■♥tr♦❞✉çã♦ à ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ■
❘♦❞r✐❣♦ ❘✐❜❡✐r♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❆♣❧✐❝❛❞❛s ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❋❡❞❡r❛❧ ❞❡ ❖✉r♦ Pr❡t♦
▼♦t✐✈❛çã♦ ✖ ✭■✮
❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦✿
❚♦❞♦ ❤♦♠❡♠ é ♠♦rt❛❧✳ ❙ó❝r❛t❡s é ✉♠ ❤♦♠❡♠✳ ▲♦❣♦✱ ❙ó❝r❛t❡s é ♠♦rt❛❧✳
❆♣ós ♣❡♥s❛r ✉♠ ♣♦✉❝♦✳✳✳
▼♦t✐✈❛çã♦ ✖ ✭■■✮
P♦ré♠ ❛♦ t❡♥t❛r r❡♣r❡s❡♥t❛r✳✳✳
P❡r❝❡❜❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ❝♦♥❡❝t✐✈♦✦ ■st♦ é✱ q✉❡ ♦ ❛r❣✉♠❡♥t♦ é ❢♦r♠❛❞♦ ❛♣❡♥❛s ♣♦r ♣r♦♣♦s✐çõ❡s ❛tô♠✐❝❛s✳
❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡ ♦ ❛r❣✉♠❡♥t♦ s❡r✐❛ r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦✿
❋ór♠✳ Pr♦♣✳
❆ ❚♦❞♦ ❤♦♠❡♠ é ♠♦rt❛❧
❇ ❙ó❝r❛t❡s é ✉♠ ❤♦♠❡♠
▼♦t✐✈❛çã♦ ✖ ✭■■■✮
❖ s❡q✉❡♥t❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❛r❣✉♠❡♥t♦ é✿
❆, ❇ ⊢ ❈
▼❛s✳✳✳
▼♦t✐✈❛çã♦ ✖ ✭■❱✮
❖ Pr♦❜❧❡♠❛ é q✉❡✳✳✳
❆ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧ ♥ã♦ é ❝❛♣❛③ ❞❡
r❡♣r❡s❡♥t❛r ❛✜r♠❛çõ❡s ❝♦♥t❡♥❞♦ ✧t♦❞♦s✧✱ ✧♣❛r❛ t♦❞♦s✧✱ ✧❛❧❣✉♥s✧✱ ✧s♦♠❡♥t❡ ✉♠✧✳✳✳
❚❛✐s ❛✜r♠❛t✐✈❛s ♦❝♦rr❡♠ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❡♠ ♠❛t❡♠át✐❝❛✿
❚♦❞♦ ♥ú♠❡r♦ ♣❛r é ❞✐✈✐sí✈❡❧ ♣♦r ✷✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■✮
❆ ▲✐♥❣✉❛❣❡♠ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
❆ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ❝♦♥s✐st❡ ❞❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧ ❛✉♠❡♥t❛❞❛ ❝♦♠ q✉❛♥t✐✜❝❛❞♦r❡s✱ ♣r❡❞✐❝❛❞♦s ❡ ✈❛r✐á✈❡✐s✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■■✮
Pr❡❞✐❝❛❞♦s
❯♠ ♣r❡❞✐❝❛❞♦ é ✉♠❛ ❛✜r♠❛çã♦ ❞❡ q✉❡ ✉♠ ♦❜❥❡t♦ ① ♣♦ss✉✐ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦♣r✐❡❞❛❞❡✳ ❚❛✐s ❛✜r♠❛t✐✈❛s ❞❡✈❡♠ s❡r ✈❡r❞❛❞❡✐r❛s ♦✉ ❢❛❧s❛s✳ ❆✜r♠❛t✐✈❛s ♣♦❞❡♠ ❡♥✈♦❧✈❡r ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♦✉ ✈❛r✐á✈❡✐s q✉❡ ♣♦❞❡♠ ❛ss✉♠✐r ✈❛❧♦r❡s ❞❡st❡ ❝♦♥❥✉♥t♦✳
❊①❡♠♣❧♦s✿
✺ é ✉♠ ♥ú♠❡r♦ ♣❛r✳ ① ❃ ②✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■■■✮
❯♥✐✈❡rs♦ ❞❡ ❉✐s❝✉rs♦
❉❛♠♦s ♦ ♥♦♠❡ ❞❡ ❯♥✐✈❡rs♦ ❞❡ ❉✐s❝✉rs♦ ❛♦ ❝♦♥❥✉♥t♦ q✉❡ é r❡❢❡r✐❞♦ ♣♦r ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s
❊①❡♠♣❧♦✿
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■❱✮
▼❛✐s s♦❜r❡ Pr❡❞✐❝❛❞♦s
◆♦r♠❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛♠♦s ♣r❡❞✐❝❛❞♦s ❞❡ ♠❛♥❡✐r❛ ❝♦♥❝✐s❛✳
❊①❡♠♣❧♦s✿
❏♦sé ❣♦st❛ ❞❡ ▼❛r✐❛✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❢❛t♦ ♣♦r✿ ❣♦st❛✭❏♦sé✱ ▼❛r✐❛✮✳ ❖♥❞❡ ❣♦st❛ é ✉♠ ♣r❡❞✐❝❛❞♦ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s✱ ♦♥❞❡ ❝❛❞❛ ✉♠ ❞❡st❡s é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠
♣❡ss♦❛s✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱✮
▼❛✐s s♦❜r❡ Pr❡❞✐❝❛❞♦s
❊①❡♠♣❧♦s✿
♣❛r✭✷✮ ❢(①,✷,❣(②))
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱■✮
◗✉❛♥t✐✜❝❛❞♦r❡s
❆ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ♣♦ss✉✐ ❞♦✐s q✉❛♥t✐✜❝❛❞♦r❡s✿
◗✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧✿ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦
∀
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱■■✮
◗✉❛♥t✐✜❝❛❞♦r ❯♥✐✈❡rs❛❧
❙❡ ❋(①) é ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✱
❝♦♥t❡♥❞♦ ❛ ✈❛r✐á✈❡❧ ①✱ ❡♥tã♦ ∀①.❋(①) é ✉♠❛
❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳
❖ s✐❣♥✐✜❝❛❞♦ ❞❡ ∀①.❋(①) é✿ P❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦
① ♣❡rt❡♥❝❡♥t❡ ❛♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦✱ ❛ ❢ór♠✉❧❛
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱■■■✮
❊①❡♠♣❧♦s
❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦
❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭N✮✳
∀♥.♥≥✵✿ ❡st❛ ❢ór♠✉❧❛ é ✈❡r❞❛❞❡✐r❛✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛✱ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡r❡s ❤✉♠❛♥♦s✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■❳✮
◗✉❛♥t✐✜❝❛❞♦r ❊①✐st❡♥❝✐❛❧
❙❡ ❋(①) é ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✱
❝♦♥t❡♥❞♦ ❛ ✈❛r✐á✈❡❧ ①✱ ❡♥tã♦ ∃①.❋(①) é ✉♠❛
❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳
❖ s✐❣♥✐✜❝❛❞♦ ❞❡ ∃①.❋(①) é✿ ❊①✐st❡ ♣❡❧♦ ♠❡♥♦s
✉♠ ❡❧❡♠❡♥t♦ ① ♣❡rt❡♥❝❡♥t❡ ❛♦ ✉♥✐✈❡rs♦ ❞❡
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳✮
❊①❡♠♣❧♦s
❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦
❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭N✮✳
∃♥.♥≤✵✿ ❡st❛ ❢ór♠✉❧❛ é ❢❛❧s❛✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛✱ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡r❡s ❤✉♠❛♥♦s✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■✮
❙✐❣♥✐✜❝❛❞♦ ❞♦s ◗✉❛♥t✐✜❝❛❞♦r❡s
❙❡❥❛ ❯ = {✉✶,✉✷, ...,✉♥}✱ ✉♠ ✉♥✐✈❡rs♦ ❞❡
❞✐s❝✉rs♦ t❛❧ q✉❡ |❯| = ♥∧ ♥6= ∞✳
❊♥tã♦ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ∀①.♣(①) é✿
♣(✉✶)∧♣(✉✷)∧...∧♣(✉♥)
❊ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ∃①.♣(①) é✿
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■■✮
❊①❡♠♣❧♦s
❈♦♥s✐❞❡r❡ ❯ = {✷,✸,✹} ❡ ♣❛r(①) ❡ ✐♠♣❛r(①)
❞♦✐s ♣r❡❞✐❝❛❞♦s s♦❜r❡ ❯✳
❊♥tã♦ ❛s ❢ór♠✉❧❛s ∀①.♣❛r(①)∨ ✐♠♣❛r(①) ❡
∃①.¬✐♠♣❛r(①)∧① < ✶ ♣♦❞❡♠ s❡r ❡①♣❛♥❞✐❞❛s
❝♦♠♦✿
(♣❛r(✷)∨✐♠♣❛r(✷))∧(♣❛r(✸)∨✐♠♣❛r(✸))∧ (♣❛r(✹)∨✐♠♣❛r(✹))
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■■■✮
❊s❝♦♣♦ ❞❡ ◗✉❛♥t✐✜❝❛❞♦r❡s
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■❱✮
❱❛r✐á✈❡✐s ❧✐✈r❡s ❡ ❧✐❣❛❞❛s
❯♠❛ ✈❛r✐á✈❡❧ é ❞✐t❛ s❡r ❧✐✈r❡ ❡♠ ✉♠❛ ❢ór♠✉❧❛ s❡ ❡st❛ ♥ã♦ ♣♦ss✉✐ ♥❡♥❤✉♠ q✉❛♥t✐✜❝❛❞♦r✳
❊①❡♠♣❧♦✿ ∀①.∃②.♣(①,②,③)✳ ◆❡st❛ ❢ór♠✉❧❛ ③ é ✉♠❛
✈❛r✐á✈❡❧ ❧✐✈r❡✳
❯♠❛ ✈❛r✐á✈❡❧ é ❞✐t❛ s❡r ❧✐❣❛❞❛ s❡ ❡st❛ ❡stá ❛ss♦❝✐❛❞❛ ❛ ✉♠ q✉❛♥t✐✜❝❛❞♦r✳
❊①❡♠♣❧♦✿ ∀①.∃②.♣(①,②,③)✳ ◆❡st❛ ❢ór♠✉❧❛ ❛s
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳❱✮
❊s❝♦♣♦ ❞❡ ✉♠ ◗✉❛♥t✐✜❝❛❞♦r
❙❡❥❛ α ✉♠❛ ❢ór♠✉❧❛ q✉❛❧q✉❡r ❞❛ ❧ó❣✐❝❛ ❞❡
♣r❡❞✐❝❛❞♦s✳ ❊♥tã♦ ♦ ❡s❝♦♣♦ ❞❡ ∀ ♥❛ ❢ór♠✉❧❛
∀①.α é t♦❞❛ ❛ ❢ór♠✉❧❛ α✳ ❚♦❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ①
❡♠ α s❡rá ❧✐❣❛❞❛ ❛♦ q✉❛♥t✐✜❝❛❞♦r ∀✳ ❆ ❞❡✜♥✐çã♦
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳❱■✮
❚r❛❞✉③✐♥❞♦ P❛r❛ ❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
Pr♦❝❡ss♦ s✐♠♣❧❡s ❡ s✐♠✐❧❛r ❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧✳ ▼❛s ❞❡✈❡♠♦s ❧❡✈❛r ❡♠ ❝♦♥t❛ ♣r❡❞✐❝❛❞♦s ❡
q✉❛♥t✐✜❝❛❞♦r❡s✳
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣❡ss♦❛s ❡ ♦❜❥❡t♦s✳
❚♦❞♦ ♠✉♥❞♦ ❣♦st❛ ❞❡ ❛❧❣✉é♠✳ ∀①.∃②.❣♦st❛(①,②)
❚♦❞♦s ❣♦st❛♠ ❞❡ q✉❡♠ ❣♦st❛ ❞❡ s✉❛ ♠ã❡✳
∀①.∀②.❣♦st❛(①,♠❛❡(②))→❣♦st❛(②,①)
❆❧✉♥♦s ♥ã♦ ❣♦st❛♠ ❞❡ ❢❛③❡r ♣r♦✈❛s✳
▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳❱■■✮
❊①❡r❝í❝✐♦
❘❡♣r❡s❡♥t❡ ❛s s❡❣✉✐♥t❡s ❢r❛s❡s✿
❙❡ ❛♠❛♥❤ã ❝❤♦✈❡r✱ ❡♥tã♦ t♦❞♦s ❞❡✈❡♠ ❧❡✈❛r ❣✉❛r❞❛✲❝❤✉✈❛s✳
❚♦❞♦s q✉❡ ❣♦st❛♠ ❞❡ ❈❛r❧♦s✱ ♥ã♦ ❣♦st❛♠ ❞❡ s❡✉s ♣r✐♠♦s✳
◆✐♥❣✉é♠ ❢♦✐ ❛ ❢❡st❛ ❞❡ ▼❛♥♦❡❧✳
❙✐♥t❛①❡ ✖ ✭■✮
❋ór♠✉❧❛s ❜❡♠ ❢♦r♠❛❞❛s
❙✐♥t❛①❡ ✖ ✭■■✮
❋ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s
❋ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ♣♦❞❡♠ ❞❡♥♦t❛r✿
❊❧❡♠❡♥t♦s ❞♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ❡♠ q✉❡stã♦✳ ❱❛❧♦r❡s ❧ó❣✐❝♦s
❙❡♥❞♦ ❛ss✐♠✱ ❛ s✐♥t❛①❡ s❡rá ❞✐✈✐❞❛ ❡♠ ❞✉❛s ❝❛t❡❣♦r✐❛s✿
❙✐♥t❛①❡ ✖ ✭■■■✮
❚❡r♠♦s
❖ ❝♦♥❥✉♥t♦ ❞❡ t❡r♠♦s T é ❞❡✜♥✐❞♦ ❝♦♠♦✿
❛✮ ❙❡❥❛ V ♦ ❝♦♥❥✉♥t♦ ❝♦♥t❡♥❞♦ t♦❞❛s ❛s ✈❛r✐á✈❡✐s
❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❊♥tã♦✱ T ⊆ V✳
❜✮ ❙❡ ❝ é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ r❡♣r❡s❡♥t❛ ✉♠
❡❧❡♠❡♥t♦ ❞♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦✱ ❡♥tã♦ ❝ ∈ T
❝✮ ❙❡❥❛ ❢ ✉♠❛ ❢✉♥çã♦ ♥✲ár✐❛ s♦❜r❡ t❡r♠♦s ❡
t✶,t✷, ...,t♥ ∈ T ✭♥ ≥ ✵✮ ❡♥tã♦
❙✐♥t❛①❡ ✖ ✭■❱✮
❚❡r♠♦s
❈♦♥s✐❞❡r❛♥❞♦ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝✐❞❛❞❡s ❡ ❡st❛❞♦s✱ t❡♠♦s q✉❡✿
❇❡❧♦ ❍♦r✐③♦♥t❡ é ✉♠ ❝♦♥st❛♥t❡ r❡♣r❡s❡♥t❛♥❞♦ ❛ ❝❛♣✐t❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s
❙❡ ❝❛♣✐t❛❧ é ✉♠❛ ❢✉♥çã♦ q✉❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ❡st❛❞♦ r❡t♦r♥❛ ❛ r❡s♣❡❝t✐✈❛ ❝❛♣✐t❛❧ ❞❡st❡✱ t❡♠♦s q✉❡
❙✐♥t❛①❡ ✖ ✭❱✮
❋ór♠✉❧❛s
❖ ❝♦♥❥✉♥t♦ F ❞❡ ❢ór♠✉❧❛s é ❞❡✜♥✐❞♦ ❝♦♠♦✿
❛✮ ⊥,⊤ ∈ F
❜✮ ❙❡ ♣ é ✉♠ ♣r❡❞✐❝❛❞♦ ♥✲ár✐♦ ❡ t✶,t✷, ...,t♥ sã♦
t❡r♠♦s✱ ❡♥tã♦ ♣(t✶,t✷, ...,t♥) ∈ F✳
❝✮ ❙❡ α ∈ F ❡♥tã♦ ¬α ∈ F
❞✮ ❙❡ α, β ∈ F ❡♥tã♦ α◦β ∈ F✱ ♦♥❞❡
◦ ∈ {→,∨,∧,↔}✳
❙✐♥t❛①❡ ✖ ✭❱■✮
Pr❡❝❡❞ê♥❝✐❛
❙❡❣✉✐r❡♠♦s ❛s ♠❡s♠❛s ❝♦♥✈❡♥çõ❡s ❞❡ ♣r❡❝❡❞ê♥❝✐❛ q✉❡ ❛❞♦t❛♠♦s ♣❛r❛ ❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧✳
❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❝♦♥s❡r❛r❡♠♦s q✉❡
q✉❛♥t✐✜❝❛❞♦r❡s ✭∃,∀✮ ♣♦ss✉❡♠ ❛ ♠❡s♠❛
❙❡♠â♥t✐❝❛ ✖ ✭■✮
❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
P❛r❛ ❞❛r s❡♠â♥t✐❝❛ ❛ ✉♠❛ ❢ór♠✉❧❛ ❞❡✈❡♠♦s ✐♥t❡r♣r❡tá✲❧❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✉♥✐✈❡rs♦ ❞❡
❞✐s❝✉rs♦ U✳
▼❛s s♦♠❡♥t❡ U é s✉✜❝✐❡♥t❡❄
❙❡❥❛U =N✳ ❈♦♠ ✐ss♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r s❡
∃✐.∀❥.▼(✐,❥) é ✈❡r❞❛❞❡✐r❛❄
❙❡♠â♥t✐❝❛ ✖ ✭■■✮
❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
❙♦♠❡♥t❡ ♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ✐♥t❡r♣r❡t❛r ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡
♣r❡❞✐❝❛❞♦s✳
❉❡✈❡ ❤❛✈❡r ❛❞✐❝✐♦♥❛♠❡♥t❡ ♠❛♥❡✐r❛s ❞❡ ✐♥t❡r♣r❡t❛r✿
❙❡♠â♥t✐❝❛ ✖ ✭■■■✮
❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
❚❡r♠♦s ❢✉♥❝✐♦♥❛✐s ♣♦❞❡♠ s❡r ♣❡♥s❛❞♦s ❝♦♠♦ ❢✉♥çõ❡s ♥✲ár✐❛s ❡♥tr❡ t❡r♠♦s✳
P❛r❛ ❝❛❞❛ ❢✉♥çã♦ ❢ :T → T✱ ❞❡✈❡ ❡①✐st✐r
❙❡♠â♥t✐❝❛ ✖ ✭■❱✮
❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s
Pr❡❞✐❝❛❞♦s ♣♦❞❡♠ s❡r ♣❡♥s❛❞♦s ❝♦♠♦ ♥✲✉♣❧❛s ❞❡ t❡r♠♦s✳
P❛r❛ ❝❛❞❛ ♣r❡❞✐❝❛❞♦ ♣:T × T✱ ❞❡✈❡ ❡①✐st✐r ✉♠
❙❡♠â♥t✐❝❛ ✖ ✭❱✮
❱♦❧t❛♥❞♦ ❛♦ ❊①❡♠♣❧♦ ❛♥t❡r✐♦r✳✳✳
❙❡❥❛ U = N✱ ❡ ❝♦♥s✐❞❡r❡ q✉❡ ♣❛r❛ ①,② ∈ N✱
▼(①,②) é ✈❡r❞❛❞❡✐r♦ s❡ ① ≥②✳
❆ ❢ór♠✉❧❛ ∃✐.∀❥.▼(✐,❥) é ✈❡r❞❛❞❡✐r❛❄
◆ã♦✦ ❆ ❢ór♠✉❧❛ r❡♣r❡s❡♥t❛ ❛ s❡❣✉✐♥t❡ s❡♥t❡♥ç❛✿ ✏❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ é ♠❛✐♦r q✉❡ t♦❞♦s ♦s ♦✉tr♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✑✳
❙❡♠â♥t✐❝❛ ✖ ✭❱■✮
❋✐♥❛❧✐③❛♥❞♦✳✳✳
■♥t❡r♣r❡t❛♠♦s ❢ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ✉t✐❧✐③❛♥❞♦✿
❯♠ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦
❙❡♠â♥t✐❝❛ ✖ ✭❱■■✮
❊①❡r❝í❝✐♦
❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ N ❡
♦s s❡❣✉✐♥t❡s ♣r❡❞✐❝❛❞♦s ❡ sí♠❜♦❧♦s ❢✉♥❝✐♦♥❛✐s✿
♣❛r(①)✿ ✈❡r❞❛❞❡✐r♦ s❡ ① é ♣❛r✳
✐♠♣❛r(①)✿ ✈❡r❞❛❞❡✐r♦ s❡ ① é í♠♣❛r✳
s : N→ N✿ ❢✉♥çã♦ s✉❝❡ss♦r✳
◗✉❛❧ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s❄
✶ ∀♥.♣❛r(♥)∨✐♠♣❛r(♥)✳
✷ ∀♥.✐♠♣❛r(♥) →♣❛r(s(♥))✳
❙❡♠â♥t✐❝❛ ✖ ✭❱■■■✮
❊①❡r❝í❝✐♦
❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❢r❛s❡✿ ✏❚♦❞♦s ♦s ✉♥✐❝ór♥✐♦s ♣♦ss✉❡♠ ✺ ♣❛t❛s✑✳
❘❡♣r❡s❡♥t❡ ❡st❛ ❢r❛s❡ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳
