?. Si-v and ?. Ret, alors R k (o) = r k?1 . Or par transitivité de ? on a également v 2 ? v ret Par la propriété 7, on a également R k?1 (V k?1 (v 2 )) = r k?1 . Donc R k (V k (v 1 )) = R k (V k (v 2 )), et la propriété

?. Sinon, est une variable v ? valable pour choose_region, et donc R k (o) = R k?1 (V k?1 (v 2 )) = R k (V k (v 2 )). Il faut noter que, par hypothèse d'induction, toutes les variables de la famille de v pointent sur des objets de la même région

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

. Selon-la-sémantique-de, Or par hypothèse d'induction la propriété 7.9 (3) est vraie en ? k?1 Elle est donc également vraie en ? k . Sinon, c'est que SF k est le cadre de sommet de pile de S k , celui qui est modifié par l'allocation v = new C. Soit SF k?1 = (V k?1 , lb k?1 , r k?1 ) le cadre de sommet de pile de S k?1 Si v 3 n'est pas la variable v, alors sa valeur n'est pas modifiée et la propriété reste vraie par hypothèse d'induction. Par contre, si v 3 est v, alors après l'allocation, v 3 pointe sur un nouvel objet o dont la région R k (o), p.1

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

. Selon-la-sémantique-de, ) est vraie en ? k?1 Elle est donc également vraie en ? k . Sinon, c'est que SF k est le cadre de sommet de pile de S k , celui qui est modifié par l'affectation v = v ? .f . Soit SF k?1 = (V k?1 , lb k?1 , r k?1 ) le cadre de sommet de pile de S k?1 Si v / ? {v 1 , v 2 }, alors V k | {v 1 ,v 2 } = V k?1 | {v 1 ,v 2 } , et la propriété reste vraie par hypothèse d'induction. Par contre, si v = v 1 (ou v 2 , par symétrie), alors après l'affectation, on a V k (v 1 ) = H k?1 (V k?1 (v ? ))(f ). Par hypothèse d'induction (propriété 7, les deux objets sont dans la même région, c'est à dire R k (V k (v 1 )) = R k?1 (V k?1 On a donc bien R k (V k (v 1 )) = R k?1 (V k?1 (v 2 )), pp.1-1

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

. Selon-la-sémantique-de, Or par hypothèse d'induction la propriété 7.9 (3) est vraie en ? k?1 Elle est donc également vraie en ? k . Sinon, c'est que SF k est le cadre de sommet de pile de S k , celui qui est modifié par l'affectation v = v ? .f . Soit SF k?1 = (V k?1 , lb k?1 , r k?1 ) le cadre de sommet de pile de S k?1 Si v 3 n'est pas la variable v, alors sa valeur n'est pas modifiée et la propriété reste vraie par hypothèse d'induction. Par contre, si v 3 est v, alors après l'affectation, on a V k (v 3 ) = H k?1 (V k?1 (v ? ))(f ). Par hypothèse d'induction (propriété 7, les deux objets sont dans la même région, c'est à dire R(V k (v 3 )) = R k?1 (V k?1 )) = r k?1, p.1

. Sinon, est que l'instruction I est responsable de la création de cette arête, avec V k (v) = o 1 et V k

. La-règle-intra, interférence de pointeurs (cf page 98) nous assure que v ? v ? . En effet, une instruction de la forme v.f = v ? implique la présence d'une arête entre v et v' dans le graphe de forme locale de la méthode (cf la règle associée à putfield page 90). Par hypothèse d'induction (propriété 7, pp.1-1

R. Comme, =. K?1, V. , and =. , V k (v 1 )) = R k (V k (v 2 )), soit,R k (o 1 ) = R k, et la propriété

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

. Sinon and S. Est-le-cadre-de-sommet-de-pile-de, et les seules variables à avoir une valeur dans V k sont les paramètres formels de la méthode m appelée Soient donc a 1 et a 2 les arguments correspondants dans la méthode appelante m ? . La règle inter-procédurale de l'analyse d'interférence de pointeurs (cf page 99) nous indique qu'ils vérifient également a 1 ? m ? a 2 . Par hypothèse d'induction, les objets correspondants sont donc dans la même région en ? k?1, Les variables v 1 et v 2 sont donc des paramètres p 1 et p 2 et le sont toujours en ? k . On a donc R k (V k (v 1 )) = R k (V k (v 2 )), et la propriété

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

. Selon-la-sémantique-de, Or par hypothèse d'induction la propriété 7.9 (3) est vraie en ? k?1 Elle est donc également vraie en ? k . Sinon, c'est que SF k est le cadre de sommet de pile de S k , et les seules variables à avoir une valeur dans V k sont les paramètres formels de la méthode m appelée. La variable v 3 est donc un certain p, et on a p ? m v ret . Soit a l'argument correspondant dans la méthode m ? appelante. De même que ci-dessus, on sait que dans cette dernière on a a ? m ? v r . Soit SF k?1 = (V k?1 , lb k?1 , r k?1 ) le cadre de sommet de pile de S k?1 ? Si on a également a ? m ? v ret , alors par hypothèse d'induction R k?1 (V k?1 (a)) = r k?1 . Par transitivité de la relation ?, on aura également v r ? m ? v ret, )) = R k?1 (V k?1 (a)) = r k?1 = r k, p.1

?. Sinon, a est en tous cas une variable v ? valable pour choose_region, et donc au moment de la création de SF k , on a bien r k = R k?1 (V k?1 (a)) = R k (V k (p))

. Dans-les-deux-cas, on a R k (V k (v 3 )) = r k , et la propriété 7

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

. Selon-la-sémantique-de, Or par hypothèse d'induction la propriété 7.9 (2) est vraie en ? k?1 Elle est donc également vraie en ? k . Sinon, c'est que SF k est le cadre de sommet de pile de S k , celui qui est modifié par l'instruction return. Soit SF k?1 = (V k?1 , lb k?1 , r k?1 ) le cadre de sommet de pile de S k?1 Si v r / ? {v 1 , v 2 }, alors leur valeur n'est pas modifiée par le return et la propriété reste vraie par hypothèse d'induction. Par contre, si v r = v 1 (ou v 2 par symétrie), montrons que R k (V k (v r )) = R k (V k (v 2 )). Par hypothèse d'induction, on sait que R k?1 (V k?1 (v ret )) = r k?1, il reste donc à montrer que r k?1 = R(V k (v 2 )). Au moment de l'appel correspondant, r k?1 a été fixée par un r k?1 = choose_region(v r , · · · , r k ), p.1

?. Sinon, v 2 est en tous cas une variable v ? valable pour choose_region, qui a donc choisi r k?1 = R(V k (v 2 ))

S. Sf-k-n, est pas le cadre de sommet de pile de S k , alors soit i ? N tel que SF k = S k (i)

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