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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡    

❙ ✉ ❝ ✉ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ✵ ✶    

❈ ♦ ✉ ✰ ❚ ❉ ✰ ❚    

❱ ❡ ✐ ♦ ♥ ✶ ✳ ✵ ✳ ✵  

▼ ✳ ■ ♠ ❛ ✐ ❧ ❍ ❆ ❉ ❏ ❆ ❉ ❏  

❖ ❝ ♦ ❜ ❡ ✷ ✵ ✶ ✸  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ 

❘ ❡ ♠ ❛ ✉ ❡ ✿  

❈ ❡ ❞ ♦ ❝ ✉ ♠ ❡ ♥ ♥ ✬ ❡ ♣ ❛ ✈ ✐ ✱ ✈ ♦ ✉ ♣ ♦ ✉ ✈ ❡ ③ ♦ ✉ ✈ ❡ ❞ ❡ ❡ ❡ ✉ ❞ ✬ ♦ ❤ ♦ ❣ ❛ ♣ ❤ ❡ ✱ ❞ ❡ ❛ ✐ ✐ ❡ ❡ ❛ ✉ ❡ ✳  

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❚ ❛ ❜ ❧ ❡ ❞ ❡ ♠ ❛ ✐ ❡      

✶ ● ♥ ❛ ❧ ✐ ✺

✶ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺  

✶ ✳ ✷ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✭ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺  

✶ ✳ ✸ ❙ ✉ ❝ ✉ ❡ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻  

✶ ✳ ✹ ❙ ♦ ✉ ✲ ♦ ❣ ❛ ♠ ♠ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻  

✶ ✳ ✹ ✳ ✶ ❛ ❛ ❣ ❡ ❞ ❡ ♣ ❛ ❛ ♠ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻  

✶ ✳ ✺ ▲ ✬ ✐ ♥ ❞ ❝ ✐ ❞ ❛ ❜ ✐ ❧ ✐ ❞ ❡ ❡ ♠ ✐ ♥ ❛ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼  

✶ ✳ ✻ ❈ ♦ ♥ ❝ ❡ ♣ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽  

✶ ✳ ✻ ✳ ✶ ❆ ♥ ❛ ❧ ② ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽  

✶ ✳ ✻ ✳ ✷ ❆ ♥ ❛ ❧ ② ❡ ❛ ❝ ❡ ♥ ❞ ❛ ♥ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽  

✶ ✳ ✻ ✳ ✸ ♠ ❧ ❛ ♥ ❣ ❡ ❞ ❡ ❞ ❡ ✉ ① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽  

✶ ✳ ✼ ❈ ♦ ♥ ❝ ❧ ✉ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽  

✶ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾  

✷ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ✬ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✶ ✵

✷ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✷ ◗ ✉ ❛ ❧ ✐ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✷ ✳ ✶ ◗ ✉ ❛ ❧ ✐ ❞ ✬ ❝ ✐ ✉ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✷ ✳ ✷ ❚ ❡ ♠ ✐ ♥ ❛ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✷ ✳ ✸ ❱ ❛ ❧ ✐ ❞ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✷ ✳ ✹ ❡ ❢ ♦ ♠ ❛ ♥ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✸ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵  

✷ ✳ ✹ ▼ ✉ ❡ ❞ ❡ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶  

✷ ✳ ✺ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ♠ ♦ ② ❡ ♥ ♥ ❡ ❡ ❡ ♥ ♣ ✐ ❡ ❞ ❡ ❝ ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶  

✷ ✳ ✺ ✳ ✶ ❚ ❡ ♠ ♣ ❧ ❡ ♣ ❧ ✉ ♠ ♦ ✉ ✈ ❛ ✐ ✭ ❡ ♥ ♣ ✐ ❡ ❞ ❡ ❝ ❛ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶  

✷ ✳ ✺ ✳ ✷ ❚ ❡ ♠ ♣ ♠ ♦ ② ❡ ♥ ♥ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶  

✷ ✳ ✻ ◆ ♦ ❛ ✐ ♦ ♥ ✉ ✐ ❧ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸  

✷ ✳ ✻ ✳ ✶ ◆ ♦ ❛ ✐ ♦ ♥ ✧  o ✧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸✷ ✳ ✻ ✳ ✷ ◆ ♦ ❛ ✐ ♦ ♥ ✧  O ✧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸

✷ ✳ ✼ ❯ ♥ ❝ ❧ ❛ ❡ ♠ ❡ ♥ ❞ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸  

✷ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹  

✷ ✳ ✽ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹  

✷ ✳ ✽ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹  

✷ ✳ ✽ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✺  

✷ ✳ ✽ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✻  

✷ ✳ ✽ ✳ ✺ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✻  

✷ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✸ 

✸ ▲ ❛ ❝ ✉ ✐ ✈ ✐ ✶ ✼

✸ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✼  

✸ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✼  

✸ ✳ ✸ ❊ ① ❡ ♠ ♣ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✽  

✸ ✳ ✸ ✳ ✶ ❊ ① ❡ ♠ ♣ ❧ ❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✽  

✸ ✳ ✸ ✳ ✷ ❊ ① ❡ ♠ ♣ ❧ ❡ ✷ ✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✽  

✸ ✳ ✹ ❘ ❣ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✾  

✸ ✳ ✺ ❘ ❝ ✉ ✐ ✈ ✐ ❝ ♦ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✾  

✸ ✳ ✺ ✳ ✶ ❊ ① ❡ ♠ ♣ ❧ ❡ ✿ ❛ ✐ ❡ ♦ ✉ ✐ ♠ ♣ ❛ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✾  

✸ ✳ ✻ ♦ ❜ ❧ ♠ ❡ ✭ ❧ ❡ ♦ ✉ ❞ ✬ ❍ ❛ ♥ ♦ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✵  

✸ ✳ ✼ ❈ ♦ ♥ ❝ ❧ ✉ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶  

✸ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✸ ✳ ✽ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✸ ✳ ✽ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✸ ✳ ✽ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✸ ✳ ✽ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✸ ✳ ✽ ✳ ✺ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷  

✸ ✳ ✾ ❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✹

✸ ✳ ✾ ✳ ✶ ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✉ ♥ ❧ ♠ ❡ ♥ ❞ ❛ ♥ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✹  

✸ ✳ ✾ ✳ ✷ ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❙ ✉ ❡ ♥ ✐ ❡ ❧ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✹  

✸ ✳ ✾ ✳ ✸ ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❞ ✐ ❝ ❤ ♦ ♦ ♠ ✐ ✉ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✹  

✸ ✳ ✾ ✳ ✹ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✺  

✹ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ✷ ✻

✹ ✳ ✶ ❚ ② ♣ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❛ ❜ ❛ ✐ ✭ ❚ ❉ ❆ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✻  

✹ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ② ♣ ❡ ❛ ❜ ❛ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✻  

✹ ✳ ✷ ✳ ✶ ❊ ① ❡ ♠ ♣ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✻  

✹ ✳ ✸ ▲ ✬ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ❉ ✬ ✉ ♥ ❚ ② ♣ ❡ ❆ ❜ ❛ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✼  

✹ ✳ ✹ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ❡ ② ♣ ❡ ❛ ❜ ❛ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✼  

✺ ❙ ✉ ❝ ✉ ❡ ❧ ✐ ♥ ❛ ✐ ❡ ✲ ▲ ❡ ▲ ✐ ❡ ✷ ✽

✺ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✽  

✺ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✽  

✺ ✳ ✸ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❛ ❜ ❛ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✽  

✺ ✳ ✸ ✳ ✶ ❊ ♥ ❡ ♠ ❜ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✽  

✺ ✳ ✸ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❛ ❜ ❛ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✾  

✺ ✳ ✸ ✳ ✸ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❛ ① ✐ ♦ ♠ ❛ ✐ ✉ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✾  

✺ ✳ ✹ ■ ♠ ♣ ❧ ✐ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✾  

✺ ✳ ✹ ✳ ✶ ■ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ❝ ♦ ♥ ✐ ❣ ✉ ✭ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✾  

✺ ✳ ✹ ✳ ✷ ■ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ❝ ❤ ❛ ♥ ❡ ✭ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ❡ ✉ ❝ ✉ ❡ ❝ ❤ ❛ ♥ ❡ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✸  

✺ ✳ ✺ ❊ ① ❡ ❝ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✼  

✺ ✳ ✺ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✶ ✿   ♦ ✐ ♥ ❡ ✉   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✼  

✺ ✳ ✺ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷ ✿   ♦ ✐ ♥ ❡ ✉   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✼  

✺ ✳ ✺ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✿   ❧ ❡ ❧ ✐ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽  

✺ ✳ ✺ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✹ ✿  ❧ ❡ ❧ ✐ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽  

✺ ✳ ✺ ✳ ✺ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✺ ✿  ❧ ❡ ❧ ✐ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽  

✺ ✳ ✺ ✳ ✻ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✻ ✿  ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ❡ ❜ ❛ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽  

✺ ✳ ✺ ✳ ✼ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✼ ✿  ❯ ♥ ♣ ❡ ✉ ❛ ✈ ❛ ♥ ❝   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✽  

✺ ✳ ✺ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✽ ✿   ❉ ✐ ✛ ❡ ♥ ❝ ❡ ❞ ❡ ❞ ❡ ✉ ① ❧ ✐ ❡ ❧ ✐ ♥ ❛ ✐ ❡ ❝ ❤ ❛ ♥ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✾  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✹ 

✺ ✳ ✺ ✳ ✾ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✾ ✿   ❘ ❛ ❧ ✐ ❡ ❧ ❡ ❝ ❤ ❛ ♥ ❛ ❣ ❡ ❛ ✐ ❡ ❞ ✬ ✉ ♥ ❡ ❧ ✐ ❡ ❞ ♦ ✉ ❜ ❧ ❡ ♠ ❡ ♥ ❝ ❤ ❛ ♥ ❡   ✳ ✳ ✳ ✳ ✳ ✸ ✾

✺ ✳ ✻ ❚ ✿ ❉ ✐ ❝ ✐ ♦ ♥ ♥ ❛ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✵  

✺ ✳ ✻ ✳ ✶ ♦ ❜ ❧ ♠ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✵  

✺ ✳ ✻ ✳ ✷ ■ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✵  

✻ ▲ ❡ ✐ ❧ ❡ ✹ ✶

✻ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶  

✻ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❛ ❜ ❛ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶  

✻ ✳ ✷ ✳ ✶ ❊ ♥ ❡ ♠ ❜ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶  

✻ ✳ ✷ ✳ ✷ ❉ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ♥ ❡ ❧ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶  

✻ ✳ ✷ ✳ ✸ ❉ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❛ ① ✐ ♦ ♠ ❛ ✐ ✉ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶  

✻ ✳ ✸ ✐ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷  

✻ ✳ ✸ ✳ ✶ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ❡ ❛ ❜ ❧ ❡ ❛ ✉ ① ✭ ■ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ❈ ♦ ♥ ✐ ❣ ✉ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷  

✻ ✳ ✸ ✳ ✷ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ❡ ♣ ♦ ✐ ♥ ❡ ✉ ✭ ■ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ❝ ❤ ❛ ♥ ❡ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✸  

✻ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✹  

✻ ✳ ✹ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷ ✿  ❆ ❧ ❣ ♦ ✳ ❞ ❡ ❜ ❛ ❡ ✉ ❧ ❡ ✐ ❧ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✹  

✻ ✳ ✹ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✿  ❯ ♥ ♣ ❡ ✉ ❛ ✈ ❛ ♥ ❝   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✹  

✻ ✳ ✹ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✿   ■ ♥ ✈ ❡ ❡ ♣ ✐ ❧ ❡ ❡ ✜ ❧ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✹  

✼ ▲ ❡ ❋ ✐ ❧ ❡ ✹ ✺

✼ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✺  

✼ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❛ ❜ ❛ ✐ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✺  

✼ ✳ ✷ ✳ ✶ ❊ ♥ ❡ ♠ ❜ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✺  

✼ ✳ ✷ ✳ ✷ ❉ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ♥ ❡ ❧ ❧ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✺  

✼ ✳ ✷ ✳ ✸ ❉ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❛ ① ✐ ♦ ♠ ❛ ✐ ✉ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✺  

✼ ✳ ✸ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✻  

✼ ✳ ✸ ✳ ✶ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ❡ ❛ ❜ ❧ ❡ ❛ ✉ ① ✭ ■ ♠ ♣ ❧ ♠ ❡ ♥ ❛ ✐ ♦ ♥ ❈ ♦ ♥ ✐ ❣ ✉ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✻  

✼ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼  

✼ ✳ ✹ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✶ ✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼  

✼ ✳ ✹ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷ ✿  ❆ ❧ ❣ ♦ ✳ ❞ ❡ ❜ ❛ ❡ ✉ ❧ ❡ ❋ ✐ ❧ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼  

✼ ✳ ✹ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✿  ❯ ♥ ♣ ❡ ✉ ❛ ✈ ❛ ♥ ❝   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼  

✼ ✳ ✹ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸ ✿   ■ ♥ ✈ ❡ ❡ ✉ ♥ ❡ ✜ ❧ ❡   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼  

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❈ ❤ ❛ ♣ ✐ ❡ ✶      

● ♥ ❛ ❧ ✐    

✶ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥      

♣ ♦ ✉ ♦ ✉ ❞ ❡ ✉ ♥ ♣ ♦ ❜ ❧ ♠ ❡ ❞ ♦ ♥ ♥ ❡ ♣ ❛ ❧ ✬ ✐ ♥ ❢ ♦ ♠ ❛ ✐ ✉ ❡ ❧ ✬ ✉ ✐ ❧ ✐ ❛ ❡ ✉ ❞ ❡ ❧ ✬ ♦ ❞ ✐ ♥ ❛ ❡ ✉ ❞ ♦ ✐ ♠ ❡ ❡ ❛ ✉  

♣ ♦ ✐ ♥ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ❡ ❧ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥ ♣ ❛ ❧ ❛ ♠ ❛ ❝ ❤ ✐ ♥ ❡  

❯ ♥ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ❡ ✉ ♥ ❡ ✉ ❝ ❝ ❡ ✐ ♦ ♥ ❧ ♦ ❣ ✐ ✉ ❡ ❡ ♦ ❞ ♦ ♥ ♥ ❡ ❞ ✬ ✐ ♥ ✉ ❝ ✐ ♦ ♥  

♦ ✉ ❝ ✐ ❡ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ✐ ❧ ❢ ❛ ✉ ✿  

✕ ❇ ✐ ❡ ♥ ❝ ♦ ♥ ♥ ❛ ❡ ❧ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ❀  

✕ ❙ ✬ ❛ ✈ ♦ ✐ ❧ ❡ ❞ ❝ ♦ ✉ ♣ ❡ ❧ ♦ ❣ ✐ ✉ ❡ ♠ ❡ ♥ ❡ ♥ ✉ ♥ ❡ ♥ ❡ ♠ ❜ ❧ ❡ ❞ ✬ ♦ ♣ ❛ ✐ ♦ ♥ ❧ ♠ ❡ ♥ ❛ ✐ ❡ ✭ ❛ ❝ ✐ ♦ ♥ ✮ ❀  

✕ ❈ ♦ ♥ ♥ ❛ ❡ ✉ ♥ ❧ ❛ ♥ ❣ ✉ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❛ ✐ ♦ ♥ ✳  

♣ ♦ ❜ ❧ ♠ ❡   −→ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡    −→ ♣ ♦ ❣ ❛ ♠ ♠ ❡  

✶ ✳ ✷ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✭ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ✮    

❯ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❡ ✉ ♥ ❡ ✧ ❙ ♣ ❝ ✐ ✜ ❝ ❛ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ❝ ❤ ♠ ❛ ❞ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❡ ♦ ✉ ❢ ♦ ♠ ❡ ❞ ✬ ✉ ♥ ❡ ✉ ✐ ❡ ✜ ♥ ✐ ❡ ❞ ✬ ♦ ♣ ❛ ✐ ♦ ♥  

❧ ♠ ❡ ♥ ❛ ✐ ❡ ✉ ♥ ❡ ♥ ❝ ❤ ❛ ♥ ❡ ♠ ❡ ♥ ❞ ❡ ♠ ✐ ♥ ✧ ✳  

❖ ✉ ❡ ♥ ❝ ♦ ❡ ✿  

▲ ❛ ❞ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❞ ❡ ❛ ♣ ❡ ✉ ✐ ✈ ❡ ♣ ♦ ✉ ❛ ❧ ✐ ❡ ✉ ♥ ❛ ✈ ❛ ✐ ❧ ✳  

❯ ♥  ♣ ♦ ❣ ❛ ♠ ♠ ❡   ❡ ✭ ❞ ♦ ♥ ❝ ✮ ❧ ❛ ❞ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ❛ ♥ ✉ ♥ ❧ ❛ ♥ ❣ ❛ ❣ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❛ ✐ ♦ ♥ ✳  

❘ ❡ ♠ ❛ ✉ ❡ ✿  

❯ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❡ ✐ ♥ ❞ ♣ ❡ ♥ ❞ ❛ ♥ ❞ ✉ ❧ ❛ ♥ ❣ ❛ ❣ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❛ ✐ ♦ ♥ ✭ ❉ ♦ ♥ ❝ ❧ ❛ ♠ ❛ ❝ ❤ ✐ ♥ ❡ ✮ ✳  

✺ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✻ 

✶ ✳ ✸ ❙ ✉ ❝ ✉ ❡ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✿  

▲ ❛ ✉ ❝ ✉ ❡ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❡ ❞ ♦ ♥ ❝ ✿  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✶ ✿  ◆ ♦ ♠ ❉ ❡ ▲ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡  

❈ ♦ ♥   ✿ ❝ ♦ ♥ ✶ ✱ ❝ ♦ ♥ ✷ ✱ ✳ ✳ ✳ ✳ ❂  

❱ ❛   ✿ ✈ ❛ ✶ ✱ ✈ ❛ ✷ ✳ ✳ ✳ ✿ ② ♣ ❡  

♦ ❝ ❞ ✉ ❡   ✶ ✭ ✳ ✳ ✳ ✳ ✮  

❞ ❜ ✉  

...

...

...

✜ ♥ 

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❢ ✶ ✭ ✳ ✳ ✳ ✳ ✮ ✿ ② ♣ ❡  

❞ ❜ ✉  

...

...

...✜ ♥ 

❞ ❜ ✉  

...P 1(...)var1 ← f 1(...)...

✜ ♥ 

✶ ✳ ✹ ❙ ♦ ✉ ✲ ♦ ❣ ❛ ♠ ♠ ❡    

❯ ♥ ❡  ❢ ♦ ♥ ❝ ✐ ♦ ♥   ❡ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✐ ♥ ❞ ♣ ❡ ♥ ❞ ❛ ♥ ✱ ❧ ✬ ❛ ♣ ♣ ❡ ❧ ❞ ❡ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❞ ❝ ❧ ❡ ♥ ❝ ❤ ❡ ❧ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥ ❞ ❡ ♦ ♥  

❜ ❧ ♦ ❝ ❞ ✬ ✐ ♥ ✉ ❝ ✐ ♦ ♥ ✳ ❯ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❡ ❡ ♠ ✐ ♥ ❡ ❡ ♥ ♦ ✉ ❛ ♥ ♦ ✉ ♥ ♦ ♥ ✉ ♥ ❡ ✈ ❛ ❧ ❡ ✉ ✳  

❯ ♥ ♣ ♦ ❝ ❞ ✉ ❡ ❡ ✉ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✉ ✐ ❡ ♦ ✉ ♥ ❡ ✈ ✐ ❞ ❡ ✳  

✶ ✳ ✹ ✳ ✶ ❛ ❛ ❣ ❡ ❞ ❡ ♣ ❛ ❛ ♠ ❡  

❛ ✈ ❛ ❧ ❡ ✉  

❖ ♥ ❞ ✐ ✉ ✬ ✉ ♥ ♣ ♦ ❝ ❞ ✉ ❡ ❢ ❛ ✐ ✉ ♥ ♣ ❛ ❛ ❣ ❡ ♣ ❛ ✈ ❛ ❧ ❡ ✉ ✱ ✬ ✐ ❧ ♥ ❡ ❝ ❤ ❛ ♥ ❣ ❡ ♣ ❛ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ❞ ❡ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ♣ ❛ ✳  

❛ ❢ ❡ ♥ ❝ ❡  

■ ❧ ✬ ❛ ❣ ✐ ❞ ❡ ♠ ♦ ❞ ✐ ✜ ❝ ❛ ✐ ♦ ♥ ❞ ✉ ✈ ❛ ❧ ❡ ✉ ❞ ✉ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ♣ ❛ ✳  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✼ 

❊ ① ❡ ♠ ♣ ❧ ❡  

♦ ❝ ❞ ✉ ❡   ✶ ✭ ① ✿ ❡ ♥ ✐ ❡ ✮  

❱ ❛   ✿ ✐ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

♣ ♦ ✉   ✐ ❂ ✶    ✶ ✵  ❢ ❛ ✐ ❡  

x ← x ∗ x✜ ♥ ♣ ♦ ✉  

✜ ♥ 

♦ ❝ ❞ ✉ ❡   ✶ ✭ ✈ ❛ ① ✿ ❡ ♥ ✐ ❡ ✮  

❱ ❛   ✿ ✐ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

♣ ♦ ✉   ✐ ❂ ✶    ✶ ✵  ❢ ❛ ✐ ❡  

x ← x ∗ x✜ ♥ ♣ ♦ ✉  

✜ ♥ 

✶ ✳ ✺ ▲ ✬ ✐ ♥ ❞ ❝ ✐ ❞ ❛ ❜ ✐ ❧ ✐ ❞ ❡ ❡ ♠ ✐ ♥ ❛ ✐ ♦ ♥  

❙ ✉ ♣ ♣ ♦ ♦ ♥ ✉ ❡ ❧ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ❞ ❡ ❧ ✬ ❛ ♦ ✐ ❞ ❝ ✐ ❞ ❛ ❜ ❧ ❡  

⇒ ∃ ✉ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥    B ✉ ✐ ❞ ❝ ✐ ❞ ❡ ✐ ✉ ♥ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ✬ ❛ ❡  ⇒  B (X   : programme) = trueB(X 2 : programme) = f alse

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✷ ✿  ❳ 

❱ ❛   ✿ ② ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

y ← 1❊ ❝ ✐ ❡ ✭ ② ✮  

✜ ♥ 

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✸ ✿  ❳ ✷ 

❱ ❛   ✿ ② ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

y ← 1 ❛ ♥ ✉ ❡   ② ❃ ✵  ❢ ❛ ✐ ❡  

y ← y  + 1✜ ♥  

❊ ❝ ✐ ❡ ✭ ② ✮  

✜ ♥ 

♦ ♣ ♦ ♦ ♥ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✿  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✽ 

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✹ ✿  ❈ 

❞ ❜ ✉  

❛ ♥ ✉ ❡   ❇ ✭ ❈ ✮   ❢ ❛ ✐ ❡  

✜ ♥  

❊ ❝ ✐ ❡ ✭ ② ✮  

✜ ♥ 

❙ ✐  C  ❜ ♦ ✉ ❝ ❧ ❡ ✐ ♥ ❞ ✐ ✜ ♥ ✐ ♠ ❡ ♥ ✱ ❛ ❧ ♦    B(C ) = false ⇒ C  ❡ ♠ ✐ ♥ ❡  ❙ ✐ ♥ ♦ ♥   B(C ) = true ⇒ C  ❜ ♦ ✉ ❝ ❧ ❡ ✐ ♥ ✜ ♥ ✐ ♠ ❡ ♥  ❆ ❧ ♦ ❈ ♦ ♥ ❛ ❞ ✐ ❝ ✐ ♦ ♥  

◆ ♦ ❡ ③ ❜ ✐ ❡ ♥   ✿ 

❉ ❛ ♥ ❝ ❡ ❝ ♦ ✉ ✱ ❖ ♥ ✉ ❞ ✐ ❡ ❛ ❡ ✉ ❧ ❡ ♠ ❡ ♥ ❞ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ♣ ♦ ✉ ❧ ❡ ✉ ❡ ❧ ✐ ❧ ❡ ① ✐ ❡ ❞ ❡ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ❝ ✐ ❞ ❛ ❜ ❧ ❡ ✳  

✶ ✳ ✻ ❈ ♦ ♥ ❝ ❡ ♣ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡      

✶ ✳ ✻ ✳ ✶ ❆ ♥ ❛ ❧ ② ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥  

❉ ❝ ♦ ♠ ♣ ♦ ❡ ❧ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ❡ ♥ ♦ ✉ ♣ ♦ ❜ ❧ ♠ ❡ ✳  

✶ ✳ ✻ ✳ ✷ ❆ ♥ ❛ ❧ ② ❡ ❛ ❝ ❡ ♥ ❞ ❛ ♥ ❡  

❯ ✐ ❧ ✐ ❡ ❧ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✱ ♣ ✐ ♠ ✐ ✐ ✈ ❡ ✱ ♦ ✉ ✐ ❧ ✱ ✳ ✳ ❞ ♦ ♥ ♦ ♥ ❞ ✐ ♣ ♦ ❡ ✱ ❧ ❡ ❛ ❡ ♠ ❜ ❧ ❡ ♣ ♦ ✉ ❡ ♥ ❢ ❛ ✐ ❡ ✉ ♥ ✉ ❝ ✉ ✐  

♦ ✉ ❞ ❡ ♥ ♦ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ✳  

✶ ✳ ✻ ✳ ✸ ♠ ❧ ❛ ♥ ❣ ❡ ❞ ❡ ❞ ❡ ✉ ①  

❖ ♥ ❢ ❛ ✐ ✉ ♥ ❡ ❛ ♥ ❛ ❧ ② ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❛ ♥ ❡ ♦ ✉ ❡ ♥ ❛ ② ❛ ♥ ❧ ✬ ❡ ♣ ✐ ❧ ❡ ♠ ♦ ❞ ✉ ❧ ❡ ❜ ✐ ❡ ♥ ❝ ♦ ♥ ✉ ✉ ✐ ❡ ① ✐ ❡ ♥ ❞ ❥ ✳  

✶ ✳ ✼ ❈ ♦ ♥ ❝ ❧ ✉ ✐ ♦ ♥      

❞ ❛ ♥ ❝ ❡ ❝ ♦ ✉ ♦ ♥ ❛ ✐ ♥ ♦ ❞ ✉ ✐ ❡ ❞ ❡ ♥ ♦ ✐ ♦ ♥ ❞ ❡ ❜ ❛ ❡ ❧ ✐ ❡ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ✳  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✾ 

✶ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡  

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❈ ❤ ❛ ♣ ✐ ❡ ✷      

❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ✬ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡    

✷ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥      

■ ❧ ❡ ① ✐ ❡ ♦ ✉ ✈ ❡ ♥ ♣ ❧ ✉ ✐ ❡ ✉ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ♣ ❡ ♠ ❡ ❛ ♥ ❞ ❡ ♦ ✉ ❞ ❡ ✉ ♥ ♠ ♠ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ✳ ❊ ① ❡ ♠ ♣ ❧ ❡ ✿ ❧ ❡  

❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ❡ ✐ ✳ ▲ ❡ ❝ ❤ ♦ ✐ ① ❞ ✉ ♠ ❡ ✐ ❧ ❧ ❡ ✉ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✐ ♠ ♣ ❧ ✐ ✉ ❡ ✉ ♥ ❡ ❛ ♥ ❛ ❧ ② ❡ ❞ ❡ ❡ ♣ ❡ ❢ ♦ ♠ ❛ ♥ ❝ ❡ ✳ ❊ ♥  

❣ ♥ ❛ ❧ ✱ ❧ ❡ ❝ ✐ ❡ ❧ ❡ ♣ ❧ ✉ ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❝ ❡ ❧ ✉ ✐ ❞ ✉ ❡ ♠ ♣ ♥ ❝ ❡ ❛ ✐ ❡ ♦ ♥ ❡ ① ❝ ✉ ✐ ♦ ♥ ✳ ❈ ❡ ❧ ✉ ✐ ❝ ✐ ❞ ♣ ❡ ♥ ❞ ❧ ❡  

♣ ❧ ✉ ♦ ✉ ✈ ❡ ♥ ❞ ❡ ❧ ❛ ✉ ❛ ♥ ✐ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❛ ✐ ❡ ✳ ❛ ❡ ① ❡ ♠ ♣ ❧ ❡ ✱ ❧ ❡ ❡ ♠ ♣ ♥ ❝ ❡ ❛ ✐ ❡ ♣ ♦ ✉ ✐ ❡ ✉ ♥ ❡ ♥ ❡ ♠ ❜ ❧ ❡  

❞ ✬ ♦ ❜ ❥ ❡ ❞ ♣ ❡ ♥ ❞ ❞ ✉ ♥ ♦ ♠ ❜ ❡ ❞ ✬ ♦ ❜ ❥ ❡ ✳  

✷ ✳ ✷ ◗ ✉ ❛ ❧ ✐ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡      

✷ ✳ ✷ ✳ ✶ ◗ ✉ ❛ ❧ ✐ ❞ ✬ ❝ ✐ ✉ ❡  

✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ♦ ✐ ❡ ✉ ❝ ✉ ✱ ✐ ♥ ❞ ❡ ♥ ✱ ♠ ♦ ❞ ✉ ❧ ❛ ✐ ❡ ✱ ❛ ✈ ❡ ❝ ❞ ❡ ❝ ♦ ♠ ♠ ❡ ♥ ❛ ✐ ❡ ♣ ❡ ✐ ♥ ❡ ♥ ✱ ❡ ❝ ✳ ■ ❧ ❢ ❛ ✉  

♣ ♦ ✉ ✈ ♦ ✐ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ❧ ❛ ✉ ❝ ✉ ❡ ❞ ✬ ✉ ♥ ❝ ♦ ✉ ♣ ❞ ✬ ♦ ❡ ✐ ❧ ❛ ♣ ✐ ❞ ❡ ✱ ❡ ♣ ♦ ✉ ✈ ♦ ✐ ❛ ✉ ✐ ❡ ✈ ❡ ♥ ✐ ❞ ❡ ✉ ✻ ♠ ♦ ✐ ♣ ❧ ✉  

❛ ❞ ❡ ❧ ❡ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ❡ ♥ ❝ ♦ ❡ ✳  

✷ ✳ ✷ ✳ ✷ ❚ ❡ ♠ ✐ ♥ ❛ ✐ ♦ ♥  

❧ ❡ ✉ ❧ ❛ ❞ ♦ ✐ ❡ ❛ ❡ ✐ ♥ ❡ ♥ ✉ ♥ ♥ ♦ ♠ ❜ ❡ ✜ ♥ ✐ ❞ ✬ ❛ ♣ ❡ ✳ ■ ❧ ♥ ❡ ❢ ❛ ✉ ❞ ♦ ♥ ❝ ♣ ❛ ❞ ❡ ❜ ♦ ✉ ❝ ❧ ❡ ✐ ♥ ✜ ♥ ✐ ❡ ✱ ✐ ❧ ❢ ❛ ✉  

✉ ❞ ✐ ❡ ♦ ✉ ❧ ❡ ❝ ❛ ♣ ♦ ✐ ❜ ❧ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ✳  

✷ ✳ ✷ ✳ ✸ ❱ ❛ ❧ ✐ ❞ ✐  

❧ ❡ ✉ ❧ ❛ ❞ ♦ ✐ ♣ ♦ ♥ ❞ ❡ ❛ ✉ ♣ ♦ ❜ ❧ ♠ ❡ ❞ ❡ ♠ ❛ ♥ ❞ ❡ ✳ ❆ ❡ ♥ ✐ ♦ ♥ ✱ ✉ ♥ ❥ ❡ ✉ ❞ ✬ ❡ ❛ ✐ ♥ ❡ ♣ ♦ ✉ ✈ ❡    ❥ ❛ ♠ ❛ ✐   ✉ ✬ ✉ ♥  

♣ ♦ ❣ ❛ ♠ ♠ ❡ ❡ ❝ ♦ ❡ ❝ ✳ ■ ❧ ♣ ❡ ✉ ❡ ✉ ❧ ❡ ♠ ❡ ♥ ♣ ♦ ✉ ✈ ❡ ✉ ✬ ✐ ❧ ❡ ❢ ❛ ✉ ① ✳  

✷ ✳ ✷ ✳ ✹ ❡ ❢ ♦ ♠ ❛ ♥ ❝ ❡  

❊ ✉ ❞ ❡ ❞ ✉ ❝ ♦ ✭ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ✮ ❡ ♥ ❡ ♠ ♣ ❡ ❡ ♥ ♠ ♠ ♦ ✐ ❡ ✳ ❖ ♥ ✬ ✐ ♥ ❡ ❡ ❞ ❛ ♥ ❝ ❡ ❝ ♦ ✉ ✉ ✬ ❛ ✉ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐  

❡ ♠ ♣ ♦ ❡ ❧ ❧ ❡ ✳  

✷ ✳ ✸ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡      

▲ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❡ ✉ ♥ ❡ ❡ ✐ ♠ ❛ ✐ ♦ ♥ ❞ ✉ ♥ ♦ ♠ ❜ ❡ ❞ ✬ ♦ ♣ ❛ ✐ ♦ ♥ ❞ ❡ ❜ ❛ ❡ ❡ ✛ ❡ ❝ ✉ ❡ ♣ ❛  

❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❡ ♥ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❞ ❡ ❧ ❛ ❛ ✐ ❧ ❧ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❡ ♥ ❡ ♥ ❡ ❞ ❡ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✳  

✶ ✵ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✶ 

▲ ❛ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❝ ✬ ❡ ♣ ♦ ✉ ✿  

✶ ✳ ❊ ✈ ❛ ❧ ✉ ❡ ❧ ❡ ❡ ♦ ✉ ❝ ❡ ✭ ♠ ♠ ♦ ✐ ❡ ❈ ❯ ✮ ✉ ✐ ❧ ❡ ✱  

✷ ✳ ❈ ♦ ♠ ♣ ❛ ❡ ❞ ❡ ✉ ① ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ♣ ♦ ✉ ❧ ❡ ♠ ♠ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ✱  

✸ ✳ ❞ ♦ ♥ ♥ ❡ ✉ ♥ ❜ ♦ ♥ ❡ ✉ ❝ ❡ ✉ ✐ ❡ ❡ ✛ ❡ ❝ ✐ ✈ ❡ ♠ ❡ ♥ ♣ ♦ ✐ ❜ ❧ ❡ ❞ ❡ ♦ ✉ ❞ ❡ ✳ ❖ ♥ ❝ ♦ ♥ ✐ ❞ ❡ ❛ ✉ ❥ ♦ ✉ ❞ ✬ ❤ ✉ ✐ ✉ ✬ ♦ ♥  

♣ ❡ ✉ ❛ ❧ ✐ ❡ ❡ ♥ ❡ ♠ ♣ ❛ ✐ ♦ ♥ ♥ ❛ ❜ ❧ ❡ ❞ ❡   260 ♦ ♣ ❛ ✐ ♦ ♥ ✱ ✉ ❛ ♥ ❞ ❧ ❛ ♠ ♠ ♦ ✐ ❡ ❡ ❞ ❡ ❧ ✬ ♦ ❞ ❡ ❞ ❡   1010

♦ ❝ ❡ ✳  

✷ ✳ ✹ ▼ ✉ ❡ ❞ ❡ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐      

♦ ✉ ♠ ❡ ✉ ❡ ❧ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❡ ♠ ♣ ♦ ❡ ❧ ❧ ❡ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✱ ♦ ♥ ✬ ✐ ♥ ❡ ❡ ♣ ❧ ✉ ❛ ✉ ① ♦ ♣ ❛ ✐ ♦ ♥ ❧ ❡ ♣ ❧ ✉  

❝ ♦ ❡ ✉ ❡ ♣ ♦ ✉ ❧ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ❞ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ♣ ❛ ✐ ❝ ✉ ❧ ✐ ❡ ❀  

✕ ❘ ❛ ❝ ✐ ♥ ❡ ❝ ❛ ❡ ✱ ▲ ♦ ❣ ✱ ❊ ① ♣ ✱ ❆ ❞ ❞ ✐ ✐ ♦ ♥ ❡ ❧ ❧ ❡ ❀  

✕ ❈ ♦ ♠ ♣ ❛ ❛ ✐ ♦ ♥ ❞ ❛ ♥ ❧ ❡ ❝ ❛ ❞ ❡ ✐ ✳ ✳ ✳  

❡ ♦ ♥ ❝ ❛ ❧ ❝ ✉ ❧ ❡ ❧ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ✬ ♦ ♣ ❛ ✐ ♦ ♥ ❢ ♦ ♥ ❞ ❛ ♠ ❡ ♥ ❛ ❧ ❡ ❡ ① ❝ ✉ ❡ ♣ ❛ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✳  

▲ ❡ ❡ ♠ ♣ ❞ ❡ ❧ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥ ❞ ♣ ❡ ♥ ❞ ❞ ❡ ❧ ❛ ❛ ✐ ❧ ❧ ❡ ❞ ❡ ❧ ✬ ❡ ♥ ❡ ✳ ❖ ♥ ✈ ❡ ✉ ❝ ♦ ♥ ✐ ❞ ❡ ❡ ✉ ❧ ❡ ♠ ❡ ♥ ❧ ❛ ❛ ✐ ❧ ❧ ❡ ❡ ❡ ♥ ✐ ❡ ❧ ❧ ❡  

❞ ❡ ❧ ✬ ❡ ♥ ❡ ✳ ❈ ❡ ❧ ❛ ♣ ❡ ✉ ❡ ♣ ❛ ❡ ① ❡ ♠ ♣ ❧ ❡ ✿  

✕ ❧ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ✬ ❧ ♠ ❡ ♥ ❝ ♦ ♠ ❜ ✐ ♥ ❛ ♦ ✐ ❡ ❞ ❛ ♥ ❧ ✬ ❡ ♥ ❡ ✱  

✕ ❧ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ❡ ❜ ✐ ♣ ♦ ✉ ❡ ♣ ❡ ♥ ❡ ❧ ✬ ❡ ♥ ❡ ✱  

✕ ✳ ✳ ✳ ❡ ❝ ✳ ✳ ✳  

✷ ✳ ✺ ❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ♠ ♦ ② ❡ ♥ ♥ ❡ ❡ ❡ ♥ ♣ ✐ ❡ ❞ ❡ ❝ ❛    

❙ ♦ ✐  n ❧ ❛ ❛ ✐ ❧ ❧ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❞ ✉ ♣ ♦ ❜ ❧ ♠ ❡ ❡    T (n) ❧ ❡ ❡ ♠ ♣ ❞ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥ ❞ ❡ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✳ ❖ ♥ ❞ ✐ ✐ ♥ ❣ ✉ ❡ ✿  

✷ ✳ ✺ ✳ ✶ ❚ ❡ ♠ ♣ ❧ ❡ ♣ ❧ ✉ ♠ ♦ ✉ ✈ ❛ ✐ ✭ ❡ ♥ ♣ ✐ ❡ ❞ ❡ ❝ ❛ ✮  

T max(n) ✉ ✐ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ❛ ✉ ❡ ♠ ♣ ♠ ❛ ① ✐ ♠ ✉ ♠ ♣ ✐ ♣ ❛ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ♣ ♦ ✉ ✉ ♥ ♣ ♦ ❜ ❧ ♠ ❡ ❞ ❡ ❛ ✐ ❧ ❧ ❡  

  n❀ 

✷ ✳ ✺ ✳ ✷ ❚ ❡ ♠ ♣ ♠ ♦ ② ❡ ♥ ♥ ❡  

T moy(n) ❡ ♠ ♣ ♠ ♦ ② ❡ ♥ ❞ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥ ✉ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❞ ❡ ❛ ✐ ❧ ❧ ❡    n ✭ ❙ ✉ ♣ ♣ ♦ ✐ ✐ ♦ ♥ ✉ ❧ ❛ ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ❞ ❡ ❞ ♦ ♥ ✲  ♥ ❡ ✮ ✳  

❙ ♦ ✐   A ✉ ♥ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡  Dn ❧ ✬ ❡ ♥ ❡ ♠ ❜ ❧ ❡ ❞ ❡ ❡ ♥ ❡ ❞ ❡ ❛ ✐ ❧ ❧ ❡    n

I  ∈ Dn ✉ ♥ ❡ ❡ ♥ ❡  ✶ ✳  CoutA(i) = ♥ ♦ ♠ ❜ ❡ ❞ ✬ ♦ ♣ ❛ ✐ ♦ ♥ ❢ ♦ ♥ ❞ ❛ ♠ ❡ ♥ ❛ ❧ ❡ ❡ ① ❝ ✉ ❡ ♣ ❛    A ✉    I ✳ 

✷ ✳ ▲ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ❡   A ❡ ♥ ♣ ✐ ❡ ❞ ❡ ❝ ❛ ✿  

T max(n) = M ax(CoutA(i)); I  ∈ Dn✸ ✳ ❧ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ❡ ❆ ❡ ♥ ♠ ♦ ② ❡ ♥ ♥ ❡ ✿  

T moy(n) =

i∈DP r[i] ∗ CoutA[i]

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✷ 

❊ ① ❡ ♠ ♣ ❧ ❡  

❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✉ ❡ ♥ ✐ ❡ ❧ ❧ ❡ ✿  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❙ ✭ ❛ ❜ ✿ ❚ ❛ ❜ ❧ ❡ ❛ ✉ ❞ ❡   n ❧ ♠ ❡ ♥ ❀ ① ✿ ❧ ♠ ❡ ♥ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ❥ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

 j ← 1 ❛ ♥ ✉ ❡   ✭ 

 j ≤ n ✮ ❡ ✭    tab[ j]  x✮  ❢ ❛ ✐ ❡   j ← j  + 1

✜ ♥  

✐   ❥ ❃ ♥  ❛ ❧ ♦  

RechercheS  ← −1 ✐ ♥ ♦ ♥  

RechercheS  ← j✜ ♥ ✐  

✜ ♥ 

❈ ♦ ♠ ❧ ❡ ① ✐ ❡ ♥ ♣ ✐ ❡ ❞ ❡ ❝ ❛ ✿  

T max(RS ) = n

❈ ♦ ♠ ♣ ❧ ❡ ① ✐ ♠ ♦ ② ❡ ♥ ♥ ❡ ✿  

✲  P r[x ∈ tab] = q ✲ ❙ ✐  x ∈ tab ❛ ❧ ♦ ♦ ✉ ❧ ❡ ♣ ❧ ❛ ❝ ❡ ♦ ♥ ✉ ✐ ♣ ♦ ❜ ❛ ❜ ❧ ❡ ♣ ♦ ✉    1 ≤ i ≤ n ♦ ✐ ✿  

I i  =  x ∈ tab❡  

I n+10 =  x /∈ tab❖ ♥ ❛ ✿  

P r[I i] = q/n ♣ ♦ ✉    1 ≤ i ≤ n ❡    CoutRS (I i) = i❡  

P r[I n+1] = 1 − q  ❡    CoutRS (I n+1) = n

T moy(RS ) =n+1

 j=1

P r[I  j ] ∗ CoutRS (I  j)

=n

i=1

q/n(i) + (1 − q )n

= q/nn

i=1

i + (1 − q )n

= q/n ∗ n(n + 1)/2 + (1 − q )n= (1 − q/2)n + q/2

✲ ❙ ✐  q  = 1 ❛ ❧ ♦    T moy(RS ) = (n + 1)/2✲ ❙ ✐  q  = 1/2 ❛ ❧ ♦    T moy(RS ) = (3n + 1)/4

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✸ 

✷ ✳ ✻ ◆ ♦ ❛ ✐ ♦ ♥ ✉ ✐ ❧ ✐ ❡  

■ ❧ ❢ ❛ ✉ ❝ ♦ ♠ ♣ ❛ ❡ ❧ ❡ ❛ ✉ ① ❞ ✬ ❛ ❝ ❝ ♦ ✐ ❡ ♠ ❡ ♥ ❞ ❡ ❞ ✐ ✛ ❡ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✉ ✐ ♠ ❡ ✉ ❡ ♥ ❧ ❡ ♣ ❡ ❢ ♦ ♠ ❛ ♥ ❝ ❡ ❞ ✬ ✉ ♥  

♣ ♦ ❣ ❛ ♠ ♠ ❡ ✳  

✷ ✳ ✻ ✳ ✶ ◆ ♦ ❛ ✐ ♦ ♥ ✧  o

✧ 

❖ ♥ ❞ ✐ ✉ ❡  

f (x) = o(g(x))♣ ♦ ✉  

  x → ∞ ✐ 

lim−→f (x)

g(x) = 0

limx→∞ ❈ ❡ ✉ ❡ ✈ ❡ ✉ ❞ ✐ ❡ ✉ ❡    f  ❝ ♦ ♣ ❧ ✉ ❧ ❡ ♥ ❡ ♠ ❡ ♥ ✉ ❡    g ✉ ❛ ♥ ❞    x ❡ ❣ ❛ ♥ ❞ ✳ ❛ ❡ ① ❡ ♠ ♣ ❧ ❡ ✿  x2 = o(x5)sin(x) = o(x)14.709

√ x =  o(x/2 + 7 cos(x))

23 log(x) = x0.002

✷ ✳ ✻ ✳ ✷ ◆ ♦ ❛ ✐ ♦ ♥ ✧  

O✧ 

❖ ♥ ❞ ✐ ✉ ❡  f (x) = O(g(x)) ✬ ✐ ❧ ❡ ① ✐ ❡    k ❡    x0 ❡ ❧ ✉ ❡ ✿  

∀x > x0 ⇒ f (x) < kg(x)▲ ❛ ♥ ♦ ❛ ✐ ♦ ♥   o ❡ ♣ ❧ ✉ ♣ ❝ ✐ ❡ ✉ ❡    O ✱ ♠ ❛ ✐    O ❡ ♣ ❧ ✉ ❢ ❛ ❝ ✐ ❧ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❡ ❡ ✉ ✣ ❛ ♥ ✳ ❛ ❡ ① ❡ ♠ ♣ ❧ ❡ ✿  

sin(x) = O(x)sin(x) = O(1)

✷ ✳ ✼ ❯ ♥ ❝ ❧ ❛ ❡ ♠ ❡ ♥ ❞ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥  

♦ ❜ ❧ ♠ ❡ ❛ ✐ ♦ ♥ ♥ ❡ ❧ ❧ ❡ ♠ ❡ ♥ ❢ ❛ ❝ ✐ ❧ ❡ ✿  

● ♦ ✉ ♣ ❡ ✵ ✶ ✿  log(log(x)), log(x), log2(x)● ♦ ✉ ♣ ❡ ✵ ✷ ✿  x0.2, x , x2, x15 log(x)

♦ ❜ ❧ ♠ ❡ ❞ ✐ ✣ ❝ ✐ ❧ ❡ ✿  

● ♦ ✉ ♣ ❡ ✵ ✸ ✿  e√ x, 1.03x, 2x

● ♦ ✉ ♣ ❡ ✵ ✹ ✿  x!, xx, xx2

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✹ 

✷ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡  

❘ ❛ ♣ ♣ ❡ ❧ ❡ ✿ ◆ ♦ ❛ ✐ ♦ ♥ ✧  O ✧ ❖ ♥ ❞ ✐ ✉ ❡  f (x) = O(g(x)) ✬ ✐ ❧ ❡ ① ✐ ❡    k ❡    x0 ❡ ❧ ✉ ❡ ✿  

∀x > x0 ⇒ f (x) < kg(x)

✷ ✳ ✽ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✶  

✶ ✳ ❛ ♠ ✐ ❧ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✉ ✐ ✈ ❛ ♥ ❡ ✱ ✉ ❡ ❧ ❧ ❡ ♦ ♥ ❝ ❡ ❧ ❧ ❡ ✉ ✐ ♦ ♥ ❧ ❡ ♠ ♠ ❡ ♦ ❞ ❡ ❞ ❡ ❣ ❛ ♥ ❞ ❡ ✉ ❄  

✕  f 1(n) = 4n3 + n✕  f 2(n) = n2 + log2(n)✕  f 3(n) = n2 × log3(n) + 6n3

✕  f 4(n) =n(n2 + 1)

2✷ ✳ ❊ ♥ ✉ ✐ ❧ ✐ ❛ ♥ ❧ ❛ ❞ ✜ ♥ ✐ ✐ ♦ ♥ ❞ ❡   O ♠ ♦ ♥ ❡ ✉ ❡    f (n) + g(n) = O(max(f (n); g(n)))

✸ ✳ ❊ ♥ ✉ ✐ ❧ ✐ ❛ ♥ ❧ ❛ ❞ ✜ ♥ ✐ ✐ ♦ ♥ ❞ ❡   O ♠ ♦ ♥ ❡ ✉ ❡    3n2 + 4n + 6 = O(n2)

✷ ✳ ✽ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷  

❈ ♦ ♥ ✐ ❞ ❡ ❧ ❡ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✉ ✐ ✈ ❛ ♥ ❡ ❛ ✈ ❡ ❝ ✉ ♥ ❡ ♠ ♣ ❞ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥   T (n) ♣ ♦ ✉ ✉ ♥ ❡ ❧ ♦ ♥ ❣ ✉ ❡ ✉ ❞ ❡ ❞ ♦ ♥ ♥ ❡  n✳ ❉ ❡ ♠ ✐ ♥ ❡ ❧ ❡ ✉ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❛ ② ♠ ♣ ♦ ✐ ✉ ❡ ❡ ♣ ❡ ❝ ✐ ✈ ❡ ✱ ❡ ❧ ❡ ❝ ❧ ❛ ❡ ③ ♣ ❛ ♦ ❞ ❡ ❞ ❡ ❣ ❛ ♥ ❞ ❡ ✉ ❝ ♦ ✐ ❛ ♥ ✳  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❆ ✶   T (n) = 3n + 2❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❆ ✷   T (n) = 6❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❆ ✸   T (n) = 4n2 + n + 2❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❆ ✹  

❊ ① ❝ ✉ ❡ ❆ ✶ ❀

❊ ① ❝ ✉ ❡ ❆ ✷ ❀

❊ ① ❝ ✉ ❡ ❆ ✸ ❀

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❆ ✺  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✺ ✿  ❆ ✺ 

❞ ❜ ✉  

♣ ♦ ✉   i ← 1     n ❢ ❛ ✐ ❡  ❊ ① ❝ ✉ ❡ ❆ ✸  

✜ ♥ ♣ ♦ ✉  

❊ ① ❝ ✉ ❡ ❆ ✶  

✜ ♥ 

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❆ ✻  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✺ 

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✻ ✿  ❆ ✻ 

❞ ❜ ✉  

♣ ♦ ✉   i ← 1     5 ❢ ❛ ✐ ❡  ❊ ① ❝ ✉ ❡ ❆ ✶  

✜ ♥ ♣ ♦ ✉  

✜ ♥ 

✷ ✳ ✽ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸  

❊ ✈ ❛ ❧ ✉ ❡ ❧ ❡ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ✉ ✐ ✈ ❛ ♥ ❡ ♥ ❞ ♦ ♥ ♥ ❛ ♥ ❧ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ✬ ✐ ♥ ✉ ❝ ✐ ♦ ♥  

❛ ❧ ✐ ❡ ✳  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✼ ✿  ❆ ✶ 

❱ ❛   ✿ ❛ ✱ ❜ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

a ← 0b ← 0Afficher(a)

Afficher(b)✜ ♥ 

❋ ♦ ♥ ❝ ✐ ♦ ♥   ♦ ♠ ♠ ❡ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ✐ ✱ ❡ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

res ← 0♣ ♦ ✉   i ← 1     n ❢ ❛ ✐ ❡  

res ← res  + i✜ ♥ ♣ ♦ ✉  

somme ←

 res

✜ ♥ 

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❢ ✶ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ✐ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

♣ ♦ ✉   i ← 1     n ❢ ❛ ✐ ❡  somme(n)

✜ ♥ ♣ ♦ ✉  

✜ ♥ 

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❢ ✷ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ✐ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

♣ ♦ ✉   i ← 1     n ❢ ❛ ✐ ❡  somme(i)

✜ ♥ ♣ ♦ ✉  

✜ ♥ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✻ 

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❢ ✸ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ✐ ✱ ❥ ✱ ❦ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

♣ ♦ ✉   i ← 1     n ❢ ❛ ✐ ❡  ♣ ♦ ✉   j ← i     n ❢ ❛ ✐ ❡  

♣ ♦ ✉   i ← 1     j ❢ ❛ ✐ ❡  inst()

✜ ♥ ♣ ♦ ✉  

✜ ♥ ♣ ♦ ✉  

✜ ♥ ♣ ♦ ✉  

✜ ♥ 

✷ ✳ ✽ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✹  

❈ ❛ ❧ ❝ ✉ ❧ ❡ ❧ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ❡ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❝ ✉ ✐ ✈ ❡ ✿  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❋ ✐ ❜ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

✈ ✉ ❡ ❡ ♥ ❝ ♦ ✉ ✳  

✷ ✳ ✽ ✳ ✺ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✺  

➱ ✉ ❞ ✐ ❡ ③ ❧ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ✬ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ✐ ❡ ♣ ❛ ❧ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✉ ✐ ✈ ❛ ♥ ❡ ❞ ❛ ♥ ❧ ❡ ♠ ❡ ✐ ❧ ❧ ❡ ✉ ❝ ❛ ✱ ❧ ❡ ♣ ✐ ❡ ❝ ❛ ✱  

♣ ✉ ✐ ❞ ❛ ♥ ❧ ❡ ❝ ❛ ♠ ♦ ② ❡ ♥ ❡ ♥ ✉ ♣ ♣ ♦ ❛ ♥ ✉ ❡ ❧ ❡ ❡ ♦ ♥ ✉ ♥ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ❞ ❡   ➼  ❞ ✬ ❡ ✈ ❛ ✐ ✳  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❢ ✶ ✭ ✿ ❛ ❜

❱ ❛   ✿ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

s ← 0♣ ♦ ✉   i ← 1    

✐  t[i] > as ← s +

✜ ♥ ✐  

✜ ♥ ♣ ♦ ✉  

f 1 ← s✜ ♥ 

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❢ ✶ ✭ ❛ ✱ ❜ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

s ← 0 ✐ 

a > b❛ ❧ ♦  

♣ ♦ ✉   i ← 1     n ❢ ❛ ✐ ❡  s ← s + a

✜ ♥ ♣ ♦ ✉  

✐ ♥ ♦ ♥  

s ← s + b✜ ♥ ✐  

f 1 ← s✜ ♥ 

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❈ ❤ ❛ ♣ ✐ ❡ ✸      

▲ ❛ ❝ ✉ ✐ ✈ ✐      

✸ ✳ ✶ ■ ♥ ♦ ❞ ✉ ❝ ✐ ♦ ♥  

❧ ❛ ❝ ✉ ❡ ♥ ❝ ❡ ✭ ✧ ✐ ♥ ❞ ✉ ❝ ✐ ♦ ♥ ✧ ✮ ❡ ✉ ♥ ♦ ✉ ✐ ❧ ♠ ❛ ❤ ♠ ❛ ✐ ✉ ❡ ❡ ❡ ♥ ✐ ❡ ❧ ✳ ❚ ♦ ✉ ✐ ♥ ❢ ♦ ♠ ❛ ✐ ❝ ✐ ❡ ♥ ❡ ❞ ♦ ✐ ❞ ❡ ❧ ❛  

♠ ❛ ✐ ❡ ✳ ❍ ❡ ✉ ❡ ✉ ❡ ♠ ❡ ♥ ✱ ❧ ❛ ❝ ✉ ❡ ♥ ❝ ❡ ❡ ✉ ♥ ❡ ❡ ❝ ❤ ♥ ✐ ✉ ❡ ✐ ♠ ♣ ❧ ❡ ✱ ❡ ♥ ❞ ♣ ✐ ❞ ❡ ❛ ♣ ✉ ✐ ❛ ♥ ❝ ❡ ✳  

▲ ❛ ❝ ✉ ✐ ✈ ✐ ❡ ✉ ♥ ❡ ♥ ♦ ✐ ♦ ♥ ✉ ✐ ❧ ✐ ❡ ❡ ♥ ♣ ♦ ❣ ❛ ♠ ♠ ❛ ✐ ♦ ♥ ✱ ❡ ✉ ✐ ♣ ❡ ♠ ❡ ❧ ✬ ❡ ① ♣ ❡ ✐ ♦ ♥ ❞ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡  

❝ ♦ ♥ ❝ ✐ ✱ ❢ ❛ ❝ ✐ ❧ ❡ ❝ ✐ ❡ ❡ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ✳ ▲ ❛ ❝ ✉ ✐ ✈ ✐ ♣ ❡ ✉ ♦ ✉ ❥ ♦ ✉ ❡ ❡ ♠ ♣ ❧ ❛ ❝ ❡ ♣ ❛ ♦ ♥ ✉ ✐ ✈ ❛ ❧ ❡ ♥  

♦ ✉ ❢ ♦ ♠ ❡ ❞ ✬ ✐ ❛ ✐ ♦ ♥ ✱ ♠ ❛ ✐ ❛ ✉ ❞ ✐ ♠ ❡ ♥ ❞ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ♣ ❧ ✉ ❝ ♦ ♠ ♣ ❧ ❡ ① ❡ ✉ ♦ ✉ ❧ ♦ ✉ ❡ ❧ ❡ ✉ ❝ ✉ ❡  

❞ ❡ ❞ ♦ ♥ ♥ ❡ ❛ ✐ ❡ ♦ ♥ ❡ ❧ ❧ ❡ ✲ ♠ ♠ ❡ ❞ ❡ ♥ ❛ ✉ ❡ ❝ ✉ ✐ ✈ ❡ ✳  

✸ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥  

✕ ❯ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✭ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✱ ♣ ♦ ❝ ❞ ✉ ❡ ✮ ❡ ❞ ✐ ❡ ❝ ✉ ✐ ❢ ❧ ♦ ✉ ✬ ✐ ❧ ✬ ❛ ♣ ♣ ❡ ❧ ❧ ❡ ❧ ✉ ✐ ✲ ♠ ♠ ❡ ✱ ❝ ✲ ✲ ❞ ✳ ✱ ❞ ✜ ♥ ✐ ✐ ♦ ♥  

✭ ♦ ♥ ❝ ♦ ❞ ❡ ✮ ❝ ♦ ♥ ✐ ❡ ♥ ✉ ♥ ❛ ♣ ♣ ❡ ❧ ❧ ✉ ✐ ✲ ♠ ♠ ❡ ✳  

✕ ❯ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✉ ✐ ♥ ✬ ❡ ♣ ❛ ❝ ✉ ✐ ❢ ❡ ❞ ✐ ✐ ❛ ✐ ❢ ✳  

◆ ♦ ❡ ③ ❇ ✐ ❡ ♥  

✕ ❖ ♥ ♣ ❡ ✉ ♦ ✉ ❥ ♦ ✉ ❛ ♥ ❢ ♦ ♠ ❡ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❝ ✉ ✐ ❢ ❡ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✐ ❛ ✐ ❢ ✳  

❧ ❛ ❢ ♦ ♠ ❡ ❣ ♥ ❛ ❧ ❡ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❝ ✉ ✐ ✈ ❡ ✿  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✽ ✿   ✭ ✳ ✳ ✳ ✮  

❞ ❜ ✉  

✐   ✜ ♥ ✭ ✮  ❛ ❧ ♦  

....✴ ✴ ♣ ❛ ❞ ✬ ❛ ♣ ♣ ❡ ❧ ❝ ✉ ✐ ❢  

✐ ♥ ♦ ♥  

...P (...) ✴ ✴ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❛ ♣ ♣ ❡ ❧ ❧ ❡ ❧ ✉ ✐ ✲ ♠ ♠ ❡ ✉ ♥ ❡  

... ✴ ✴ ♦ ✉ ♣ ❧ ✉ ✐ ❡ ✉ ❢ ♦ ✐  

✜ ♥ ✐  

✜ ♥ 

✶ ✼ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✽ 

✸ ✳ ✸ ❊ ① ❡ ♠ ♣ ❧ ❡    

✸ ✳ ✸ ✳ ✶ ❊ ① ❡ ♠ ♣ ❧ ❡ ✶  

❈ ❛ ❧ ❝ ✉ ❧ ❞ ❡ ❧ ❛ ❢ ❛ ❝ ♦ ✐ ❡ ❧ ❧ ❡ ❞ ✬ ✉ ♥ ♥ ♦ ♠ ❜ ❡ ✳  

❉ ✜ ♥ ✐ ✐ ♦ ♥ ✐ ❛ ✐ ✈ ❡ ✿  

n! = F (n) = n ∗ (n − 1) ∗ (n − 2) ∗ ... ∗ 2 ∗ 1❙ ♦ ✐ ❡ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ✿  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✾ ✿  ❢ ❛ ❝ ♦ ✐ ❡ ❧ ❧ ❡ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❱ ❛   ✿ ✐ ✱ ❢ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

f  ← 1♣ ♦ ✉   ✐ ❂ ✶    ♥  ❢ ❛ ✐ ❡  

f  ←  f  ∗ i✜ ♥ ♣ ♦ ✉  

factorielle ← f ✜ ♥ 

❉ ✜ ♥ ✐ ✐ ♦ ♥ ❝ ✉ ✐ ✈ ❡ ✿  

F (0) = 1F (n) = n ∗ F (n − 1); ❙ ♦ ✐ ❡ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ✿  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✶ ✵ ✿  ❢ ❛ ❝ ❘ ❡ ❝ ✉ ✐ ✈ ❡ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

✐  n = 0 ❛ ❧ ♦  factorielle ← 1

✐ ♥ ♦ ♥  

factorielle ← factorielle(n − 1) ∗ n✜ ♥ ✐  

✜ ♥ 

✸ ✳ ✸ ✳ ✷ ❊ ① ❡ ♠ ♣ ❧ ❡ ✷ ✿  

▲ ❛ ✉ ✐ ❡ ❞ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ❡ ❋ ✐ ❜ ♦ ♥ ❛ ❝ ❝ ✐ ❡ ❞ ✜ ♥ ✐ ❝ ♦ ♠ ♠ ❡ ✉ ✐ ✿  

f 0 = 0f 1  = 1f n =  f n−1 + f n−2 ✐   n > 1❖ ♥ ♣ ❡ ✉ ❢ ♦ ♠ ✉ ❧ ❡ ❝ ❡ ❡ ✉ ✐ ❡ ♦ ✉ ❢ ♦ ♠ ❡ ❞ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✭ ♥ ❃ ❂ ✵ ✮ ✿  

fibonacci(n) = n ✐   n  1

♦ ✐ ❡ ♥ ❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ✿  

❆ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ✶ ✶ ✿  ✜ ❜ ♦ ♥ ❛ ❝ ❝ ✐ ❘ ❡ ❝ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

✐  n

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✶ ✾ 

✸ ✳ ✹ ❘ ❣ ❧ ❡    

✶ ✳ ❚ ♦ ✉ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❝ ✉ ✐ ❢ ❞ ♦ ✐ ❞ ✐ ✐ ♥ ❣ ✉ ❡ ♣ ❧ ✉ ✐ ❡ ✉ ❝ ❛ ✱ ❞ ♦ ♥ ❧ ✬ ✉ ♥ ❛ ✉ ♠ ♦ ✐ ♥ ♥ ❡ ❞ ♦ ✐ ♣ ❛ ❝ ♦ ♠ ♣ ♦ ❡  

❞ ✬ ❛ ♣ ♣ ❡ ❧ ❝ ✉ ✐ ❢ ✳  

✕ ▲ ❡ ❝ ❛ ♥ ♦ ♥ ❝ ✉ ✐ ❢ ❞ ✬ ✉ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❝ ✉ ✐ ❢ ♦ ♥ ❛ ♣ ♣ ❡ ❧ ❝ ❛ ❞ ❡ ❜ ❛ ❡ ✳  

✕ ▲ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✉ ❡ ❞ ♦ ✐ ✈ ❡ ♥ ❛ ✐ ❢ ❛ ✐ ❡ ❧ ❡ ❞ ♦ ♥ ♥ ❡ ❞ ❛ ♥ ❝ ❡ ❝ ❛ ❞ ❡ ❜ ❛ ❡ ♦ ♥ ❛ ♣ ♣ ❡ ❧ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥  

❞ ❡ ❡ ♠ ✐ ♥ ❛ ✐ ♦ ♥ ✳  

✷ ✳ ❚ ♦ ✉ ❛ ♣ ♣ ❡ ❧ ❝ ✉ ✐ ❢ ❞ ♦ ✐ ❡ ❢ ❛ ✐ ❡ ❛ ✈ ❡ ❝ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ♣ ❧ ✉ ♣ ♦ ❝ ❤ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❛ ✐ ❢ ❛ ✐ ❛ ♥ ✉ ♥ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥  

❞ ❡ ❡ ♠ ✐ ♥ ❛ ✐ ♦ ♥ ✳  

✸ ✳ ✺ ❘ ❝ ✉ ✐ ✈ ✐ ❝ ♦ ✐ ❡      

❉ ❛ ♥ ❧ ❛ ❝ ✉ ✐ ✈ ✐ ❝ ♦ ✐ ❡ ✱ ❧ ❡ ❛ ♣ ♣ ❡ ❧ ❝ ✉ ✐ ❢ ♦ ♥ ♣ ♦ ✈ ♦ ✉ ♣ ❛ ❧ ✬ ❡ ① ❝ ✉ ✐ ♦ ♥ ❞ ✬ ❛ ✉ ❡ ♣ ♦ ❝ ❞ ✉ ❡ ♦ ✉  

❢ ♦ ♥ ❝ ✐ ♦ ♥ ✳  

♦ ❝ ❞ ✉ ❡   ✶ ✭ ① ✮  

❞ ❜ ✉  

...Q(f (x))...

✜ ♥ 

♦ ❝ ❞ ✉ ❡   ◗ ✶ ✭ ① ✮  

❞ ❜ ✉  

...P (g(x))...

✜ ♥ 

✸ ✳ ✺ ✳ ✶ ❊ ① ❡ ♠ ♣ ❧ ❡ ✿ ❛ ✐ ❡ ♦ ✉ ✐ ♠ ♣ ❛ ✐ ❡  

❖ ♥ ♣ ❡ ✉ ❢ ♦ ♠ ✉ ❧ ❡ ❧ ❡ ♥ ♦ ✐ ♦ ♥ ❝ ✉ ✐ ✈ ❡ ✉ ✐ ✈ ❛ ♥ ✿  

✕ ▲ ❡ ✵ ❡ ♣ ❛ ✐ ❡ ❡ ♥ ♦ ♥ ✐ ♠ ♣ ❛ ✐ ❡ ✳  

✕ ❯ ♥ ♥ ♦ ♠ ❜ ❡ ♥ ❡ ♣ ❛ ✐ ❡ ✐ ♥ ✲ ✶ ❡ ✐ ♠ ♣ ❛ ✐ ❡ ✳  

✕ ❯ ♥ ♥ ♦ ♠ ❜ ❡ ♥ ❡ ✐ ♠ ♣ ❛ ✐ ❡ ✐ ♥ ✲ ✶ ❡ ♣ ❛ ✐ ❡ ✳  

❙ ♦ ✐ ❡ ♥ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ✿  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❛ ✐ ❡ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

✐  n = 0 ❛ ❧ ♦  Paire ← vrai

✐ ♥ ♦ ♥  

Paire ← ImPaire(n − 1)✜ ♥ ✐  

✜ ♥ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✷ ✵ 

❋ ✐ ❣ ✉ ❡   ✸ ✳ ✶ ✕ ♦ ✐ ✐ ♦ ♥ ❞ ❡ ❞ ♣ ❛ ❞ ❡ ♦ ✉ ❞ ✬ ❍ ❛ ♥ ♦ ✐  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ■ ♠ ❛ ✐ ❡ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❡ ♥ ✐ ❡  

❞ ❜ ✉  

✐  n = 0 ❛ ❧ ♦  ImPaire ← false

✐ ♥ ♦ ♥  

ImPaire ← Paire(n − 1)✜ ♥ ✐  

✜ ♥ 

✸ ✳ ✻ ♦ ❜ ❧ ♠ ❡ ✭ ❧ ❡ ♦ ✉ ❞ ✬ ❍ ❛ ♥ ♦ ✮    

▲ ❡ ✐ ❡ ❞ ✉ ♣ ♦ ❜ ❧ ♠ ❡ ✈ ✐ ❡ ♥ ❞ ❡ ❧ ✬ ❤ ✐ ♦ ✐ ❡ ❛ ❝ ♦ ♥ ❡ ❤ ❛ ❜ ✐ ✉ ❡ ❧ ❧ ❡ ♠ ❡ ♥ ✱ ✉ ✐ ❡ ❝ ❡ ❧ ❧ ❡ ❞ ❡ ♠ ♦ ✐ ♥ ❡ ❜ ♦ ✉ ❞ ❞ ❤ ✐ ❡  

❡ ♥ ❆ ✐ ❡ ❞ ✉ ✉ ❞ ✲ ❡ ✉ ✐ ❣ ♥ ❡ ♥ ❧ ❡ ❡ ♠ ♣ ❡ ♥ ❛ ♥ ❢ ❛ ♥ ❞ ❡ ❞ ✐ ✉ ❡ ✱ ♦ ✉ ❞ ❡ ❛ ✐ ❧ ❧ ❡ ❞ ✐ ✛ ❡ ♥ ❡ ✱ ✉ ✉ ♥ ❥ ❡ ✉  

❞ ❡ ♦ ✐ ♣ ✐ ✉ ❡ ✭ ✈ ♦ ✐ ✜ ❣ ✉ ❡ ✶ ✮ ✳ ❉ ❛ ♥ ❝ ❡ ❡ ✜ ❣ ✉ ❡ ✐ ❧ ♥ ✬ ② ❛ ✉ ❡ ❝ ✐ ♥ ❞ ✐ ✉ ❡ ✱ ♠ ❛ ✐ ❧ ❛ ❛ ❞ ✐ ✐ ♦ ♥ ✈ ❡ ✉ ✉ ❡ ❧ ❡  

♠ ♦ ✐ ♥ ❡ ❥ ♦ ✉ ❡ ♥ ❛ ✈ ❡ ❝ ✻ ✹ ✳ ▲ ✬ ❤ ✐ ♦ ✐ ❡ ❡ ✉ ♥ ❡ ✐ ♥ ✈ ❡ ♥ ✐ ♦ ♥ ❞ ✉ ❞ ✐ ① ✲ ♥ ❡ ✉ ✈ ✐ ♠ ❡ ✐ ❝ ❧ ❡ ✱ ▲ ✉ ❝ ❛ ❧ ❛ ♣ ❧ ❛ ❛ ♥ ❇ ♥ ❛  

✭ ❡ ♥ ■ ♥ ❞ ❡ ✮ ✳ ❖ ♥ ♥ ❡ ❛ ✐ ♣ ❛ ❝ ♦ ♠ ♠ ❡ ♥ ❡ ❧ ❧ ❡ ✬ ❡ ❡ ✐ ✉ ❡ ❍ ❛ ♥ ♦ ✳ ✳ ✳  

▲ ❡ ❥ ❡ ✉ ❝ ♦ ♥ ✐ ❡ ❛ ♥ ❢ ❡ ❧ ❛ ♣ ✐ ❧ ❡ ❞ ❡ ❞ ✐ ✉ ❡ ❞ ✉ ♣ ✐ ✉ ❡ ❆ ✈ ❡ ❧ ❡ ♣ ✐ ✉ ❡ ❇ ✱ ❡ ♥ ✉ ✐ ❧ ✐ ❛ ♥ ❈ ❝ ♦ ♠ ♠ ❡  

♣ ✐ ✉ ❡ ❞ ❡ ♠ ❛ ♥ ♦ ❡ ✉ ✈ ❡ ✱ ♦ ✉ ❡ ♥ ❡ ♣ ❡ ❝ ❛ ♥ ❧ ❡ ❞ ❡ ✉ ① ❣ ❧ ❡ ✉ ✐ ✈ ❛ ♥ ❡ ✿  

✕ ✉ ♥ ❞ ✐ ✉ ❡ ♥ ❡ ♣ ❡ ✉ ♣ ❛ ❡ ♣ ♦ ✉ ♣ ❧ ✉ ♣ ❡ ✐ ✉ ❡ ❧ ✉ ✐ ✱  

✕ ♦ ♥ ♥ ❡ ❞ ♣ ❧ ❛ ❝ ❡ ✉ ✬ ✉ ♥ ❞ ✐ ✉ ❡ ❧ ❛ ❢ ♦ ✐ ✳  

▲ ❛ ♦ ❧ ✉ ✐ ♦ ♥ ❧ ❛ ♣ ❧ ✉ ✐ ♠ ♣ ❧ ❡ ✈ ✐ ❡ ♥ ❞ ❡ ❧ ❛ ♣ ♦ ♥ ❡ ❧ ❛ ✉ ❡ ✐ ♦ ♥ ✉ ✐ ✈ ❛ ♥ ❡ ✿ ✧ ✐ ❥ ❡ ❛ ✈ ❛ ✐ ❛ ♥ ❢ ❡ ♥ ✲ ❧ ❞ ✐ ✉ ❡ ✱  

❛ ✉ ❛ ✐ ✲ ❥ ❡ ❛ ♥ ❢ ❡ ♥ ❄ ✧ ✳ ▲ ❛ ♣ ♦ ♥ ❡ ❡ ♦ ✉ ✐ ✱ ❝ ❛ ❛ ✉ ❡ ♠ ❡ ♥ ♥ ♦ ✉ ♥ ✬ ❛ ✉ ✐ ♦ ♥ ♣ ❛ ♣ ♦ ❧ ❛ ✉ ❡ ✐ ♦ ♥ ✳ ❆ ✐ ♥ ✐ ✱  

♣ ♦ ✉ ❛ ♥ ❢ ❡ ♥ ❞ ✐ ✉ ❡ ❞ ❡ ❆ ❇ ✱ ♦ ♥ ❝ ♦ ♠ ♠ ❡ ♥ ❝ ❡ ♣ ❛ ❧ ❡ ❛ ♥ ❢ ❡ ❞ ❡ ♥ ✲ ❧ ❞ ✐ ✉ ❡ ❞ ❡ ❆ ❈ ✱ ✉ ✐ ✈ ✐ ❞ ✉  

❞ ♣ ❧ ❛ ❝ ❡ ♠ ❡ ♥ ❞ ✉ ❞ ❡ ♥ ✐ ❡ ❞ ✐ ✉ ❡ ✭ ❧ ❡ ♣ ❧ ✉ ❣ ♦ ✮ ❞ ❡ ❆ ❇ ✱ ✉ ✐ ✈ ✐ ❞ ✉ ❛ ♥ ❢ ❡ ❞ ❡ ♥ ✲ ❧ ❞ ✐ ✉ ❡ ❞ ❡ ❈ ❇ ✳  

❈ ❡ ❧ ❛ ❞ ♦ ♥ ♥ ❡ ❧ ❛ ♣ ♦ ❝ ❞ ✉ ❡ ✿  

♦ ❝ ❞ ✉ ❡   ❤ ❛ ♥ ♦ ✐ ✭ ♥ ✿ ❡ ♥ ✐ ❡ ❀ ❛ ✱ ❜ ✿ ❡ ♥ ✐ ❡ ✮  

❞ ❜ ✉  

✐  n = 1 ❛ ❧ ♦  deplacement(a, b)

✐ ♥ ♦ ♥  

hanoi(n − 1, a, 6 − a − b)deplacement(a, b)hanoi(n − 1, 6 − a − b, b)

✜ ♥ ✐  

✜ ♥ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✷ ✶ 

❉ ❛ ♥ ❝ ❡ ❡ ♣ ♦ ❝ ❞ ✉ ❡ ✱ ♥ ♦ ✉ ❛ ✈ ♦ ♥ ✉ ♣ ♣ ♦ ✉ ❡ ❧ ❡ ♣ ✐ ✉ ❡ ♦ ♥ ♥ ✉ ♠ ♦ ✶ ✱ ✷ ✱ ✸ ✳ ❉ ♣ ❧ ❛ ❝ ❡ ✉ ♥ ❞ ✐ ✉ ❡  

❡ ❢ ❛ ✐ ♣ ❛ ❧ ❛ ♣ ♦ ❝ ❞ ✉ ❡ ❞ ♣ ❧ ❛ ❝ ❡ ♠ ❡ ♥ ✳ ♦ ✉   n =  l ❧ ❡ ❞ ♣ ❧ ❛ ❝ ❡ ♠ ❡ ♥ ❡ ✐ ♠ ♠ ❞ ✐ ❛ ✱ ❛ ✉ ❡ ♠ ❡ ♥ ♦ ♥ ❛ ♣ ♣ ❧ ✐ ✉ ❡  ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ❝ ✐ ❝ ✐ ✲ ❞ ❡ ✉ ✱ ✻ ✲ ❛ ✲ ❜ ❛ ♥ ❧ ❡ ♥ ✉ ♠ ♦ ❞ ✉ ♦ ✐ ✐ ♠ ❡ ♣ ✐ ✉ ❡ ✭ ✻ ❂ ✶ ✰ ✷ ✰ ✸ ✱ ❞ ♦ ♥ ❝ ❡ ♥ ♦ ✉ ❛ ② ❛ ♥  

❞ ❡ ✉ ① ❞ ❡ ♦ ✐ ♣ ♦ ✐ ❜ ✐ ❧ ✐ ❞ ❡ ✻ ♦ ♥ ♦ ❜ ✐ ❡ ♥ ❧ ❛ ♦ ✐ ✐ ♠ ❡ ✮ ✳  

✸ ✳ ✼ ❈ ♦ ♥ ❝ ❧ ✉ ✐ ♦ ♥  

❈ ❡ ❛ ✐ ♥ ♣ ♦ ❜ ❧ ♠ ❡ ♣ ❡ ✉ ✈ ❡ ♥ ❡ ♦ ❧ ✉ ♣ ❧ ✉ ❧ ♦ ❣ ✐ ✉ ❡ ♠ ❡ ♥ ❡ ♥ ✉ ✐ ❧ ✐ ❛ ♥ ❧ ❛ ❝ ✉ ✐ ✈ ✐ ✳ ▲ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❡  

♦ ♥ ♣ ❧ ✉ ❝ ♦ ♠ ♣ ❛ ❝ ✱ ♣ ❧ ✉ ❢ ❛ ❝ ✐ ❧ ❡ ❝ ✐ ❡ ❡ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ✳ ❙ ♦ ♥ ✉ ❛ ❣ ❡ ❡ ♥ ❛ ✉ ❡ ❧ ✉ ❛ ♥ ❞ ❧ ❡ ♣ ♦ ❜ ❧ ♠ ❡  

❛ ✐ ❡ ♣ ❡ ✉ ❡ ❞ ❝ ♦ ♠ ♣ ♦ ❡ ❡ ♥ ❞ ❡ ✉ ① ♦ ✉ ♣ ❧ ✉ ♦ ✉ ✲ ♣ ♦ ❜ ❧ ♠ ❡ ✐ ❞ ❡ ♥ ✐ ✉ ❡ ❛ ✉ ♣ ♦ ❜ ❧ ♠ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ♠ ❛ ✐ ❛ ✈ ❡ ❝ ❞ ❡  

✈ ❛ ❧ ❡ ✉ ❞ ❡ ♣ ❛ ❛ ♠ ❡ ❞ ✐ ✛ ❡ ♥ ❡ ✳ ❘ ❡ ❢ ✉ ❡ ❧ ❛ ❝ ✉ ✐ ✈ ✐ ❞ ❛ ♥ ❝ ❡ ❞ ❡ ♥ ✐ ❡ ❝ ❛ ♦ ❜ ❧ ✐ ❣ ❡ ❧ ✬ ✉ ✐ ❧ ✐ ❛ ❡ ✉ ❣ ❡  

❧ ✉ ✐ ✲ ♠ ♠ ❡ ✉ ♥ ❡ ♣ ✐ ❧ ❡ ❞ ❡ ❞ ✐ ✛ ❡ ♥ ❡ ✈ ❛ ❧ ❡ ✉ ❞ ❡ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✱ ❝ ❡ ✉ ❡ ❧ ❡ ② ♠ ❡ ❢ ❛ ✐ ❛ ✉ ♦ ♠ ❛ ✐ ✉ ❡ ♠ ❡ ♥ ❧ ♦ ❞ ❡  

❧ ✬ ✉ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ❡ ❧ ❛ ❝ ✉ ✐ ✈ ✐ ✳  

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✷ ✷ 

✸ ✳ ✽ ❊ ① ❡ ❝ ✐ ❝ ❡  

✸ ✳ ✽ ✳ ✶ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✶  

❊ ❝ ✐ ❡ ✉ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❝ ✉ ✐ ✈ ❡ ✉ ✐ ❝ ❛ ❧ ❝ ✉ ❧ ❡ ❧ ❡ ❝ ❛ ❞ ✬ ✉ ♥ ❡ ♥ ✐ ❡   n2 = n × n :❋ ♦ ♥ ❝ ✐ ♦ ♥ ❝ ❛ ✭ ♥ ✿ ❊ ♥ ✐ ❡ ✮ ✿ ❊ ♥ ✐ ❡  

❖ ♥ ♣ ♦ ✉ ❛ ✉ ✐ ❧ ✐ ❡ ❧ ❛ ❡ ❧ ❛ ✐ ♦ ♥ ✉ ✐ ✈ ❛ ♥ ❡ ✿   (n + 1)2 = n2 + 2n + 1.

✸ ✳ ✽ ✳ ✷ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✷  

❊ ❝ ✐ ❡ ✉ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❝ ✉ ✐ ✈ ❡ ✉ ✐ ♣ ❡ ♥ ❞ ❡ ♥ ❛ ❣ ✉ ♠ ❡ ♥ ✉ ♥ ❡ ❝ ❤ ❛ ♥ ❡ ❞ ❡ ❝ ❛ ❛ ❝ ❡ ❡ ✉ ✐ ❡ ♥ ✈ ♦ ✐ ❡ ✉ ♥  

❜ ♦ ♦ ❧ ❡ ♥ ✐ ♥ ❞ ✐ ✉ ❛ ♥ ✬ ✐ ❧ ✬ ❛ ❣ ✐ ❞ ✬ ✉ ♥ ♣ ❛ ❧ ✐ ♥ ❞ ♦ ♠ ❡ ♦ ✉ ♥ ♦ ♥ ✳  

✸ ✳ ✽ ✳ ✸ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✸  

❖ ♥ ❞ ✜ ♥ ✐ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✉ ✐ ✈ ❛ ♥ ❡ ✿  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ▼ ❝ ❈ ❛ ② ✭ ♥ ✿ ❊ ♥ ✐ ❡ ✮ ✿ ❊ ♥ ✐ ❡  

❞ ❜ ✉  

✐  n > 100 ❛ ❧ ♦  McCarthy ← n − 10

✐ ♥ ♦ ♥  

McCarthy ← McCarthy(McCarthy(n + 11))✜ ♥ ✐  

✜ ♥ 

✶ ✳ ◗ ✉ ❡ ❧ ❧ ❡ ❡ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ❞ ❡ ▼ ❝ ❈ ❛ ❤ ② ✭ ♥ ✮ ♣ ♦ ✉ ♥ ❃ ✶ ✵ ✵ ❄  

✷ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❡ ▼ ❝ ❈ ❛ ❤ ② ✭ ✾ ✽ ✮ ✱ ▼ ❝ ❈ ❛ ❤ ② ✭ ✾ ✾ ✮ ❡ ▼ ❝ ❈ ❛ ❤ ② ✭ ✶ ✵ ✵ ✮ ✳  

✸ ✳ ❊ ♥ ❞ ❞ ✉ ✐ ❡ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ❞ ❡ ▼ ❝ ❈ ❛ ❤ ② ✭ ♥ ✮ ♣ ♦ ✉ ♥ ✲ ✶ ✵ ✵ ✳ ❊ ① ♣ ❧ ✐ ✉ ❡ ✳  

✸ ✳ ✽ ✳ ✹ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✹  

❊ ❝ ✐ ❡ ✉ ♥ ❡ ♣ ♦ ❝ ❞ ✉ ❡ ❝ ✉ ✐ ✈ ❡ ✉ ✐ ❡ ♣ ❡ ♥ ❡ ❧ ❡ ❞ ♣ ❧ ❛ ❝ ❡ ♠ ❡ ♥ ❞ ✬ ✉ ♥ ❝ ❛ ✈ ❛ ❧ ✐ ❡ ✉ ✉ ♥ ❝ ❤ ✐ ✉ ✐ ❡ ♣ ❛ ✐  

❞ ❡ ❧ ❛ ♣ ♦ ✐ ✐ ♦ ♥ ✭  X 0, Y 0 ✮ ✳ 

✸ ✳ ✽ ✳ ✺ ❊ ① ❡ ❝ ✐ ❝ ❡ ✵ ✺  

❙ ♦ ✐ ✉ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❝ ♦ ♥ ✐ ♥ ✉ ❡   f  ❞ ✜ ♥ ✐ ❡ ✉ ✉ ♥ ✐ ♥ ❡ ✈ ❛ ❧ ❧ ❡    [a; b]✳ ❖ ♥ ❝ ❤ ❡ ❝ ❤ ❡ ♦ ✉ ✈ ❡ ✉ ♥ ③ ♦ ❞ ❡    f ✱ ❝ ✬ ❡ ✲   ✲ ❞ ✐ ❡ ✉ ♥ ❡ ❧   x ∈   [a; b] ❡ ❧ ✉ ❡    f (x) = 0✳ ❙ ✐ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❛ ❞ ♠ ❡ ♣ ❧ ✉ ✐ ❡ ✉ ③ ♦ ✱ ♥ ✬ ✐ ♠ ♣ ♦ ❡ ❧ ❡ ✉ ❡ ❧ ❢ ❡ ❛  ❧ ✬ ❛ ✛ ❛ ✐ ❡ ✳ ❙ ✬ ✐ ❧ ♥ ✬ ② ❡ ♥ ❛ ♣ ❛ ✱ ✐ ❧ ❢ ❛ ✉ ❞ ❛ ❧ ❡ ✐ ❣ ♥ ❛ ❧ ❡ ✳  

❉ ❛ ♥ ❧ ❡ ❝ ❛ ♦   f (a).f (b)   <   0✱ ♦ ♥ ❡ ❞ ❡ ❧ ❛ ♣ ❡ ♥ ❝ ❡ ❞ ✬ ✉ ♥ ③ ♦ ✳ ▲ ♦ ✉ ❡    f (a).f (b)   >   0✱ ✐ ❧ ❢ ❛ ✉   ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✉ ♥ ♦ ✉ ✲ ✐ ♥ ❡ ✈ ❛ ❧ ❧ ❡   [α; β ]✱ ❡ ❧ ✉ ❡    f (α).f (β ) 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✷ ✸ 

❝ ✐ ✈ ❡ ③ ❞ ❡ ❢ ❛ ♦ ♥ ❝ ✉ ✐ ✈ ❡ ❧ ✬ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ❞ ❡ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❞ ✬ ✉ ♥ ③ ♦ ✱   ε ♣ ✱ ❞ ❡ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥    f ✳ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✷ ✹ 

✸ ✳ ✾ ❚  

▲ ✬ ♦ ❜ ❥ ❡ ❝ ✐ ✈ ❡ ❞ ❡ ❝ ❡ ❚ ❡ ❞ ✬ ❡ ① ♣ ✐ ♠ ❡ ♥ ❡ ❧ ❛ ❝ ✉ ✐ ✈ ✐ ❡ ❧ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ❡ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡  

◆ ♦ ❡ ③ ❜ ✐ ❡ ♥ ✿  

✕ ❯ ♥  ❛ ♣ ♣ ♦ ✭ ✸ ♣ ❛ ❣ ❡ ♠ ❛ ① ✮ ❡ ❝ ♦ ❞ ❡ ♦ ✉ ❝ ❡   ❞ ♦ ✐ ❡ ❡ ♠ ✐ ❛ ✈ ❛ ♥ ❧ ❡ ✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳  

✸ ✳ ✾ ✳ ✶ ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✉ ♥ ❧ ♠ ❡ ♥ ❞ ❛ ♥ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉  

▲ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❝ ♦ ♠ ♣ ❛ ❡ ❧ ❡ ❞ ✐ ✛ ❡ ♥ ❧ ♠ ❡ ♥ ❞ ✬ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉ ❧ ❛ ❝ ❧ ❡ ❢ ❡ ❤ ❡ ❝ ❤ ❡ ✳  

❡ ① ❡ ♠ ♣ ❧ ❡ ✿  

▲ ❛ ✈ ❛ ❧ ❡ ✉ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❡ ✼ ✳  

▲ ❛ ❝ ❛ ❡ ❝ ♦ ❧ ♦ ❡ ❡ ❧ ✬ ❧ ♠ ❡ ♥ ❡ ♥ ❝ ♦ ✉ ❞ ❡ ❛ ✐ ❡ ♠ ❡ ♥ ✳  

✽  ✺ ✷ ✼ ✶ ✸ ✵ ✹  

✽  ✺  ✷ ✼ ✶ ✸ ✵ ✹  

✽ ✺  ✷  ✼ ✶ ✸ ✵ ✹  

✽ ✺ ✷  ✼  ✶ ✸ ✵ ✹  

❚ ❛ ❜ ❧ ❡   ✸ ✳ ✶ ✕ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❞ ❛ ♥ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉  

▲ ❛ ✈ ❛ ❧ ❡ ✉ ❛ ♦ ✉ ✈ ❡ ❀ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ❡ ♦ ✉ ♥ ❡ ❧ ❛ ♣ ♦ ✐ ✐ ♦ ♥ ❞ ❡ ❝ ❡ ❡ ✈ ❛ ❧ ❡ ✉ ✱ ✉ ✐ ❞ ❛ ♥ ♥ ♦ ❡ ❝ ❛ ❡ ✹ ✳  

✸ ✳ ✾ ✳ ✷ ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❙ ✉ ❡ ♥ ✐ ❡ ❧ ❧ ❡  

▲ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✉ ❡ ♥ ✐ ❡ ❧ ❧ ❡ ❝ ♦ ♥ ✐ ❡ ♣ ❛ ❝ ♦ ✉ ✐ ❧ ❡ ❛ ❜ ❧ ❡ ❛ ✉ ✉ ♥ ♣ ❛ ✉ ♥ ❥ ✉ ✉ ✬ ♦ ✉ ✈ ❡ ❧ ❡ ❝ ❧ ❡ ❢ ✳  

✶ ✳ ❊ ❝ ✐ ✈ ❡ ③ ✉ ♥ ♦ ✉ ✲ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ♣ ❛ ❝ ❛ ❧ ❡ ♣ ♦ ✉ ♦ ✉ ❞ ❡ ❧ ❡ ♣ ♦ ❜ ❧ ♠ ❡ ❞ ❡ ❧ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✉ ❡ ♥ ✐ ❡ ❧ ❧ ❡ ✳  

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❙ ✭ ❆ ✿ ❛ ❜ ❧ ❡ ❛ ✉ ✱ ❡ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❊ ♥ ✐ ❡  

✷ ✳ ❉ ♦ ♥ ♥ ❡ ③ ✉ ♥ ❡ ① ❡ ♠ ♣ ❧ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ♣ ✐ ♥ ❝ ✐ ♣ ❛ ❧ ❡ ✉ ✐ ✉ ✐ ❧ ✐ ❡ ♥ ❝ ❡ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ✳  

✸ ✳ ❙ ✐ ❧ ❡ ❛ ❜ ❧ ❡ ❛ ✉ ❡ ♦ ❞ ♦ ♥ ♥ ❡ ❀ ♠ ♦ ❞ ✐ ✜ ❡ ③ ❧ ❛ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ✉ ✉ ♣ ♣ ♦ ❡ ❝ ❡ ❝ ❤ ❛ ♥ ❣ ❡ ♠ ❡ ♥ ✳  

✹ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❡ ❧ ❡ ♥ ♦ ♠ ❜ ❡ ❞ ❡ ❝ ♦ ♠ ♣ ❛ ❛ ✐ ♦ ♥ ♣ ♦ ✉ ✉ ♥ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❞ ✬ ✉ ♥ ❧ ♠ ❡ ♥ ❧ ❛ ✜ ♥ ❞ ✉ ❛ ❜ ❧ ❡ ❛ ✉ ❞ ❡ ✻ ✹  

❧ ♠ ❡ ♥ ✳ ◗ ✉ ✬ ❡ ❝ ❡ ✉ ❡ ❡ ♣ ❡ ♥ ❡ ❝ ❡ ♥ ♦ ♠ ❜ ❡ ✳  

✸ ✳ ✾ ✳ ✸ ❘ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❞ ✐ ❝ ❤ ♦ ♦ ♠ ✐ ✉ ❡  

❈ ❡ ❛ ❧ ❣ ♦ ✐ ❤ ♠ ❡ ♦ ♣ ❡ ✉ ✉ ♥ ❡ ♥ ❡ ♠ ❜ ❧ ❡ ♦ ❞ ♦ ♥ ♥ ❡ ❡ ❡ ❞ ❡ ❧ ✬ ♦ ❞ ❡ ♣ ♦ ✉ ❞ ✐ ✐ ❣ ❡ ❧ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✳ ▲ ❡ ♠ ♦  

❞ ✐ ❝ ❤ ♦ ♦ ♠ ✐ ❡   ✈ ✐ ❡ ♥ ❞ ✉ ❣ ❡ ❝ ✉ ✐ ✐ ❣ ♥ ✐ ✜ ❡ ✿   ❝ ♦ ✉ ♣ ❡ ❡ ♥ ❞ ❡ ✉ ①  ✳ 

▲ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❞ ✐ ❝ ❤ ♦ ♦ ♠ ✐ ✉ ❡ ❞ ❛ ♥ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉ ✐ ❡ ♣ ♦ ❡ ✉ ❧ ❡ ♣ ✐ ♥ ❝ ✐ ♣ ❡ ✉ ✐ ✈ ❛ ♥ ✿ ✐ ❧ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡  

✬ ❡ ✛ ❡ ❝ ✉ ❡ ❡ ♥ ❡ ❧ ❡ ✐ ♥ ❞ ✐ ❝ ❡   bi ❡    bs ♦ ♥ ❝ ♦ ♠ ♣ ❛ ❡ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ❡ ❝ ❤ ❡ ❝ ❤ ❡    x ❛ ✈ ❡ ❝ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ✉ ✐ ❡ ❣ ❛ ❧ ❡  

❞ ✐ ❛ ♥ ❝ ❡ ❞ ❡ ❜ ♦ ♥ ❡  

bi❡  

  bs✱ ♦ ✐ ❧ ✬ ✐ ♥ ❞ ✐ ❝ ❡  

  m =

bi + bs

2✕ ❙ ✐  x ❡ ❣ ❛ ❧ ❡ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ❡ ♥    m✱ ❛ ❧ ♦ ❧ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ✬ ❛ ❡ ♣ ♦ ✐ ✐ ✈ ❡ ♠ ❡ ♥ ✱ ✐ ♥ ♦ ♥  ✕ ❙ ✐  x ❡ ✐ ♥ ❢ ✐ ❡ ✉ ❧ ❛ ✈ ❛ ❧ ❡ ✉ ❡ ♥    m ❛ ❧ ♦ ♦ ♥ ❝ ♦ ♥ ✐ ♥ ✉ ❡ ❧ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❡ ♥ ❡ ❧ ❡ ✐ ♥ ❞ ✐ ❝ ❡    bi ❡    m − 1 ✐ ♥ ♦ ♥  ✕ ❖ ♥ ❝ ♦ ♥ ✐ ♥ ✉ ❡ ❧ ❛ ❡ ❝ ❤ ❡ ❝ ❤ ❡ ❡ ♥ ❡ ❧ ❡ ✐ ♥ ❞ ✐ ❝ ❡   m + 1 ❡    bs ✳ 

✶ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❡   log2(64).

✷ ✳ ❊ ❝ ✐ ❡ ✉ ♥ ❡ ❢ ♦ ♥ ❝ ✐ ♦ ♥   RechDicho

❋ ♦ ♥ ❝ ✐ ♦ ♥   ❘ ❡ ❝ ❤ ❉ ✐ ❝ ❤ ♦ ✭ ❆ ✿ ❛ ❜ ❧ ❡ ❛ ✉ ✱ ❡ ✿ ❡ ♥ ✐ ❡ ✮ ✿ ❊ ♥ ✐ ❡  

✉ ✐ ❛ ♣ ♦ ✉ ♣ ❛ ❛ ♠ ❡ ✉ ♥ ❛ ❜ ❧ ❡ ❛ ✉ ✉ ❡ ❧ ✬ ♦ ♥ ✉ ♣ ♣ ♦ ❡ ✐ ✱ ✉ ♥ ❡ ♥ ✐ ❡   x ❡ ✉ ✐ ❡ ♥ ✈ ♦ ✐ ❡ ❧ ✬ ✐ ♥ ❞ ✐ ❝ ❡ ❞ ❡    x ✐ x ❡ ❞ ❛ ♥ ❧ ❡ ❛ ❜ ❧ ❡ ❛ ✉ ✱    −1 ✐ ♥ ♦ ♥ ✱ ❡ ❧ ♦ ♥ ❧ ❡ ♣ ✐ ♥ ❝ ✐ ♣ ❡ ♥ ♦ ♥ ❝ ❝ ✐ ✲ ❞ ❡ ✉ ✳  

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❈ ❤ ❛ ♣ ✐ ❡ ✹      

❙ ✉ ❝ ✉ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡    

✹ ✳ ✶ ❚ ② ♣ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ❛ ❜ ❛ ✐ ✭ ❚ ❉ ❆ ✮    

❯ ♥ ❚ ❉ ❆ ❡ ✉ ♥ ❡ ♥ ❡ ♠ ❜ ❧ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ♦ ❣ ❛ ♥ ✐ ❞ ❡ ♦ ❡ ✉ ❡ ❧ ❡ ♣ ❝ ✐ ✜ ❝ ❛ ✐ ♦ ♥ ❞ ❡ ♦ ❜ ❥ ❡ ❡ ❞ ❡  

♦ ♣ ❛ ✐ ♦ ♥ ✉ ❝ ❡ ♦ ❜ ❥ ❡ ✭ ✐ ♥ ❡ ❢ ❛ ❝ ❡ ✮ ♦ ✐ ❡ ♥ ♣ ❛ ❡ ❞ ❡ ❧ ❛ ❡ ♣ ❡ ♥ ❛ ✐ ♦ ♥ ✐ ♥ ❡ ♥ ❡ ❞ ❡ ♦ ❜ ❥ ❡ ❡ ❞ ❡ ❞ ❡ ❧ ❛  

♠ ✐ ❡ ❡ ♥ ♦ ❡ ✉ ✈ ❡ ❞ ❡ ♦ ♣ ❛ ✐ ♦ ♥ ✳ ❊ ① ❡ ♠ ♣ ❧ ❡ ❞ ❡ ❚ ❉ ❆ ✿ ❧ ❡ ② ♣ ❡ ❡ ♥ ✐ ❡ ♠ ✉ ♥ ✐ ❞ ❡ ♦ ♣ ❛ ✐ ♦ ♥ ✰ ❀ ✲ ❀ ✯ ❀ ✪ ❀ ❂ ❀ ❃ ❀  

❁ ❀ ❁ ❂ ❀ ❃ ❂ ❀ ❂ ❂ ❡ ✉ ♥ ❚ ❉ ❆ ✳  

❯ ♥ ❡ ♠ ✐ ❡ ❡ ♥ ♦ ❡ ✉ ✈ ❡ ❞ ✬ ✉ ♥ ❚ ❉ ❆ ❡ ❧ ❛ ✉ ❝ ✉ ❡ ❞ ❡ ❞ ♦ ♥ ♥ ❡ ♣ ❛ ✐ ❝ ✉ ❧ ✐ ❡ ❡ ❧ ❛ ❞ ✜ ♥ ✐ ✐ ♦ ♥ ❞ ❡ ♦ ♣ ❛ ✐ ♦ ♥  

♣ ✐ ♠ ✐ ✐ ✈ ❡ ❞ ❛ ♥ ✉ ♥ ❧ ❛ ♥ ❣ ❛ ❣ ❡ ♣ ❛ ✐ ❝ ✉ ❧ ✐ ❡ ✳  

▲ ❡ ❛ ✈ ❛ ♥ ❛ ❣ ❡ ❞ ❡ ❚ ❉ ❆ ♦ ♥ ✿  

✕ ♣ ✐ ❡ ❡ ♥ ❝ ♦ ♠ ♣ ❡ ❞ ❡ ② ♣ ❡ ❝ ♦ ♠ ♣ ❧ ❡ ① ❡ ✳  

✕ ♣ ❛ ❛ ✐ ♦ ♥ ❞ ❡ ❡ ✈ ✐ ❝ ❡ ❡ ❞ ✉ ❝ ♦ ❞ ❛ ❣ ❡ ✳ ▲ ✬ ✉ ✐ ❧ ✐ ❛ ❡ ✉ ❞ ✬ ✉ ♥ ❚ ❉ ❆ ♥ ✬ ❛ ♣ ❛ ❜ ❡ ♦ ✐ ♥ ❞ ❡ ❝ ♦ ♥ ♥ ❛ ❡ ❧ ❡ ❞ ❛ ✐ ❧  

❞ ✉ ❝ ♦ ❞ ❛ ❣ ❡ ✳  

✕ ❝ ✐ ✉ ❡ ❞ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❡ ♠ ♦ ❞ ✉ ❧ ❛ ✐ ❡ ✳  

✹ ✳ ✷ ❉ ✜ ♥ ✐ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ② ♣ ❡ ❛ ❜ ❛ ✐    

❯ ♥ ② ♣ ❡ ❛ ❜ ❛ ✐ ❡ ❞ ❝ ✐ ♣ ❛ ❛ ✐ ❣ ♥ ❛ ✉ ❡ ✉ ✐ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ✿  

✕ ✉ ♥ ❡ ❞ ❝ ❧ ❛ ❛ ✐ ♦ ♥ ❞ ❡ ❡ ♥ ❡ ♠ ❜ ❧ ❡ ❞ ✜ ♥ ✐ ❡ ✉ ✐ ❧ ✐ ❀  

✕ ✉ ♥ ❡ ❞ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ♥ ❡ ❧ ❧ ❡ ❞ ❡ ♦ ♣ ❛ ✐ ♦ ♥ ❀  

✕ ✉ ♥ ❡ ❞ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❛ ① ✐ ♦ ♠ ❛ ✐ ✉ ❡ ❞ ❡ ❧ ❛ ♠ ❛ ♥ ✐ ✉ ❡ ❞ ❡ ♦ ♣ ❛ ✐ ♦ ♥ ✳  

✹ ✳ ✷ ✳ ✶ ❊ ① ❡ ♠ ♣ ❧ ❡  

❉ ❝ ❧ ❛ ❛ ✐ ♦ ♥ ✿  

EntierNaturel.0 ∈ EntierNaturel❉ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❢ ♦ ♥ ❝ ✐ ♦ ♥ ♥ ❡ ❧ ❧ ❡ ✿  

succ :  EntierNaturel → EntierNaturel+ : EntierNaturel + EntierNaturel → EntierNaturel× :  EntierNaturel × EntierNaturel → EntierNaturel❉ ❡ ❝ ✐ ♣ ✐ ♦ ♥ ❛ ① ✐ ♦ ♠ ❛ ✐ ✉ ❡  

✶ ✳  ∀x ∈ EntierNaturel; ∃x′; succ(x) = x′✷ ✳  ∀x; x0 ∈ EntierNaturel; x = x0succ(x) = succ(x0)✸ ✳  ∄x ∈ EntierNaturel; succ(x) = 0

✷ ✻ 

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❆ ❧ ❣ ♦ ✐ ❤ ♠ ✐ ✉ ❡ ❡ ❙ ✉ ❝ ✉ ❡ ❞ ❡ ❉ ♦ ♥ ♥ ❡ ■ ♠ ❛ ✐ ❧ ❍ ❛ ❞ ❥ ❛ ❞ ❥   ✷ ✼ 

✹ ✳  ∀x ∈ EntierNaturel; x + 0 = x✺ ✳  ∀x; y ∈ EntierNaturel; x + succ(y) = succ(x + y)✻ ✳  ∀x ∈ EntierNaturel; x × 0 = 0✼ ✳  ∀x; y ∈ EntierNaturel; x × succ(y) = x + xy

✹ ✳ ✸ ▲ ✬ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ❉ ✬ ✉ ♥ ❚ ② ♣ ❡ ❆ ❜ ❛ ✐    

▲ ✬ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ❡ ❧ ❛ ❢ ❛ ♦ ♥ ❞ ♦ ♥ ❧ ❡ ② ♣ ❡ ❛ ❜ ❛ ✐ ❡ ♣ ♦ ❣ ❛ ♠ ♠ ❞ ❛ ♥ ✉ ♥ ❧ ❛ ♥ ❣ ❛ ❣ ❡ ♣ ❛ ✐ ❝ ✉ ❧ ✐ ❡ ✳  

♦ ✉ ✉ ♥ ② ♣ ❡ ❛ ❜ ❛ ✐ ❞ ♦ ♥ ♥ ✱ ♣ ❧ ✉ ✐ ❡ ✉ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ♣ ♦ ✐ ❜ ❧ ❡ ♣ ❡ ✉ ✈ ❡ ♥ ❡ ❞ ✈ ❡ ❧ ♦ ♣ ♣ ❡ ✳ ▲ ❡ ❝ ❤ ♦ ✐ ①  

❞ ✬ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ❞ ✉ ② ♣ ❡ ❛ ❜ ❛ ✐ ✈ ❛ ✐ ❡ ❛ ❡ ❧ ♦ ♥ ❧ ✬ ✉ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ✉ ✐ ❡ ♥ ❡ ❢ ❛ ✐ ❡ ❡ ❛ ✉ ❛ ✉ ♥ ❡ ✐ ♥ ✢ ✉ ❡ ♥ ❝ ❡ ✉ ❧ ❛  

❝ ♦ ♠ ♣ ❧ ❡ ① ✐ ❞ ❡ ♦ ♣ ❛ ✐ ♦ ♥ ✳  

✹ ✳ ✹ ❯ ✐ ❧ ✐ ❛ ✐ ♦ ♥ ❞ ❡ ② ♣ ❡ ❛ ❜ ❛ ✐    

✉ ✐ ✉ ❡ ❧ ❛ ❞ ✜ ♥ ✐ ✐ ♦ ♥ ❞ ✬ ✉ ♥ ② ♣ ❡ ❛ ❜ ❛ ✐ ❡ ✐ ♥ ❞ ♣ ❡ ♥ ❞ ❛ ♥ ❡ ❞ ❡ ♦ ✉ ❡ ✐ ♠ ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ♣ ❛ ✐ ❝ ✉ ❧ ✐ ❡ ✱ ❧ ✬ ✉ ✐ ❧ ✐ ❛ ✲  

✐ ♦ ♥ ❞ ✉ ② ♣ ❡ ❛ ❜ ❛ ✐ ❞ ❡ ✈ ❛ ❡ ❢ ❛ ✐ ❡ ❡ ① ❝ �