map rev L X l X - PDF document

★ ➒ ⑨ ❘ ✛ ⑦ ✑ ☛ ⑦ ❬ ❾ ★ ⑨ ➢ ❘ ➫ ⑦ ➢ ❾ ♠ ❩ ➔ ❾ ➒ ⑦ ❩ ✄ ◗ → ★ ➂ ☛ ➑ ◗ ❘ ➢ ⑨ ① ➒ s ❾ ✌ ☛ ➢ ☎ ✗ ➫ ➢ ❿ ☛ ✍ ☞ ☛ ➢ ✞ ✝ ✆ ☎ ↔ ☞ ⑦ ➫ ➒ ❫ ➂ ➢ ❩ ◗ ➫ ✛ ★ ★ ◗ ❩ ✑ ⑦ ✑ ➒ ❾ �✂✁ ✟✡✠ ✍✏✎ ✑✓✒✔✒✖✕ ✍✘✎ ✍✚✙✜✛✣✢✥✤ ★✪✩✫✩✜✬ ∗ ✗✧✦ ✑ ∗ ✭✯✮✰✮✲✱✴✳✶✵✷✵✹✸✰✸✷✸✻✺✽✼✿✾❁❀❂✾❃✺❅❄✷❆❇✾❉❈❋❊✫✺●✾❍✮ (*)DISI - Univ. of Genova, Italy, ✭✯✮✰✮✰✱✻✳✶✵✷✵✹✸✷✸✰✸✻✺■❀✲❏▲❑✲❀▼✺◆❄✷✮❋❀✪✺❖❊❁✼❁❄✴✺❖P✹❄ (+)SCS - Univ. of Tech. Sydney, Australia, ★❙❘❚✬❱❯ ❘❚✬✔❲✻✩ ✑❇❳❨✛ ✒✶✑▼❬ map rev ❬ L • ❭ X → l ❭ X ❪❴❫❛❵❝❜❡❞❣❢✷❜✐❤✹❥❋❜❧❦♥♠✷♦♣♦rq ❢❂t✉❤✈❦♣❥❋❜●❤✡t✇❤②①✇❜❧❦♣❥❝③ ❵④①⑤❤②①❉⑥ ✑❛✒■✛ ✑❙⑧ ❬✴❶❸❷❡❹✻❺④❺❼❻❋❽❛❾✹❿➁➀❨❻❋❽❚➂✔❾❋❶➄➃ ❹✴➀❨❻❛➅❂❾➆➃ ➇➈❻❋➉✰➂✔❾❂❶➋➊➌❹➍❻❋➎✷➂✔❾ • ⑦✰⑩ ❢❂❥ ❤➆❵ ♠❚q ❶➄➏ ❻❋➉➐❾✓➏ ❻❋➉✰➂✔❾❙❶➄➏ ❻❋➉ ➑✻✛ ✬●➒ �➓☞ ❬❛➃ • . . . ➣↕↔✇➙ ❜➝q➐❞➞❤②① ①✖➟➡➠❝❤✹s ♠✰① ✒❱✢ ❲➜➛✿❘✲★ ✬●➒ ⑦▲❩ ❜●❤✡t✇s ❜➝q➐❞➞❤➓♠✰①➡①⑤❦➤③❋❥④s➥❤✹❥❋❜ t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥ ↔✇➙ ➑✻✛ �➧☞ ❲➨➛➩❘✲★ • ❵➐❥✓➟⑤❜✉❢❂t✇① ↔✇➙ ♠❚q ➑✻✛ ✑✓✒✽✢ ❲➨➛➩❘✲★ ❶➄➏ ❻❋➉ t✇❤✹❞❛t✉❤②①⑤❤✹❥❋❜➭♠✹❜❧❦♣❢❂❥④①❡❢ ①✖➠➐♠✷❞➞❤✹♦rq ❵❼❥✓➟⑤❜✉❢❂t✇① ❪✶♠✰①✚❤❍➯✓❞➞❤➆➟⑤❜✐❤✲➦➲⑥ ❘✲✬ �➧☞ ❶❸❷➳➏❋➃ ❻❂➵✯➂✔❬ • ❩➜⑦ ❜➸q➐❞➲❤✈❦♣❥ ❤✡t✇❤✹❥✓➟➺❤➻♠✷♦➤③❋❢❂t➡❦r❜➸➠❛s ①➭❵➼❫❼➽●❤➆➟❉❜➾t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥ ➠➼❵❛t✉➟❍➠❝➚ ❢➪①➡①❉❤➆t➶❞❛t✉❢✲❞➞❤➆t✐❜➝q ❾➐➊ ①❧❜➡t✉❢❂❥❝③ ❥❛❢❂t✇s ♠✷♦♣❦➤➹❚♠✹❜❧❦♣❢❂❥ ❪❴❥❛❢➴➘➼➯❃⑥ 1

✠ ◗ ➣ ❘ ➛ ✛ ❲ ✃ ✑ ↔ ✑ ➫ ⑥ ❾ ➫ ⑥ ⑥ Ð ➫ ❬ ➇ ➊ ➒ ➛ ✛ ★ ✠ ➬ ➱ ➫ ➬ ➢ ✛ ◗ ❩ ➬ ★❙❘❚✬❱❯ ❘❚✬✔❲✻✩ ✑❇❳❨✛ ✒✶✑ ❪ X → Y ⑥ → FX → FY ❬ ∀ F ❬②➅ . ∀ X, Y. • map ➦④♠✹❜➭♠➴❞❣❢✯♦rq➐s ❢❂t✇❞❛➠❛❦➷①✖s ➘➼➯✓❤✲➦ ♠✷t●q ❬ F ❾ X, Y ➣➻①➭➠➐♠✷❞➲❤✈❞❣❢✯♦rq➐s ❢❂t✉❞❛➠❛❦➷①✖s ♠✷t❍❦♣❤②① ➘➼➯✓❤✲➦ ❬ F ❾ X, Y ❜➸➠❝❤✈♦♣❦➮①❧❜ ❵❼❥✓➟⑤❜✉❢❂t ❬❁➅ • L ❬ ∀ X.LX → LX t✉❤ ❤✡t❧①⑤❦♣❥❝③ ♠↕♦♣❦➮①✇❜ • rev ➑✻✛ ✬✔❲ ✑❙⑧✏❐ ⑦✯⑦ ❵④♦♣♦rq ❤➡➯✓❞➐♦♣❦♣➟➺❦r❜ • ❪ rev X ❬ L map L ❒ LX LX ❭ X → L ❭ X ❪ rev LX ❬ L map L L ❒ X L ❒ X ❭ X → L ❭ X ❜➝q➐❞➞❤❮❦♣s ❞➐♦♣❦❰➟✹❦r❜ • ❬ L map L ❒ rev ❭ X → L ❭ X ❬ L map L rev ❭ X → L ❭ X ❵④♦♣♦rq ❦♣s ❞❼♦♣❦♣➟➺❦r❜ ❬ L map rev ❭ X → L ❭ X ➑➍✑ ❩✻❩➨⑦ ➠➐♠✷t➡❦r❜➝q s↕❵④①✇❜➳❫➞❤✂❤❍➯✓❞❼♦♣❦♣➟➺❦r❜ ❬ F • ♠✰①✖Ï➪❤✹♦♣♦➞❜➸q➐❞➲❤❮➟✹♦♥♠✰①➡①⑤❤②① ➟✹❢✿➦❇❤ ❢❂t map ①✖❵❛❞❝❞❼♦♣❦♣❤✲➦ ❫➼q ❵➐①❉❤➆t • ❪ GX ⑥ � ❪ F ◦ G ⑥➺❪ X ❢❂❥❛♦rq ❦➷①✖❢❂s ❢❂t✇❞❛➠❛❦♣➟▲Ñ ❹✥➃ ➔✥❬ F • 2

Ò ⑥ ❬ ⑥ ⑥ ➫ ❢ ❬ ⑥ ➫ ⑥ ❒ ⑥ ❒ ➟ ➂ ⑥ ⑥ ⑥ ❾ ⑥ ✬ Ð ✑❇✢ ⑧✻✑➲✛ ❪ GX ⑥ � ❪ F ◦ G ⑥➺❪ X ❢❂❥❛♦rq ❦➷①✖❢❂s ❢❂t✇❞❛➠❛❦♣➟▲Ñ • F ❪ GX ❵❼❥✓➟⑤❜✉❢❂t ➦④♠✹❜➭♠ ❬ F ❾ GX • F ❪ X ❪ F ◦ G ❵➐❥✓➟⑤❜✉❢❂t ➦④♠✹❜➭♠ ❾ X • F � G � ❤✹❥ ❵➐❥❛➟❉❜✉❢❂t❧① ➟➺❢❂s ❞❃❢➪①❍❦r❜❧❦♣❢❂❥➥❵❛❞Þ❜➭❢ß❦➮①✖❢à❪✶♠✰① ❶❸❷ ➵➼Ý❚➂ ➫④➫ • Ó②Ô➸Õ✹Ö❝×✐Ø●Ù✓Ú❋Û❁Ü ①✖❢✷♦♣❵❝❜✐❦♣❢❂❥④①✏❜✉❢à➦④❢❂s ♠✷❦♣❥ ❤➆á✿❵➲♠✹❜❧❦♣❢❂❥④①⑤⑥ãâ ➟✡♠✷❥❛❢❂❥❛❦❰➟❁♠✷♦▼❦➷①✖❢➪①➌♠✰①❡➦❛♠✹❜➭♠➴➦④❦➮①❧❜➡t➡❦♣❫❛❵✿❜❧❦♣❢❂❥ ♠❚q ❶➄➏ ❻❋➉ • ❪ GX ❪ L ❪ LX ⑥✉⑥ map rev ❪ L ❪ LX ⑥✉⑥ ✲ L F L dex dex dex ❪ X ❪ LX ❪ LX ❄ ❄ ❄ ✲ L F � G � L map rev ❪ X ❪ Y map f ✲ H H 3

� ✛ ❬ ➫ ◗ ë ⑧ ➙ ✢ ➢ ⑦ ✛ ⑦ ➙ � Ð é ➢ å ⑥ ➬ ⑥ Ð ➅ → ❦ ❾ ⑥ ❾ ⑥ Ð ❦ ❾ Ð å é Ð ❲ ◗ å ✗ ➒ ✑ ✛ ❩ ➣ ➫ ✢ ❘ ✗ ➒ ⑦ ❢ ➫ Ð Ð ❲➨➛➩❘✲★ ➒✰➛ ➑➍✑ ❵❼❥✓➟⑤❜✉❢❂t✇①➳ä ♠✷t➡❦r❜➝q m ❬ m ⊢ F ❬②❬ i | F � G � n | µ m F m F, G X | C | ❜➝q➐❞➞❤②①❡ä ❬✖æ • ⊢ τ ❪ τ ❬②❬ τ X | τ ç → τ ❒ | . . . | F ①➭➟❍➠❝❤✹s ♠èä • ⊢ σ ❬ m.σ ❬②❬ σ τ | ∀ X.σ | ∀ X ä , X ä , X ➠❝❤➆t✉❤✂ä ❬ m | ❜➝q➐❞➞❤✂➟✹❢❂❥➪❜✐❤➡➯➐❜❚â ❬❁❬ ❬➭æ | ❲➜➛✿❘✲★ ➒❼ê ✬✽❘ ✑➲✛ ❲➜✬■❲✻✩✜ì ➟➺❢❂❥④①✇❜✉♠✷❥❋❜ ❵➐❥✓➟⑤❜✉❢❂t✇① • ❵➐❥❛❦r❜✏❜➝q➐❞➞❤ ➅▲❬❍❽ ❫❛❦♣❥➐♠✷t●q ❞❛t✇❢❝➦✿❵❇➟⑤❜➈♠✷❥❇➦ ①✖❵❼s →í❬⑤➎ × , ➠❝❤➆t✉❤ i ∈ m ➣Þ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥ ❬ m m i ❪ X ⑥ ∼ m X i i ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ F � G � n ♠✷❥❇➦ G i ∈ m ➣↕➟➺❢❂s ❬ n ❬ m ❬ n ➫ F ❪ X ⑥ ∼ ❪ G i ❪ X F � G � n F i ∈ m ❦♣❥❇➦✿❵❇➟⑤❜❧❦ ❤ µ m F ❬ m ❬ m ➫ F • ❪ X ⑥ ∼ ❪ X, Y µ m F µY.F 4

➱ ➬ ➔ ☎ ❬ ➃ ➔ ø é ❾ é û ✺ ➫ ➫ ✄ ☎ ❬ ➫ ➇ â ➬ ✔ ✕ ★ ❲ ➃ ➫ ✄ ➫ Ø ❐ ❬ ☛ î ✑ ➒ ➫ é ❾ ⑦ ✛ ❘ ⑦ ↔✇➙ ➑✻✛ �➓☞ ❲➜➛✿❘✲★ ➣è♠➴t✇❤❉➘➐❥❝❤➆sà❤➆❥➪❜➾❢ ❜➝q➐❞➞❤❉➚■➟✹❢❂❥➐①✇❜➡t✇❵❇➟❉❜✉❢❂t❧① ❦r❜➸➠ ♠✯➦❛➦④❦r❜✐❦♣❢❂❥➐♠✷♦❃①✇❜➡t✇❵❇➟⑤❜❧❵❛t✉❤②ð✰❞❛t✇❢✲❞➲❤➆t✐❜✐❦♣❤②① Ú❋ï ❤❂âñ③➞â ❵➐❥❛➟❉❜✉❢❂t❧① ①✖➠➐♠✷❞➞❤✹♦rq ❵❼❥✓➟⑤❜✉❢❂t✇① ❾ . . . ❵❼❥✓➟⑤❜✉❢❂t✇① ❲❨★▼❘ • ❫❛❵✿❜➳➟➺❢❂❵❛♦➷➦ ❫➞❤ ♠✰① ①✇❜➡t✇❵❇➟⑤❜❧❵❛t✉❤②① Ø②òóÕ✹Ú✯ô➐Ø✹ô ➠❝❤✹❥ F ❬ m ➠❝❤➆t✉❤ ❬✡õ❃ö❂÷ F ∗ õ❃ö❂÷✣ù✜ú õ❃ö❂÷ �✂✁ ✆✞✝ ✟✡✠ ÿ ☞☛ ❼ü✓ý❛þ▼ÿ ✝✍✌ ü✓ý❛þ▼ÿ ✆✖�✂✁✑✗✙✘✚�✜✛✢✌✣✗✙✘✚�✂✁ ✄✍✗✙✘✚�✜✛ ✎✑✏✓✒ ÿ ✦✥✓✧★☎ ✤✙✤✙✤ ❫❛❵✿❜ ❦♣❥ ❤✡t❧①❉❤ ô④Ø✹Õ✹Ú✯ô➼Ô❱ò❛Ù ♠✰①✖Ï➪❤✹♦♣♦✘ð ✪✩ ➓❢ ❤➆t✏❜➝q➐❞➞❤✈➟➺♦♥♠✰①➡①⑤❤②① ❲❨★▼❘ • ❫❛❵✿❜✈➟✹❢❂❵④♦➷➦ ❫➞❤ ♠✰①❮❤✹♦♣❤✹s➥❤✹❥❋❜✇①✈❢ ♠➥❜➸q➐❞➲❤↕➟➺♦♥♠✰①➡①➥❪ ①⑤♠✷s➥❤ Ø❁òóÕ➆Ú✷ô④Ø➺ô ❦➷➦❝❤✲♠í♠✰①➾♠❁❫❃❢ ❤②⑥ãâ ✫✭✬✯✮✱✰✳✲✵✴ ✶✵✷✹✸ ✻✽✼✿✾❁❀❃❂❅❄❇❆❉❈❊✻❋❄❇❆❍● ■✡✾ ✲❑❏▼▲❖◆✍P✓✲✵◗❘◆ ❙❁■❚❂❯■❚❂✜❱❁❲❨❳ ❄✡✻❩❂✂❱❁❲❨❳ ❀❬❄❇✾❭❈❪❂❃❆❍✼✑❀❃❂❅❄❇❆❉❈❴❫ 5

map rev L X l X - PDF document

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