❚é❝♥✐❝❛s ♣❛r❛ ❉❡♠♦♥str❛çã♦ ❞❡
❚❡♦r❡♠❛s ✖ P❛rt❡ ■■■
▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ■
❘♦❞r✐❣♦ ❘✐❜❡✐r♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❆♣❧✐❝❛❞❛s ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❋❡❞❡r❛❧ ❞❡ ❖✉r♦ Pr❡t♦
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✹
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∀①.♣(①)✿
❯s❡ ✉♠❛ ♥♦✈❛ ✈❛r✐✁✈❡❧ ① ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠ ♦❜❥❡t♦ ❛r❜✐trár✐♦ ❡ ♣r♦✈❡ ♣(①)✳ ❙❡ ❛ss✐♠ ♦ ✜③❡r✱
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✹
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∀①.♣(①)✿
❘❛s❝✉♥❤♦ ❛♥t❡s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... ∀①.♣(①)
❘❛s❝✉♥❤♦ ❞❡♣♦✐s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... ♣(①)
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■■■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✹
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∀①.♣(①)✿
❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿
❙❡❥❛ ① ❛r❜✐trár✐♦✳
❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ♣(①)❪✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❱✮
❊①❡♠♣❧♦
❚❡♦r❡♠❛ ✹✿ ❙✉♣♦♥❤❛ q✉❡ ❆,❇ ❡ ❈ sã♦ ❝♦♥❥✉♥t♦s
t❛✐s q✉❡ ❆\❇ ⊆ ❈✳ ❊♥tã♦ ❆\❈ ⊆ ❇✳
❚❡♦r❡♠❛ ✺✿ ❙✉♣♦♥❤❛ q✉❡ ❆ ❡ ❇ sã♦ ❝♦♥❥✉♥t♦s✳ ❙❡ ❆∩❇ = ❆ ❡♥tã♦ ❆ ⊆ ❇✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✺
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∃①.♣(①)✿
❊♥❝♦♥tr❡ ✉♠ ✈❛❧♦r ♣❛r❛ ① ♣❛r❛ ♦ q✉❛❧ s❡❥❛ ♣♦ssí✈❡❧ ♣r♦✈❛r ♣(①)✳ ❙❡ ❛ss✐♠ ♦ ✜③❡r✱ ♣♦❞❡rá
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✺
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∃①.♣(①)✿
❘❛s❝✉♥❤♦ ❛♥t❡s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... ∃①.♣(①)
❘❛s❝✉♥❤♦ ❞❡♣♦✐s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... ♣(①)
① = ✈❛❧♦r ❡s❝♦❧❤✐❞♦
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✺
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∃①.♣(①)✿
❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿
❙❡❥❛ ① = ✭✈❛❧♦r ❡s❝♦❧❤✐❞♦✮✳
❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ♣(①)❪✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■■✮
❊①❡♠♣❧♦
❚❡♦r❡♠❛ ✻✿ ❙✉♣♦♥❤❛ ① ∈ R✱ ① > ✵ ❛r❜✐trár✐♦✳
❊♥tã♦ ❡①✐st❡ ② t❛❧ q✉❡ ②(② + ✶) =①✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❳✮
❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✸
P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛
∃①.♣(①)✿
■♥tr♦❞✉③❛ ✉♠❛ ♥♦✈❛ ✈❛r✐á✈❡❧ ①✵ q✉❡ r❡♣r❡s❡♥t❛
✉♠ ♦❜❥❡t♦ ♣❛r❛ ♦ q✉❛❧ ♣(①✵) é ✈❡r❞❛❞❡✐r♦✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳✮
❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✹
P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛
∀①.♣(①)✿
❙❡❧❡❝✐♦♥❡ ✉♠ ✈❛❧♦r ❛ ♣❛r❛ ① ❡ ✉s❡ ✐ss♦ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ ♣(❛) é ✈❡r❞❛❞❡✐r♦✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■✮
❊①❡r❝í❝✐♦
❚❡♦r❡♠❛ ✼✿ ❙✉♣♦♥❤❛ ❇ ✉♠ ❝♦♥❥✉♥t♦ ❡ F ✉♠❛
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✻
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ∧◗✿
Pr♦✈❡ s❡♣❛r❛❞❛♠❡♥t❡ P ❡ ◗
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✻
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ∧◗✿
❘❛s❝✉♥❤♦ ❛♥t❡s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... P ∧◗
❘❛s❝✉♥❤♦ ❞❡♣♦✐s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... P
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❱✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✻
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ∧◗✿
❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿
❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ P❪✳ ❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ◗❪✳
❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ P ∧◗
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■✮
❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✺
P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ P ∧◗✿
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■✮
◆♦t❛çã♦
①|② q✉❡r ❞✐③❡r ① ❞✐✈✐❞❡ ②✱ ✐st♦ é✿ ∃③.③ ∈ Z∧ (❦① = ②)
❊①❡♠♣❧♦
❚❡♦r❡♠❛ ✽✿ P❛r❛ t♦❞♦s ✐♥t❡✐r♦s ❛✱ ❜ ❡ ❝✱ s❡ ❛|❜ ❡ ❜|❝ ❡♥tã♦ ❛|❝✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■■✮
❘❡♣r❡s❡♥t❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✉s❛♥❞♦ ❧ó❣✐❝❛
∀❛.∀❜.∀❝.❛ ∈ Z → ❜ ∈ Z → ❝ ∈ Z →
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❳✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶
❍✐♣ót❡s❡s Pr♦✈❛r
∀❛.∀❜.∀❝.❛ ∈ Z →
❜ ∈ Z → ❝ ∈ Z →
(❛|❜∧❜|❝ →❛|❝)
❈♦♠♦ ❛ ❢ór♠✉❧❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ t❡♦r❡♠❛ é ❞❛ ❢♦r♠❛ ∀①.♣(①)✱ ❛ ❡str❛té❣✐❛ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡
❢ór♠✉❧❛✱ ❢❛③❡♥❞♦ q✉❡ ❛,❜ ❡ ❝ s❡❥❛♠ ✈❛❧♦r❡s
❛r❜✐trár✐♦s✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✷
❍✐♣ót❡s❡s Pr♦✈❛r ❛ ∈ Z →
❜ ∈ Z →❝ ∈ Z →
(❛|❜∧❜|❝ →❛|❝)
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸
❍✐♣ót❡s❡s Pr♦✈❛r
❛ ∈ Z (❛|❜∧❜|❝ → ❛|❝)
❜ ∈ Z
❝ ∈ Z
❆♣❧✐❝❛♠♦s ♠❛✐s ✉♠❛ ✈❡③ ❛ ❡str❛té❣✐❛ ♣❛r❛ P →◗✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✹
❍✐♣ót❡s❡s Pr♦✈❛r ❛ ∈ Z ❛|❝
❜ ∈ Z
❝ ∈ Z
❛|❜∧❜|❝
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✺
❍✐♣ót❡s❡s Pr♦✈❛r ❛ ∈ Z ❛|❝
❜ ∈ Z
❝ ∈ Z
❛|❜
❜|❝
❊s❝r❡✈❡♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ①|②✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■❱✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✻
❍✐♣ót❡s❡s Pr♦✈❛r
❛ ∈ Z ∃❦.❦ ∈ Z∧(❦❛ = ❝)
❜ ∈ Z
❝ ∈ Z
∃♥.♥ ∈ Z∧ (♥❛ = ❜) ∃♠.♠ ∈ Z∧(♠❜ = ❝)
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✼
❍✐♣ót❡s❡s Pr♦✈❛r
❛ ∈ Z ∃❦.❦ ∈ Z ∧(❦❛ = ❝)
❜ ∈ Z
❝ ∈ Z
♥✵ ∈ Z∧(♥✵❛ = ❜)
♠✵ ∈ Z ∧(♠✵❜ = ❝)
❉✐✈✐♥❞♦ ❤✐♣ót❡s❡s ❞❛ ❢♦r♠❛ P ∧◗
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✽
❍✐♣ót❡s❡s Pr♦✈❛r
❛ ∈ Z ∃❦.❦ ∈ Z∧(❦❛ = ❝)
❜ ∈ Z
❝ ∈ Z
♥✵ ∈ Z
♠✵ ∈ Z
♥✵❛ = ❜
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✾
❍✐♣ót❡s❡s Pr♦✈❛r
❛ ∈ Z ❦ ∈ Z∧ (❦❛ = ❝)
❜ ∈ Z
❝ ∈ Z
♥✵ ∈ Z
♠✵ ∈ Z
♥✵❛ = ❜
♠✵❜ = ❝
❯s❛♥❞♦ ❛ ❡str❛té❣✐❛ ♣❛r❛ ♣r♦✈❛r q✉❡ P ∧◗
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶✵
❍✐♣ót❡s❡s Pr♦✈❛r ❛ ∈ Z ❦ ∈ Z
❜ ∈ Z ❦❛ = ❝
❝ ∈ Z
♥✵ ∈ Z
♠✵ ∈ Z
♥✵❛ = ❜
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■❳✮
❊s❝r❡✈❡♥❞♦ ♦ t❡①t♦ ❞❛ ♣r♦✈❛✳✳✳
Pr✐♠❡✐r❛ ❡str❛té❣✐❛ ✉s❛❞❛✿ ∀①.♣(①) ❡ P →◗
❙❡❥❛♠ ❛,❜ ❡ ❝ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s
❬Pr♦✈❛ ❞❡ ❛|❜∧❜|❝ →❛|❝❪
❈♦♠♦ ❛,❜ ❡ ❝ sã♦ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s ✐♥t❡✐r♦s
❛r❜✐trár✐♦s✳✳✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳✮
❊s❝r❡✈❡♥❞♦ ♦ t❡①t♦ ❞❛ ♣r♦✈❛✳✳✳
◆♦✈❛♠❡♥t❡ P →◗✳✳✳
❙❡❥❛♠ ❛,❜ ❡ ❝ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s
❙✉♣♦♥❤❛ q✉❡ ❛|❜ ❡ ❜|❝
❬Pr♦✈❛ ❞❡ ❛|❝❪
▲♦❣♦✱ s❡ ❛|❜ ❡ ❜|❝ ❡♥tã♦ ❛|❝
❈♦♠♦ ❛,❜ ❡ ❝ sã♦ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s ✐♥t❡✐r♦s
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■✮
❊s❝r❡✈❡♥❞♦ ♦ t❡①t♦ ❞❛ ♣r♦✈❛✳✳✳
❊s❝r❡✈❡♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❛|❝✳
❙❡❥❛♠ ❛,❜ ❡ ❝ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s
❙✉♣♦♥❤❛ q✉❡ ❛|❜ ❡ ❜|❝
❬Pr♦✈❛ ❞❡ ∃❦.❦ ∈ Z∧(❦❛ = ❝)❪
▲♦❣♦✱ s❡ ❛|❜ ❡ ❜|❝ ❡♥tã♦ ❛|❝
❈♦♠♦ ❛,❜ ❡ ❝ sã♦ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s ✐♥t❡✐r♦s
❛r❜✐trár✐♦s✳✳✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■■✮
❊s❝r❡✈❡♥❞♦ ♦ t❡①t♦ ❞❛ ♣r♦✈❛✳✳✳
◆♦✈❛♠❡♥t❡ P →◗✳✳✳
❙❡❥❛♠ ❛,❜ ❡ ❝ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s
❙✉♣♦♥❤❛ q✉❡ ❛|❜ ❡ ❜|❝
❙❡❥❛ ❦ = ...
❬Pr♦✈❛ ❞❡ ❦ ∈ Z ∧(❦❛ = ❝)❪
❆ss✐♠ t❡♠♦s q✉❡ ❛|❝✳
▲♦❣♦✱ s❡ ❛|❜ ❡ ❜|❝ ❡♥tã♦ ❛|❝
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■■■✮
❊①❡♠♣❧♦ ✖ ❚❡①t♦ ✜♥❛❧
Pr♦✈❛✿ ❙❡❥❛♠ ❛,❜ ❡ ❝ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s✳
❙✉♣♦♥❤❛ q✉❡ ❛|❜ ❡ ❜|❝✳ ❈♦♠♦ ❛|❜ t❡♠♦s q✉❡
❡①✐st❡ ✉♠ ✐♥t❡✐r♦ ♥ t❛❧ q✉❡ ♥❛= ❜ ❡
s✐♠✐❧❛r♠❡♥t❡ ❡①✐st❡ ♠ t❛❧ q✉❡ ♠❜ = ❝✳ P❛r❛
♠♦str❛r q✉❡ ❛|❝ ❜❛st❛ ❡♥❝♦♥tr❛r ✉♠ ✐♥t❡✐r♦ ❦ t❛❧
q✉❡ ❦❛ = ❝✳ ❙❡❥❛ ❦ = ♥♠✳ ❈♦♠♦ ❝ = ♠❜ ❡
❜ = ♥❛ t❡♠♦s q✉❡ ❝ = ♠♥❛ ❝♦♥❢♦r♠❡ r❡q✉❡r✐❞♦✳
▲♦❣♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ❛|❜ ❡ ❜|❝ ❡♥tã♦
❛|❝✳ ❈♦♠♦ ❛,❜ ❡ ❝ sã♦ ✐♥t❡✐r♦s ❛r❜✐trár✐♦s
✐♥t❡✐r♦s ❛r❜✐trár✐♦s t❡♠♦s q✉❡ ♣❛r❛ t♦❞♦s ✐♥t❡✐r♦s ❛✱ ❜ ❡ ❝✱ s❡ ❛|❜ ❡ ❜|❝ ❡♥tã♦ ❛|❝✳
