+ C8C 7Mq O0~021 ?+2@6+SW
$V+ 1<C K 0\= + C 9 O ] K {021 @l7 3 9 1<102ZD021 5 = 7 c K 0 ^9 C8C402] ;0Y;02Cx3802C8C 9 O<3(=E02C_C8O C + 1 5 02C4t K 02C ^9 C8C 7 c 7MKM7:5l02/ C 96€ 043 5 02C
043(ZD0 5Q5 021 5 ;|{ O 587`KM7 CQ043 K {021<CQ02Zdc K 0r;BO KM+ 1 -[+6- 0 uW
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$V+ 1<C K {0 T 02Z BK 0u;0 KM+ A - O380 2k W ž t K { 9*'04/ 3 +6587:9 1 foo 380 5Q9 O3(1<0 5 3 9[7 Ca3 02/ C(O K:58+65 CfZ +L7 C'10 38021B;
B+ C};0 B+ 3 + Ze0 ] 5 3802C WL*,+\5 3 + ;<OB= 587:9 1d;<0#=E0 5Q5 0 9*'04/ 3 +6587:9 1eg‡A - 2k W nm j +L7:5,+G+ 3 +  5 380#O<1dC802O K 3 02/ C8O K:58+65 t
=E02/ = 7 KM0258C+65V5 7 Z ^9 C /0 / 9 B+ 3 K 02T Cu= 5\9 1<5 =E9[0 7<5 C ?+2K @6*9 + 'tG/ 0 +65 58;7:9 02O >T uGW + 3 + Z¥0] 5 3802C q O 7 3 02/ C8O K:5 021 5 ;0 KM+p5 3 + ;BO<= 587:9 1P;02C
3 02C8O C~1lO<Z 043 ;02O 0 3 C\;0 { 043 1
'&
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%
IMPLEMENTATION
M imp
CONCRETE VARIABLES
var1,
var2
INVARIANT
var1 INT
var2 INT
INITIALISATION
var1 := 10 ;
var2 := 20
OPERATIONS
bb test(v) BEGIN bb := BOOL(var1 = v ) END ;
augmenter BEGIN var2 := var2 + 1 END ;
v1,v2,bb foo BEGIN v1 := var1 ; v2 := var2 ; bb := BOOL(var1 = var2) END
END
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j 9 3(ZD0Dz
r
op(p) PRE P THEN S END
9 O ] 02C 5 O<10 KM7 C 5 0y;0 G+ 3 + Z¥0] 5 3802CYj 9 3(Zp0 K C~;|{021 5 3 04/ 0h0 5 r 02C 5 O<10 KM7 C 5 0r;0 B+ 3 + Z¥0] 5 3802Cuj 9 3(Zp0 K C
;0uC K 9 *9 3 #587 / 0 W?+6* 587:0 9 3 02/ ; 7 = +6/ 5 7:5 02C 5#KM+y 3 02/ = 9 1<; 9 7:587:9 1p0 5 5¤SKM+JKM+ KM7 C8O5 cBC 587M5 O B587:+ 9 1p+ = 9 3 5 C_;0 € K { 9*#58047 / 3 +6587:9 1 W29 0 581 7 + 0 K
;KM7 0 5 { 043 @U+ 7`+ 1 K opm C2{ 025 =E3 KM+ez K`v7 5 op(e)G+ O ]+ v 025 C € C 5807 ;<02C 5 3 / Z¥0] 3802C¤KM0 7 5 02= j‡C¤T'; 0sC 7:3 9 0.m giW¬O<*,1+0
/C 02C Z 0¤+ 1;<580 7Mq O0h3 ;<c 0 02K {C +0 0 K e ;|{ 9*#04C/ 3 +60p587:;9 021 C ; + 1B3 C Z¥K 0 0] 3802b aCr9l0 9 02= UjiCy02C;|5 {02; 1 04/ AB3 041 0P7 0 giOBB1+ 03 KM+ C 320D0] -[;|K 0D{0 ;0h38C802OC(CcBC 581<7:5 C O 587:9 1
C8O 7M@U+ 1 5 0Dz
g nm z v op(e) 32t z @ t0 PRE P THEN S END
)&
L­
class M ...
boolean test(int v) return (var1 == v) ;
void augmenter() var2 := var2 + 1 ;
int foo(BInteger v2, BBoolean bb) v1 = var1 ;
v2.setValue (var2) ;
bb.setValue (var1==var2) ;
return (v1) ;
§#¨i©_ª 2k W £« 3 + ;<O<= 587:9 1>;BO B+ C8C +6- 0y;<0 B+ 3 + Ze0] 5 3802CY021 [+n@U+
* 02CN3802C 5 3 7 = 587:9 1<CNC8O3 K { + 0 K C 9 1 5Pq O<0 v 02C 5 O<10 K`7 C 5 0Ÿ;0 @U+ 3 7`+ c K 02C>C 7 Z BK 02CŸg‡0 5 1 9 1H;<02C
3 04/ j 04/ 5 3(021<=E02CY9 + ] ;+LK`027MC + 0 / 7K¦02/ Zp- m 02W1 *,5 +C ;0 / 58+ c 7:58K 7M0 9 + O¬t Bnm + 3}0 5~T 02+ Z +B<K 0 5lm / 0 5}+Dq KMO+0' =E02C @6@ + 3 7M+ c +LK 027 C 1|{ +5lBB+ +3 +L7 9*C8C8#02/ 1 5&+658B7:+9 Cx; + K`1<K C
C 1 ; 04AB1 1.g 02C ; 040e] 3802O 0Lt<Z Cf1|{02C C 043 1<10 0
P 0 S gi1 1
^9 O3 KM+y5 3 + ;<O<= 587:9 1 W 1s0 € 0 5 tO<1 + 0 K ;|{ 9*'04/ 3 +6587:9 1¥021 ; 9[7M5†^9 O @L9[7 3'CQ0 5 3 + ;<O 7 380 B+ 3fO<1 + 0 K
+ OŸ= 9 ;0;<0p=E0 5Q5 0 *9 '04/ 3 +6587M9 1 W,*,+ 340] -[K 0= 7 bˆ;02C8C8O<C£1¬{02C 5 ; 7 3(02= 5 02ZD021 5h+BKM7 = + c K 0 q O0C8O3 K 02CyC8Ocb
CK 587:5 O+ 587:9 1<C~7;BONB1 K¦7:/ @ 0 + OJ58+6Z 587:+ 9 =(F 7 1q0Lt 7q O 7 C #@ 02/ = 7 5 A<0h9 O<1N5 =vF 7 + 1 - 02ZD02/ q 1 5 ;|{ 0 / 58+65 ;<5 0dZ 9 + 1 7 04] 3(0 7M5I+6/ 5Q9 +6Z 587:97Mq O0 W7MK $V++ 1<C5
0£587M= K`7 C\;02KM+ C Z -[K 02ZD027:@61 + 5 1<C4t O 02O 021 = 1 021 3Y;02CfC 0 O<021<=E02ZD021 C O>;02C 043 1<C4t j O
O CQ043 320] 0yC(O 1 0Dz
g k[m z v op(e) PRE z 0 P THEN VAR p,r IN p := e ;S ;v :=r END END
q O 7 = 9 3(3802C ^9 1<; 0 'T BKM7 = 7M5 02ZD021 5h+ O B+ C(C +6- 0 G+ 3h= *9 B7 0 W,* 02Cd3802C 5 3 7 = 587:9 1<CrC(O3 KM+ C 5 3(O<= 5 O<3 +6587:9 1
;02Cf02C B+ =E02C~;0 @6+ 3 7M+ c K 02C'021 j 9 1 5~q O0 KM+ 320] -[K 0sg k[m 02C 5e0 / q O 7M@U+LK 021 5 0s+D] KM+ j 9 3(ZD0z
g m z v op(e) PRE P THEN VAR p IN p := e ; S END END
*,+ +L7 9
587 K`K K / q 7M@U+LK
5 k[m 5 m 5pq K @6+ 7M+ K K`+ KM7 5
+1|{ G+ 3 3 +L7 C8C CQ021”1 b5 02BC8C8+ 02Cf1 ; + 0 1<C 0eS;W0 7MK {C40 {021<O C(O 7:5'02q 1BO=E00eK 02021 C G380„+ 3 g+ Z¥0] 0 5 3802gCf3 02/ C802O C K:58+65 OC 0 0202O C @ 021 3 5 0 c 5 3(020rCDC8O;<cB0 C 587M5 O 02/C C4t0 =Ev0
q O 7 = 9 383802C ^9 1<;N+ ] O<1 B+ C8C +6- 0 B+ 3’3 04/ j 04/ 38021<=E0 W š {02C 5 =E0 5Q5 0Y340] -[K 0 q O 7 02C 5 O 587MKM7 C 04/ 0 B+ 3 K 02C 5 3 + ;<O<= 5 02O<3(C
;0 K { ?'5 0 KM7 043 uW 0 1.;043v1 7 043~3 02/ C(O K:58+65 02C 5£q O<0 K 02C B+ 3 + Z¥0] 5 3802C£;|{021 5 3 04/ 0 02O @ 021 5V+ O<C8C 7 0 5 380 B+ C(C 02/ C
B+ 3\3 04/ j 04/ 38021<=E0¤C 9 O<C K`+ = 9 1<; 7:587:9 1.z
« K`+KM7 C 5 0y;02C B+ 3 + Ze0] 5 3802Cu0 € 02= 587 j‡CV;|{021 5 3 04/ 0D10d= 9 1 587 021 5u+ OB=4O<10 9 =4=4O383(021<=E0d;02C B+ 3 + Ze0] 5 3(02C
0 € 02= 587 jiC\3 02/ C8O K:58+65 C4tB1 7|+ OB=4O<10 9 =4=4O3838021<=E0d;02C @6+ 3 7M+ c K 02C'3 04/ j 04/ 38021<= 04/ 02C£; + 1<C K { *9 #04/ 3 +6587:9 1 W
l9 OBC}=E02C_F?R ^9L5 F^02] CQ02C4t 7MfK 02O 5 0  5 380\Z 9 1 5 3 0 / q O0 KM+ 340 ] -[K 0rg nm 02C 5’+ BK`7 = + c K 0_C(O3 K 02CxC8OcBC 587M5 O 587:9 1<C
+ ;<Z 7 CQ02C¬; + 1<C K 02C 7 Z BK¦02/ ZD021 58+6587:9 1<C W * l+ 3(02O @ 0’02C 5 ; 0 / 58+L7MKMK¦04/ 0'; + 1<C 6ž •W š 0 5Q5 0 *9 <587 Z 7 C +6587:9 1 043(ZD0 5
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; 1<?C r0 5 = C~;0 5 7:35£K Z¥0] +3802Cu;K 0 R 9*#0 / +6c587:9 0 O K`3f+ 0 - 02Z 0L2tBk W= k 3 9 = 5r7 0r5 02C 7:5 0 W’?Y04@60 + 5rKM+ / +6587M9
O 3802ZD021 ; 02C 0 Cu;¬{ 043 1Ÿ;<0 A O380 C 1 1 043(; C 1 =E3 0 1
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op (x, x)
#§ ¨‡©aª 2k W k¤« ? 0 K Cf;|{ 9*'04/ 3 +6587M9 1 7 1 5 043(; 7:5 C
;02C 9 O 587MK C#3 0 / +LKM7 C 02/ CY; + 1<C K 0r= + ;3(0r;<O 3 9U¡ 0 5 & t K 0 B+ C8C +6- 0y;<02C 58+ c K 0 + O T g‡A - 2k W m CQ0rj +L7 C +L7:5
B+ 3'= 9*B7 0u0 5#7`Kf0 / 58+L7:5f7 Z ^9 C8C 7 c K 0f;0VZ 9 ; 7 A<043#O<1 58+ c K 0 + O G+ C8C 0£/ = 9 ZpZD0 B+ 3 + Ze0 ] 5 380£;|{021 5 3 04/ 0 W<w}9 O3
0 5 380 9 GK O<C¤3 0K / +LKM7 CB5 + 0 + @S7 CN5+ ] @S7 Cp58+ ;02K C + + T BKM7 = +69 587M9 51<BC¤+ 021 / 5 04B3(ZD+ 0J;/ 0N/ ; 0 / @ 0 K:9* - 02ZD2k 02W 1 5 m 0 5 ;|{ 02/ = 9 1 9 Z 58+ 7 0P5¤;K 02C
3802-[C8C K O3(7 =E02C 7Mq 02C / ¡ 3 Ze5 0] +238@602+ C 54W c 0 O CQ043 1 C8C 02C 3h3 04j 0438021B=E0Ÿg‡A
021„3802C 02= 1 02C
320] 02C 1<; O 0402C O<C 0 1
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MACHINE
...
OPERATIONS
T6, T5 op(T4) PRE T4 : 1..4
BOOL
THEN
/* T4 n’est pas accessible en écriture
T4(2) := TRUE */
T5 := T4 T6 := T4
END
END
IMPLEMENTATION
...
OPERATIONS
T6, T5 op(T4) BEGIN
/* T4 n’est pas accessible en écriture
T4(2) := TRUE */
T5 := T4 ;
T6 := T4
END
END
§#¨i©_ª 2k W « 0 >1 0 T 02Z GK r0 ;0 B+ (C C +6- 0y;|{ O<1 G+ 3 + Z¥0] 5 380 85 + c K 0 + O B+ \3 3 04/ j 04/ 38021<=E0
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class ... ...
boolean[] op( boolean[] T4, boolean[] T5) boolean[] T6 = new boolean[T4.length];
/* T4 n’est pas accessible écriture
T4[2-(1)] = true ;
*/
System.arraycopy (T4, 0, T5, 0 ) ;
System.arraycopy (T4, 0, T6, 0) ;
return (T6) ;
§#¨‡©aª 2k W « 3 + ;<OB= 587:9 1¥021 ?+2@6+ |; { O<1 B+ 3 + Z¥0] 5 380 85 + c K 0 + O B+ 8C C 0 / B+ 3f3 04/ j 04/ 38021B=E0
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0 10JC(OcBC 587:5 O 587:9 1 02C 5 O<10J1 9L58+6587:9 1Z +65 F 02/ Z +6587Mq O0 043(Zp0 5Q58+ 1 5 ;0JZ 9 ; 0 / KM7 CQ043 KM+Ÿ5 3 + 1<CQj 9 3Qb
Z +6587:589 7:1"5 ;|587:{ 9 0 / 58+6W 5 ? C G+ 37:@ O<1+ 0 5 3 + 1B CQžj 9 3vK Z +6587:9 1–587:;5 0 58 7:9 3 02/ ; 7 = 9+65 C 5yW +$u+ 1BCK¦/ K 02C 7 9*'5 04/ 3 +658587M7M9 9 1<C4t K 0>+ = 9 3 KM+sCD5 02C 5 7 O<19 0
C8K:9LO-[cG7 C O š K 1 + O›^1 9 0 O 9;BO 5 K 02CrKM7 C8OcG+2C @ O 1BKMCY+ C -[1 +6- 0 04029LC - 1<+ C 3(O<+6= 587:9 1<W|CV* ; 1<C 58047M5 3(Z 587:91 b
0D;0 0 3 R O3VZ 1 38043 0 021 02=O<1 1 0p;G0 + 5837 3 Zpq Z KMK 1 029 C£C8OcB9 C O 1<5C
3 02/ CQ021 5I04/ 02C\= 7 bˆ;02C(C 9 O<C_= 9 3(3802C ^9 1<;021 5a+ O T C8O<cBC 587:5 O 587M9 1<CY+
] 3 3a;<02C O<0 02C’;<O= ;<0uC O<3(=E0 02O
50 380 -}02/ 1 04/ 3 0 / W
1<C 5 3(OB= 587:9 5 1 58z¬7:9 z $ @S7 5 _-[+LK
1<C 3v5 O<= 587:1 9 ?0 021 K # +6587:9
1<C 3vO<= 1 0 043 1
1<C 55 3vO<= 58587:7:99 1 ' 0 K:9q O021<=E0
1<C 5 3vO<= 587:9 1 = 587:5
1<C 5 3vO<= 587:9 1 !’;+ 021 7`+ K0 *¬9 +LK
1<C 3vO<= 1 3 c 0 = 0
1<C 55 3vO<= 58587:7:99 1 šš 9+ 1B; 7:587:9 1<1<0 KMK 0
1<C 5 3vO<= 587:9 1 + C 5
1<C 5 3vO<= 587:9 1 ? 1 587MO9 0
1<C 3vO<= 1 C8CQ043 1
* 0 58+ c K 0 + O 2k W ® 380 3 02/ CQ021 5 0 KM+s5 3 + ;<O<= 587M9 1P;<02CuC 5 3(O<= 5 O3802CV;0¤= 9 1 5 3U9[ K 0p;0 ž 021 ?+2@6+SW $V+ 1<C
8
5
=E0 / + 7 c K+60 5 + O S, S1, S2,'T S3 0 7:95 Si C 9 1 5 ;7 02CfC8O58cG7 C + 587:5 5 O 587:9 @61B+ C4t 7MT1+ K 0 5 5Ti ;02C @67 +LK 025 O3vC4t587:C,9 P,q P1,$7 P2 0 5 5 V ; 02ž C
3 02; = C4t O<10’0 3802C8C 1|t O<1 ;<021 A 1 ;<0 3 c 0 0 OB10 1BC 3(O<= 1 O 043vZD0 021
;0 5 5 R 043 K 02EC @6+ 37 7M+ c 5 K 02C 58KM97:9 = +LK 025~C XXW KM+ 9 0 585 7:9 5 ; +02/ 1 9L5 9 021 5 +63858027:C 9 02=I587M@ 02ZD9 021 5 5\K`+ 58j 7:9 9 1<= 587:9 1e@ ;¬{0 Tl[5 +n3 @U+ +S= 58W 7M9 1
;<O R 0y;|{ OB10 1<C 3(OB= 1¥0 j 1<= 1>;<0 3 1<CQj 3(Z 1N;02C\= 1<C 3vO<= 1<C 043(C
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ª 2k W ¤® « 3 + ;BO<= 587:9 1>;02C~C 5 3(O<= 5 O3(02C\;0r= 9 1 5 3U9[ K 0 ž 021 [+2@6+
…B† † ‰\Ž U—! [˜ [— [˜ˆŒ ™
*,+ C(OcBC 587:5 O 587:9 1›z g m 3(02Z BKM+ =E0¤O<10 @U+ 3 7M+ c K 0 B+ 3rO<10d0 fT 3(02C8C 7:9 1 W _K`K 0d= 9 383(02Cb
^9 1<;e; 9 1<=+ ] O<10 +U€ 02= 58+6587M9 1 W* 02C\; 7 €~04/ 38021 5 02C ^9 C8C 7 c 7MKM7:5l02/ C'C 9 1 5~K 02C\C8O 7:@U+ 1 5 02CVz
« +U€ 02= 58+6587:9 1P;|{ O<10£; 9 1B1 04/ 0y021 587 04] 380Lt
« +U€ 02= 58+6587:9 1P;|{ O<10£; 9 1B1 04/ 0yc 9I9[K¦04/ 021B10+p] K { +L7 ;<0V;¬{ O<10£= 9 1<; 7:587M9 1|t
« +U€ 02= 58+6587:9 1P;|{ O<1 0 / K¦02/ ZD021 5 ;<0 58+ c K 0 + O|t
« +U€ 02= 58+6587:9 1P;|{ O<10£; 9 1B1 04/ 0 58+ c K 0 + O W
1<C 5 3(OB!’= 58+ 7:9 7M1 + $ K 0 @l7 021 5 _-[+LK 'T z¬z 7:9 +LKM+L7
3 + cK + 0 z 'T 3802C8C 7:19 = + K 38+0
c0O z
3802C(C 1 c 0 O
$V+ K + 7 BK KM+d5 + 587:9
[+n@U+ @S7 5 +
BK`+
K B+ 2W $V+ K +
58+ c K 0 + O 1<T C + ] 0VO<= 1C'0sC ; Z 7 ZD020L1Bt C 7:9 1 3 9 1„;<O<O = 587MKM7 C81043 02+e1 KM+ j 9 1<38= 0 587:9 021 1 ;0] =3802*9 ZG7 0Pg +=E043833 + RS0d= z*9 R m 3;0 KM+ = KM1<+ C8C CQ0 0V = RSC'C 5 ;0202Z C
g‡A - K`2+k W ® 0 5 7 2k W ­[7 m W $V+ 7:1<9 C K 0y5Y= KM+ + CY;<9 02C 58587:+ 9 c K 0 + O T 9*+G] 7 ;<02O T + ; 7 ZD587M02KM7 1<C / 7:9 1<^C49 tSO<1KM0+ 7:5l04/ 3 +6TS587 7M9 1JCQ043 7 + 0 € 027:= 9 5 O W04/ 0
C8O3 3802Z 04] 380d; ZD021<C 10 j 1B= 1>;0y= 0rCQ043 O C 040 O3 ;02O 02] ZD0d; ZD021<C 1
…B† )( x— ! [˜ [— [˜•Œ,™¢Ž^‹_‹fŠB“£
‘ "MŒ¬‹ Š< [Ž[˜•Œ ™
aKMK 0 043vZD0 5 ;|{ + BKM7`q O043 KM+ C8OcBC 587:5 O 587:9 1;|{ O<10 *9 '04/ 3 +6587:9 1¢021 3802Z BKM+ = ) + 1 5eK 02C B+ 3 + Ze0] 5 3(02C
j 9 T'3( ZD0 K C 7M9G+ 3Y;02+LC KM+LB7 + 3 + Ze0] 5 385802+ CY0 K € + 02= Ty587 j‡BCh+ g‡A - / 2Bk + W m / W|/ * 02C B+ 3 5 + Ze0] 5 3802Cu+ ;|{021 KM5 7:3 5Q045l/ 0L/ t+LC2K { 7MK W R¥* 021 B+ + tC + 9 1 5 5 ;02C
0 389 02C858C 7 1</ @C¬C8= 5 K 3802C49t2;502C c @U0 + O 7`+ K C8C 02C +LKM+L7 3 3 04j 9 0438021<=E0#580+ ;K 02+ C,=(W F `102C 043 02C 02C 3 Ze0] 3(02C
;0rC 3 0 0 021 O<0 C~C 1 ;02C 3 c 02CfC8= 3802C ONOB1 c 0 O
1<C 5 3vO<= 587:9!’1 + ?7M+ K0 K # 043 +6587:+ 9 1 K + z|z
+ c K0 + O
3 c0
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MACHINE
Tableaux
CONCRETE VARIABLES
T1, T2, T3
INVARIANT
T1 -5..5
INT
T2 -5..5
INT
T3 (1..2)*(1..2)
BOOL
INITIALISATION
INT T1 : -5..5
T2 : -5..5
INT T3 : (1..2)*(1..2)
BOOL
...
IMPLEMENTATION
Tableaux imp
REFINES
Tableaux
INITIALISATION
T1 := (-5..5)*1 ;
T2 := T1 ;
T3 := 1 1 TRUE,1 2
...
OPERATIONS
...
var := T3(1 2) ;
T1(2) := T2(5) ;
...
END
FALSE,2
1
FALSE,2
2
TRUE §#¨i©_ª 2k W ®¤« OcGC 587:5 O 587:9 1 '^9 O3 K 02C 58+ c K 0 + O T
# 043 +6587:9 1 ou+ ZD0
+ c K 0 + O l5 3 7 1 {`gX{ ’fT 'T 3802C8C 7:9 7:91 = +L+LKM+LK`+L7 387 0
+ c K 0 + O l5 3 7 1 - { m {
{:t:{ 3802C(C 1 = 380
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5 3 + ;<O 1 7:5B+ 3 t^=EW 0 _5 0pO<; C(O7:9 cB1<C C_C8OO 3 q O1>0 =KMq O38380202C_C 0 T 021BZ ;”G] K 02O<C 1 K 02Ca; 7 €~0 04/ 38;020¤1 5 Z 02C 0 ^F 9 C8;C 7 0 c 7MKM7:5l0 02/ C\g‡A - 3v2C k W O¬{ 0 0 5 2C k W 0h[m02C W
* { + 0 K ;0Nj 9 1<= 587:9 1C8Oc 7M5pK 02CZe02 ZD02C¥= 9 1 5 3 +L7 1 5 02C q O0 KM+ ; 04/ AB1 7:587:9 1–;0JZ 0 / 5 F 9 ;<02C.gi= K`+ O<CQ0
K 5Q9 K:587tGK 02C W?wx+ 3~= 9 1<C 0 / q O021 5 t 7MK 02C 5 1 02/ =E02C8C +L7 3(0r;0
m q 7 9
OPERATIONS tI021e=E0 O = 1<=E043(1<0 02C'380 O3(CfZhO
j +L/ 7 380¥KM58O<+615 0= 9 1 +@ 043(C 7MK¦9 / 1 ;7`K 02Cd= + K`+ C8CQ02C¤021 @ / 0 9K:9*^ 9 025C @ T_04/ 3vC 9 K 02C 5 /R @S 7:5 02C K 3 7 Z 7:587 ji/ C£KM;+ 0 +6[58+n7M9 @U+SW? 7 O”1 7:@ @U0 + + O„7M+ ;02K C
3 025C8O 587MKM7 CV/ 021<B= K C87 O 02C2t 9[7 R 9 OB1J1lO<Z / 043 K:58+65 C *9 A #/0 +6587:9 O<3 W 0 043 02Cu3802; 02= 3 1<CrC O<10 3 c 0
02C O C 040 O<C 02O3vC_j Cf= ZpZD0y3 02C8O ;¬{ 043 1
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class Tableaux int[] T1 = new int[5-(-5)+1];
int[] T2 = new int[5-(-5)+1];
boolean[][] T3 = true, false , false, true ;
void INITIALISATION() Arrays.fill (T1, 1) ;
System.arraycopy (T1, 0, T2, 0) ;
...
...
var = T3[1-(1)][2-(1)];
T1[2-(-5)] = T2[5-(-5)] ;
...
§#¨‡©aª 2k W ­h« 3 + ;BO<= 587:9 1¥021 ?+2@6+ ;0 KM+ C(OcBC 587:5 O 587:9 1 '9 O<3 K 02C 58+ c K 0 + O T
…B†
x—! [˜ [— [˜•Œ,™ *,+ C8OcGC 587:5 O 587:9 1 = 9 3(3802C ^9 1<;+J] K { +BK`7 = +6587:9 1 021 C 0 / q O021B=E0¥;<0;02O T C8O<cBC 587:5 OSb
587:9 1<C W 3 + ;<O 7 380 K 0DC 0 / q O021B=E02ZD021 5 02C 5 C 7 Z GK 0z 7MK C8O 5 ;<0 5 3 + ;<O 7 380 K 02C£;02O T C8OcGC 587:5 O 587:9 1BC q O|{ 7MK
= 9 1 587 021 5 t5 ; + 1<58C 7:9 K { 9 3v;q 380r;|{ + G+ 3 7:587:9 1 g‡A - 5 2k 58W L7:9 œ 0 5 2k W ž?m W 5 587M9
1<C 3vO<= 1 0 O021<=E0 z|z 1BC 3(O<= 1 { :{ 1<C 3(O<= 1
…B† x— ! [˜ [— [˜•Œ,™ aKMK 0JC8043 5 +„] - 3 9 O 043 BK OBC 7 02O3(CC8OcBC 587M5 O 587:9 1<C q O 7 CQ043 9 1 5 0 € 02= 5 O 04/ 02CN021C 0 / q O021<=E0 W aKMK 0PCQ043 5
B
;+ 0 K:+ 5838+602581 7:@ 5 F|02] CQ+ 0Lt|021 K q O+ 0 KMq O0dC 9 3 5 0 W _587:KM5 K 0d58027:9 C 5 1 02/ =E7M02q C8C +LW7 380p; + 1<C K 0¤= + Cr;0 BK OBC 7 02O3(CYC8OcBC 587:5 O 587:9 1<C~0 5
j =4O 5 0y; 587:1<9 C '0rK:9 = CY;|{ O<10£C8OcGC O 1¥5 O<1 587MO<9 0
1<+ C 3vK O<= 7 1 7t = 5yz|zK BEGIN K 1<C KM9 3(O<= 1 / { 5I:{ / B1<+ C 5 3vO<= 587:9 9 1 BK END
[
2
+
6
@
+ 9[KM+
m
9C 1 1 5£9 c KM7:t -[+60 5Q9[7 338021<C = q O<0d0 02^C 9 O3£0pO<Ze1P02 ZDc KM0J9 =hz ;|02{ CV+ O c Z =49[C47 t|1B3(C£0 ;023 02O CQTJ021 7 1B02C C 5 3(O<= 583d7:9 O<1B1 C W = š O 0 020p1B;¬; {+ 1 =45 = t|; + ;1B02CuCOBg 1 C 9 t}O<1= 0 7
;0r= KM+ 3 5I0L/ t 7MK 02C 5 3 04/ j 04/ 3 + c K 0y;0rZD0 5Q5 380hCQRSC 5I02/ Z +6587Mq O02ZD021 5YK 02C + =4= 9[KM+ ;<02C W
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x—! [˜ [— [˜•Œ,™
š 5Q5 587M5 587:9 +L7:5 7 W aKMK 5 587MK`7 / B+ T GK 9 7 7Mq q 58+L7
+c<3 1B=(0 F02CD0;|C8O{ OBcB1C 0 7 O1<C 5 3(1POB= 1580¤7:9 1j IF3 9 02O 1 CASE0h02C 10eO Z 9 ;C 047 A<0 021 53VB0 + C 02Z K 02C 0@U+ 3 7`+ O<c 3 K 02C1<W ; _K`OK 0043 02C 5DO0D5 3 =E+ 04;<3 O 7:5 0e102C C 7
1 02/ =E02C8C +L7 385 0r021 587:¡89 +2@6'+ K:B9 + 3YO<1¥c K:9 = @l7 ;0 W
1<C 3vO<= 1 = z|z Skip
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IMPLEMENTATION
M imp
CONCRETE VARIABLES
res1,res2, res3
IMPORTS
i.M2
INVARIANT
res1 INT
res2 0..4
INT
res3 BOOL
INITIALISATION
res1 = 10 ;
res2 = (0..4)* -1 ;
res3 = TRUE
OPERATIONS
foo BEGIN
res1 i.op1(res2) ;
i.op2 (1, TRUE) ;
res2,res3 i.op3
END
...
§#¨i©_ª 2k W « * + C(OcBC 587:5 O 587:9 1 + 0 K ;|{ 9*#04/ 3 +6587:9 1N021 …B†
x—! [˜ [— [˜•Œ,™ $V+ 1<C K 02C 9*#04/ 3 +6587:9 1<C2t 7MK 02C c5 B+ 3(j 9[7 C 1 02/ =E02C(C +L7 380~;¬{ O 587MKM7 C8043x;<02C @U+ 3 7M+ c K 02C q O 7 10\C 9 1 5B+ C’; 04/ AG1 7 02C
+ O1 7:@ 0 + O;0 KM+ = KM+ O<CQ0 CONCRETE VARIABLES W * + C8OcBC 587M5 O 587:9 1 @6+ 3 7M+ c K 0 K:9 = +LK 0 043(ZD0 5 t
=š 9 ZpZD@6+ 0p7MC+ 9 1 K 1 9 Z K9 { 7 1<5 ; B7Mq+ O<0L5 tB';0D/ ; 04/K:AB9 1 7 3£O<1<K 0 KM7 C 5 0h/ KM;<+ 02C +658@U7:+9 3 7`+ c +LK 027 C K:K:99 = +LK 02CK +¥] K { 9* #04/ 3 +6587 7:9 1"+Ug‡A€ - 582+6k 58W 7:9 nm WW
02C 3 c 02Cx10\C 1 C R 0402C 3(C ;0 02O<3’; 02= 3 1pZ C 3(C ;<0 02O3 3802Z 04] 3(0 02= 1
1<C 5 3(OB= 587:9 1 ' K:9 = 58z7 z
;5 021 587:9 A<043 {:t:{ ;021 587 A<043
VAR
1 END
IN 1<C 3(O<=
* +s5 3 + ;BO<= 587:9 1J021 ?+2@6+ CQ0hj +L7:#5 B+ 3£O<1Jc K:9 =r0 5£K 02C @6+ 3 7M+ c K 02C K:9 = +LK 02CVC 9 1 5 ; 02/ = K`+ 3 04/ 02C + O ; 04/ cBO 5 ;0
=E0rc KM9 =pg‡A - 2k W k[m W
…B†
x— ! [˜ [— [˜•Œ,™ *,+ C(OcBC 587:5 O 587:9 1= 9 1B; 7:587:9 1<1<0 KMK 0 IF ; 04/ AB1 7M5 O<1 9 O BK O<C 7 02O3vC’= 9 Z ^9 3 5 02ZD021 5 Cf021sj 9 1B= 587:9 1e;0 KM+
@6+LKM7 ; 7:5l0r/ ;|{ OB10 9 O BK O<C 7 02O3(C#= 9 1<; 7:587M9 1<C W S+ CQRS1 58+UT 0r02C 5~KM+ C8O 7M@U+ 1 5 0Dz
1<C 5 3(OB= 587:9 š 1 9 š 9 1<7:58; 7M97:587:9 1<10 K`K 0 z¬z 5 587:9
1<; 1 THEN 1BC 3(O<= 1
IF
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class M int res1 ;
int[] res2 = new int[4-(0)+1] ;
M2 i ;
...
void foo() res1 = i.op1(res2) ;
i.op2(1, true) ;
BBoolean res3 Object = new BBoolean() ;
res2 = i.op3(res3 Object) ;
res3 = res3 Object.getValue();
...
§#¨i©_ª 2k W ¤« 3 + ;<O<= 587:9 1e;0 KM+ C8OcBC 587:5 O 587:9 1 + 0 K ;|{ 9*'04/ 3 +6587M9 1>021 ? +2@6+
'&
$
tmp := x ;
x := y ;
y := tmp ;
§#¨i©_ª 2k W Lœ « * 0rC 0 / q O021<=E02Zp021 5<
'&
tmp = x ;
x = y;
y = tmp ;
§#¨‡©aª 2k W ž¤« 3 + ;<O<= 587M9 1¥021 [ +n@U+ ;<O )(
$
(
VAR bb, ii IN
bb := unBooleen ;
ii := 2 + unEntier ;
...
END
§#¨i©_ª 2k W £« * 02C~C8OcBC 587M5 O 587:9 1<C
š 9 1<; 7:587:9 1 THEN 1<C 5 3vO<= 587:9 1
1BC 5 3(O<= 587:9 1
)(
-
ELSIF
ELSE
*
.
021
&
œ
(
boolean bb ;
int ii ;
bb = unBooleen ;
ii = 2 + unEntier ;
...
§#¨i©_ª 2k W ¤k « 3 + ;<O<= 587:9 1e;<02C~C8OcGC 587:5 O 587:9 1BC
)(
$
! 021 [ +n@U+
END
aKMK O 5l 38021<;<380 BK O<C 7 02O3(Caj 9 3(Zp02C£z
KM+ j 9 3(ZD0rZ 7 1 7 Z +LK 0 W* + C(OcBC 587:5 O 587:9 1 s 02C 5~+BK`7Mq O 04/ 0£C 7 p 02C 5\@’04/ 3 7 A 0 / W
« IF0 p02THEN
s END z
« IF p THEN s1 ELSE s2 END z KM+ C(OcBC 587:5 O 587:9 1 s1 02C 5u+BKM7Mq O 04/ 0dC 7 p 02C 5u@’04/ 3 7 A 0 / W$V+ 1<C K 0
+ 9 5 +L7 5uKM+ 587:5 587:9 q 7 5~+BKM7Mq O 04/ 0 W
« = CY= 1 3 380LtB=6{02C C8OcGC O 1 s2 O 02C
!
IF p1 THEN s1 ELSIF p2 THEN s2 ... ELSIF pn THEN sn END : cette forme est
équivalente à imbriquer plusieurs substitutions IF. La substitution s1 est appliquée
si p1 est vérifié, s2 si p2 est vérifié, etc...
– IF p1 [THEN s1 ELSIF p2 THEN s2 ... ELSIF pn THEN sn ELSE sd END
zfZe02 ZD0
; 04/ 5~AG1+7:58B7:9 KM7`1 q q O/ 0 + 3 02/ = + 02/ ;902ZZD+ 021 5 t’+] K {/ 0 T 7 =E0+6<5 587M9 1 5fq O¬@’{ / 7MK 7 0 TS/ 7W C 5 0yO<10rC(OcBC 587:5 O 587:9 1 B+ 3Y; 04/ j + O 5~q O 7
9 02O<C C K 02C K`+ 1 -[O +604- 0 02C’O>;0 = C 3 9L- O ] 3 + ZpO<=4Z O<+61 587:9 3 1 02; ^9 = C8C402] ;<1¬02{021 C 5 O<1<040'3 CA 5 0 3(O<= 5 O380f;0\=vF 9[7 TŸ0 / q O 7M@U+LK 021 5 0 q O0 K { 9 1
02O 5f@L9[7 3~C(O3 K 02C\;02O T 0 T 02Z BK 02C\C(O 7:@6+ 1 5 C\;BO 58+ c K 0 + O 2k W ­ W
$
IF a b THEN
a := b
END
IF a b THEN
b := 0
ELSIF a b THEN
a := 0
ELSE
a := b/2
END
b) ª 2k W ­h« O<cBC 587:5 O 587M9 1
021 g8+D] -[+ OB=(F0 m 0 5 021 [+n@U+ g8+ ] ;3 9[7M5 0 m
19L5 + 02O 5 387|02K Z + B3 q + O<043 5 q O0Y;029 C + KM=47M-[= +69[5QKM+ 9[;7 02C'; + 0 / KM5Q7 Z 9 7:5 021 5aK 02 Cac / K:9 7 =4+6C_5 ZeW 02 ZD0VC 7 =E0u1|{02C 5†B+ C#1 02/ =E02C8C +L7 380 W
1 043 O<C(C 02C 3(021 F|02] CQ02C c 3802C O O3u;02C 3 02; = C
243
?
if (a b) a = b;
if (a b) b = 0
else if (a
a = 0;
else a = b/2 ;
Ϩ
B… †
x—! [˜ [— [˜•Œ,™ š 0 5Q5 0£;02O TS7 02] ZD0£C8O<cBC 587:5 O 587M9 1p= 9 1<; 7:587M9 1<10 KMK 0 043vZD0 5 ;<0Y=(F 9[7 C 7 3#021 5 3(0V; 7:@ 043(C#= 9 Z ^9 3 5 02ZD021 5 C
0521sj 9 1B= 587:9 9[17 T ;<0 TKM7+d5 @6+LK 02+ O3';|KM+
{ O<B1K 0YG0 Tf+ 5 3(02C8C 7:9 KM+ 1.g‡-[A +6- - 2k W m W 9L- 9 O + 5 = 9 Zp+658Zp7:9 0 K`+ C8O[cG+2C @658+ 7:5 O 587:9 15 K IF7 t K`5 + C 5 3v58O<7:9 =¦b
O380’;<0’=(F - 02k W C 0_m W; 1BC O 3 ;02C 1 02C,;0 3 3 ZpZ 1Dz4021 t =6{02C { 1<C 3(O<= 1
switch g‡A
1<C 5 3(OB= 587:9 1 š 9 1<'T ; 7:587:9 1<7:19 0 K` K 0 +LKM+Lz¬7 z
3802C8C š 1 9 = 58+ 5 380 +LOFK`+L7
CASE
1<C5 1 +L0 KM+L7= 380 THEN
1<C 5 3v+LO<KM= +L587 7:9 1
EITHER
š
š
9
8
5
+
9
8
5
+
5
1BC 5 1 580 7M9 = 380 {:t:{ 1<C 1 0 = 380 THEN 1<C 5 3(O<= 587M9 1
OR
1
ELSE 1<C 3(O<=
)(
&
$
)( (
!
-
*
*
.
END
END
CASE x/5+3 OF
EITHER 3 THEN
y := 0
OR 4,5,6 THEN
y := x
OR 10 THEN
y := 10
ELSE
y := -x
END
END
#§ ¨i©_ª 2k W « * + C(OcBC 587:5 O 587:9 1
021 …B† ˆ… x—! [˜ [— [˜ˆŒ ™ š 0 5Q5 0eC8O<cBC 587:5 O 587M9 1›3 0 / +LKM7 C80sO<1<0 7:5l04/ 3 +6587:9 1”C8O<3hO<1 c K:9 = W,* 02C 7 1<C 5 3(OB= 587:9 1<Cy= 9 1 5 021lO02CD; + 1<Ch=E0
cBK:K 9 ='C@ 9 1+L57 0 € K 02= T_5 O / 04/ 0258C 7M958+ 1 5aq ^O|9 { OB10f= 7:9 5'1<+2; @ 7:587:9 K 1e7 giO<51 3 5802/ 7:; 9 7 = +65 m 7:@602C+ 5 5'@’04/K`3+ 7 A 04/9 0 WI* K 9 3(C q - O0 2k KM+ W = ®[9 m 1<W ; 7:587M9 1D1|{02C 5
OBC 3 0Lt {0 02=4O 1NCQ0 O3(C(O 02= { 1<C 3vO<= 1¥C8O 1 c OB= 0g‡A
1<C 5 3(OB= 587:9 1 + 1 š 5 9 O<0 7:587M9 z|z
1<; w 1 7 +6DO
WHILE
3802; = 5
INVARIANT
'T 3802C8C 7M9 1
VARIANT
)(
&
)(
$
!
END
* 02C MK + 1 -[+6- 02C 7 1j 9 3(Z +6587Mq O02C ^9 C8CE02] ;021 5\5Q9 O<C~OB10£C 5 3(O<= 5 O3(0 0 / q O 7M@U+LK 021 5 0 W
$V+ 1<C K { 0 T 02Z BK 0N= 7 bˆ;02C8C8O<CPg‡A - 2k W ­[m t\1 9 O<C @L9 R 9 1<C q O<0 K 02C B+ 3 587 02C VARIANT 0 5 INVAMK + (C OcBC 587:5 O 587:9 11|{ 7 1 O021 5B+ C'C8O<3 K`+y5 3 + ;BO<= 587:9 1 W 1sj +L7:5 tl=E02Cf;<02O T = KM+ O<C802C#C8043 @ 021 5 + ]
RIANT ;0
KM+ 3(02O @ 0Lt ^9 O3 ’@ 04/ 3 7 A<043~1 9L58+ ZpZp021 5uq O0 K`+ c 9 OB= K 0£CQ0 5 043(Z 7 10Vj 9 3(= 02/ ZD021 54W
)(
"!
œ­
switch (x/5+3) case 3 :
y = 0;
break ;
case 4 :
case 5 :
case 6 :
y = x;
break ;
case 10 :
y = 10 ;
break ;
default :
y = -x ;
§#¨‡©aª 2k W « 3 + ;<O<= 587M9 1¥021 [+n@U+ ; 0 MK + 8C OcGC 587:5 O 587:9 1
)( (
$
;<0
i := 0 ;
WHILE i 5 DO
var := var * 2 ;
i := i + 1
INVARIANT
i NAT
var NAT
VARIANT
5-i
END
§#¨‡©aª 2k W ®h« *,+ C8OcBC 587:5 O 587:9 1
)(
1
021
i = 0;
while (i 5) var = var * 2 ;
i = i + 1;
#§ ¨‡©aª 2k W ­D« 3 + ;<O<= 587M9 1e;0 KM+ C8OcGC 587:5 O 587:9 1 021 [+n@U+
…B† ˆ…¬… x—! [˜ [— [˜ˆŒ ™ $V+ 1<C =E0 5Q5 0 7 1BC 5 3(O<= 587:9 1d; 9 1 5xK`+ CQRS1 58+UT 0'02C 5 ; 9 1<1 04/ 0f= 7 bˆ;<02C8C 9 O<C4t6CQ02O K 0 KM+B+ 3 587 0 1<C 5 3vO<= 587:9 1
02C 5.5 3 + ;<O 7:5 0 G+ 3›O<1Hc K:9 = [+n@U+ = + 3 9 1 j +L7:5 K { FIR ^9L5 F|02] CQ0 q O0 K 02C = 9 Z ^9 C + 1 5 C !+ O3 9 1 5H0 / 5I0 /
)(
$
1
3 0 / +LKM+ c K 02ZD021 5 3 9 O @’02/ C W
1<C 5 3(OB= 587:9 1 ? C8CQ04w 3 58+ 7:9 1 $ š z|? z
ASSERT
œ
1<C 5 3vO<= 587:9 1 END
+ 20 Z + 3 q O0Nz 9 1›02C 5¤KM7 Z 7M5I0 / B+ 3 K 02Cd= + 3 + = 5I04/ 3 7 C 587`q O02C¤;<O KM+ 1 -[+6- 0s= 7 c K 0Z +LK:- 3 0 / K 02Ch; 7 C 9 C 7:587 jiC
5 3 9 O @’02/ C ^9 O3 B+LK`KM7 043 K 0rZ + 1 q O0r;0 0 TS7 c 7MKM7:5l0£/ ;<O KM+ 1 -[+6- 0 [+n@U+G+ 3\3 +^9 3 5 + ] uW
!
$
THEN
œ
! !¢
% (% ) š +B7:5 @l7 + TBKM7`q K 7 7B+LK 7 €~/
5
5 + 587:9 +n@U+ 58+
+; 3(; 0>0 5 =vF O<1<0 3(5 0 3 + ;<O<C80›= 587:] 9 1 0 ^9 O3.O04O<3 1002C BKM+63 5 1<0Eb•= j 9 3(ZD020 Cy;02Zdc 0438+ 023 1Bq =EO 0204/ Cp0–02;<1 O 380>5 R O< 100 +n3 @U+;<š OB=+ 3v; Wr1o~ 9 OBC C= 9 Z¤1Sbb
ZD029 1 = 9 1BC B/ + 3#9 ; 7 0 / 58/ +L7MKMK 043 K 02C_C '02T / = 7 AG= G7:K5I02/ C#7;<0 BKM+K BK`+65 0E^b•9j 9 3(5 Zp09 +n@U+ +Lš 7 + 3v/ ; K W 5 1<9 C8O 7M7:q 5 0\1 W’9 oYO<C’9 Z 9 1 '5 3 / 9 1<7 C K { 9 + b
3 =(F<0 3 02= 1 C 040NC8O<3¤OB1”0 02Z 0eC Z 0;0 3 0EbˆZ 1<1 0 0 02= 3 1 O0 O<CDC 02= A<043 1<C
=EoY0 9 5 0 T 02Z BK/ 0e0215 9 O 7 C¤5 1 9 T O<Cp; G9 K 1B1049 3 9 1<C KM+ KM+.9 5 3 + ;BO<= 587:9 1 ^+9 BOK3D5.O<1/ 0 7:B5 K`+65 0Eb•+ j 9 3(ZpKM0K +2@6+ 5C ^589 + 1B; + 3(; W
+ n@UO<+ Cyš 3(+ 0 3(; 3 W 02CQ0211 AB1e043 1 1<9 C¤OBC =E00 / 5 O<0 ; 7 0204Z3 9 1<C\0= C 9 ZpOBCZD021 5Vj 7 3(1 Zp5l0 / 0- 38;|04{ 3 O<1<K 0 0 ^9 3 5 0E0 bˆZ 029 =E1B3 1 +L0¥7 0 Z 0 / K 021l= O5 30 9 1 027MZDq O021 0yC '02/ = O7 3hA 0d/ O<0211 0
021–Z 9 ; 7 A + 1 5¤š K { + BK 0 5 ^9 O3DO<10 BK`+65 0Eb•j 9 3(Zp0 +2@6+ C 58+ 1B; + 3(; ^9 O3 9 c 5 021 7 3pOB10 BKM+65 0Eb•j 9 3(ZD0
02Zdc + 3 q O 04/ 0 +n@U+ + 3(; W
% &
"!$# &%(' *) ,+ .- .… ( ‡† ˆ… ‰\.Š U‹ .Š 0˜ / ˜  .Š e‘_Š„“ˆŽ–‹a“ˆŽ62Š 143Œ¬6 58
Š 7,Ž ,^:Ž 9Ž^?‘
!
!
§#¨‡©aª; lW; « $>0 / @ 0 K:9< 02ZD021 5 ;|{ OB10 + BK 0 5 +2@6+ š + 3(;
&
;2œLœ
k6žLž
#
MACHINE
BWallet
CONSTANTS
MAX BALANCE, MAX TRANSACTION AMOUNT, DEFAULT BALANCE
PROPERTIES
MAX BALANCE NAT MAX BALANCE 50000
MAX TRANSACTION AMOUNT NAT DEFAULT BALANCE NAT
DEFAULT BALANCE MAX BALANCE
CONCRETE VARIABLES
balance
INVARIANT
balance 0..MAX BALANCE
INITIALISATION
balance := DEFAULT BALANCE
OPERATIONS
...
setBalance (balanceInit) PRE
balanceInit NAT
balanceInit 0..MAX BALANCE
THEN
balance := balanceInit
END ;
debit (debitAmount) PRE
debitAmount NAT
(debitAmount 0)
(debitAmount MAX TRANSACTION AMOUNT)
(balance - debitAmount 0)
THEN
balance := balance - debitAmount
END ;
credit (creditAmount) PRE
creditAmount NAT (creditAmount 0 )
(creditAmount MAX TRANSACTION AMOUNT)
((balance + creditAmount) MAX BALANCE)
THEN
balance := balance + creditAmount
END
END
§#¨‡©aª; lW kh« # 20 / = 7 AB= +6587:9 1 + cBC 5 3 +L7:5 0 ;BO ^ 9 3 5 0EbˆZ 9 1<1 +L7 0 0 / K 20 = 5 3 9 1 7Mq O0
&
k6ž ;
IMPLEMENTATION
BWallet imp
...
VALUES
MAX BALANCE = 10000 ;
MAX TRANSACTION AMOUNT = 100 ;
DEFAULT BALANCE = 0
...
END
#§ ¨i©_ª; lW « Z BKv02/ ZD021 58+6587:9 1;<O ^9 3 5 0EbˆZ 9 1<1 +L7 0 0 / K 02= 5 3 9 1 7Mq O<0
* #/ 7 7:5l/ KM+ GKM+65 9 +2@6+ š + ® ; k B+ +^9 5 +”KM+ GKM+65 0Eb•j 9 3vZD0 +2@6+
5C 8+ 1<; 02+ Cs3(;>C C 029 = 1 AB5\= K 02C\02CeC8O ;7:@U0 + 1 5 02C£z 0Eb•j 3vZD0 3(; t 33 3 ]
« K 1|{R +$B+ C\;<0y=(F + 3 - 02ZD021 5 ;RS1 + Z 7Mq O0V;0y= KM+ C8C802C W
« 1 02O 5 =E3 04/ 043r;Rl1 + Z 7Mq O02ZD021 5 ;02C 9 c ¡ 0 5 C¤gi1<0
m 021.Z 02/ Z 9[7 380 043(Z + 1021 5 0¤Z +L7 C 7MK 1|{R +
G+ C ;0a3 + Z + C8CQ0EbˆZ 7 0 5Q5 0LtU1 7 ;0aZ 9 RL021 9 O<3 ; 02/ C +LKMK:9 O<043 0 TBKM7 = 7:5 02Zp021 5 K 02C 9 c ¡ 0 5 C W6* + Z 02/ Z 9[7 380
+LK`K:9 O 04/ 0u1<0 02O 5 ; 9 1B=y0 5 380u3 02/ =4O '04/ 3 04/ 0 9 3 KM+yq O + 1 587:5I0V/ ;<0~Z 02/ Z 9[7 380u02C 5 j +L7 c K 0 W K R +d5Q9 O 5 04j 9[7 C
K`+ ^9 C8C 7 c 7MKM7:5l0_/ ;<0#=E3 04/ 043_;<02C 9 c ¡ 0 5 C 5 3 + 1BC 7:5Q9[7 3802Cfg‡021 &.m 021 G+ C8C + 1 &5 G+ 3 K { w ;<OhCQRlC 5 02] Zp0 W
« * 02C 5 R 02C';0uc + CQ0 + O 5Q9 3 7 C 02/ CfC 9 1 5'K 02C byte g c 7:5 C m t K 02C short g ; ­ c 7:5 C m 0 5'K 02C boolean W
* 0 5 R 0 int g k c 7:5 C m 02O 5 0 5 380dC(O ^9 3 5I0r/ ;0£j + = 9 1 <9 <587:9 1B10 KMK 0 W
« 1>10r; 7 C 9 C80 q O0y;02C 58+ c K 0 + O T + ] O<10£; 7 ZD021<C 7:9 1 W
« K 1|{R $+ B+ C\;<0 5 F380 + ;<C W
« *,$+ ^9[KM7:587Mq O0u;0yC 02/ =4O3 7M5I0d/ 02C 5~7 1B= 9 3 ^9 3 04/ 0r; + 1<C KM+ Z + =(F 7 1<0 @l7 3 5 O0 KMK 0 W
« * 02C + BKM7 = +6587:9 1BC ;02C 587 1 04/ 02C'+ ] =E0 5Q5 0 BKM+65 0Eb•j 9 3(ZD0aC 9 1 5 ;02C +BK 0 5 C +n@U+ š + 3v;zUO<10a;<02C,= KM+ C(CQ02C
0m 5h5 7 Z BK¦02/ ZD021 5 043 ^+9 O Z 5 7 1 +L77:Zh5 O<Z K K 02Cy9 j 9 1<= +587M9 1<C install
; ^9[9 7:5 F 04/K 3 7 7:5 043¤58+L;KMKM0 +658KM7:+ 9 = K`+ C8CQ0 Applet
+
B
K
5
587 1 04/ 02C
g+¥K +O 3 BK { 1B5 m C W|w}9 -[1 + ;|+ { O<5817 0 9 0 580 7M9 process
g
O
3
3
4
0
3
2
0
h
C
=
p
Z
Z
<
1
;
2
0
¤
C
;
2
0
C
] { 5 + 0 + 587:9 O3 + 3 1 3rK O<1P7:@ j + 1<= m W 0 1<102ZD/ 021 + 5 7= 9 F 04/ 38021 5d- 7MK 58R 7M9 + O<1PBZ + 02/ = 58+ +61 - 7 C8ZD0p9 ;<¡ 0 5 - 02C 587:5 9 1
;<+ 02CBK 3 5 1<C T7= 5 1</ CV-[+Lg8K] O<1sCQ5402W O 1 0 O 1¥Z 02= 1 C(ZD0u;<0 02C 1s;<0 3 0£;|{ c 0 C#021 380
« *,+ = 90 ZpCfZd0 OB1 C 7 =0 +60 587:9 1¥0202Zp1 025 1380 KM+ = + 3 5 0£0 5\K 0 K 02= 5 02O<3f02C 5 3 0 / +LKM7 C 04/ 0 B+ 3\OB1 3 9L5Q9 = 9[K 0rO 587MKM7 C + 1 5 ;02C
G+Lq O0 5 C $Vw 0dW
K j + O 5h+ O<C8C 7 3802Z + 3 q O043 q O0 KM+ BKM+65 0Eb•j 9 3(ZD0 +2@6+ š + 3(;„02C 5 021 0 / @L9[K O 587:9 1„3 + B7 ;<0p0 5hq O0s;02C
5
385 02C 3 7 =9[58K:9L7:9 -[1<7MqC B+ 3¤53 + ^KM9 +3 5 + ] KMKM+ 7 +6B58KM7:+69 5 0Eb•j 9 3(ZD0 9 @ +2@6KM+ K C 9 1 '5p/ K 70 @’04/ +60258C 7:9 + O”W jiO3¤0 5 + ] ZD02C8O380¥;02C +n@U+ 1<= 04/ 02C
02=(FB1 O02C\0 ;0 Oc = 1¥;02C\1 O 0 02CfC 02= AB= 1<C
.… ( ‡† † ,L?!— lL—_[Š ‘ "`—_™_Š Ž^‹_‹a“ˆŠN‹fŒ —_ 7,Ž ,^:Ž 9Ž^?‘
*,+ 5 5
+ GK 5 ^9 +n@U+ š + 5 9 9 / 5 9[7 B+ 587 - ; lW ; m z
« OBC 1e3(Z O<= 9 ;BO<O 38K0r0£;;020rC = 9 ZpZh0 O<C 1 7 = +6O5837M9 1|t 3v;>02C = Z C 040h;0 3 C 3 02Cyg‡A
« K 02C 3 02/ = 9 1<; 7M587:9 1<Cf;02C <9 '04/ 3 +6587:9 1<C 5 3 + ;<O 7:5 02Cf021 +2@6+ t
« K 0r= 9 ;0d;02C 7 Z GKv02/ ZD021 58+6587M9 1<Cf021 ž W
* 0yZ 9 ;<O K 0£;0r= 9 ZpZhO<1 7 = +6587:9 1 - 04] 380 K 0 BO T ;0 B+Lq O0 5 C w$ 0 021 5 3(0 K 0 K 02= 5 02O<3\0 5YKM+ = + 3 5 0¥+ ]
O<=E0 W<* 0 3 9L5Q9 = 9[K 0yO 587MKM7 C 0 / 043(ZD0 5 ;|{ 7 ;021 587 A<043 K { + BK 0 5fq O 7 = 9 ZpZhO<1 7Mq O0 B+ 3\OB1 7 ;021 587 A + 1 54W
&
'&
&
&
k6ž[k
#
IMPLEMENTATION
UsingBWallet imp
REFINES
UsingBWallet
IMPORTS
BWallet, BT IO
OPERATIONS
main VAR balanceAmount, creditAmount, balanceFuture IN
balanceAmount
getBalance ;
INT WRITE (balanceAmount) ;
creditAmount := 100 ;
balanceFuture := balanceAmount + creditAmount ;
IF (creditAmount 0) (creditAmount MAX TRANSACTION AMOUNT)
(balanceFuture MAX BALANCE)
THEN
credit (creditAmount)
END ;
getBalance ;
balanceAmount
INT WRITE (balanceAmount)
END
END
§#¨‡©aª; lW « *,+ C '02/ = 7 AB= +6587M9 1 ;0 K { 0 T 02Z B K r0 ;BO  9 3 5 0EbˆZ 9 1<1 +L7 0
&
)(
# # k6ž
&
§#¨i©_ª; lW ®¤« 3 + <; O<= 587:9 1e;<0 K { +BK 0 5 BWallet
% ) $! # ! &% ) ƒ # ….( ( ˆ…  Š/ ŠG™ LŽ[˜•Œ ™
* T GK ^9 5 9 +L7 / K 5 9 7Mq ^9 5 K 5 9[7 9 587M9 +LKM7:5l/ 7M@U+ 5 K
/; 04c 7:5 {t 0 K 0r02Z=E3 02/ ; 0d7:5 ;<0 O 5~K`+¤3 @’04/0E3 bˆ7 Z AG= +61<581 7:9 1N0 ;<0 Oe02=C 9[3 K ;10r;<OO<0h^C89 O 3 5 0EbˆZ 3 9 0 1B1 02+LC 7 0 W 3 C~j 1<= 1<1 02CVC8O 1 02Cyz 0
….( ( † ‰\.Š 5Ž *_˜ ™’.Š * 0 3 9U¡ 0 5 Hg‡A - ; lW k 0 .5 ; lW m tS021 587 04] 3802Zp021 5 3 9 O @’0L/ tI= 9 Z 38021<;pO<1<0~Z + =(F 7 10 + cBC 5 3 +L7:5 0\0 5 C 9 1
7 Z GKv02/ ZD021 58+6587M9 1 380 3 02/ CQ021 58+ 1 5yK 0 ^9 3 5 0EbˆZ 9 1<1 +L7 0 0 / K 02= 5 3 9 1 7Mq O0D0 5 O<1<0pZ + =(F 7 10 + cBC 5 3 +L7:5 00 5 C 9 1
&
$
&
&
)(
k6ž
(
#
/**** class BWallet.java * generated on 20 févr. 2003 19 :06 :28 ****/
package wallet ;
class BWallet static final int MAX BALANCE = 10000 ;
static final int MAX TRANSACTION AMOUNT = 100 ;
static final int DEFAULT BALANCE = 0 ;
static int balance ;
static void INITIALISATION() balance = 0 ;
public static void setBalance ( int balanceInit) balance = balanceInit ;
public static void debit ( int debitAmount) balance = balance - debitAmount ;
public static void credit (int creditAmount) balance = balance + creditAmount ;
public static int getBalance () int amount = 0 ;
amount = balance ;
return amount ;
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/*** class UsingBWallet.java * generated on 20 févr. 2003 19 :06 :28 ****/
package wallet ;
class UsingBWallet static void INITIALISATION() BWallet.INITIALISATION();
BT IO.INITIALISATION();
public static void main ( String Args[]) INITIALISATION();
int balanceAmount ;
int creditAmount ;
int balanceFuture ;
balanceAmount = BWallet.getBalance();
BT IO.writeInteger(balanceAmount);
creditAmount = 100 ;
balanceFuture = balanceAmount + creditAmount ;
if (creditAmount = 0
&& creditAmount = BWallet.MAX TRANSACTION AMOUNT
&& balanceFuture = BWallet.MAX BALANCE) BWallet.credit(creditAmount);
balanceAmount = BWallet.getBalance();
BT IO.writeInteger(balanceAmount);
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k6ž package wallet ;
import javacard.framework.*;
public class Wallet extends Applet ...
final static byte WALLET = 0xAA ;
// codes of INS byte in the command APDU header
final static byte GET = 0x01 ;
final static byte CREDIT = 0x02 ;
final static byte DEBIT = 0x03 ;
final static short MAX BALANCE = 10000 ;
final static short MAX TRANSACTION AMOUNT = 100 ;
private short balance ;
public Wallet() balance = 0 ; register() ; public static void install( APDU apdu ) new Wallet() ; public void process( APDU apdu ) byte[] buffer = apdu.getBuffer();
if ( buffer[ISO.OFFSET CLA] != WALLET )
ISOException.throwIt(ISO.SW CLA NOT SUPPORTED) ;
switch (buffer[ ISO.OFFSET INS]) case GET : getBalance(buffer); break ;
case CREDIT : credit(buffer); break ;
case DEBIT : debit(buffer); break ;
default : ISOException.throwIt(ISO.SW INS NOT SUPPORTED) ;
...
void credit( Byte[] buffer) byte byteRead = (byte) apdu.setIncomingAndReceive();
if (byteRead != 1) ISOException. throwIt( ISO.SW WRONG LENGTH) ;
short creditAmount = (buffer[ISO.OFFSET CDATA]
8)
buffer[ISO.OFFSET CDATA+1] ;
// process data
if ((creditAmount
MAX TRANSACTION AMOUNT) (creditAmount
0 ))
ISOException.throwIt (INVALID TRANSACTION AMOUNT) ;
// check the new balance
if ((balance - creditAmount)
MAX BALANCE)
ISOException.throwIt(EXCEED MAXIMUM BALANCE) ;
balance = balance - creditAmount ;
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package wallet ;
import javacard.framework.*;
public class Wallet extends Applet ...
BWallet bw ;
public Wallet() bw = new BWallet() ; register() ; public static void install( APDU apdu ) new Wallet() ; public void process( APDU apdu ) ...
void credit(Byte[] buffer) ...
bw.credit(creditAmount);
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aa.FEU
BASIC_IO
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bb.FEU
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MACHINE Feu
SETS ETAT = on, off CONCRETE VARIABLES rr, oo, vv
INVARIANT
rr ETAT oo ETAT vv ETAT (rr = on (oo = off vv = off))
(oo = on (rr = off vv = off)) (vv = on (rr = off oo = off))
INITIALISATION rr, oo, vv := off, on, off
OPERATIONS
estRouge
estIlRouge BEGIN estRouge := bool (rr = on) END ;
estOrange
estIlOrange BEGIN estOrange := bool (oo = on) END ;
estIlVert BEGIN estVert := bool (vv = on) END ;
estVert
devientRouge BEGIN rr, oo, vv := on, off, off END ;
devientOrange BEGIN rr, oo, vv := off, on, off END ;
devientVert BEGIN rr, oo, vv := off, off, on END ;
END
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REFINES Feu
INITIALISATION rr := off ; oo := on ; vv := off
OPERATIONS
estRouge
estIlRouge BEGIN estRouge := bool (rr = on) END ;
estOrange
estIlOrange BEGIN estOrange := bool (oo = on) END ;
estVert
estIlVert BEGIN estVert := bool (vv = on) END ;
devientRouge BEGIN rr := on ; oo := off ; vv := off END ;
devientOrange BEGIN rr := off ; oo := on ; vv := off END ;
devientVert BEGIN rr := off ; oo := off ; vv := on END ;
END
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SETS COUL= rouge, orange, vert ABSTRACT CONSTANTS suiv
PROPERTIES
suiv COUL COUL suiv(rouge) = vert
suiv(vert) = orange suiv(orange) = rouge
VARIABLES feu1, feu2
INVARIANT
feu1 COUL feu2 COUL ((feu1=orange feu2=orange) or
(feu1=rouge feu2 rouge) or (feu1 rouge feu2=rouge))
INITIALISATION feu1, feu2 := orange, orange
OPERATIONS
MiseEnService SELECT feu1 = orange feu2 = orange THEN feu1, feu2 := rouge, vert END ;
ChangerFeux
SELECT feu1 = vert feu2 = rouge THEN feu1 := orange
WHEN feu1 = rouge feu2 = vert THEN feu2 := orange
WHEN feu1 = rouge feu2 = orange THEN feu1, feu2 := vert, rouge
WHEN feu1 = orange feu2 = rouge THEN feu1, feu2 := rouge, vert
END
END
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CARREFOUR
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IMPLEMENTATION CARREFOUR imp
REFINES CARREFOUR
SEES BASIC IO
IMPORTS aa.Feu, bb.Feu
INVARIANT
(aa.vv = on
(feu1 = vert) ) (aa.oo = on (feu1 = orange))
(aa.rr = on (feu1 = rouge)) (bb.vv = on (feu2 = vert))
(bb.oo = on (feu2 = orange)) (bb.rr = on (feu2 = rouge))
INITIALISATION aa.devientOrange ; bb.devientOrange
OPERATIONS
MiseEnService
VAR aaOrange, bbRouge IN
aaOrange
aa.estIlOrange ; bbOrange
bb.estIlOrange
IF aaOrange = TRUE bbOrange = TRUE THEN aa.devientRouge ; bb.devientVert END
END ;
ChangerFeux
VAR aaRouge, aaVert, aaOrange, bbRouge, bbVert, bbOrange IN
aa.estIlRouge ; aaVert
aa.estIlVert ;
aaRouge
aaOrange
aa.estIlOrange ; bbRouge bb.estIlRouge ;
bbVert
bb.estIlVert ; bbOrange
bb.estIlOrange ;
IF aaVert = TRUE bbRouge = TRUE THEN aa.devientOrange
ELSIF aaRouge = TRUE bbVert = TRUE THEN bb.devientOrange
ELSIF aaRouge = TRUE bbOrange = TRUE THEN aa.devientVert ; bb.devientRouge
ELSIF aaOrange = TRUE bbRouge = TRUE THEN aa.devientRouge ; bb.devientVert
END ;
aaRouge
aa.estIlRouge ;
IF aaRouge = TRUE THEN STRING WRITE (”feu 1 Rouge”) END ;
aa.estIlVert ;
aaVert
IF aaVert = TRUE THEN STRING WRITE (”feu 1 Vert”) END ;
aaOrange
aa.estIlOrange ;
IF aaOrange = TRUE THEN STRING WRITE (”feu 1 Orange”) END ;
bbRouge
bb.estIlRouge ;
IF bbRouge = TRUE THEN STRING WRITE (”, feu 2 Rouge n”) END ;
bbVert
bb.estIlVert ;
IF bbVert = TRUE THEN STRING WRITE (”, feu 2 Vert n”) END ;
bbOrange
bb.estIlOrange ;
IF bbOrange = TRUE THEN STRING WRITE (”, feu 2 Orange n”) END
END
END
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OPERATIONS
main BEGIN skip END
END
IMPLEMENTATION MAIN imp
REFINES MAIN
IMPORTS CARREFOUR, BASIC IO
OPERATIONS
main
VAR ii IN
STRING WRITE(”Deux feux a un carrefour n”) ;
STRING WRITE(”Mise en service des feux n”) ;
MiseEnService ; ii := 20 ;
WHILE ii 0 DO
STRING WRITE (”Changer les feux n”) ;
ChangerFeux ;
ii := ii - 1
INVARIANT ii NAT
VARIANT ii
END
END
END
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CARREFOUR
context CARREFOUR inv :
suiv forAll(c1,c2 : E suiv c1 c2 implies c1.e X c2.e X)
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and suiv size()= E COUL size()
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FEU
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'9+#5 ,+> ˆ: #M;\#&, $ 0% 0%'!&0%]'9,]0 +2 7/!&0%<!0%G<;!!O 5';& ,],C0%3 53 ,B j ?_>
ˆP#X$ ,2 + … +/!&0%
T #&, A,2 '+/!N' 2 … !5GN#&,E5, 5!&0%E'9,8I#,;95
! 0O+^+2 ,;xt*8I/5MaU#&, feuACARREFOUR
+ feuB 0%3< 5';&!],C#E0%G 5MG PB j ?A,2 ,GA+2 … E%;95Ec >c v >
j + E0%3 53 05!, $0%' CR #P †R E%#E'9BE, !0% J ';-+ M +('9, M /7E,+>
?U, !0 5^+2 A,2 C0%
) #& tO,*'9-# C 38I/5MGP'9-#B,2 + … +/!&0% Feu 0!5,$0%'9,3 LR #B †R E%#&
'9 E, !&0% d ';K+ M +',6 M /7 E,+> #M#&,6' 2 … ! ,36#,(!,! 5!&0%:'9,68I#,;95:! 0O+7+2 ,F'9,
8//5a U#&, rr, oo + vv >? C [ 2 ^A;&8/#&,Eq `,; +O 5" 5'; "'&!, ,)*,31,[B j ? C , $0%;9N^3;90%
0%X;9!M# '9,x; t-) #M+/!&0% $0%;9< E[,2 +2 !+#*0%G 7G N'9"# … E%;!Cc >cc/>
$ &' ',
dd ‰
context CARREFOUR inv :
feuA = COUL Value : :orange and feuB = COUL Value : :orange and
feuA = COUL Value : :rouge and not(feuB = COUL Value : :rouge) and
not(feuA = COUL Value : :rouge) and feuB = COUL Value : :rouge
c >c v p j 0%G 7G S!,#/!8=;xt 505^+2 3,2 P0%) #& tO,'=# C 38//53E'9=#MK,2 + … +/!&0%
.0/13
CARREFOUR
X
X
\s\ and
\
\s\ and
\ and
\s\ and
\ and
context Feu inv :
rr = ETAT Value : :on implies oo = ETAT Value : :off and vv = ETAT Value : :off
oo = ETAT Value : :off and vv = ETAT Value : :off implies rr = ETAT Value : :on
oo = ETAT Value : :on implies rr = ETAT Value : :off and vv = ETAT Value : :off
rr = ETAT Value : :off and vv = ETAT Value : :off implies oo = ETAT Value : :on
vv = ETAT Value : :on implies rr = ETAT Value : :off and oo = ETAT Value : :off
rr = ETAT Value : :off and oo = ETAT Value : :off implies vv = ETAT Value : :on
XsX
X
XsX
X
XsX
X
X
\
\
\
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.0/13
Feu
ˆ_ !0NA + T #A, _33+2 5,!!3'9O.2 ,2 5&!6, B j ?X#, `0%! 0%'&!&0%]'9,\0+2 5/!&0% Mi+
';*0%) $0%!3
X … E4c >c d +(c >c J \]+/0%) $0%!G
seEnService ChangerFeux
0%) $0! E# C G8//531#& #;'+2 8,#02 > CARREFOUR -'& / Z # $)+# ?U,:'aE5)C)*,' C +2 !/! 7!&!&0% =R tO 5&5N 0%3F,;xt%!0H8,3F'9,,2 + … +/!0%
?
+ CARREFOUR >aˆ#M !,)C +R 5(,2 + … +/!0% T Feu T … 4' C ##^†2 E+#&('MaE5)C)* … E%;9!Ec Feu
>c M
90%;(,80%(!,) #.E2 #,18I#,;95 on + off !,$`,!8,)*,G / O + F >
5'nX SELECT feu1 = orange feu2 = orange THEN feu1, feu2 := rouge, vert END\
XsX 7I ' ,;@XsX c I ,;Uc 070E5 E I , ; d I ,; d 070E5\ E\ 7' X I ,I ;U,;Uc;c o T I ,!; 0%d ;Eo !0%I ,;; Ed T o 8+58N+\ !N\s\
XsX I ,;@c 07E I ,; d 07E\ X I ,;@c C 50%;E I ,; d C 8+5N\s\
?U40%G 5MG 4B j ? "' ,2 ';&'9"+ `0%! 0%'&!&0%yo
Context CARREFOUR : :MiseEnService() post :
((feu1@pre = COUL Value : :orange and feu2@pre = COUL Value : :orange) and
feu1 = COUL Value : :rouge and feu2 = COUL Value : :vert
X
\s\
\c >c d pq tA 7!&0%K'9"#M`0%! 0%'&!&0%-'"# C 0+2 5/!&0%
.0/13
MiseEnService
d Jav
$% |?1 5'nX SELECT feu1 = vert feu2 = rouge THEN feu1 := orange
WHEN feu1 = rouge feu2 = vert THEN feu2 := orange
WHEN feu1 = rouge feu2 = orange THEN feu1, feu2 := vert, rouge
WHEN feu1 = orange feu2 = rouge THEN feu1, feu2 := rouge, vert
END
\
5'nXsX I ,;@c
8 +! I ,; d !0%; E\ I ,;@c`o 07E X I ,;@c !0%;E I ,; d 8+!N\ I ,; d o 05E X I ,;@c !0%;E I ,; d 05E\ I ,;@c T I ,; d o 8+! T !0%; E X I ,;@c 05 E I ,; d !0%;E\ I ,;@c T I ,; d o !0%;E T 8+5N\
5' XsX I ,;@c 8+! I ,; d !0%;E\ I ,;Uc`o 05E\
5' XsX I ,;@c !0%; E I ,; d 8+!N\ I ,; d o 05E\
5' XsX I ,;@c !0%; E I ,; d 05E\ I ,;@c T I ,; d o 8+! T !0%;E\
5' XsX I ,;@c 05 E I ,; d !0%;E\ I ,;@c T I ,; d o !0%; E T 8+!N\
XsX I ,;@c 8+! I ,; d !0%; E \ 5' X I ,;@c o 05E\s\
XsX I ,;@c !0%; E I ,; d 8+!N \ 5' X I ,; d o 05 E \s\
XsX I ,;@c !0%; E I ,; d 05 E \ 5' X I ,;@c T I ,; d o 8+! T !0%; E \s\
XsX I ,;@c 05 E I ,; d !0%; E \ 5' X I ,;@c T I ,; d o !0%; E T 8+!N \s\
XsX I ,;@c 8+! I ,; d !0%; E \ 5' X I ,;@c o 05 E \s\
XsX I ,;@c !0%; E I ,; d 8+!N \ 5' X I ,; d o 05 E \s\
XsX I ,;@c !0%; E I ,; d 05 E \ 5' X I ,;@c o 8+! I ,; d o !0%; E \s\
XsX I ,;@c 05 E I ,; d !0%; E \ 5' X I ,;@c o !0%; E I ,; d o 8+!N \s\
XsX I ,;@c 8+! I ,; d !0%; E \ I ,;@c C 07 E \
XsX I ,;@c !0%; E I ,; d 8+!N \ I ,; d C 07 E \
XsX I ,;@c !0%; E I ,; d 05 E \ X I ,;@c C 8+! I ,; d C !0%; E \s\
XsX I ,;@c 05 E I ,; d !0%; E \ X I ,;@c C !0%; E I ,; d C 8+!N \s\
?U40%G 5MG 4 B j ? "',2 ';&'9"+ `0%! 0%'&!&0%y o
Context CARREFOUR : :ChangerFeux() post :
((feu1@pre = COUL Value : :vert and feu2@pre = COUL Value : :rouge) and
feu1 = COUL Value : :orange) or
((feu1@pre = COUL Value : :rouge and feu2@pre = COUL Value : :vert) and
feu2 = COUL Value : :orange) or
((feu1@pre = COUL Value : :rouge and feu2@pre = COUL Value : :orange) and
(feu1 = COUL Value : :vert and feu2 = COUL Value : :rouge)) or
((feu1@pre = COUL Value : :orange and feu2@pre = COUL Value : :rouge) and
(feu1 = COUL Value : :rouge and feu2 = COUL Value : :vert))
\c >c J p q tA 5!0%K'9"#$0% 0%'&!&0%-'9"# C 0+2 5/!&0%
.0/13
ChangerFeux
% $
$ &' ',
dJc
Feu
rr,jj,vv
F,O,F
DevientRouge
DevientVert
DevientJaune
DevientJaune
DevientRouge
O,F,F
F,F,O
DevientVert
\c >c M p ˆaE5)C)CE' C +2 !/! 55&!&0%( tA 5('9"#*,2 + … +/!&0%
.0/21(3
Feu
ˆ4'aE5)4)*C' C +2 !/! 55&!&0% X … E c >c M \ T 0% $,;9F8/#'9+"#&C0%)
`05 +5)*,3E'9
#S,2 + … +/!0%B'; !,#/!&8,)*,3F;L+ M &+'9, M /7E,,>qBF$,;9!U8I#M'9+6#,6 †R E%#,*c
+(!;! 0%;9 d ,2 90%.+2 Feu
,'(#* ,!&0%bc > d > d >
J 0%;9P#&,E){,O )C,7 0% 3^ ;9S'# … %E ;!Kc >c M 90%;,80%N!,) .#S2 #&,E8I#,;95 rouge,
+
vert
orange
!,$`,!8,)*,G /
R, V
+ J >
CARREFOUR
FeuA,FeuB
MiseEnService
J,J
R,V
ChangerFeu
ChangerFeu
R,J
J,R
ChangerFeu
ChangerFeu
V,R
\c >c[Z p ˆaE5)C)CE' C +2 !/! 55&!&0%( tA 5('9"#*,2 + … +/!&0% CARREFOUR
D ;E='MaE5)C)*C' C +2 !/! 7!&!&0% X … E c >c[Z]\F0% $,;9F8/#'9+"#X †R E%#& J '94#X!,!&0%
c >d >d >
? -'&( <Z (#( ?U<'a E5)4)*]'9<A[ 2 ^A;9,<'9_#M … E%;9!c >c 0!!, $0%' R # C tA 5!0%"'9 ;MD# ,2 + … +/!&0%
> b # $+5)*+1'9F8O!;#M +#,1 $,#1'9,0 +2 5/!&0%(s rO) U$0%#M3,2 /'9,)*, CARREFOUR imp
!a E,(,G !#&,#‚ E%,1'9F8A"'9,1K0 U1A +!+>
4
? -'& Z ?UQ'a E7)C)*Q' C !&8A3,2 V'9B# … E%;9! c >c H , = tA 7&S'Q# ,2 + … +/!&0% MAIN imp >
? C M 5;!0% WHILE '9(#E ,2 + … +/!&0%4, <5 A,2 ,33+2 F 0%;_# I 05)*(' C ;9< U$0%;+#&(' C +2 !/!F X '9
.0/21(3
dJd
CARREFOUR
$% |?1 bb.Feu
aa.Feu
% $
BASIC_IO
estIlRouge
aaRouge
estIlVert
aaVert
estIlJaune
aaJaune
estIlRouge
bbRouge
estIlVert
bbVert
estIlJaune
bbJaune
[aaRouge=TRUE et bbVert=TRUE]devientJaune
[aaRouge=TRUE et bbJaune=TRUE]devientRouge
[aaJaune=TRUE et bbRouge=TRUE]devientVert
[aaVert=TRUE et bbRouge=TRUE]devientJaune
[aaRouge=TRUE et bbJaune=TRUE]devientVert
[aaJaune=TRUE et bbRouge=TRUE]devientRouge
estIlRouge
aaRouge
[aaRouge=TRUE]STRING_WRITE("feu1 Rouge")
estIlVert
aaVert
[aaVert=TRUE]STRING_WRITE("feu1 Vert")
estIlJaune
aaJaune
\c >c p ˆaE5)C)CE'9"A[2 ^A;9,E tO 5&'E#*,2 + … +/!&0%
.0/21(3
E7
R
[aaJaune=TRUE]STRING_WRITE("feu1 Jaune")
estIlRouge
bbRouge
[bbRouge=TRUE]STRING_WRITE("feu2 Rouge")
estIlVert
bbVert
[bbVert=TRUE]STRING_WRITE("feu2 Vert")
estIlJaune
bbJaune
[bbJaune=TRUE]STRING_WRITE("feu2 Jaune")
E11
CARREFOUR imp
\7> j<M K^3;Q+2 !/:! O,2 !,G E#M … X' C t<,2 +;9!&0%K'" M K^A;9E 5;!&0%@>
2 3 _7 C8$;_=?>@ @'7 7
ˆ :+ 7';!&0% X … EQc >c ƒ T c >c ‰ +Nc > d v \6#&" r$'9 U ;!#AF2 , :#&E|r$
+/1#&,8I#&,;5',<8//5aU #,;9!M#A+2 ,@X rr, vv, oo + ii\_'9,) .#7,2 )*,G!/!0% 0%G0%) 5M ,short
'
# C 3 +!8/##& d T d >wˆC#&-+*0%; #L8I/7aU#& , 4;9!#M3+2 -'HG(+2 5,)C)*,3 X 90%) U!
' C &A+2 5/!0%-,2 J #&+8_H *K2 ƒ '9J #iH `UIH0%;+#&\ #@!+5=,2 ,5!&!N' C +O !ii5;'9,3 ^A;G 4R # C ;!#!/!0%-';X|r$
'9;U X # C 0%;!#\'9!,;98P'9+8A5 +O ! /5) +2 O2 /F* r`$0%;9F#&,0% !3 , MININT +
7\ >
MAXINT
dJJ
&]%
$ N 1
MAIN
CARREFOUR
BASIC_IO
/STRING_WRITE("Deux feux à un carrefour\n")
E0
E1
E2
/STRING_WRITE("Mise en service des feux\n")
E3
E4
/MiseEnService
E5
E6
/ii = 20
[else]
E7
[ii>0]/STRING_WRITE("Changer les feux\n")
E8
E9
/ii=ii-1
/ChangerFeux
E10
E11
\c >c H p ˆaE7)C)*E' C !8A&A,2 ( tA 5('9"#*,2 + … +/!&0%
.0/13
MAIN imp
d JNM
$% |?1 import java.util.Arrays;
class MAIN
static void INITIALISATION() BASIC IO.INITIALISATION();
CARREFOUR.INITIALISATION();
X
\
public static void main String Args[] INITIALISATION();
short ii ;
BASIC IO.STRING WRITE("Deux feux a un carrefour n") ;
BASIC IO.STRING WRITE("Mise en service des feux n") ;
CARREFOUR.MiseEnService();
!
ii = short short 20
while (ii short 0 BASIC IO.STRING WRITE("Changer les feux n") ;
CARREFOUR.ChangerFeux();
!
ii = short ii - short 1
X
\ XsX
X
X
\ \
\ \
\X
X
\ \
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.0/13
% $
dJZ
&]%
$ N 1
import java.util.Arrays;
class CARREFOUR static Feu aa ; static Feu bb ; static boolean hh ;
static void INITIALISATION() bb = new Feu() ; aa = new Feu() ;
aa.devientOrange(); bb.devientOrange();
public static void MiseEnService () aa.devientRouge(); bb.devientVert();
public static void ChangerFeux () short aaRouge ; short aaVert ; short aaOrange ;
short bbRouge ; short bbVert ; short bbOrange ;
aaRouge = aa.estIlRouge(); aaVert = aa.estIlVert();
aaOrange = aa.estIlOrange(); bbRouge = bb.estIlRouge();
bbVert = bb.estIlVert(); bbOrange = bb.estIlOrange();
if (aaVert == 1 && bbRouge == 1) aa.devientOrange(); else if (aaRouge == 1 && bbVert == 1) bb.devientOrange();
else if (aaRouge == 1 && bbOrange == 1) aa.devientVert(); bb.devientRouge();
else if (aaOrange == 1 && bbRouge == 1)
aa.devientRouge(); bb.devientVert();
aaRouge = aa.estIlRouge();
if (aaRouge == 1) BASIC IO.STRING WRITE("feu 1 Rouge") ; aaVert = aa.estIlVert();
if (aaVert == 1) BASIC IO.STRING WRITE("feu 1 Vert") ; aaOrange = aa.estIlOrange();
if (aaOrange == 1) BASIC IO.STRING WRITE("feu 1 Orange") ; bbRouge = bb.estIlRouge();
if (bbRouge == 1) BASIC IO.STRING WRITE(", feu 2 Rouge n") ; bbVert = bb.estIlVert();
if (bbVert == 1) BASIC IO.STRING WRITE(", feu 2 Vert n") ; bbOrange = bb.estIlOrange();
if (bbOrange == 1) BASIC IO.STRING WRITE(", feu 2 Orange n") ;
\c >c ‰ p W 7';!&0%X';-0%) $0%!G
.0/21(3
CARREFOUR
,%H8I
dJ
$% |?1 % $
import java.util.Arrays;
class Feu
short rr ; short oo ; short vv ;
public Feu () INITIALISATION(); void INITIALISATION() !
rr = short ETAT.off oo = short ETAT.on
X
\X
\
X
\X
\
X
!
\X
\
!
vv = short ETAT.off
public short estIlRouge () short estRouge = 0 ;
if rr == ETAT.on estRouge = 1 ; else estRouge = 0 ;
return estRouge ;
XsX
\s\
public short estIlOrange () short estOrange = 0 ;
if oo == ETAT.on estOrange = 1 ; else estOrange = 0 ; return estOrange ;
XsX
\s\
public short estIlVert () short estVert = 0 ;
if vv == ETAT.on estVert = 1 ; else estVert = 0 ; return estVert ;
XsX
\s\
X
\X
\
X
\X
X
\X
\
X
\X
X
\X
\
X
\X
\
public void devientRouge () !
rr = short ETAT.on oo = short ETAT.off
public void devientOrange () !
rr = short ETAT.off oo = short ETAT.on
\
!
!
public void devientVert () !
rr = short ETAT.off oo = short ETAT.off
X
\X
\
X
\X
\
!
vv = short ETAT.off
\
vv = short ETAT.off
!
X
\X
vv = short ETAT.on
\
!
!
public void changerCouleur () short cc ;
cc = short (rr) ; rr = short (oo) ; oo = short (vv) ; vv = short (cc) ;
X
\
X
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.0/21(3
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