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❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥✿ ❛ Pr❛❝t✐❝❛❧ ❘❡❛❧✐③❛t✐♦♥

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r

▲❛❜♦r❛t♦✐r❡ ❞✬■♥❢♦r♠❛t✐q✉❡ ❞❡ ●r❡♥♦❜❧❡✴❈◆❘❙ ❈❆PP t❡❛♠ ✲ ❆❙❆P ♣r♦❥❡❝t ✭❆◆❘✲✵✾✲❇▲❆◆✲✵✹✵✼✲✵✶✮

❋❘❖❈❖❙ ✷✵✶✸ ✲ ❙❡♣t❡♠❜❡r ✷✵✶✸ ✲ ◆❛♥❝②

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

■♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡

❧❡♥❣t❤❴❛t❴❧❡❛st(l, n) ⇔ n = ✵ ∨ ∃x, l′, n′ (l = cons(x, l′) ∧ n = s(n′) ∧❧❡♥❣t❤❴❛t❴❧❡❛st(l′, n′)) ♥t❤(x, l, n) ⇔ ∃l′ l = cons(y, l′) ∧ (n = s(✵) ∧ x = y) ∨ ∃n′ (n = s(n′) ∧ ♥t❤(x, l′, n′)) ❈❤❡❝❦ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ∀n ∈ N, ∀l (❧❡♥❣t❤❴❛t❴❧❡❛st(l, n) ∧ n = ✵ ⇒ ∃x ♥t❤(x, l, n))

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

■♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡ ✭✷✮

❚❤✐s ♣r♦❜❧❡♠ ❝❛♥♥♦t ❜❡ st❛t❡❞ ✐♥ ✜rst✲♦r❞❡r ❧♦❣✐❝ ✭n ∈ N✮ ❆♥ ✐♥❞✉❝t✐✈❡ ♣r♦♣❡rt② ♦❢ t❤❡ ❢♦r♠ ∀n, l ∃x φ ▼✉st ❝♦♠❜✐♥❡✿

❙t❛♥❞❛r❞ ❡q✉❛t✐♦♥❛❧ r❡❛s♦♥✐♥❣ ✇✐t❤ ✉♥✐✜❝❛t✐♦♥ t♦✿

✶

❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ x ✭✇✳r✳t✳ n, l✮

✷

❈❤❡❝❦ t❤❛t ✐t ✐♥❞❡❡❞ ❢✉❧✜❧❧s t❤❡ ❞❡s✐r❡❞ ♣r♦♣❡rt②

■♥❞✉❝t✐✈❡ r❡❛s♦♥✐♥❣ ♦♥ n

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

■♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡ ✭✸✮

❙tr❛✐❣❤t❢♦r✇❛r❞ ❛♣♣r♦❛❝❤✿ ✉s❡ st❛♥❞❛r❞ ♣r♦♦❢ ♣r♦❝❡❞✉r❡s ❢♦r ✜rst✲♦r❞❡r ❧♦❣✐❝ t♦❣❡t❤❡r ✇✐t❤ ❡①♣❧✐❝✐t ✐♥❞✉❝t✐♦♥ s❝❤❡♠❡s (ψ(✵) ∧ ∀n ψ(n) ⇒ ψ(s(n))) ⇒ ∀n ψ(n) ❢♦r s♦♠❡ ✏✇❡❧❧✲❝❤♦s❡♥✑ ❢♦r♠✉❧❛ ψ ❖✉r ❛♣♣r♦❛❝❤✿ tr② t♦ ❞✐s❝♦✈❡r ❛✉t♦♠❛t✐❝❛❧❧② s✉❝❤ ✐♥❞✉❝t✐✈❡ ❧❡♠♠❛t❛✱ ❜② ❞❡t❡❝t✐♥❣ ❝②❝❧❡s ✐♥ t❤❡ s❡❛r❝❤ s♣❛❝❡

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

P❧❛♥ ♦❢ t❤❡ t❛❧❦

❚❤❡ ❧❛♥❣✉❛❣❡ ❆ ♣r♦♦❢ ♣r♦❝❡❞✉r❡✿ s✉♣❡r♣♦s✐t✐♦♥ ✰ ❧♦♦♣ ❞❡t❡❝t✐♦♥ ❆ ❝②❝❧❡ ❞❡t❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ❊①♣❡r✐♠❡♥t❛t✐♦♥s

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚❤❡ ❧❛♥❣✉❛❣❡

❈❧❛✉s❛❧ ✭✜rst✲♦r❞❡r✮ ❧♦❣✐❝ ✰ ❛ ✭✉♥✐q✉❡✮ ❛r✐t❤♠❡t✐❝ ♣❛r❛♠❡t❡r n ❚✇♦ s♦rts ι ✭st❛♥❞❛r❞ t❡r♠s✮ ❛♥❞ ω ✭♥❛t✉r❛❧ ♥✉♠❜❡rs✮✱ ✇✐t❤ ✵ : ω, s : ω → ω ❆ s♣❡❝✐❛❧ ❝♦♥st❛♥t s②♠❜♦❧ n ❞❡♥♦t✐♥❣ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r ❚❡r♠s✱ ✭❡q✉❛t✐♦♥❛❧✮ ❧✐t❡r❛❧s ❛♥❞ ❝❧❛✉s❡s ❛r❡ ❞❡✜♥❡❞ ❛s ✉s✉❛❧ ❞♦ ♥♦t ❝♦♥t❛✐♥ t❤❡ s♣❡❝✐❛❧ s②♠❜♦❧ n n✲❝❧❛✉s❡s✿ ❝♦♥str❛✐♥❡❞ ❝❧❛✉s❡s ♦❢ t❤❡ ❢♦r♠ [C | X] ✇❤❡r❡✿

C ✐s ❛ ❝❧❛✉s❡ X ✐s ♦❢ t❤❡ ❢♦r♠ k

i=✶ n = ti✱ ✇❤❡r❡ t✶, . . . , tk ✭k ≥ ✵✮ ❛r❡

t❡r♠s ♦❢ s♦rt ω

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❙❡♠❛♥t✐❝s

❚❤❡ s♣❡❝✐❛❧ s②♠❜♦❧ n ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❛ t❡r♠ ♦❢ t❤❡ ❢♦r♠ sm(✵) ✭m ∈ N✮ ✵ ❛♥❞ s ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ✵ ❛♥❞ s✉❝❝❡ss♦r ❢✉♥❝t✐♦♥ ❚❤❡ ♦t❤❡r s②♠❜♦❧s ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ✉s✉❛❧ [C | k

i=✶ n = ti] ❤♦❧❞s ✐♥ I ✐✛ ❢♦r ❡✈❡r② s✉❜st✐t✉t✐♦♥ σ s✉❝❤

t❤❛t I(n) = tiσ✱ Cσ ❤♦❧❞s ✐♥ I

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚❤❡ ❧❛♥❣✉❛❣❡ ✭✷✮

❘❡♠❛r❦s✿ ❆ str✐❝t ❡①t❡♥s✐♦♥ ♦❢ ✜rst✲♦r❞❡r ❧♦❣✐❝ ❚❤❡ ❝♦♥st❛♥t n ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ t❤❡ ❝❧❛✉s❡s ❆ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❢♦r♠ f (n) = a ♠✉st ❜❡ ✇r✐tt❡♥✿ [f (x) = a | n = x] ❊①t❡♥s✐♦♥ t♦ ❢♦r♠✉❧æ ✇✐t❤ s❡✈❡r❛❧ ♣❛r❛♠❡t❡rs

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❆ t❤❡♦r❡t✐❝❛❧ ❧✐♠✐t❛t✐♦♥

❚❤❡♦r❡♠ ❚❤❡ s❡t ♦❢ s❛t✐s✜❛❜❧❡ s❡ts ♦❢ n✲❝❧❛✉s❡s ✐s ♥❡✐t❤❡r r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ✭♦❢ ❝♦✉rs❡ ✦✮ ♥♦r ❝♦✲r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ❉❡♣❛rt ❢r♦♠✿ ❋✐rst✲♦r❞❡r ❧♦❣✐❝ ✭✉♥s❛t✐s✜❛❜✐❧✐t② ✐s s❡♠✐✲❞❡❝✐❞❛❜❧❡✮ ❘❡✇r✐t❡✲❜❛s❡❞ ✐♥❞✉❝t✐✈❡ t❤❡♦r❡♠ ♣r♦✈✐♥❣ ✭♥♦♥✲♣r♦✈❛❜✐❧✐t② ✐s s❡♠✐✲❞❡❝✐❞❛❜❧❡✮

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚❤❡ ❧❛♥❣✉❛❣❡ ✭✸✮

Pr♦♣♦s✐t✐♦♥ ❊✈❡r② ✭♥♦♥✲t❛✉t♦❧♦❣✐❝❛❧✮ n✲❝❧❛✉s❡ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛♥ n✲❝❧❛✉s❡ ♦❢ t❤❡ ❢♦r♠ [C | ⊤] ♦r [C | n = t] Pr♦♦❢✿ k

i=✶ n = ti ⇔ n = t✶ ∧ k i=✷ t✶ = ti✱ t❤✉s

[C |

k

• i=✶

n = ti] ⇔ [Cσ | n = t✶σ] ✇❤❡r❡ σ = ♠❣✉(t✶, . . . , tk) ❛♥❞ [C |

k

• i=✶

n = ti] ⇔ ⊤ ✐❢ t✶, . . . , tk ❛r❡ ♥♦t ✉♥✐✜❛❜❧❡

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❊①❛♠♣❧❡

[f (x, y) = a | n = s(z) ∧ n = x ∧ n = y] − → [f (s(z), s(z)) = a | n = s(z)] [f (x, y) = a | n = s(x) ∧ n = ✵] − → ⊤

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚❤❡ ❧❛♥❣✉❛❣❡ ✭✹✮

✸ ❦✐♥❞s ♦❢ n✲❝❧❛✉s❡s✿

✶ ❙t❛♥❞❛r❞ ✜rst✲♦r❞❡r ❝❧❛✉s❡s✿ ❡①♣r❡ss ✉♥✐✈❡rs❛❧ ♣r♦♣❡rt✐❡s✱ ♥♦t

❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ n

✷ [C | n = sk(✵)]✿ ❡①♣r❡ss❡s ❛ ♣r♦♣❡rt② t❤❛t ❤♦❧❞s ♦♥❧② ✐❢ n ❤❛s

s♦♠❡ s♣❡❝✐✜❝ ✈❛❧✉❡ ✭n = k✮

✸ [C[x] | n = sk(x)]✿ ❡①♣r❡ss❡s ❛ ♣r♦♣❡rt② C t❤❛t ❤♦❧❞s ❢♦r

x = n − k

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚❤❡ ❧❛♥❣✉❛❣❡ ✭✹✮

✸ ❦✐♥❞s ♦❢ n✲❝❧❛✉s❡s✿

✶ ❙t❛♥❞❛r❞ ✜rst✲♦r❞❡r ❝❧❛✉s❡s✿ ❡①♣r❡ss ✉♥✐✈❡rs❛❧ ♣r♦♣❡rt✐❡s✱ ♥♦t

❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ n r❛♥❦ ⊥

✷ [C | n = sk(✵)]✿ ❡①♣r❡ss❡s ❛ ♣r♦♣❡rt② t❤❛t ❤♦❧❞s ♦♥❧② ✐❢ n ❤❛s

s♦♠❡ s♣❡❝✐✜❝ ✈❛❧✉❡ ✭n = k✮ ♥♦ r❛♥❦

✸ [C[x] | n = sk(x)]✿ ❡①♣r❡ss❡s ❛ ♣r♦♣❡rt② C t❤❛t ❤♦❧❞s ❢♦r

x = n − k r❛♥❦ k S[i] ❞❡♥♦t❡s t❤❡ s❡t ♦❢ n✲❝❧❛✉s❡s ♦❢ r❛♥❦ i ✐♥ S

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

Pr♦♦❢ Pr♦❝❡❞✉r❡✿ ❈♦♥str❛✐♥❡❞ s✉♣❡r♣♦s✐t✐♦♥ ❝❛❧❝✉❧✉s

❙✉♣❡r♣♦s✐t✐♦♥✿ [C ∨ t ⊲ ⊳ s | X], [D ∨ u = v | Y] [C ∨ D ∨ t[v]p ⊲ ⊳ s | X ∧ Y]σ ■❢ ⊲ ⊳∈ {=, =}✱ σ = ♠❣✉(u, t|p)✱ uσ ≤ vσ, tσ ≤ sσ✱ t|p ✐s ♥♦t ❛ ✈❛r✐❛❜❧❡✱ (t ⊲ ⊳ s)σ < Cσ✱ (u = v)σ < Dσ✳

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

Pr♦♦❢ Pr♦❝❡❞✉r❡✿ ❈♦♥str❛✐♥❡❞ s✉♣❡r♣♦s✐t✐♦♥ ❝❛❧❝✉❧✉s ✭✷✮

❘❡✢❡❝t✐♦♥✿ [C ∨ t = s | X] [C | X]σ ■❢ σ = ♠❣✉(t, s)✱ (t = s)σ < Cσ ❋❛❝t♦r✐s❛t✐♦♥✿ [C ∨ t = s ∨ u = v | X] [C ∨ s = v ∨ t = s | X]σ ■❢ σ = ♠❣✉(t, u)✱ tσ < sσ✱ uσ < vσ✱ (t = s)σ < Cσ✳

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

Pr♦♦❢ Pr♦❝❡❞✉r❡

❘❡♠❛r❦s✿ ❚❤❡ ♣❛r❛♠❡t❡r n ✐s ❛❜str❛❝t❡❞ ❛✇❛② ❢r♦♠ t❤❡ ❝❧❛✉s❡s✿ f (n) = a − → [f (x) = a | n = x] ❆❧❧♦✇s ❢♦r ❛ ❧❛③② ✐♥st❛♥t✐❛t✐♦♥ ♦❢ t❤✐s ♣❛r❛♠❡t❡r✿ [f (x) = a | n = x], f (✵) = a ⊢ [ | n = ✵] ✏❲❡❛❦❧②✑ ❝♦♠♣❧❡t❡✿ ✐❢ S | = n = k ✭❢♦r s♦♠❡ k ∈ N✮ t❤❡♥ S ⊢ [ | n = k] ✭♠♦❞✉❧♦ s✉❜s✉♠♣t✐♦♥✮ ◆♦t ❝♦♠♣❧❡t❡✿ ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ❞❡r✐✈❡❞ ✐♥ ✜♥✐t❡ t✐♠❡ ✭❛❧♠♦st ♥❡✈❡r t❡r♠✐♥❛t❡s✮

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡

Pr♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ p(✵) ∧ ∀x p(x) ⇒ p(s(x)) | = ∀n ∈ N p(n)

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ✭✷✮

❯s❡ t❤❡ s✉♣❡r♣♦s✐t✐♦♥ ❝❛❧❝✉❧✉s✿ ✶ p(✵) = tr✉❡ ✷ p(x) = tr✉❡ ∨ p(s(x)) = tr✉❡ ✸ [p(x) = tr✉❡ | n = x] ✹ [ | n = ✵] ✭s✉♣❡r♣♦s✐t✐♦♥✱ ✶✱ ✸✮ ✺ [p(x) = tr✉❡ | n = s(x)] ✭s✉♣❡r♣♦s✐t✐♦♥✱ ✷✱ ✸✮ ✻ [ | n = s(✵)] ✭s✉♣❡r♣♦s✐t✐♦♥✱ ✶✱ ✺✮ ✳ ✳ ✳ ✳ ✳ ✳ [ | n = sk(✵)]

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❯♥❝♦♠♣❧❡t❡ ❝❛❧❝✉❧✉s

■❢ S ✐s ✉♥s❛t✐s✜❛❜❧❡✱ ✇❡ ❤❛✈❡✿ ∀k ∈ N S ⊢ n = k ❜✉t ♥♦t✿ S ⊢ ∀k ∈ N n = k (≡ ⊥)

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ✭✸✮

❆ ✏❝②❝❧❡✑ ✐♥ t❤❡ s❡❛r❝❤ s♣❛❝❡✿ ❈❧❛✉s❡ ✺ : [p(x) = tr✉❡ | n = s(x)] ✐s ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ ❈❧❛✉s❡ ✸ : [p(x) = tr✉❡ | n = x]✱ ✉♣ t♦ ❛ tr❛♥s❧❛t✐♦♥ ♦♥ n✳ ❈❧❛✉s❡ ✸ ≡ p(n) ❈❧❛✉s❡ ✺ ≡ p(n − ✶) ■❞❡❛✿ ❞❡t❡❝t t❤♦s❡ ❝②❝❧❡s ❛♥❞ ✉s❡ t❤❡♠ t♦ ♣r✉♥❡ t❤❡ s❡❛r❝❤ s♣❛❝❡

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

❋✐rst st❡♣✿ ❋♦r♠❛❧✐③❡ t❤❡ ♥♦t✐♦♥ ♦❢ tr❛♥s❧❛t✐♦♥ S↓i≡ S{n ← n − i} [C | n = t] − → [C | n − i = t]

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

❋✐rst st❡♣✿ ❋♦r♠❛❧✐③❡ t❤❡ ♥♦t✐♦♥ ♦❢ tr❛♥s❧❛t✐♦♥ S↓i≡ S{n ← n − i} [C | n = t] − → [C | n − i = t]

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

❋✐rst st❡♣✿ ❋♦r♠❛❧✐③❡ t❤❡ ♥♦t✐♦♥ ♦❢ tr❛♥s❧❛t✐♦♥ S↓i≡ S{n ← n − i} [C | n = t] − → [C | n = t + i]

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

❋✐rst st❡♣✿ ❋♦r♠❛❧✐③❡ t❤❡ ♥♦t✐♦♥ ♦❢ tr❛♥s❧❛t✐♦♥ S↓i≡ S{n ← n − i} [C | n = t] − → [C | n = si(t)]

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

❙❡❝♦♥❞ st❡♣✿ ❈②❝❧❡ ❉❡t❡❝t✐♦♥ ❘✉❧❡ ■❢ t❤❡r❡ ❡①✐sts Sind ⊆ S s✉❝❤ t❤❛t✿

✶ Sind |

= n = l✱ ❢♦r ❡✈❡r② l ∈ [i, i + j[

✷ ❛♥❞ Sind |

= Sind↓j✱ t❤❡♥ S | = n < i ✭✐✳❡✳ S | = [ | n = si(x)]✮ Pr♦♦❢✿ ❜② ✏❞❡s❝❡♥t❡ ✐♥✜♥✐❡✑

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

■♥ ♣r❛❝t✐❝❡✿ S ✐s t❤❡ ✇❤♦❧❡ s❡❛r❝❤ s♣❛❝❡ ✭s❡t ♦❢ ❣❡♥❡r❛t❡❞ n✲❝❧❛✉s❡s✮ Sind ⊆ S ❉❡❝✐❞❛❜❧❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❡❞❡❞

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

■♥ ♣r❛❝t✐❝❡✿ S ✐s t❤❡ ✇❤♦❧❡ s❡❛r❝❤ s♣❛❝❡ ✭s❡t ♦❢ ❣❡♥❡r❛t❡❞ n✲❝❧❛✉s❡s✮ Sind ⊆ S ❉❡❝✐❞❛❜❧❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❡❞❡❞

❈♦♥❞✐t✐♦♥ ✶✿ ❈❤❡❝❦ t❤❛t [ | n = l] ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind ❈♦♥❞✐t✐♦♥ ✷✿ ❈❤❡❝❦ t❤❛t s♦♠❡ s❡t ♦❢ n✲❝❧❛✉s❡s Sloop ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind✱ ✇✐t❤ Sloop = Sind↓j

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

■♥ ♣r❛❝t✐❝❡✿ S ✐s t❤❡ ✇❤♦❧❡ s❡❛r❝❤ s♣❛❝❡ ✭s❡t ♦❢ ❣❡♥❡r❛t❡❞ n✲❝❧❛✉s❡s✮ Sind ⊆ S ❉❡❝✐❞❛❜❧❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❡❞❡❞

❈♦♥❞✐t✐♦♥ ✶✿ ❈❤❡❝❦ t❤❛t [ | n = l] ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind ❈♦♥❞✐t✐♦♥ ✷✿ ❈❤❡❝❦ t❤❛t s♦♠❡ s❡t ♦❢ n✲❝❧❛✉s❡s Sloop ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind✱ ✇✐t❤ Sloop s✉❜s✉♠❡s Sind↓j

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

■♥ ♣r❛❝t✐❝❡✿ S ✐s t❤❡ ✇❤♦❧❡ s❡❛r❝❤ s♣❛❝❡ ✭s❡t ♦❢ ❣❡♥❡r❛t❡❞ n✲❝❧❛✉s❡s✮ Sind ⊆ S ❉❡❝✐❞❛❜❧❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❡❞❡❞

❈♦♥❞✐t✐♦♥ ✶✿ ❈❤❡❝❦ t❤❛t [ | n = l] ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind ❈♦♥❞✐t✐♦♥ ✷✿ ❈❤❡❝❦ t❤❛t s♦♠❡ s❡t ♦❢ n✲❝❧❛✉s❡s Sloop ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind✱ ✇✐t❤ Sloop s✉❜s✉♠❡s Sind↓j ✭♦r ❛♥② ❞❡❝✐❞❛❜❧❡ ❡♥t❛✐❧♠❡♥t r❡❧❛t✐♦♥✮

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥

■♥ ♣r❛❝t✐❝❡✿ S ✐s t❤❡ ✇❤♦❧❡ s❡❛r❝❤ s♣❛❝❡ ✭s❡t ♦❢ ❣❡♥❡r❛t❡❞ n✲❝❧❛✉s❡s✮ Sind ⊆ S ❉❡❝✐❞❛❜❧❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❡❞❡❞

❈♦♥❞✐t✐♦♥ ✶✿ ❈❤❡❝❦ t❤❛t [ | n = l] ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind ❈♦♥❞✐t✐♦♥ ✷✿ ❈❤❡❝❦ t❤❛t s♦♠❡ s❡t ♦❢ n✲❝❧❛✉s❡s Sloop ❤❛s ❜❡❡♥ ❞❡r✐✈❡❞ ❢r♦♠ Sind✱ ✇✐t❤ Sloop s✉❜s✉♠❡s Sind↓j ✭♦r ❛♥② ❞❡❝✐❞❛❜❧❡ ❡♥t❛✐❧♠❡♥t r❡❧❛t✐♦♥✮ ❆ ❢✉rt❤❡r r❡str✐❝t✐♦♥✿ ❛ss✉♠❡ t❤❛t ❛❧❧ n✲❝❧❛✉s❡s ✐♥ Sind ❤❛✈❡ t❤❡ s❛♠❡ r❛♥❦ i ✭♦r ⊥✮

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❊①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

✶ p(✵) = tr✉❡ ✷ p(x) = tr✉❡ ∨ p(s(x)) = tr✉❡ ✸ [p(x) = tr✉❡ | n = x] ✹ [ | n = ✵] ✭s✉♣❡r♣♦s✐t✐♦♥✱ ✶✱ ✸✮ ✺ [p(x) = tr✉❡ | n = s(x)] ✭s✉♣❡r♣♦s✐t✐♦♥✱ ✷✱ ✸✮ ✻ [ | n = s(✵)] ✭s✉♣❡r♣♦s✐t✐♦♥✱ ✶✱ ✺✮ ✳ ✳ ✳ ✳ ✳ ✳ [ | n = sk(✵)] Sind = {✶, ✷, ✸}✱ Sloop = {✶, ✷, ✺}✱ i = ✵✱ j = ✶ [ | n = x] ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❯♥s❛t✐s✜❛❜✐❧✐t② ✐s ❞❡t❡❝t❡❞

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❈②❝❧❡ ❉❡t❡❝t✐♦♥ ✐♥ Pr❛❝t✐❝❡

❍♦✇ t♦ ❣❡♥❡r❛t❡ ❡✛❡❝t✐✈❡❧② t❤❡ ♥✉♠❜❡rs i, j ❛♥❞ t❤❡ s❡ts Sind, Sloop ❄ ❆♥ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ Sind, Sloop ✭❢♦r ✜①❡❞ i, j) Pr♦♣❡rt✐❡s✿ ❙♦✉♥❞✿ t❤❡ ❝♦♠♣✉t❡❞ s❡ts Sind, Sloop s❛t✐s❢② t❤❡ ❞❡s✐r❡❞ ♣r♦♣❡rt② ❈♦♠♣❧❡t❡✿ ✐❢ s♦♠❡ s❡ts Sind, Sloop s❛t✐s❢② t❤❡ ❞❡s✐r❡❞ ♣r♦♣❡rt②✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ s✉❝❝❡❡❞s ✭❜✉t ♥♦t ♥❡❝❡ss❛r✐❧② ✇✐t❤ ♦✉t♣✉t Sind, Sloop✮ ❊✣❝✐❡♥t✿ ♣♦❧②♥♦♠✐❛❧ ✇✳r✳t✳ t❤❡ s✐③❡ ♦❢ t❤❡ s❡t S ❇❛s❡❞ ♦♥ ❛ ❣r❡❛t❡st ✜①♣♦✐♥t ❝♦♠♣✉t❛t✐♦♥

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚❤❡ ❆❧❣♦r✐t❤♠

S✵ ← {n = k, k ∈ [i, i + j[} Sind ← S[i] ✐❢ Sind ⊢ S✵ t❤❡♥ r❡t✉r♥ ❢❛❧s❡ ❡♥❞ ✐❢ Sloop ← {D ∈ S[i + j] | Sind ⊢ {D}} ✇❤✐❧❡ ∃C ∈ Sind | Sloop ⊇ {C↓j} ❞♦ Sind ← Sind \ {C} ✐❢ Sind ⊢ S✵ t❤❡♥ r❡t✉r♥ ❢❛❧s❡ ❡♥❞ ✐❢ ❘❡♠♦✈❡ ❢r♦♠ Sloop ❛❧❧ t❤❡ n✲❝❧❛✉s❡s D s✳t✳ Sind ⊢ {D} ❡♥❞ ✇❤✐❧❡ r❡t✉r♥ tr✉❡

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❊①♣❡r✐♠❡♥t❛t✐♦♥s

■♠♣❧❡♠❡♥t❡❞ ✐♥ Pr♦✈❡r✾ ❯s❡ n✲❝❧❛✉s❡s t♦ ♠♦❞❡❧ s❝❤❡♠❛t❛ ♦❢ ❢♦r♠✉❧æ ❋♦r♠✉❧æ ❞❡♣❡♥❞✐♥❣ ♦♥ s♦♠❡ ♣❛r❛♠❡t❡r n ❈♦♥str✉❝t❡❞ ✉s✐♥❣ s♣❡❝✐❛❧ ❝♦♥♥❡❝t✐✈❡s b

i=a φ ❛♥❞ b i=a φ

❊①❛♠♣❧❡✿ n✲❜✐t ❛❞❞❡r ❙✉♠i(p, q, c, r)

❞❡❢

= ri ⇔ (pi ⊕ qi) ⊕ ci ❈❛rr②i(p, q, c)

❞❡❢

= ci+✶ ⇔ (pi ∧ qi) ∨ (ci ∧ pi) ∨ (ci ∧ qi) ❆❞❞❡r(p, q, c, r)

❞❡❢

=

n

• i=✶

❙✉♠i(p, q, c, r) ∧

n

• i=✶

❈❛rr②i(p, q, c) ∧ ¬c✶

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚r❛♥s❧❛t✐♦♥ ✐♥t♦ ❝❧❛✉s❛❧ ❢♦r♠ ✭✶✮

n

• i=✵

φ − → p(n) ✇✐t❤✿ p(✵) ⇔ φ{i → ✵} ∀x p(x + ✶) ⇔ φ{i → x + ✶} ∨ p(x)

n+b

• i=a

φ − →

n

• i=✵

(φ ∧ qi) ∨ φ{i → n + ✶} ∨ . . . ∨ φ{i → n + b} ✇✐t❤✿ ¬q(✵) ∧ . . . ∧ ¬q(a − ✶) ∧ q(a) ∀x q(x) → q(s(x))

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❚r❛♥s❧❛t✐♦♥ ✐♥t♦ ❝❧❛✉s❛❧ ❢♦r♠ ✭❊✮

❊❧✐♠✐♥❛t❡ t❡r♠s ♦❢ t❤❡ ❢♦r♠ s(t) ✇❤❡r❡ t ✐s ♥♦t ❛ ✈❛r✐❛❜❧❡✿ ps(t) − → p′

t

✇✐t❤✿ ∀x ps(x) ⇔ px

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❊①♣❡r✐♠❡♥t❛t✐♦♥s

❊①❛♠♣❧❡ ❚✐♠❡ ★ ♦❢ ❝❛❧❧s ★ ❝❧❛✉s❡s t♦ ❈②❝❧❡✷ ❘✐♣♣❧❡✲❝❛rr② ❛❞❞❡r ✭A + ✵ = A✮ ✵✳✹✽ ✸✸✻ ✸✸✽✸✸ ❘✐♣♣❧❡✲❝❛rr② ❛❞❞❡r ✭❝♦♠♠✉t❛t✐✈✐t②✮ ✵✳✵✸ ✶✵✷ ✷✵✵✸ ❘✐♣♣❧❡✲❝❛rr② ❛❞❞❡r ✭❛ss♦❝✐❛t✐✈✐t②✮ ✵✳✵✾ ✷✵✼ ✶✵✶✺✹ ❯♥✐❝✐t② ♦❢ t❤❡ r❡s✉❧t ✭r✐♣♣❧❡✲❝❛rr②✮ ✵✳✼ ✶✺✵ ✺✵✾✵✶ ❈❛rr②✲♣r♦♣❛❣❛t❡ ❛❞❞❡r ✭❝♦♠♠✉t❛t✐✈✐t②✮ ✵✳✵✷ ✶✹ ✶✾✽✵ ❈❛rr②✲♣r♦♣❛❣❛t❡ ❛❞❞❡r ✭❛ss♦❝✐❛t✐✈✐t②✮ ✵✳✵✶ ✷✵ ✸✾✼✷ ❊q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ r✐♣♣❧❡✲❝❛rr② ❛♥❞ t❤❡ ❝❛rr②✲♣r♦♣❛❣❛t❡ ❛❞❞❡rs ✵✳✵✸ ✶✹ ✶✾✽✵ ❚♦t❛❧✐t② ♦❢ < ✭n✶ ≥ n✷ ∨ n✶ < n✷✮ ✵✳✵✶ ✹✼ ✶✽✺

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❙✉♠♠❛r②

❆ t❡❝❤♥✐q✉❡ t♦ ❝♦♠❜✐♥❡ s✉♣❡r♣♦s✐t✐♦♥ ❝❛❧❝✉❧✉s ❛♥❞ ✐♥❞✉❝t✐✈❡ t❤❡♦r❡♠ ♣r♦✈✐♥❣ ❆✉t♦♠❛t❡❞ ❞✐s❝♦✈❡r② ♦❢ ✭s♦♠❡✮ ✐♥❞✉❝t✐✈❡ ✐♥✈❛r✐❛♥ts ❈♦♠♣❧❡t❡♥❡ss ❝❛♥ ❜❡ ❡♥s✉r❡❞ ✐♥ s♦♠❡ ❝❛s❡s ✭❈❆❉❊✮✱ ❡✳❣✳ ✐❢ t❤❡ ❢♦r♠✉❧æ ❝♦♥t❛✐♥ ♥♦ ♥♦♥✲❛r✐t❤♠❡t✐❝ ✈❛r✐❛❜❧❡ ✭s❝❤❡♠❛t❛ ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❢♦r♠✉❧æ✮ ❆♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❜❛s❡❞ ♦♥ Pr♦✈❡r✾

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥

❋✉t✉r❡ ❲♦r❦

■♥❝r❡♠❡♥t❛❧ ❧♦♦♣ ❞❡t❡❝t✐♦♥ ❍❡✉r✐st✐❝s t♦ ✏❣✉❡ss✑ t❤❡ ✈❛❧✉❡s ♦❢ i ❛♥❞ j ♦r t♦ tr✐❣❣❡r t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❧♦♦♣ ❞❡t❡❝t✐♦♥ r✉❧❡ ■♠♣r♦✈❡ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ♠♦r❡ ❡①♣❡r✐♠❡♥t❛t✐♦♥s

❆❜❞❡❧❦❛❞❡r ❑❡rs❛♥✐ ❛♥❞ ◆✐❝♦❧❛s P❡❧t✐❡r ❈♦♠❜✐♥✐♥❣ ❙✉♣❡r♣♦s✐t✐♦♥ ❛♥❞ ■♥❞✉❝t✐♦♥