title: Normal bisimulations in process calculi with passivation. [Research Report] RR-6664, INRIA. 2008, pp.102. &#x27E8;inria-00330565&#x27E9;
text: ◦
  τ
  a
  s✉❝❤ t❤❛t
  Q=
  ⇒ hR′ iS ′′ ✱ S ′′ =
  ⇒ S ′ ✱ T ≈ R′ ✱ ❛♥❞ 0 ≈ S ′ ✳ ❲❡ ❤❛✈❡ R (≈)•c R′
  . • ′
  ❛♥❞ 0 (≈)c S ✉s✐♥❣ ❖♣❡♥ r✐❣❤t ❛♥❞ r❡✢❡①✐✈✐t②✳
  .◦
  
  ▲❚❙✲❙✉♠✳
  
  a
  
  α
  
  ′
  
  ′
  
  α
  
  P = U +V −
  → A ✇✐t❤ U −
  → A✳ ❇② ❞❡✜♥✐t✐♦♥ t❤❡r❡ ❡①✐sts U ′ , V ′
  .•
  .•
  .◦
  ′
  ′
  ′
  s✉❝❤ t❤❛t U ≈ U ✱ V ≈ V ❛♥❞ U + V ′ ≈ Q✳ ❲❡ ❤❛✈❡ t❤r❡❡ ❝❛s❡s t♦ ❝♦♥s✐❞❡r
  ❢♦r A✿
  l
  
  ❼ A ✐s ❛ ♣r♦❝❡ss T ✿ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ U −
  → T ✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❡①✐sts T ′
  . •
  l
  ′ l
  ′
  ′
  s✉❝❤ t❤❛t U =
  ⇒ T ❛♥❞ T (≈)c T ✳ ❇② r✉❧❡ ▲❚❙✲❙✉♠ ✇❡ ❤❛✈❡ U ′ + V ′ =
  ⇒
  .◦
  
  l
  
  ⇒ Q′ ❛♥❞
  T ′ ✱ ❛♥❞ s✐♥❝❡ U ′ + V ′ ≈ Q✱ t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q =
  ′ .◦
  ′
  T ≈ Q ✳ ❲❡ ❤❛✈❡ t❤❡ r❡q✉✐r❡❞ r❡s✉❧t ❜② ❖♣❡♥ r✐❣❤t✳
  a.
  ❼ A ✐s ❛♥ ❛❜str❛❝t✐♦♥ F ✿ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ U −
  → F ✳ ▲❡t T (≈)•c T ′ ❜❡
  b
  
  ❝❧♦s❡❞ ♣r♦❝❡ss❡s✳ ❇② ✐♥❞✉❝t✐♦♥✱
  t❤❡r❡ ❡①✐sts G✱ U ′′ s✉❝❤ t❤❛t U ′ =
  ⇒ G✱.
   τ
  G ◦ T′ =
  ⇒ U ′′ ✱ ❛♥❞ F ◦ T (≈)•c U ′′ ✳
  .◦
  a
  ⇒ G✱ ❛♥❞ s✐♥❝❡ U ′ + V ′ ≈ Q✱ t❤❡r❡ ❡①✐sts
  ❇② ▲❚❙✲❙✉♠✱ ✇❡ ❤❛✈❡ U ′ + V ′ =
  .◦
  τ
  a
  F ′ ✱ Q′′ s✉❝❤ t❤❛t Q =
  ⇒ F ′✱ F ′ ◦ T ′ =
  ⇒ Q′′ ✱ ❛♥❞ G ◦ T ′ ≈ Q′′ ✳ ❲❡ ❤❛✈❡
  .◦
  τ
  τ
  G ◦ T′ =
  ⇒ U ′′ ✱ ❤❡♥❝❡ t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q′′ =
  ⇒ Q′ ❛♥❞ U ′′ ≈ Q′ ✳
  . •
  ❲❡ t❤❡♥ ❤❛✈❡ F ◦ T (≈)c Q′ ❜② ❖♣❡♥ r✐❣❤t✳
  a
  
  → hRiS ✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡
  ❼ A ✐s ❛ ❝♦♥❝r❡t✐♦♥ hRiS ✿ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ U −.
  τ
  ′′
  ′′
  ′′
  ′ a
  ′′
  ′′
  ❡①✐sts
  hR iU ✱ S s✉❝❤ t❤❛t U =
  ⇒ hR iU ✱ U ′′ =
  ⇒ S ′′ ✱ R (≈)•c R′′ ✱ ❛♥❞
  . • ′′
  S (≈)c S ✳
  ❘❘ ♥➦ ✻✻✻✹
  
  ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐
  
  ✼✻
  
  .◦
  
  a
  
  ❇② ▲❚❙✲❙✉♠ ✇❡ ❤❛✈❡ U ′ +V ′ =
  ⇒ hR′′ iU ′′ ✱ ❛♥❞ s✐♥❝❡ ✇❡ ❤❛✈❡ U ′ +V ′ ≈ Q✱
  .◦
  a
  τ
  ′
  ′′
  ′
  t❤❡r❡ ❡①✐sts hR iQ ✱ Q s✉❝❤ t❤❛t Q =
  ⇒ hR′ iQ′′ ✱ Q′′ =
  ⇒ Q′ ✱ R′′ ≈ R′ ✱ ❛♥❞
  .◦
  τ
  τ
  ⇒ S ′′ ✱ ❤❡♥❝❡ t❤❡r❡ ❡①✐sts S ′ s✉❝❤ t❤❛t Q′ =
  ⇒ S′
  U ′′ ≈ Q′ ✳ ❲❡ ❤❛✈❡ U ′′ =.
   ◦
  ❛♥❞ S ′′ ≈ S ′ ✳ ❋✐♥❛❧❧② ✇❡ ❤❛✈❡ R (≈)•c R′ ❛♥❞ S (≈)•c S ′ ❜② ❖♣❡♥ r✐❣❤t✳
  α
  
  α
  
  ▲❚❙✲❘❡♣❧✐❝✳ P =!U −
  → A |!U ✇✐t❤ U −
  → A✳ ❇② ❞❡✜♥✐t✐♦♥ t❤❡r❡ ❡①✐sts U ′ s✉❝❤
  ′ .◦
  
  .•
  
  t❤❛t U ≈ U ′ ❛♥❞ !U ≈ Q✳ ❲❡ ❤❛✈❡ t❤r❡❡ ❝❛s❡s t♦ ❝♦♥s✐❞❡r ❢♦r A✿
  l
  
  ❼ A ✐s ❛ ♣r♦❝❡ss T ✿ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ U −
  → T ✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❡①✐sts.
  l
  T ′ s✉❝❤ t❤❛t U ′ =
  ⇒ T ′ ❛♥❞ T (≈)•c T ′ ✳ ❇② ▲❚❙✲❘❡♣❧✐❝ ❛♥❞ s❡✈❡r❛❧
  .◦
  
  l
  
  ⇒ T ′ |!U ′ ✱ ❛♥❞ s✐♥❝❡ !U ′ ≈ Q✱
  ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ▲❚❙✲P❛r✱ ✇❡ ❤❛✈❡ !U ′ =
  .◦
  l
  t❤❡r❡ ❡①✐sts Q′ s✉❝❤
  t❤❛t Q =
  ⇒ Q′ ❛♥❞ T ′ |!U ′ ≈ Q′ ✳ ❯s✐♥❣ ❈♦♥❣ t✇✐❝❡✱
  .•.
  ✇❡ ❤❛✈❡ T |!U ≈ T ′ |!U ′ ✱ s♦ ❜② ❖♣❡♥ r✐❣❤t ✇❡ ❤❛✈❡ a[T ] (≈)•c Q′ ❛s
  r❡q✉✐r❡❞✳.
  a
  
  ❼ A ✐s ❛♥ ❛❜str❛❝t✐♦♥ F ✿ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ U −
  → F ✳ ▲❡t T (≈)•c T ′ ❜❡
  b
  
  ❝❧♦s❡❞ ♣r♦❝❡ss❡s✳ ❇② ✐♥❞✉❝t✐♦♥✱. t❤❡r❡ ❡①✐sts G✱ U ′′ s✉❝❤ t❤❛t U ′ =
  ⇒ G✱
  τ
  G ◦ T′ =
  ⇒ U ′′ ✱ ❛♥❞ F ◦ T (≈)•c U ′′ ✳ ❇② ❈♦♥❣ ✉s❡❞ t✇✐❝❡✱ ✇❡ ❤❛✈❡.
   (F |!U ) ◦ T (≈)•c U ′′ |!U ′ ✳
  a
  
  ⇒
  ❇② ▲❚❙✲❘❡♣❧✐❝ ❛♥❞ s❡✈❡r❛❧
  ❛♣♣❧✐❝❛t✐♦♥ ♦❢ r✉❧❡ ▲❚❙✲P❛r✱ ✇❡ ❤❛✈❡ !U ′ =
  a
  ′
  ′ .◦
  ⇒ F ′✱
  G |!U ✱ ❛♥❞ s✐♥❝❡ !U ≈ Q✱ t❤❡r❡ ❡①✐sts F ′ ✱ Q′′ s✉❝❤ t❤❛t Q =
  .◦
  τ
  τ
  F′ ◦ T′ =
  ⇒ Q′′ ✱ ❛♥❞ (G |!U ′ ) ◦ T ′ ≈ Q′′ ✳ ❲❡ ❤❛✈❡ G ◦ T ′ =
  ⇒ U ′′ ✱ s♦ ❜②
  τ
  ⇒ U ′′ |!U ′ ✳ ❍❡♥❝❡ t❤❡r❡ ❡①✐sts Q′ s✉❝❤
  ▲❚❙✲P❛r ✇❡ ❤❛✈❡ (G |!U ′ ) ◦ T ′ =.
  ′′ τ
  ′
  ′′
  ′ .◦
  ′
  t❤❛t Q =
  ⇒ Q ❛♥❞ U |!U ≈ Q ✳ ❲❡ t❤❡♥ ❤❛✈❡ F ◦ T (≈)•c Q′ ❜② ❖♣❡♥
  r✐❣❤t✳
  a
  
  → hRiS ✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡
  ❼ A ✐s ❛ ❝♦♥❝r❡t✐♦♥ hRiS ✿ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ U −.
   a
  τ
  ❡①✐sts
  hR′′ iU ′′ ✱ S ′′ s✉❝❤ t❤❛t U ′ =
  ⇒ hR′′ iU ′′ ✱ U ′′ =
  ⇒ S ′′ ✱ R (≈)•c R′′ ✱ ❛♥❞
  . • ′′
  . • ′
  ′
  S (≈)c S ✳ ❇② ❈♦♥❣ ✉s❡❞ t✇✐❝❡✱ ✇❡ ❤❛✈❡ S |!U (≈)c S |!U ✳
  a
  
  ❇② ▲❚❙✲❘❡♣❧✐❝ ❛♥❞ s❡✈❡r❛❧ ❛♣♣❧✐❝❛t✐♦♥
  ♦❢ r✉❧❡ ▲❚❙✲P❛r✱ ✇❡ ❤❛✈❡ !U ′ =
  ⇒
  ′′
  ′′
  ′
  ′ .◦
  hR iU |!U ✱ ❛♥❞ s✐♥❝❡ ✇❡ ❤❛✈❡ !U ≈ Q✱ t❤❡r❡ ❡①✐sts hR′ iQ′′ ✱ Q′ s✉❝❤
  .◦
  .◦
  τ
  a
  t❤❛t Q =
  ⇒ hR′ iQ′′ ✱ Q′′ =
  ⇒ Q′ ✱ R′′ ≈ R′ ✱ ❛♥❞ U ′′ |!U ′ ≈ Q′ ✳ ❲❡ ❤❛✈❡
  τ
  τ
  U ′′ =
  ⇒ S ′′ ✱ s♦ ❜② ▲❚❙✲P❛r ✇❡ ❤❛✈❡ U ′′ |!U ′ =
  ⇒ S ′′ |!U ′ ✳ ❍❡♥❝❡ t❤❡r❡ ❡①✐sts.
  ′
  ′ τ
  ′
  ′′
  ′ .◦ ′
  ⇒ S ❛♥❞ S |!U ≈ S ✳ ❋✐♥❛❧❧② ✇❡ ❤❛✈❡ R (≈)•c R′ ❛♥❞
  S s✉❝❤ t❤❛t Q =
  . • ′
  S |!U (≈)c S ❜② ❖♣❡♥ r✐❣❤t✳
  ❆s ❛ ❝♦r♦❧❧❛r②✱ ✇❡ ❤❛✈❡
  ▲❡♠♠❛ ✸✶✳
  
  . •
  ■❢ P (≈)
  c Q t❤❡♥✿
  
  ⇒ P ′ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′ ❛♥❞ P ′ (≈)•c Q′ ✳
  ❼ ■❢ P =
  l
  
  l.
   ■◆❘■❆
  
  ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s
  
  ✼✼
  
  ❼ ■❢ P =
  ⇒ F ✱ ❢♦r ❛❧❧ R✱ t❤❡r❡ ❡①✐sts F ′ ✱ Q′ s✉❝❤ t❤❛t Q =
  ⇒ F ′✱ F ′ ◦ R =
  ⇒ Q′
  . •
  ′
  ❛♥❞ F ◦ R (≈)c Q
  a
  
  a
  
  τ
  
  ❼ ■❢ P =
  ⇒ hRiS ✱ t❤❡r❡ ❡①✐sts R′ , S ′ , S ′′ s✉❝❤ t❤❛t Q =
  ⇒ hR′ iS ′′ ✱ S ′′ =
  ⇒ S′✱
  . •
  . • ′
  ′
  R (≈)c R ✱ ❛♥❞ S (≈)c S ✳
  a
  
  τ
  
  a
  
  α
  ⇒ A✳ ■♥ t❤❡ ❛❜str❛❝t✐♦♥ ❝❛s❡✱ ✇❡ ❛❧s♦ ✉s❡ t❤❡ r❡✢❡①✲
  Pr♦♦❢✳ ❇②. ✐♥❞✉❝t✐♦♥ ♦♥ P =.
  ✐✈✐t② ♦❢ (≈)•c ✭✇❡ ❤❛✈❡ R (≈)•c R ❢♦r ❛❧❧ ❝❧♦s❡❞ ♣r♦❝❡ss R✮✳
  ❲✐t❤ t❤✐s r❡s✉❧t ✇❡ ❝❛♥ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✿
  
  ▲❡♠♠❛ ✸✷✳
  
  ✉❧❛t✐♦♥✳.
  ❚❤❡ tr❛♥s✐t✐✈❡ ❛♥❞ r❡✢❡①✐✈❡ ❝❧♦s✉r❡ ♦❢ (≈)•c ✐s ❛♥ ❡❛r❧② ✇❡❛❦ ❜✐s✐♠✲.
  Pr♦♦❢✳ ❲✐t❤ ♣r♦♣❡rt② ❙②♠♠
  ✐t ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ t❤❛t ((≈)•c )∗ ✐s ❛. s✐♠✉❧❛t✐♦♥✳.
  ▲❡t P, Q s✉❝❤ t❤❛t P ((≈)•c )∗ Q✳ ❚❤❡r❡ ❡①✐sts k ≥ 0 s✉❝❤ t❤❛t P ((≈)•c )k Q
  ✳ ❚❤❡.
  ♣r♦♦❢ ✐s ❜② ✐♥❞✉❝t✐♦♥ ♦♥ k✳ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ s❤♦✇ ❢♦r k = 0 s✐♥❝❡ ((≈)•c )∗ ✐s
  r❡✢❡①✐✈❡✳.
  ❲❡ s✉♣♣♦s❡ t❤❡ r❡s✉❧t ❤♦❧❞s ✉♣ t♦ k✳ ❲❡ ❤❛✈❡ P (≈)•c P1 . . . Pk−1 (≈)•c Q
  a
  ❛♥❞ P −
  → A✳ ❲❡ ❤❛✈❡ t❤r❡❡ ❝❛s❡s t♦ ❝♦♥s✐❞❡r✳
  l
  
  ′
  ′
  s✉❝❤ t❤❛t Pk−1 =
  ⇒ Pk−1
  ❼ A ✐s ❛ ♣r♦❝❡ss P ′ ✳ ❇② ✐♥❞✉❝t✐♦♥ t❤❡r❡ ❡①✐sts Pk−1.
   l
  
  ′
  ❛♥❞ P ′ ((≈)•c )∗ Pk−1
  ✳ ❇② ❧❡♠♠❛ ✸✶✱ t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′ ❛♥❞
  . •
  . • ∗ ′
  ′
  ′
  ′
  Pk−1 (≈)c Q ✳ ❲❡ ❤❛✈❡ P ((≈)c ) Q ❜② tr❛♥s✐t✐✈✐t②✳
  
  ❼ A ✐s ❛♥ ❛❜str❛❝t✐♦♥ F ✳ ▲❡t R ❜❡ ❛ ❝❧♦s❡❞ ♣r♦❝❡ss✳ ❇② ✐♥❞✉❝t✐♦♥ t❤❡r❡
  τ
  a
  ′
  ′
  ❡①✐sts Fk−1 ✱ Pk−1
  s✉❝❤ t❤❛t Pk−1 =
  ⇒ Fk−1 ✱ Fk−1 ◦ R =
  ⇒ Pk−1
  ❛♥❞ F ◦
  . • ∗ ′
  a
  ′
  ′′
  ⇒ F ′✱
  R((≈)c ) Pk−1 ✳ ❇② ❧❡♠♠❛ ✸✶✱ t❤❡r❡ ❡①✐sts F ✱ Q s✉❝❤ t❤❛t Q =.
  τ
  τ
  ′
  F′ ◦ R =
  ⇒ Q′′ ❛♥❞ Fk−1 ◦ R (≈)•c Q′′ ✳ ❙✐♥❝❡ Fk−1 ◦ R =
  ⇒ Pk−1
  ✱ t❤❡r❡.
  τ
  ′
  ⇒ Q′ ❛♥❞ Pk−1
  (≈)•c Q′ ✳ ❲❡ ❤❛✈❡ F ◦ R((≈)•c )∗ Q′
  ❡①✐sts Q′ s✉❝❤ t❤❛t Q′′ =
  
  ❜② tr❛♥s✐t✐✈✐t②✳
  
  ′
  ❼ A ✐s ❛ ❝♦♥❝r❡t✐♦♥ hRiS ✳ ❇② ✐♥❞✉❝t✐♦♥ t❤❡r❡ ❡①✐sts Rk−1 , Sk−1 , Sk−1
  s✉❝❤.
  τ
  
  a.
  ′
  ′
  t❤❛t Pk−1 =
  ⇒ hRk−1 iSk−1
  ✱ Sk−1
  ⇒ Sk−1 ✱ R((≈)•c )∗ Rk−1 ✱ ❛♥❞ S((≈)•c
  =
  a
  ⇒ hR′ iQ′′ ✱
  )∗ Sk−1 ✳ ❇② ❧❡♠♠❛ ✸✶✱ t❤❡r❡ ❡①✐sts hR′ iQ′′ ✱ Q′ s✉❝❤ t❤❛t Q =.
  τ
  τ
  ′
  ′
  Q′′ =
  ⇒ Q′ ✱ Rk−1 (≈)•c R′ ✱ ❛♥❞ Sk−1
  (≈)•c Q′ ✳ ❙✐♥❝❡ Sk−1
  ⇒ Sk−1 ✱ t❤❡r❡
  =
  . • ′.
  ′
  ′ τ
  ′
  ⇒ S ❛♥❞ Sk−1 (≈)c S ✳ ❲❡ ❤❛✈❡ R((≈)•c )∗ R′ ❛♥❞
  ❡①✐sts
  S s✉❝❤ t❤❛t Q =
  . • ∗ ′
  S((≈)c ) S ❜② tr❛♥s✐t✐✈✐t②✳.
   ❚♦ ♣r♦✈❡ t❤❛t ≈ ✐s ❛ ❝♦♥❣r✉❡♥❝❡✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✿
  ▲❡♠♠❛ ✸✸✳
  
  .•
  
  .◦
  
  ≈ =≈.
  .◦
  
  Pr♦♦❢✳. •❇② ❧❡♠♠❛
  ✸✷ ✇❡ ❤❛✈❡ (≈)•c ⊆≈✱ s♦. ✇❡ ❤❛✈❡ (≈)•c ⊆≈ ✳ ❲❡ ♥♦✇ ♣r♦✈❡.
  ◦
  •
  ∗
  
  ∗◦
  
  t❤❛t ≈ ⊆(≈)•c . ✳ ▲❡t P, Q s✉❝❤ t❤❛t P ≈ Q✳ ❋♦r ❛❧❧ σ ✇❤✐❝❤ ❝❧♦s❡s P ✱ Q✱
  •
  ✇❡ ❤❛✈❡ P σ ≈ Qσ ❜② ❙✉❜st✳ ❙✐♥❝❡ t❤❡ ❝♦♥s✐❞❡r❡❞ ♣r♦❝❡ss❡s ❛r❡ ❝❧♦s❡❞✱
  ❘❘ ♥➦ ✻✻✻✹
  
  ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐
  
  ✼✽.
  ◦
  
  ✇❡
  ❤❛✈❡
  P σ (≈)•c Qσ ✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡ P (≈)•c Q✳ ❍❡♥❝❡ ✇❡ ❤❛✈❡
  .•
  . •◦.
  .◦
  .•
  .◦
  ◦∗
  ≈ ⊆(≈)c ⊆(≈)•c ⊆≈ ✱ ✐✳❡✳ ≈ ⊆≈ ✳ ❚❤❡ r❡✈❡rs❡ ✐♥❝❧✉s✐♦♥ ✐s ❣✐✈❡♥ ❜② ❖♣❡♥✱
  .•
  .◦
  s♦ ✇❡ ❤❛✈❡ ≈ =≈ ✳
  
  ❊ ❈♦♠♣❧❡t❡♥❡ss ♣r♦♦❢s ❢♦r ❍❖P
  ❊✳✶
  
  ❙tr♦♥❣ ❝❛s❡
  
  α
  ✐s ✐♠❛❣❡✲✜♥✐t❡✳
  ▲❡♠♠❛ ✸✹✳ ❋♦r ❛❧❧ ❛❝t✐♦♥s α✱ t❤❡ r❡❧❛t✐♦♥ −→
  
  Pr♦♦❢✳ ❇② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ s❤❛♣❡ ♦❢ P ✱ ❛s ✐♥ ❍❖πP✳ ❲❡ ❥✉st ❛❞❞ t❤❡ ❝❛s❡
  P = Q + R✳ ❇② ▲❚❙✲❙✉♠✱ t❤❡ ♣♦ss✐❜❧❡ r❡❞✉❝t✐♦♥s ❢r♦♠ P ❛r❡ t❤❡ ♦♥❡ ❢r♦♠ Q
  ❛♥❞ t❤❡ ♦♥❡ ❢r♦♠ R✱ ✇❤✐❝❤ ❛r❡ ✜♥✐t❡ ❜② ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s✳
  
  ❉❡✜♥✐t✐♦♥ ✸✹✳ ❚❤❡ r❡❧❛t✐♦♥ ∼. ω ✐s ❞❡✜♥❡❞ ♦♥ ❝❧♦s❡❞ ♣r♦❝❡ss❡s ❜②✿
  ✶✳ ❲❡ ❤❛✈❡ P ∼. 0 Q ❢♦r ❛❧❧ ♣r♦❝❡ss❡s P, Q✳
  ✷✳ P ∼. k+1 Q ✐✛
  ❼ ■❢ P −
  → P ′ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q −
  → Q′ ❛♥❞ P ′ ∼k Q′ ✱ ❛♥❞
  l
  → Q′ ✳
  ❝♦♥✈❡rs❡❧② ✐❢ Q −
  l.
  l
  
  → F ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❧♦s❡❞ ♣r♦❝❡ss❡s R✱ t❤❡r❡ ❡①✐sts F ′ s✉❝❤ t❤❛t
  ❼ ■❢ P −
  a.
   a
  Q−
  → F ′ ❛♥❞ F ′ ◦ R ∼k F ◦ R✱ ❛♥❞ ❝♦♥✈❡rs❡❧② ✐❢ Q −
  → F✳
  a
  
  → hRiS ✱ t❤❡r❡ ❡①✐sts R′ , S ′ s✉❝❤ t❤❛t Q −
  → hR′ iS ′ ❛♥❞ R ∼k
  ❼ ■❢ P −.
  a
  R′ , S ∼k S ′ ✱ ❛♥❞ ❝♦♥✈❡rs❡❧② ✐❢ Q −
  → C✳
  T.
  ✸✳ ∼ω = k ∼k
  a
  
  a.
  ▲❡♠♠❛ ✸✺✳ ❚❤❡ r❡❧❛t✐♦♥s ∼. ❛♥❞ ∼. ω ❝♦✐♥❝✐❞❡✳
  ❚❤❡ ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ t❤❡ ❍❖π P ♦♥❡✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ✇r✐t❡ τ.P + s ❢♦r
  
  (τ.P ) + s.0✳
  
  ▲❡♠♠❛ ✸✻✳ ▲❡t P, Q t✇♦ ❝❧♦s❡❞ ♣r♦❝❡ss❡s✳ ❋♦r ❛❧❧ ✐♥t❡❣❡rs k✱ ✐❢ P 6∼. k Q t❤❡♥
  t❤❡r❡ ❡①✐sts ❛ ❝♦♥t❡①t K s✉❝❤ t❤❛t K{P } ≁b K{Q}✳
  Pr♦♦❢✳ ❲❡ ♣r♦❝❡❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ k✳ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ♣r♦✈❡ ❢♦r n = 0✳
  
  ❆ss✉♠❡ t❤❡ ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ❛❧❧ k ≤ n✳ ❲❡ ♥♦✇ ♣r♦✈❡ ✐t ❢♦r n + 1✳ ❲❡
  ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s ✿
  τ
  
  τ.
   τ
  
  ❼ P −
  → P ′ ✳ ❋♦r ❛❧❧ Q′ s✉❝❤ t❤❛t Q −
  → Q′ ✱ ✇❡ ❤❛✈❡ P ′ ∼
  6 k Q′ ✳ ❙✐♥❝❡ −
  →
  τ
  ′
  ′
  ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t {Qi |Q −
  → Qi } ✐s ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡
  ❝♦♥t❡①ts Ki s✉❝❤ t❤❛t Ki {P ′ } ≁b Ki {Q′i } ❢♦r ❛❧❧ i✳ ❲❡ ❞❡✜♥❡✿
  K = a[✷] | a(X)
  
  X
  
  τ.(τ.Ki {X} + di )
  
  i
  
  ■◆❘■❆
  
  ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s
  
  ✼✾
  
  ✇❤❡r❡ (di )i , a ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ❆ss✉♠❡ t❤❛t K{P } ∼b K{Q}✳ ❙✐♥❝❡
  P −→ P ′ ✱ ✇❡ ❤❛✈❡
  K{P } −→ a[P ′ ] | a(X)
  
  X
  
  τ.(τ.Kj {X} + dj ) = R1
  
  j
  
  ❚❤✐s r❡❞✉❝t✐♦♥ ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜②
  K{Q} −→ a[Q′i ] | a(X)
  
  X
  
  τ.(τ.Kj {X} + dj ) = S1
  
  j
  
  ❢♦r s♦♠❡ i✳ ❲❡ ♥♦✇ ❤❛✈❡
  R1 −→
  
  X
  
  τ.(τ.Kj {P ′ } + dj ) = R2
  
  j
  
  ✇❤✐❝❤ ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜②
  S1 −→
  
  X
  
  τ.(τ.Kj {Q′i } + dj ) = S2
  
  j
  
  s✐♥❝❡ ¬R2 ↓a ✳ ❲❡ ❤❛✈❡ R2 −→ τ.Ki {P ′ } + di = R3 ✱ ✇❤✐❝❤ ❝❛♥ ♦♥❧②
  ❜❡ ♠❛t❝❤❡❞ ❜② S3 −→ τ.Ki {Q′i } + di = S3 ✱ s✐♥❝❡ R3 ↓di ✳ ❋✐♥❛❧❧② ✇❡
  ❤❛✈❡ R3 −→ Ki {P ′ }✱ ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜② S3 −→ Ki {Q′i }✳ ❍❡♥❝❡ ❛
  ❝♦♥tr❛❞✐❝t✐♦♥✱ s✐♥❝❡ Ki {P ′ } ≁b Ki {Q′i }✱ s♦ ✇❡ ❤❛✈❡ K{P } ≁b K{Q} ❛s
  r❡q✉✐r❡❞✳
  m
  
  m.
   l
  
  ❼ P −→ P ′ ❋♦r ❛❧❧ Q′ s✉❝❤ t❤❛t Q −→ Q′ ✱ ✇❡ ❤❛✈❡ P ′ ∼
  6 k Q′ ✳ ❙✐♥❝❡ −
  →
  m
  ′
  ′
  ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t {Qi |Q −→ Qi } ✐s ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡
  ❝♦♥t❡①ts Ki s✉❝❤ t❤❛t Ki {P ′ } ≁b Ki {Q′i } ❢♦r ❛❧❧ i✳ ❲❡ ❞❡✜♥❡ ✿
  K = a[✷] | m.a(X)
  
  X
  
  τ.(τ.Ki {X} + di )
  
  i
  
  ✇❤❡r❡ (di )i , a ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ❆ss✉♠❡ t❤❛t K{P } ∼b K{Q}✳ ❲❡
  ❤❛✈❡
  X
  K{P } −→ a[P ′ ] | a(X)
  
  τ.(τ.Kj {X} + dj ) = R1
  
  j
  
  ❙✐♥❝❡ R1 ↓a ✱ ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜②
  K{Q} −→ a[Q′i ] | a(X)
  
  X
  
  τ.(τ.Kj {X} + dj ) = S1
  
  j
  
  ❋r♦♠ ❤❡r❡✱ t❤❡ ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ❝❛s❡✳
  m
  
  ❼ P −→ P ′ ❙✐♠✐❧❛r t♦ t❤❡ ❝❛s❡ ❛❜♦✈❡✱ ✇✐t❤ m ✐♥st❡❛❞ ♦❢ m ✐♥ K✳
  
  ❘❘ ♥➦ ✻✻✻✹
  
  ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐
  
  ✽✵
  a
  
  ❼ P −
  → F = (X)V ✳ ❚❤❡r❡ ❡①✐sts ❛ ♣r♦❝❡ss T s✉❝❤ t❤❛t ❢♦r ❛❧❧ F ′ s✉❝❤ t❤❛t
  a.
   a
  → F ′ ✱ ✇❡ ❤❛✈❡ F ◦ T 6∼k F ′ ◦ T ✳ ❙✐♥❝❡ −
  → ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t
  Q −
  a
  {Fi′ |Q −
  → Fi′ } ✐s ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡ ❝♦♥t❡①ts Ki s✉❝❤ t❤❛t
  Ki {Fi′ ◦ T } ≁b Ki {F ◦ T }✳ ❲❡ ❞❡✜♥❡✿
  X
  
  K = b[✷] | ahT ib(X)
  
  τ.(τ.Ki {X} + di )
  
  i
  
  ✇❤❡r❡ b, (di ) ❛r❡ ❛❧❧ ❞✐st✐♥❝t✱ ❛♥❞ ❞♦ ♥♦t ♦❝❝✉r ✐♥ T, P, Q✳ ❲❡ ❤❛✈❡
  K{P } −→ b[F ◦ T ] | b(X)
  
  X
  
  τ.(τ.Kj {X} + dj ) = R1
  
  j
  
  ❙✐♥❝❡ R1 ↓b ✱ ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜② ❛
  K{Q} −→ b[Fi′ ◦ T ] | b(X)
  
  X
  
  τ.(τ.Kj {X} + dj ) = S1
  
  j
  
  ❢♦r s♦♠❡ i✳ ◆♦✇ ✇❡ ❤❛✈❡
  R1 −→
  
  X
  
  τ.(τ.Kj {F ◦ T } + dj ) = R2
  
  j
  
  ❙✐♥❝❡ ¬R2 ↓b ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜② ✿
  S1 −→
  
  X
  
  τ.(τ.Kj {Fi′ ◦ T } + dj ) = S2
  
  j
  
  ■♥ t✉r♥✱ ✇❡ ❤❛✈❡
  R2 −→ τ.Ki {F ◦ T } + di = R3
  
  ❙✐♥❝❡ R3 ↓di ✱ ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜② t❤❡ r❡❞✉❝t✐♦♥
  S2 −→ τ.Ki {Fi′ ◦ T } + di = S3
  
  ❋✐♥❛❧❧② ✇❡ ❤❛✈❡ R3 −→ Ki {F ◦ T }✱ ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜② S3 −→ Ki {Fi′ ◦
  T }✳ ❍❡♥❝❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ s✐♥❝❡ ❜② ❛ss✉♠♣t✐♦♥ ✇❡ ❤❛✈❡ Ki {F ◦ T } ≁b
  Ki {Fi′ ◦ T }✳ ❙♦ ✇❡ ❤❛✈❡ K{P } ≁b K{Q} ❛s r❡q✉✐r❡❞✳
  a
  
  a.
  → hRiS ✳ ❋♦r ❛❧❧ hR′ iS ′ s✉❝❤ t❤❛t Q −
  → hR′ iS ′ ✱ ✇❡ ❤❛✈❡ R 6∼k R′
  ❼ P −.
   a
  a
  ♦r S 6∼k S ′ ✳ ❙✐♥❝❡ −
  → ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t {Ci |Q −
  → Ci } ✐s ✜♥✐t❡✳
  ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❡①✐sts ❝♦♥t❡①ts Ki s✉❝❤ t❤❛t Ki {R} ≁b Ki {Ri } ♦r
  Ki {S} ≁b Ki {Si } ❢♦r ❛❧❧ i✳ ❲❡ ❞❡✜♥❡ t❤❡♥ ✿
  K = b[✷] | a(X)b(Y )
  
  X
  
  τ.(τ.(τ.Ki {X} + e) + τ.(τ.Ki {Y } + f ) + di )
  
  i
  
  ✇❤❡r❡ b, (di ), e, f ❛r❡ ❛❧❧ ❞✐st✐♥❝t ❛♥❞ ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ❲❡ ❤❛✈❡✿
  
  ■◆❘■❆
  
  ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s
  
  ✽✶
  
  K{P } −→ b[S] | b(Y )
  
  X
  
  τ.(τ.(τ.Kj {R} + e) + τ.(τ.Kj {Y } + f ) + dj ) = P2
  
  j
  ❙✐♥❝❡
  
  P2 ↓b ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜②
  
  K{Q} −→ b[Si ] | b(Y )
  
  X
  
  τ.(τ.(τ.Kj {Ri }+e)+τ.(τ.Kj {Y }+f )+dj ) = Q2
  
  j
  
  i✳ ◆♦✇ ✇❡ ❤❛✈❡
  X
  P2 −→
  τ.(τ.(τ.Kj {R} + e) + τ.(τ.Kj {S} + f ) + dj ) = P3
  
  ❢♦r s♦♠❡
  
  j
  ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜②✿
  
  Q2 −→
  
  X
  
  τ.(τ.(τ.Kj {Ri } + e) + τ.(τ.Kj {Si } + f ) + dj ) = Q3
  
  j
  s✐♥❝❡
  
  ¬P3 ↓b ✳ ❲❡ ❤❛✈❡ ♥♦✇✿
  P3 −→ τ.(τ.Ki {R} + e) + τ.(τ.Ki {S} + f ) + di = P4
  
  ❙✐♥❝❡
  
  P4 ↓di ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜②✿
  Q3 −→ τ.(τ.Ki {Ri } + e) + τ.(τ.Ki {Si } + f ) + di = Q4
  
  ❲❡ s✉♣♣♦s❡ t❤❛t ❢♦r t❤✐s ♣❛rt✐❝✉❧❛r
  
  i ✇❡ ❤❛✈❡ Ki {R} ≁b Ki {Ri }✳ ❲❡ ❤❛✈❡✿
  
  P4 −→ τ.Ki {R} + e = P5
  s✐♥❝❡
  
  P5 ↓e ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜②✿
  Q4 −→ τ.Ki {Ri } + e = Q5
  
  P5 −→ Ki {R}✱ ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜② Q5 −→ Ki {Ri }✳ ❲❡ ❤❛✈❡
  Ki {R} ≁b Ki {Ri }✱ ❤❡♥❝❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳
  
  ❲❡ ❤❛✈❡
  
  ❲❡ s✉♣♣♦s❡ ♥♦✇ t❤❛t
  
  Ki {S} ≁b Ki {Si }✳ ❲❡ ❤❛✈❡✿
  P4 −→ τ.Ki {S} + f = P5
  
  s✐♥❝❡
  
  P5 ↓f ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜②✿
  Q4 −→ τ.Ki {Si } + f = Q5
  
  ❲❡ ❤❛✈❡ P5 −→ Ki {S}✱ ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜② Q5 −→ Ki {Si }✳ ❲❡ ❤❛✈❡
  Ki {S} ≁b Ki {Si }✱ ❤❡♥❝❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳
  
  ❘❘ ♥➦ ✻✻✻✹
  
  ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐
  
  ✽✷
  
  ❊✳✷
  
  ❲❡❛❦ ❝❛s❡
  
  ❉❡✜♥✐t✐♦♥ ✸✺✳ ❚❤❡ r❡❧❛t✐♦♥ ≈ω ✐s ❞❡✜♥❡❞ ♦♥ ❝❧♦s❡❞ ♣r♦❝❡ss❡s ❜②✿.
   ✶✳ ❲❡ ❤❛✈❡ P ≈0 Q ❢♦r ❛❧❧ ♣r♦❝❡ss❡s P, Q✳.
  ✷✳ P ≈k+1 Q ✐✛.
  ❼ ■❢ P −
  → P ′ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′ ❛♥❞ P ′ ≈k Q′ ✱ ❛♥❞
  l
  → Q′ ✳
  ❝♦♥✈❡rs❡❧② ✐❢ Q −
  l
  
  l
  
  ❼ ■❢ P −
  → F ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❧♦s❡❞ ♣r♦❝❡ss❡s R✱ t❤❡r❡ ❡①✐sts F ′ ✱ Q′ s✉❝❤.
  a
  τ
  ⇒ F ′✱ F ′ ◦ R =
  ⇒ Q′ ❛♥❞ Q′ ≈k F ◦ R✱ ❛♥❞ ❝♦♥✈❡rs❡❧② ✐❢
  t❤❛t Q =
  a
  → F✳
  Q−
  a
  
  ❼ ■❢ P −
  → hRiS ✱ t❤❡r❡ ❡①✐sts R′ , S ′ , S ′′ s✉❝❤ t❤❛t Q =
  ⇒ hR′ iS ′′ ✱ S ′′ =
  ⇒ S′✱.
  a
  R ≈k R′ ✱ S ≈k S ′ ✱ ❛♥❞ ❝♦♥✈❡rs❡❧② ✐❢ Q −
  → C✳
  T.
  ✸✳ ≈ω = k ≈k
  a
  
  τ
  
  a
  
  ▲❡♠♠❛ ✸✼✳ ❚❤❡ r❡❧❛t✐♦♥ ≈ ❛♥❞ ≈ω ❝♦✐♥❝✐❞❡ ♦♥ ✐♠❛❣❡✲✜♥✐t❡ ♣r♦❝❡ss❡s✳.
  Pr♦♦❢✳ ❋r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥. ♦❢ ≈ω ✱ ✇❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❛t ≈⊂≈ω ✳ ❲❡ s❤♦✇ t❤❡.
  ❝♦♥✈❡rs❡ ❜② ♣r♦✈✐♥❣ t❤❛t
  ≈ω ✐s ❛ ✇❡❛❦ ❜✐s✐♠✉❧❛t✐♦♥✳ ▲❡t P, Q ❜❡ ✐♠❛❣❡✲✜♥✐t❡.
  ♣r♦❝❡ss❡s s✉❝❤ t❤❛t P ≈ω Q✳ ❙✐♥❝❡ t❤❡ r❡❧❛t✐♦♥ ✐s s②♠♠❡tr✐❝❛❧✱ ✇❡ ♠❛❦❡ t❤❡
  ♣r♦♦❢ ❢♦r t❤❡ tr❛♥s✐t✐♦♥s ❢r♦♠ P ♦♥❧②✳ ❲❡ ❤❛✈❡ t❤r❡❡ ❝❛s❡s t♦ ❝❤❡❝❦✿
  l
  
  l
  
  ❼ ❆ss✉♠❡ P −
  → P ′ ✳ ❋♦r ❛❧❧ ✐♥t❡❣❡rs k ✱ t❤❡r❡ ❡①✐sts Qk s✉❝❤ t❤❛t Q =
  ⇒ Qk.
  l
  ′
  ❛♥❞ P ≈k Qk ✳ ❙✐♥❝❡ Q ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t {Qi |Q =
  ⇒ Qi } ✐s ✜♥✐t❡✳
  l
  
  ❲❡ ♥♦✇ ♣r♦✈❡ ❜② ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t t❤❡r❡ ❡①✐sts Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′
  l
  ′.
   ′
  ❛♥❞ ❢♦r ❛❧❧ k✱ P ≈k Q ✳ . ❆ss✉♠❡ t❤❛t ❢♦r
  ❛❧❧ .Qi s✉❝❤ t❤❛t Q =
  ⇒ Qi ✱ t❤❡r❡.
  6 ki Qi ✳ ❙✐♥❝❡ ≈m ⊂≈l ✐❢ l ≤ m✱ ❢♦r ❛❧❧ m ≥ ki ✱
  ❡①✐sts ki s✉❝❤ t❤❛t P ′ ≈.
  l
  6 m Qi ✳ ❙✐♥❝❡ {Qi |Q =
  ⇒ Qi } ✐s ✜♥✐t❡✱ t❤❡ s❡t {ki } ✐s ✜♥✐t❡
  ✇❡ ❤❛✈❡ P ′ ≈
  l
  ❛♥❞ ❤❛s ❛ ❣r❡❛t❡st ❡❧❡♠❡♥t M ✳ ❋♦r ❛❧❧ Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′ ✱ ✇❡ ❤❛✈❡.
  l
  6 m Q′ ❢♦r ❛❧❧ m ≥ M ✳ ❇✉t ❢♦r ❛❧❧ k ✱ t❤❡r❡ ❡①✐sts Qk s✉❝❤ t❤❛t Q =
  ⇒ Qk
  P′ ≈
  ′.
  ′
  ❛♥❞ P ≈k Qk ✱ ❤❡♥❝❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡r❡❢♦r❡ t❤❡r❡ ❡①✐sts Q s✉❝❤ t❤❛t.
  l
  ⇒ Q′ ❛♥❞ ❢♦r ❛❧❧ k ✱ P ′ ≈k Q′ ✱ ✐✳❡✳ P ′ ≈ω Q′ ❛s r❡q✉✐r❡❞✳
  Q=
  a
  
  ❼ ❆ss✉♠❡ P −
  → F ✳ ▲❡t R ❜❡ ❛ ❝❧♦s❡❞ ♣r♦❝❡ss✳ ❋♦r ❛❧❧ k ✱ t❤❡r❡ ❡①✐sts Fk ✱ Qk.
  a
  τ
  s✉❝❤ t❤❛t Q =
  ⇒ Fk ✱ F k ◦ R =
  ⇒ Qk ❛♥❞ F ◦ R ≈k Qk ✳ ❙✐♥❝❡ Q ✐s ✐♠❛❣❡✲
  a
  a
  τ
  ✜♥✐t❡✱ t❤❡ s❡ts {Fi |Q =
  ⇒ Fi } ❛♥❞ {Qi |∃Fi , Q =
  ⇒ Fi ❛♥❞ Fi ◦ R =
  ⇒ Qi } ❛r❡
  ′
  ✜♥✐t❡✳ ❇② ❝♦♥tr❛❞✐❝t✐♦♥ ✇❡ ❝❛♥ s❤♦✇ t❤❛t. t❤❡r❡ ❡①✐sts F ′ ✱ Q
  s✉❝❤ t❤❛t
  a
  τ
  ′
  ′
  ′
  ′
  ′.
   Q=
  ⇒F ✱F ◦R=
  ⇒ Q ❛♥❞ ❢♦r ❛❧❧ k ✱ Q ≈k F ◦ R✱ ✐✳❡✳ Q ≈ω F ◦ R ❛s
  
  r❡q✉✐r❡❞✳
  
  a
  
  → C = hRiS ✳ ❋♦r ❛❧❧ k ✱ t❤❡r❡ ❡①✐sts Ck = hRk iSk′ ✱ Sk s✉❝❤ t❤❛t
  ❼ ❆ss✉♠❡ P −
  τ
  
  a.
  Q=
  ⇒ Ck ✱ Sk′ =
  ⇒ Sk ✱ R ≈k Rk ❛♥❞ S ≈k Sk ✳ ❙✐♥❝❡ Q ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡
  a
  
  a
  
  τ
  
  ⇒ Ci } ❛♥❞ {Si |∃hRi iSi′ , Q =
  ⇒ hRi iSi′ ❛♥❞ Si′ =
  ⇒ Si } ❛r❡ ✜♥✐t❡✳
  s❡ts {Ci |Q =
  
  ■◆❘■❆
  
  ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s
  
  ✽✸
  
  ❇② ❝♦♥tr❛❞✐❝t✐♦♥ ✇❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts C ′ = hR′ iS ′′ ✱ S ′ s✉❝❤ t❤❛t.
  τ
  a
  Q=
  ⇒ C ′ ✱ S ′′ =
  ⇒ S ′ ❛♥❞ ❢♦r ❛❧❧ k ✱ R ≈k R′ ✱ S ≈k S ′ ✱ ✐✳❡✳ R ≈ω R′ ✱ S ≈ω S ′
  ❛s r❡q✉✐r❡❞✳
  ▲❡♠♠❛ ✸✽✳ ▲❡t P, Q t✇♦ ❝❧♦s❡❞ ✐♠❛❣❡✲✜♥✐t❡ ♣r♦❝❡ss❡s✳ ❋♦r ❛❧❧ ✐♥t❡❣❡rs k ✱ ✐❢.
  P ≈
  6 k Q t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t❡①t K ❛♥❞ ❛ ♥❛♠❡ d s✉❝❤ t❤❛t K{P } + d 6≈b
  K{Q} + d✳
  
  Pr♦♦❢✳ ❲❡ ♣r♦❝❡❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ k✳ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ♣r♦✈❡ ❢♦r n = 0✳
  ❆ss✉♠❡ t❤❡ ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ❛❧❧ k ≤ n✳ ❲❡ ♥♦✇ ♣r♦✈❡ ✐t ❢♦r n + 1✳ ❲❡
  ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s ✿
  τ.
   τ
  
  ❼ P −
  → P ′ ✳ ❋♦r ❛❧❧ Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′ ✱ ✇❡ ❤❛✈❡ P ′ ≈
  6 k Q′ ✳ ❙✐♥❝❡ Q
  τ
  ′
  ′
  ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t {Qi |Q =
  ⇒ Qi } ✐s ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡
  ❝♦♥t❡①ts Ki ❛♥❞ ♥❛♠❡s di s✉❝❤ t❤❛t Ki {P ′ } + di 6≈b Ki {Q′i } + di ❢♦r ❛❧❧ i✳
  
  ❲❡ ❞❡✜♥❡✿
  
  K = a[✷] | a(X)(s +
  
  X
  
  τ.(Ki {X} + di ))
  
  i
  
  ✇❤❡r❡ a, s ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ▲❡t t ❜❡ ❛ ❢r❡s❤ ♥❛♠❡✳ ❆ss✉♠❡ t❤❛t
  K{P } + t ≈b K{Q} + t✳ ❙✐♥❝❡ P −→ P ′ ✱ ✇❡ ❤❛✈❡
  K{P } + t −→ a[P ′ ] | a(X)(s +
  
  X
  
  τ.(Kj {X} + dj )) = R1
  
  j
  
  ❙✐♥❝❡ ¬R1 ↓t ❛♥❞ ¬R1 ↓a ✭t❤❡ ♣❛ss✐✈❛t✐♦♥ ♦❢ ❧♦❝❛❧✐t② a ✐s ♥♦t tr✐❣❣❡r❡❞✮✱
  ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜②
  K{Q} + t =⇒ a[Q′l ] | a(X)(s +
  
  X
  
  τ.(Kj {X} + dj )) = S1
  
  j
  
  ❢♦r s♦♠❡ l✳ ❲❡ ♥♦✇ ❤❛✈❡
  R1 −→ s +
  
  X
  
  τ.(Kj {P ′ } + dj ) = R2
  
  j
  
  ❙✐♥❝❡ ✇❡ ❤❛✈❡ ¬R2 ↓a ❛♥❞ R2 ↓s ✱ ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜②✿
  S1 =⇒ s +
  
  X
  
  τ.(Kj {Q′i } + dj ) = S2
  
  j
  
  ✇✐t❤ Q′l =⇒ Q′i ❢♦r s♦♠❡ i✳ ❲❡ ❤❛✈❡ R2 −→ Ki {P ′ } + di = R3 ✱ ✇❤✐❝❤
  ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜② S2 =⇒ Ki {Q′i } + di = S3 ✱ s✐♥❝❡ R3 ↓di ✳ ❍❡♥❝❡ ❛
  ❝♦♥tr❛❞✐❝t✐♦♥✱ s✐♥❝❡ Ki {P ′ } + di 6≈b Ki {Q′i } + di ✱ s♦ ✇❡ ❤❛✈❡ K{P } + t 6≈b
  K{Q} + t ❛s r❡q✉✐r❡❞✳
  ❘❘ ♥➦ ✻✻✻✹
  
  ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐
  
  ✽✹
  m.
   m
  
  ❼ P −→ P ′ ❋♦r ❛❧❧ Q′ s✉❝❤ t❤❛t Q =
  ⇒ Q′ ✱ ✇❡ ❤❛✈❡ P ′ ≈
  6 k Q′ ✳ ❙✐♥❝❡ Q
  m
  ′
  ′
  ⇒ Qi } ✐s ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡
  ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡t {Qi |Q =
  ❝♦♥t❡①ts Ki ❛♥❞ ♥❛♠❡s di s✉❝❤ t❤❛t Ki {P ′ } + di 6≈b Ki {Q′i } + di ❢♦r ❛❧❧ i✳
  
  ❲❡ ❞❡✜♥❡ ✿
  
  K = a[✷] | m.a(X)(s +
  
  X
  
  τ.(Ki {X} + di ))
  
  i
  
  ✇❤❡r❡ s, a ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ▲❡t t ❜❡ ❛ ❢r❡s❤ ♥❛♠❡✳ ❆ss✉♠❡ t❤❛t
  K{P } + t ≈b K{Q} + t✳ ❲❡ ❤❛✈❡
  K{P } + t −→ a[P ′ ] | a(X)(s +
  
  X
  
  τ.(Kj {X} + dj )) = R1
  
  j
  
  ❙✐♥❝❡ R1 ↓a ✱ ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜②
  K{Q} + t =⇒ a[Q′l ] | a(X)(s +
  
  X
  
  τ.(Kj {X} + dj )) = S1
  
  j
  
  ❋r♦♠ ❤❡r❡✱ t❤❡ ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ❝❛s❡✳
  m
  
  ❼ P −→ P ′ ❙✐♠✐❧❛r t♦ t❤❡ ❝❛s❡ ❛❜♦✈❡✱ ✇✐t❤ m ✐♥st❡❛❞ ♦❢ m ✐♥ K✳
  a
  
  ❼ P −
  → F = (X)V ✳ ❚❤❡r❡ ❡①✐sts ❛ ♣r♦❝❡ss T s✉❝❤ t❤❛t ❢♦r ❛❧❧ F ′ ✱ Q′ s✉❝❤.
   a
  τ
  ⇒ F ′ ❛♥❞ F ′ ◦ T =
  ⇒ Q′ ✱ ✇❡ ❤❛✈❡ F ◦ T 6≈k Q′ ✳ ❙✐♥❝❡ Q ✐s ✐♠❛❣❡✲
  t❤❛t Q =
  a
  a
  τ
  ✜♥✐t❡✱ t❤❡ s❡ts {Fi′ |Q =
  ⇒ Fi′ } ❛♥❞ {Q′i |∃Fi , Q =
  ⇒ Fi ❛♥❞ Fi ◦ T =
  ⇒ Q′i }
  ❛r❡ ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡ ❝♦♥t❡①ts Ki ❛♥❞ ♥❛♠❡s (di ) s✉❝❤ t❤❛t
  Ki {Q′i } + di 6≈b Ki {F ◦ T } + di ✳ ❲❡ ❞❡✜♥❡✿
  K = b[✷] | ahT ib(X)(s +
  
  X
  
  τ.(Ki {X} + di ))
  
  i
  
  ✇❤❡r❡ b, s ❛r❡ ❞✐st✐♥❝t✱ ❛♥❞ ❞♦ ♥♦t ♦❝❝✉r ✐♥ T, P, Q✳ ▲❡t t ❜❡ ❛ ❢r❡s❤ ♥❛♠❡✳
  ❲❡ ❤❛✈❡
  K{P } + t −→ b[F ◦ T ] | b(X)(s +
  
  X
  
  τ.(Kj {X} + dj )) = R1
  
  j
  
  ❙✐♥❝❡ R1 ↓b ✱ ✐t ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ ❜② ❛
  K{Q} + t =⇒ b[Q′l ] | b(X)(s +
  
  X
  
  τ.(Kj {X} + dj )) = S1
  
  j
  τ
  
  ❢♦r s♦♠❡ l✳ ❋r♦♠ ❤❡r❡✱ t❤❡ ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ t❤❡ P −
  → P ′ ❝❛s❡✳
  a
  
  a
  
  τ
  
  ❼ P −
  → hRiS ✳ ❋♦r ❛❧❧ hR′ iS ′′ ✱ S ′ s✉❝❤ t❤❛t Q =
  ⇒ hR′ iS ′′ ❛♥❞ S ′′ =
  ⇒ S ′ ✱ ✇❡.
   a ′ ′ ❤❛✈❡ R ≈ 6 k R ♦r S ≈ 6 k S ✳ ❙✐♥❝❡ Q ✐s ✐♠❛❣❡✲✜♥✐t❡✱ t❤❡ s❡ts {Ci |Q = ⇒ Ci } τ a ⇒ hRi′ iSi′ ❛♥❞ Si′ = ⇒ Si,j } ❛r❡ ✜♥✐t❡✳ ❇② ✐♥❞✉❝t✐♦♥✱ ❛♥❞ {Si,j |∃hRi′ iSi′ , Q = ■◆❘■❆ ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s ✽✺ ❢♦r i ∈ I s✉❝❤ t❤❛t R 6≈b Ri ✱ t❤❡r❡ ❡①✐sts ❝♦♥t❡①ts Ki ❛♥❞ ♥❛♠❡s di s✉❝❤ t❤❛t Ki {R} + di 6≈b Ki {Ri } + di ✳ ❖t❤❡r✇✐s❡✱ ❢♦r i ∈ J s✉❝❤ t❤❛t R ≈b Ri ✱ t❤❡r❡ ❡①✐sts ❝♦♥t❡①ts Ki,j ❛♥❞ ♥❛♠❡s di,j s✉❝❤ t❤❛t Ki,j {S} + di,j 6≈b Ki,j {Si,j } + di,j ❢♦r ❛❧❧ j ✳ ❲❡ ❞❡✜♥❡✿ K = b[✷] | a(X)b(Y )(s + X τ.(Ki {X} + di ) + i∈I XX i∈J τ.(Ki,j {Y } + di,j )) j ✇❤❡r❡ b, s ❛r❡ ❞✐st✐♥❝t ❛♥❞ ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ▲❡t t ❜❡ ❛ ❢r❡s❤ ♥❛♠❡✳ ❲❡ ❤❛✈❡✿ K{P } + t −→ b[S] | b(Y )(s + ✇✐t❤ K = ′ P i∈J X τ.(Ki {R} + di ) + K ′ ) = P2 i∈I P j τ.(Ki,j {Y } + di,j )✳ ❙✐♥❝❡ P2 ↓b ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜② K{Q} + t =⇒ b[Sl,m ] | b(Y )(s + X τ.(Ki {Rl } + di ) + K ′ ) = Q2 i∈I ❢♦r s♦♠❡ l, m✳ ◆♦✇ ✇❡ ❤❛✈❡✿ P2 −→ s + X τ.(Ki {R} + di ) + i∈I XX τ.(Ki,j {S} + di,j ) = P3 j i∈J ❙✐♥❝❡ P3 ↓s ✱ ✐t ✐s ♠❛t❝❤❡❞ ❜②✿ Q2 =⇒ s + X τ.(Ki {Rl } + di ) + i∈I XX i∈J τ.(Ki,j {Sl,n } + di,j ) = Q3 j ❢♦r s♦♠❡ n s✉❝❤ t❤❛t Sl,m =⇒ Sl,n ✳ ❲❡ s✉♣♣♦s❡ t❤❛t l ∈ I ✱ ✐✳❡✳ ✇❡ ❤❛✈❡ R 6≈b Rl ✳ ❲❡ ❤❛✈❡ P3 −→ Kl {R} + dl = P4 ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜②✿ Q3 =⇒ Kl {Rl } + dl = Q4 s✐♥❝❡ P4 ↓dl ✳ ❲❡ ❤❛✈❡ P4 6≈b Q4 ❜② ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s✱ ❤❡♥❝❡ ✇❡ ❤❛✈❡ K{P } + t 6≈b K{Q} + t ❛s r❡q✉✐r❡❞✳ ❲❡ s✉♣♣♦s❡ ♥♦✇ t❤❛t l ∈ J ✱ ✐✳❡✳ ✇❡ ❤❛✈❡ R ≈b Rl ✳ ❲❡ ❤❛✈❡ P3 −→ Kl,n {S} + dl,n = P4 ✇❤✐❝❤ ✐s ♠❛t❝❤❡❞ ❜②✿ Q3 =⇒ Kl,n {Sl,n } + dl,n = Q4 s✐♥❝❡ P4 ↓dl,n ✳ ❲❡ ❤❛✈❡ P4 6≈b Q4 ❜② ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s✱ ❤❡♥❝❡ ✇❡ ❤❛✈❡ K{P } + t 6≈b K{Q} + t ❛s r❡q✉✐r❡❞✳ ❘❘ ♥➦ ✻✻✻✹ ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐ ✽✻ ❋ ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥ ❋✳✶ ❙tr♦♥❣ ❝❛s❡ α α A✳ ❚❤❡♥ E{P } − → E′ {A} ▲❡t E ❜❡ ❛♥ ❡✈♦❧✉t✐♦♥ ❝♦♥t❡①t ❛♥❞ P −→ ′ ❛♥❞ t❤❡ ❤♦❧❡ ✐♥ E ✐s ♥♦t ✉♥❞❡r ❛ r❡♣❧✐❝❛t✐♦♥ ♦r ❝❤♦✐❝❡ ♦♣❡r❛t♦r✳ Pr♦♦❢✳ ■♠♠❡❞✐❛t❡ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ E✱ ❛♥❞ ❝♦♥s✐❞❡r✐♥❣ t❤❡ r✉❧❡s ▲❚❙✲P❛r✱ ▲❚❙✲ ▲❡♠♠❛ ✸✾✳ ▲♦❝✱ ▲❚❙✲❘❡♣❧✐❝✱ ▲❚❙✲❙✉♠✳ ▲❡♠♠❛ ✹✵✳ ▲❡t P, Q s✉❝❤ t❤❛t ❢✈(P, Q) ⊆ {X} ❛♥❞ m, n t✇♦ ♥❛♠❡s ✇❤✐❝❤ ❞♦ ♥♦t ♦❝❝✉r ✐♥m P, Q✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ P {m.n.0/X} ∼. ′
  Q {m.n.0/X}{n.0/Y } = Qn ✇✐t❤ Pn ∼l Qn ✳ ❖♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿.
   ❼ ❚❤❡r❡ ❡①✐sts P1 , Q1 s✉❝❤ t❤❛t Pn = n.0 | P1 ✱ Qn = n.0 | Q1 ✇✐t❤ P1 ∼l Q1 ✳
  ❼ ❚❤❡r❡ ❡①✐sts a1 , . . . ak ✱ P1 . . . Pk+1 ✱ Q1 . . . Qk+1 s✉❝❤ t❤❛t
  
  Pn = a1 [. . . ak−1 [ak [n.0 | Pk+1 ] | Pk ] | Pk−1 . . .] | P1
  
  ❛♥❞
  
  Qn = a1 [. . . ak−1 [ak [n.0 | Qk+1 ] | Qk ] | Qk−1 . . .] | Q1
  
  ❛♥❞ ❢♦r ❛❧❧ 1 ≤ j ≤ k + 1✱ Pj ∼. l Qj ✳
  n
  tr❛♥s✐t✐♦♥✱ ✇❡ ❝❛♥ ❞❡t❡❝t ✐❢ n.0 ✐s ✐♥ ❛
  Pr♦♦❢✳ ❙✐♥❝❡ Pn ❝❛♥ ♦♥❧② ♣❡r❢♦r♠ ♦♥❡ −→
  a
  
  → hR′ niSn′ ❢♦r s♦♠❡ a s✉❝❤ t❤❛t
  ❧♦❝❛❧✐t② ♦r ♥♦t✿ ✐❢ t❤❡r❡ ❡①✐sts ❛ tr❛♥s✐t✐♦♥ Pn −
  n
  ′
  Rn ♠❛② ♣❡r❢♦r♠ ❛ tr❛♥s✐t✐♦♥ −
  →✱ t❤❡♥ t❤❡ tr❛♥s✐t✐♦♥ ✐s ❛ ♣❛ss✐✈❛t✐♦♥ ❛♥❞ t❤❡
  ♣r♦❝❡ss n.0 ✐s ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥ Pn ✳ ❖t❤❡r✇✐s❡✱ n.0 ✐s ♥♦t ✐♥ ❛ ❧♦❝❛❧✐t②✳
  ❇② ❧❡♠♠❛ ✸✾✱ n.0 ✐s ♦♥❧② ✉♥❞❡r ❧♦❝❛❧✐t✐❡s ❛♥❞ ♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥s ✐♥ Pn
  ❛♥❞ Qn ✳
  ❲❡ s❤♦✇ t❤❛t ✐❢ n.0 ✐s ♥♦t ✉♥❞❡r ❛ ❧♦❝❛❧✐t② ✐♥ Pn ✱ ✐t ✐s ❛❧s♦ ♥♦t ✉♥❞❡r ❛
  ❧♦❝❛❧✐t② ✐♥ Qn ✳ ❙✉♣♣♦s❡ n.0 ✐s ♥♦t ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥ Pn ❛♥❞ ✐s ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥
  a
  Qn ✳ ❲❡ ❤❛✈❡ Qn −
  → hE{n.0}iQ′′ ❢♦r s♦♠❡ a, E, Q′′ ✳ ❚❤❡s❡ tr❛♥s✐t✐♦♥s ❝❛♥ ♦♥❧②
  ❜❡ ♠❛t❝❤❡❞ ❜② ❛ ♣❛ss✐✈❛t✐♦♥ ♦❢ n.0 ✐♥ Pn ✱ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡ ❜② ❤②♣♦t❤❡s✐s✱
  ❤❡♥❝❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❲❡ ❤❛✈❡ t❤❡ s❛♠❡ r❡❛s♦♥✐♥❣ ✐❢ n.0 ✐s ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥ Pn
  ❛♥❞ ♥♦t ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥ Qn ✳ ❚❤❡r❡❢♦r❡ ✐❢ n.0 ✐s ♥♦t ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥ Pn ✱ ✐t ✐s
  ♥♦t ✐♥ ❛ ❧♦❝❛❧✐t② ✐♥ Qn ✳ ❈♦♥s❡q✉❡♥t❧② ✐♥ t❤✐s ❝❛s❡✱ t❤❡r❡ ❡①✐sts P1 , Q1 s✉❝❤ t❤❛t
  n
  Pn = n.0 | P1 ❛♥❞ Qn = n.0 | Q1 ✳ ❍❡♥❝❡ ✇❡ ❤❛✈❡ Pn −
  → P1 ✱ ✇❤✐❝❤ ❝❛♥ ♦♥❧② ❜❡
  n.
   → Q1 ✱ s♦ ✇❡ ❤❛✈❡ P1 ∼l Q1 ✳
  ♠❛t❝❤❡❞ ❜② Qn −
  ❲❡ s✉♣♣♦s❡ ♥♦✇ t❤❛t n.0 ✐s ✉♥❞❡r ❛ ❧♦❝❛❧✐t② ✐♥ Pn ❛♥❞ Qn ✳ ❲❡ ♣r♦✈❡ t❤❛t
  n.0 ✐s ✉♥❞❡r t❤❡ s❛♠❡ ❤✐❡r❛r❝❤② ♦❢ ❧♦❝❛❧✐t✐❡s ✐♥ Pn , Qn ✱ ❛♥❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢
  t❤❡ ♣❛✐r✇✐s❡ ❜✐s✐♠✐❧❛r ♣r♦❝❡ss❡s ❞❡✜♥❡❞ ✐♥ t❤❡ ❧❡♠♠❛✳ ❙✉♣♣♦s❡ n.0 ✐s ✉♥❞❡r
  k ❧♦❝❛❧✐t✐❡s a1 , . . . ak ✐♥ Pn ❛♥❞ ✉♥❞❡r l ❧♦❝❛❧✐t✐❡s b1 , . . . bl ✐♥ Qn ✱ ✇✐t❤ k >
  a1
  bi
  l✳ ❲❡ ❤❛✈❡ Pn −→
  hP1′ {n.0/Xi }iP1 ✱ s♦ t❤❡r❡ ❡①✐sts Q1 , Q′1 s✉❝❤ t❤❛t Qn −→
  
  ■◆❘■❆
  
  ✽✼
  
  ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s.
  hQ′1 {n.0/Xj }iQ1 ✇✐t❤ a1 = bi ❛♥❞ P1′ {n.0/Xi } ∼l Q′1 {n.0/Xj }✳ ❚❤❡ ♣r♦❝❡ss ✐s
  ✉♥❞❡r k − 1 ❧♦❝❛❧✐t✐❡s ✐♥ P1′ ❛♥❞ ✉♥❞❡r l − i ❧♦❝❛❧✐t✐❡s ✐♥ Q′1 ✱ ✇✐t❤ i ≥ 1✳ ❆❢t❡r l
  ♣❛ss✐✈❛t✐♦♥✱ ✇❡ ❤❛✈❡ Pl′ s✉❝❤ t❤❛t t❤❡ ♣r♦❝❡ss n.0 ✐s ✉♥❞❡r k − l ❧♦❝❛❧✐t✐❡s✱ ❛♥❞ ❛.
   ♣r♦❝❡ss Q′l s✉❝❤ t❤❛t t❤❡ ♣r♦❝❡ss n.0 ✐s ♥♦t ✉♥❞❡r ❛ ❧♦❝❛❧✐t② ❛♥❞ ✇✐t❤ Pl′ ∼l Q′l ✱
  ✇❤✐❝❤ ✐s ♥♦t ♣♦ss✐❜❧❡ ✭s❛♠❡ ♣r♦♦❢ ❛s ✐♥ t❤❡ ✜rst ❝❛s❡✮✳ ■❢ k < l✱ ✇❡ ❤❛✈❡ ❛ s✐♠✐❧❛r
  ❝♦♥tr❛❞✐❝t✐♦♥ ❜② r❡❛s♦♥✐♥❣ ♦♥ Q✱ ❝♦♥s❡q✉❡♥t❧② ✇❡ ❤❛✈❡ k = l✳
  ❚❤❡r❡❢♦r❡ t❤❡r❡ ❡①✐sts a1 . . . ak ✱ P1 . . . Pk ✱ Q1 . . . Qk ✱ s✉❝❤ t❤❛t Pn =
  a1 [. . . ak−1 [ak [n.0 | Pk+1 ] | Pk ] | Pk−1 . . .] | P1 ❛♥❞ Qn = a1 [. . . ak−1 [ak [n.0 |
  Qk+1 ] | Qk ] | Qk−1 . . .] | Q1 ✳ ▲❡t Pi′ ✭r❡s♣ Q′i ✮ ❜❡ t❤❡ ♣r♦❝❡ss ✐♥s✐❞❡ t❤❡ ❧♦✲
  a
  
  n
  
  1
  →✱ ✇❤✐❝❤ ✐s
  hP1′ iP1 ✱ ✇✐t❤ P1′ −
  ❝❛❧✐t② ai ✐♥ Pn ✭r❡s♣ Qn ✮✳ ❲❡ ❤❛✈❡ Pn −→
  ai.
  ′
  ′
  ′
  ♠❛t❝❤❡❞ ❜② ❛ ♣❛ss✐✈❛t✐♦♥ Qn −→ hQi iQ s✉❝❤ t❤❛t P1 ∼l Q ✱ P1′ ∼l Q′i ❛♥❞
  n
  Q′i −
  →✳ ■❢ i 6= 1✱ ✇❡ ❤❛✈❡ t❤❡ ♣r♦❝❡ss ✉♥❞❡r k − 1 ❧♦❝❛❧✐t✐❡s ✐♥ P1′ ❛♥❞ ✐♥.
  k − i < k − 1 ❧♦❝❛❧✐t✐❡s ✐♥ Q′i ✱ ✇✐t❤ Pi′ ∼l Q′i ✿ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❍❡♥❝❡ ✇❡ ❤❛✈❡.
  i = 1✱ P1 ∼l Q′ = Q1 ❛♥❞ P1′ ∼l Q′1 ✳ ❇② ✐♥❞✉❝t✐♦♥ ♦♥ 1 ≤ j ≤ k ✱ ✇❡ ❤❛✈❡.
  Pj ∼l Qj ❛♥❞ Pk′ = n.0 | Pk+1 ∼l n.0 | Qk+1 = Q′k ✳ ❙✐♥❝❡ t❤❡ r❡❞✉❝t✐♦♥
  n
  n.
   Pk′ −
  → Pk+1 ❝❛♥ ♦♥❧② ❜❡ ♠❛t❝❤❡❞ Q′k −
  → Qk+1 ✱ ✇❡ ❤❛✈❡ Pk+1 ∼l Qk+1 ✱ ❝♦♥s❡✲
  q✉❡♥t❧② ✇❡ ❤❛✈❡ t❤❡ r❡q✉✐r❡❞ r❡s✉❧t✳
  
  ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ✇r✐t❡ Xi t❤❡ i✲t❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ✐♥ ❛ ♣r♦❝❡ss P ✳
  ▲❡♠♠❛ ✹✶✳ ▲❡t P, Q t✇♦ ♦♣❡♥ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t ❢✈(P, Q) ⊆ {X} ❛♥❞ m, n
  t✇♦ ♥❛♠❡s ✇❤✐❝❤ ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ▲❡t R, R′ t✇♦ ❝❧♦s❡❞ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t.
  m
  R ∼l R′ ✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ P {m.n.0/X} ∼l Q{m.n.0/X} ❛♥❞ P {m.n.0/X} −→
  
  m
  
  P ′ {m.n.0/X}{n.0/Xi } = Pn ✐s ♠❛t❝❤❡❞ ❜② t❤❡ tr❛♥s✐t✐♦♥ Q{m.n.0/X} −→.
  Q′ {m.n.0/X}{n.0/Xj } = Qn ✭✇✐t❤ Pn ∼l Qn ✮✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤❡ r❡❧❛t✐♦♥.
  ′
  ′
  ′
  P {m.n.0/X}{R/Xi } ∼l Q {m.n.0/X}{R /Xj }✳
  Pr♦♦❢✳ ❇② ❧❡♠♠❛ ✹✵✱ ✇❡ ❤❛✈❡ t✇♦ ❝❛s❡s t♦ ❝♦♥s✐❞❡r✿.
  ❼ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ Pn = n.0 | P1 ✱ Qn = n.0 | Q1 ✇✐t❤ P1 ∼l Q1 ✳ ❙✐♥❝❡.
  P1 ∼l Q1 ✱ R ∼l R′ ❛♥❞ ∼l ✐s ❛ ❝♦♥❣r✉❡♥❝❡ ✇❡ ❤❛✈❡ R | P1 ∼l R′ | Q1 ❜②
  
  tr❛♥s✐t✐✈✐t②✱ ❝♦♥s❡q✉❡♥t❧② t❤❡ r❡s✉❧t ❤♦❧❞s✳
  
  ❼ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ Pn = a1 [. . . ak−1 [ak [n.0 | Pk+1 ] | Pk ] | Pk−1 . . .] | P1
  ❛♥❞ Qn = a1 [. . . ak−1 [ak [n.0 | Qk+1 ] | Qk ] | Qk−1 . . .] | Q1 ❛♥❞ ❢♦r ❛❧❧.
  1 ≤ j ≤ k + 1✱ Pj ∼l Qj ✳ ❙✐♥❝❡ Pk+1 ∼l Qk+1 ✱ R ∼l R′ ✱ ∼l ✐s ❛.
  ❝♦♥❣r✉❡♥❝❡ ❛♥❞ ✐s tr❛♥s✐t✐✈❡✱ ✇❡ ❤❛✈❡ R | Pk+1 ∼l R′ | Qk+1 ✳ ❙♦ ✇❡ ❤❛✈❡.
   ak [R | Pk+1 ] | Pk ∼l ak [R′ | Qk+1 ] | Qk ✳ ❇② ✐♥❞✉❝t✐♦♥ ♦♥ 1 ≤ j ≤ k ✱ ✇❡.
  ❤❛✈❡ aj [. . . ak [R | Pk+1 ] | Pk . . .] | Pj ∼l aj [. . . ak [R′ | Qk+1 ] | Qk . . .] | Qj ✱
  s♦ ✇❡ ❤❛✈❡ t❤❡ r❡q✉✐r❡❞ r❡s✉❧t ✇✐t❤ j = 1✳
  
  ▲❡t P, Q t✇♦ ♦♣❡♥ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t ❢✈(P, Q) ⊆ {X} ❛♥❞ m, n.
  t✇♦ ♥❛♠❡s ✇❤✐❝❤ ❞♦ ♥♦t ♦❝❝✉r ✐♥ P, Q✳ ■❢ P {m.n.0/X} ∼l Q{m.n.0/X}✱ t❤❡♥.
  ❢♦r ❛❧❧ ❝❧♦s❡❞ ♣r♦❝❡ss❡s R✱ ✇❡ ❤❛✈❡ P {R/X} ∼l Q{R/X}
  ❚❤❡♦r❡♠ ✶✷✳
  
  ❘❘ ♥➦ ✻✻✻✹
  
  ▲❡♥❣❧❡t✱ ❙❝❤♠✐tt ✫ ❙t❡❢❛♥✐
  
  ✽✽
  
  Pr♦♦❢✳ ❲❡ s❤♦✇ t❤❛t t❤❡ r❡❧❛t✐♦♥ R= {(P {R/X}, Q{R/X}), P {m.n.0/X} ∼. l
  
  Q{m.n.0/X}, m, n ♥♦t ✐♥ P, Q} ✐s ❛ str♦♥❣ ❜✐s✐♠✉❧❛t✐♦♥✳ ❙✐♥❝❡ t❤❡ r❡❧❛t✐♦♥ ✐s
  s②♠♠❡tr✐❝❛❧✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❛t ✐t ✐s ❛ s✐♠✉❧❛t✐♦♥✳
  
  ❲❡ ♠❛❦❡ ❛ ❝❛s❡
  
  ❛♥❛❧②s✐s ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ P {R/X}✿
  α
  
  ❚❤❡ tr❛♥s✐t✐♦♥ ❝♦♠❡s ♦♥❧② ❢r♦♠ P ✳
  
  α
  
  → A✳
  P −
  
  α
  
  ❲❡ ❤❛✈❡ P {R/X} −
  → A{R/X} ✇✐t❤
  
  ❍❡♥❝❡ ✇❡ ❤❛✈❡ P {m.n.0/X} −
  → A{m.n.0/X}✳ ❲❡ ❞✐st✐♥❣✉✐s❤ t❤❡
  
  t❤r❡❡ ❝❛s❡s ❢♦r A✿.
  ❼ Pr♦❝❡ss ❝❛s❡ P ′ ✳ ❙✐♥❝❡ P {m.n.0/X} ∼l Q{m.n.0/X}✱ t❤❡r❡ ❡①✐sts Q′ s✉❝❤
  α.
   → Q′ ❛♥❞ P ′ {m.n.0/X} ∼l Q′ ✳ ❙✐♥❝❡ m ❞♦❡s ♥♦t ♦❝❝✉r
  t❤❛t Q{m.n.0/X} −
  α
  ✐♥ P, Q✱ ✇❡ ❤❛✈❡ α 6= m✱ s♦ t❤❡ tr❛♥s✐t✐♦♥ Q{m.n.0/X} −
  → Q′ ❝♦♠❡s ♦♥❧②
  ′
  ′
  ′′
  ❢r♦♠ Q✳ ❚❤❡r❡❢♦r❡ Q ❝❛♥ ❜❡ ✇r✐tt❡♥ Q = Q {m.n.0/X} ❢♦r s♦♠❡ Q′′ ✱
  α
  ❛♥❞ ✇❡ ❤❛✈❡ Q{R/X} −
  → Q′′ {R/X}✳ ❲❡ ❤❛✈❡ P ′ {R/X} R Q′′ {R/X}✱
  ❤❡♥❝❡ t❤❡ r❡s✉❧t ❤♦❧❞s✳.
  ❼ ❆❜str❛❝t✐♦♥ ❝❛s❡ F ✳ ❙✐♥❝❡ P {m.n.0/X} ∼l Q{m.n.0/X}✱ t❤❡r❡ ❡①✐sts
  α.
  F ′ s✉❝❤ t❤❛t Q{m.n.0/X} −
  → F ′ ❛♥❞ (F {m.n.0/X}) ◦ T ∼l F ′ ◦ T
  ❢♦r ❛❧❧ ♣r♦❝❡ss❡s T ✳ ❙✐♥❝❡ t❤❡ tr❛♥s✐t✐♦♥ ✐s ♦♥ ❛ ❤✐❣❤❡r✲♦r❞❡r ♥❛♠❡✱ ✇❡
  α
  → F ′ ❝♦♠❡s ♦♥❧② ❢r♦♠ Q✳
  ❤❛✈❡ α 6= m✱ s♦ t❤❡ tr❛♥s✐t✐♦♥ Q{m.n.0/X} −
  ❚❤❡r❡❢♦r❡ F ′ ❝❛♥ ❜❡ ✇r✐tt❡♥ F ′ = F ′′ {m.n.0/X} ❢♦r s♦♠❡ F ′′ ✱ ❛♥❞ ✇❡ ❤❛✈❡
  α
  Q{R/X} −
  → F ′′ {R/X}✳ ❙✐♥❝❡ T ✐s ❛ ❝❧♦s❡❞ ♣r♦❝❡ss✱ ✇❡ ❤❛✈❡ (F {R/X}) ◦
  T = (F ◦ T ){R/X} R (F ′′ ◦ T ){R/X} = (F ′′ {R/X}) ◦ T ✱ ❤❡♥❝❡ t❤❡
  r❡s✉❧t ❤♦❧❞s✳.
  ❼ ❈♦♥❝r❡t✐♦♥ ❝❛s❡ C = hT iS ✳ ❙✐♥❝❡ P {m.n.0/X} ∼l Q{m.n.0/X}✱ t❤❡r❡
  α.
  ❡①✐sts C ′ = hT ′ iS ′ s✉❝❤ t❤❛t Q{m.n.0/X} −
  → C ′ ✱ T {m.n.0/X} ∼l.
   ′
  ′
  T ❛♥❞ S{m.n.0/X} ∼l S ✳ ❲❡ ❤❛✈❡ α 6= m✱ s♦ t❤❡ tr❛♥s✐t✐♦♥
  α
  → C ′ ❝♦♠❡s ♦♥❧② ❢r♦♠ Q✳ ❚❤❡r❡❢♦r❡ T ′ , S ′ ❝❛♥ ❜❡ ✇r✐t✲
  Q{m.n.0/X} −
  ′
  ′′
  t❡♥ T = T {m.n.0/X} ❛♥❞ S ′ = S ′′ {m.n.0/X} ❢♦r s♦♠❡ T ′′ , S ′′ ✱ ❛♥❞ ✇❡
  α
  → (hT ′′ iS ′′ ){R/X}✳ ❲❡ ❤❛✈❡ T {R/X} R T ′′ {R/X} ❛♥❞
  ❤❛✈❡ Q{R/X} −
  S{R/X} R S ′′ {R/X}✱ ❤❡♥❝❡ t❤❡ r❡s✉❧t ❤♦❧❞s✳
  ❚❤❡ tr❛♥s✐t✐♦♥ ❝♦♠❡s ♦♥❧② ❢r♦♠ R✳
  
  t❡①t ❛♥❞ ♣❡r❢♦r♠ ❛ tr❛♥s✐t✐♦♥✳
  
  ❝♦♣② ♦❢ R ♣❡r❢♦r♠s t❤❡ tr❛♥s✐t✐♦♥✳
  α
  
  ❆ ❝♦♣② ♦❢ R ✐s ✐♥ ❛♥ ❡✈❛❧✉❛t✐♦♥ ❝♦♥✲
  
  ❲❡ ✇r✐t❡ Xi t❤❡ ♦❝❝✉rr❡♥❝❡ ♦❢ X ✇❤❡r❡ t❤❡
  α
  
  → P ′ {R/X}{A/Xi }
  ❲❡ ❤❛✈❡ P {R/X} −
  m
  
  ✇✐t❤ R −
  → A✳ ❙✐♥❝❡ Xi ✐s ✐♥ ❛♥ ❡✈❛❧✉❛t✐♦♥ ❝♦♥t❡①t✱ ✇❡ ❤❛✈❡ P {m.n.0/X} −→.
  ′
  
  P {m.n.0/X}{n.0/Xi }✳ ❙✐♥❝❡ ✇❡ ❤❛✈❡ P {m.n.0/X} ∼l Q{m.n.0/X}✱ t❤❡r❡
  m
  ❡①✐sts ❛ tr❛♥s✐t✐♦♥ Q{m.n.0/X} −
  → Q′ {m.n.0/X}{n.0/Xj } ✭❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢.
  X ✱ ♥♦t❡❞ Xj ✱ ✐s ✐♥ ❛♥ ❡✈❛❧✉❛t✐♦♥ ❝♦♥t❡①t ✐♥ Q✮ ✇✐t❤ P ′ {m.n.0/X}{n.0/Xi } ∼l
  α
  Q′ {m.n.0/X}{n.0/Xj }✳ ❈♦♥s❡q✉❡♥t❧② ✇❡ ❤❛✈❡ Q{R/X} −
  → Q′ {R/X}{A/Xj }✳
  ❲❡ ❞✐st✐♥❣✉✐s❤ t❤r❡❡ ❝❛s❡s ❢♦r A✿.
  ❼ Pr♦❝❡ss ❝❛s❡ R′ ✳ ❲❡ ❤❛✈❡ P ′ {m.n.0/X}{R′ /Xi } ∼l Q′ {m.n.0/X}{R′ /Xj }
  ❜② ❧❡♠♠❛ ✹✶✱ s♦ ✇❡ ❤❛✈❡ P ′ {R/X}{R′ /Xi } R Q′ {R/X}{R′ /Xj } ❛s r❡✲
  q✉✐r❡❞✳
  
  ■◆❘■❆
  
  ◆♦r♠❛❧ ❜✐s✐♠✉❧❛t✐♦♥s
  
  ✽✾.
   ❼ ❆❜str❛❝t✐♦♥ ❝❛s❡ F ✳ ❇② ❧❡♠♠❛ ✹✶✱ ✇❡ ❤❛✈❡ P ′ {m.n.0/X}{F ◦ T /Xi } ∼l
  Q′ {m.n.0/X}{F ◦ T /Xj } ❢♦r ❛❧❧ T ✳ ❲❡ ❤❛✈❡ (P ′ {R/X}{F/Xi }) ◦ T =
  P ′ {R/X}{F ◦ T /Xi } R Q′ {R/X}{F ◦ T /Xi } = (Q′ {R/X}{F/Xj }) ◦ T
  
  ❛s r❡q✉✐r❡❞✳.
  ❼ ❈♦♥❝r❡t✐♦♥ ❝❛s❡ hSiT ✳ ❇② ❧❡♠♠❛ ✹✶✱ ✇❡ ❤❛✈❡ P ′ {m.n.0/X}{T /Xi } ∼l
  Q′ {m.n.0/X}{T /Xj }✱ s♦ ✇❡ ❤❛✈❡ P ′ {R/X}{T /Xi } R Q′ {R/X}{T /Xj }✳.
  ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ S ∼l S ✱ ❛♥❞ s✐♥❝❡ ∼l ⊆R ✭✇✐t❤ P, Q ❝❧♦s❡❞ ♣r♦❝❡ss❡s✮✱
  ′
  ✇❡ ❤❛✈❡ S R S ❛♥❞ P {R/X}{T /Xi } R Q′ {R/X}{T /Xj } ❛s r❡q✉✐r❡❞✳
  
  ❆ ❝♦♣②
  ♦❢ R ✐s ✐♥ ❛♥ ❡✈❛❧✉❛t✐♦♥ ❝♦♥t❡①t ❛♥❞ ❝♦♠♠✉♥✐❝❛t❡ ✇✐t❤ ❛ s✉❜✲♣r♦❝❡ss P ′ ♦❢ P ✳
  ❲❡ ❤❛✈❡ t✇♦ ❝❛s❡s t♦ ❝♦♥s✐❞❡r✳
  
  ❆ ❤✐❣❤❡r✲♦r❞❡r ❝♦♠♠✉♥✐❝❛t✐♦♥ t❛❦❡s ♣❧❛❝❡ ❜❡t✇❡❡♥ R ❛♥❞ P ✳
  
  a
  
  a
  
  → hT {R/X}iS{R/X} ❢♦r s♦♠❡ a✳
  ❚❤❡ ✜rst ♣♦ss✐❜✐❧✐t② ✐s R −
  → F ❛♥❞ P ′ −
  ❲❡ ❤❛✈❡ t❤❡ tr❛♥s✐t✐♦♥
  τ
  
  P {R/X} −
  → E1,R {E2,R {F ◦ (T {R/X})} | E3,R {S{R/X}}}
  
  ❢♦r s♦♠❡ ❡✈❛❧✉❛t✐♦♥ ❝♦♥t❡①ts E1,R , E2,R , E3,R ✭t❤❡ s✉❜s❝r✐♣t R ♠❡❛♥s t❤❛t ♦❝✲
  ❝✉rr❡♥❝❡s ♦❢ X ✐♥ t❤❡ ❝♦♥t❡①t ❛r❡ ✜❧❧❡❞ ✇✐t❤ R✮✳ ❲❡ ❤❛✈❡
  m a
  
  P {m.n.0/X} −→−
  →
  hT {m.n.0/X}iE1,m.n.0 {E2,m.n.0 {n.0} | E3,m.n.0 {S{m.n.0/X}}}
  
  s♦ ❜② ❜✐s✐♠✐❧❛r✐t② ❤②♣♦t❤❡s✐s✱ t❤❡r❡ ❡①✐sts T ′ , E′ s✉❝❤ t❤❛t ✇❡ ❤❛✈❡
  m a
  
  Q{m.n.0/X} −→−
  → hT ′ {m.n.0/X}iE′ m.n.0 {n.0}
  
  ❛♥❞ t❤❡ ♠❡ss❛❣❡s ❛♥❞ ❝♦♥t✐♥✉❛t✐♦♥s ❛r❡ ❜✐s✐♠✐❧❛r✱ ✐✳❡✳ ✇❡ ❤❛✈❡.
  T {m.n.0/X} ∼l T ′ {m.n.0/X}
  
  ❛♥❞.
  E1,m.n.0 {E2,m.n.0 {n.0} | E3,m.n.0 {S{m.n.0/X}}} ∼l E′ m.n.0 {n.0}
  
  ❋r♦♠ t❤❡ r❡❧❛t✐♦♥ ♦♥ ♠❡ss❛❣❡s✱ ✇❡ ❤❛✈❡.
  