switchのexhaustiveness/redundancy チェック 理論と実装 わいわいswiftc #8 @ukitaka

swiftcがいかにしてswitchの網羅性チェックをしているかを、背景にあるSpaceという概念を元にしたパターンマッチの網羅性チェックの理論とその実装について、lib/Sema/TypeCheckSwitchStmt.cppとソースコード中で紹介されている論文を元に紹介します。

Transcript

1. Θ͍Θ͍swiftc #8 @ukitaka
   switchͷexhaustiveness/redundancy νΣοΫ
   ཧ࿦ͱ࣮૷
   

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4. switchͷ໢ཏੑ/৑௕ੑνΣοΫ
   ࠓ೔ͷ࿩
   ▸ swiftc͕͍͔ʹͯ͠switchͷ໢ཏੑνΣοΫΛ͍ͯ͠Δ͔Λɺഎܠʹ͋ΔSpaceͱ
   ͍͏֓೦Λݩʹͨ͠ύλʔϯϚονͷ໢ཏੑνΣοΫͷཧ࿦ͱͦͷ࣮૷ʹ͍ͭ
   ͯɺlib/Sema/TypeCheckSwitchStmt.cppͱιʔείʔυதͰ঺հ͞Ε͍ͯΔ࿦
   จΛݩʹ঺հͯ͠Έ͍ͨͱࢥ͍·͢ɻ
   

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5. switchͷ໢ཏੑ/৑௕ੑνΣοΫ
   ࠓ೔ͷ࿩
   

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6. switchΛ஌Δ
   

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7. switch
   ໢ཏੑνΣοΫ
   

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8. switch
   ଍Γͳ͍caseͷ௥Ճ
   

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9. switch
   ΋͏গ͠ෳࡶͳέʔεͰ΋Մ
   

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10. switch
    ଍Γͳ͍caseͷ௥Ճ
    

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11. switch
    Unreachableͳέʔε͸লུՄೳ
    

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12. switch
    where۟Λ࢖͏ͱ࣮ࡍʹ͸໢ཏ͞Ε͍ͯͯ΋Τϥʔʹ
    

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13. switch
    IntͳͲ͸໢ཏ͞Ε͍ͯͯ΋Τϥʔʹ
    

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14. switch
    ৑௕ͳέʔε͕͋ΔͱwarningʹͳΔ
    

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15. switch
    SwitchͷεϖοΫ
    ▸ έʔε͕໢ཏ͞Ε͍ͯΔ͜ͱΛݕࠪͰ͖ɺ໢ཏ͞Ε͍ͯͳ͍৔߹͸Τϥʔʹ͢Δ
    ▸ ͦͷࡍ(ิ׬͢ΔͨΊʹ)Ͳͷέʔε͕଍Γͳ͍͔Λ೺ѲͰ͖͍ͯΔඞཁ͕͋Δ
    ▸ ॏෳ͢Δέʔε΋ݕग़Ͱ͖Δඞཁ͕͋Δ
    → ͜ΕΛ࣮ݱ͍ͯ͠Δͷ͕Space (Engine)ͱݺ͹ΕΔ࢓૊Έ
    

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16. Space EngineΛ༻
    ͍ͨ໢ཏੑνΣοΫ
    

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17. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    TypeCheckSwitchStmt.cpp
    

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18. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    TypeCheckSwitchStmt.cpp
    

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19. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    TypeCheckSwitchStmt.cpp
    

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20. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    TypeCheckSwitchStmt.cpp
    

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21. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    ࢀߟจݙ
    ▸ Maranget, Luc. "Warnings for pattern matching." Journal of Functional
    Programming 17.3 (2007): 387-421.
    ▸ ͜Ε͕ԼͷPaperͷϕʔεͱͳͬͨ࿦จ
    ▸ (GenericͰͳ͍) ୅਺తσʔλܕΛѻ͑ΔܕγεςϜʹ͓͚Δ໢ཏੑνΣοΫͷΞϧΰϦζϜ
    ▸ A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala
    16’, 2017
    ▸ ͕ͬͪ͜SpaceΛ࢖ͬͨ໢ཏੑݕࠪʹؔ͢Δ࿦จɺ͜ΕΛԠ༻ͯ͠৑௕ੑݕࠪ΋ߦ͏
    ▸ ಛఆͷܕγεςϜʹґଘ͠ͳ͍ΞϧΰϦζϜ ( GADTs΋ѻ͑Δ )
    

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22. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    A Generic Algorithm for Checking Exhaustivity of Pattern Matching
    ▸ ݩʑ͸Dottyͱ͍͏ScalaͷίϯύΠϥ (Scala 3.0~)Ͱ࠾༻͞Ε͍ͯΔ΋ͷΛSwift
    ʹҠ২͖ͯͨ͠΋ͷɻ
    https://fengy.me/
    

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23. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    A Generic Algorithm for Checking Exhaustivity of Pattern Matching
    ▸ ໢ཏੑνΣοΫͷΞϧΰϦζϜΛಛఆͷܕγεςϜ͔Β੾Γ཭͠ɺͲͷΑ͏ͳݴ
    ޠ(ͷܕγεςϜ)Ͱ΋࢖͑ΔΑ͏ͳҰൠతͳΞϧΰϦζϜΛఏڙ͢Δ
    ▸ ྫͱͯ͠Scala (Dotty) ͷܕγεςϜ΁ͷԠ༻Λ͍ࣔͯ͠Δ
    ▸ ϑΥʔϚϧͳূ໌͕͋ΔΘ͚Ͱ͸ͳ͍͚Ͳɺ͋Δఔ౓ෳࡶͳܕγεςϜΛ΋ͭ
    ScalaͰͷ࠾༻࣮੷͕͋Δ
    ▸ ͪͳΈʹFengyun Liuࢯͷࢦಋڭһ͸Oderskyઌੜ
    

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24. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    ͦ΋ͦ΋SpaceҎલͷ࣮૷
    ▸ ݕࠪ͢Δର৅ͷܕ (Subject type) ͔Β֤έʔεΛ
    औΓআ͍ͯߦͬͯ࢒Γ͕ͳ͍͔ݟΔɺΈ͍ͨͳ࣮
    ૷
    ▸ ࣮͸SpaceΛ༻͍࣮ͨ૷ʹ͍ۙ͜ͱΛ͍ͯ͠Δ
    ͕ɺ͍͔ͭ͘໰୊͕͋ͬͨͨΊScala (Dotty)Ͱͷ
    ࣮੷ͷ͋ΔํࣜΛվΊͯ࠾༻
    

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25. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    ݁Ռ
    

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26. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    (ࢀߟ) SR-483
    

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27. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ
    (ࢀߟ) SR-2293
    

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28. SpaceΛ༻͍ͨ
    “໢ཏ”ͷఆࣜԽ
    

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29. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    Space Engineͷ࣮૷ΛಡΉʹ͋ͨͬͯ
    ▸ SwiftͰͷ໢ཏੑνΣοΫͷ࣮૷Λཧղ͢ΔͨΊʹΞϧΰϦζϜΛ஌Δඞཁ͕͋Δ
    ▸ ΞϧΰϦζϜΛཧղ͢ΔͨΊʹ͸ Spaceʹؔ͢Δ༻ޠ΍ͦͷఆٛΛཧղ͓ͯ͘͠
    ඞཁ͕͋Δ
    ▸ SpaceʹΑͬͯఆࣜԽ͞Εͨ “໢ཏ͍ͯ͠Δ” ͜ͱΛද͍ͯ͠ΔࣜΛཧղ͢Δͷ͕
    ࠷ॳͷ໨ඪ
    (T) ⪯ (p1
    ) ∣ (p2
    ) ∣ . . .
    ↑ ͜Ε͕໢ཏ͍ͯ͠Δ͜ͱΛදࣜ͢
    

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30. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    ఆٛ (Space)
    ▸ ܕ΍ύλʔϯ͕औΓ͏Δ஋ͷू߹ΛSpaceͱ͍͏
    ▸ Space͸࠶ؼతʹҎԼͷΑ͏ʹఆٛ͞ΕΔ
    1. ΛۭSpace (empty space)
    2. ΛܕTͷSpace (type space)
    3. ͕Spaceͷͱ͖ɺ Λ union space
    4. ͕SpaceͰɺK͕Constructor Typeͷͱ͖
    (T)
    
    s1
    , s2
    , . . . s1
    |s2
    | . . .
    s1
    , s2
    , . . . , sn
    (K, s1
    , s2
    , . . . , sn
    ) Λconstructor space
    

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31. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SwiftͰͷΠϝʔδ:
    (Bool) = {true, false}
    (Never) =
    

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32. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SwiftͰͷΠϝʔδ:
    (Bool)|(Never) = {true, false}
    

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33. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SwiftͰͷΠϝʔδ:
    (Optional . some, (Bool)) = {.some(true), . some(false)}
    Optional.someͷྫ
    Type constructor
    

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34. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    ·ͩఆ͍ٛͯ͠ͳ͍͕ɺ㲈 ू߹ͷแؚؔ܎
    

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35. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    

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36. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    

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37. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    (Bool?) = {.some(true), . some(false), . none}
    

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38. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    (Optional . none) = {.none}
    

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39. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    s1
    |s2
    |s3
    s1
    s2
    s3
    

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40. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    (Bool?) ⪯ s1
    |s2
    |s3
    

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41. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    ΑΓ௚؍తʹ͸
    (Bool?) ⪯ s1
    |s2
    |s3
    ֤case͕Χόʔ͍ͯ͠Δൣғͷ΄͏͕
    େ͖͍ΑɺΈ͍ͨͳΠϝʔδ
    “େ͖͍” Λఆٛ͢Δํ๏ → Subspaceؔ܎
    

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42. Subspaceؔ܎
    

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43. Subspaceؔ܎
    ఆٛ (Subspaceؔ܎)
    ▸ ΛSpace্ͷೋ߲ؔ܎ Subspaceؔ܎ ͱ͢Δ
    ⪯
    s1
    ⪯ s2
    ▸ ௚؍తʹ͸ू߹ͷแؚؔ܎͕ͩɺཁૉʹ͸௚઀ݴٴͤͣʹ ܕ΍ܕߏஙࢠ (case)
    ͷؔ܎Λ࢖ͬͯఆ͍ٛͯ͘͠ɻৄࡉͳϧʔϧ͸ޙड़ɻ
    

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44. Subspaceؔ܎
    ఆٛ (projection)
    ▸ Λύλʔϯͱ͢Δɻ͜ͷͱ͖ ͕Χόʔ͢ΔSpaceΛ ͱද͢
    ▸ ύλʔϯ = case ͱࢥͬͯࠩ͠ࢧ͑ͳ͍ (ਖ਼֬Ͱ͸ͳ͍)
    p (p)
    p
    ͬͪ͜͸type constructor
    ͬͪ͜͸pattern
    

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45. Subspaceؔ܎
    SwiftͰͷΠϝʔδ:
    (.some(true)) = {.some(true)}
    (.some(_)) = {.some(true), some(false)}
    

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46. Subspaceؔ܎
    ໢ཏͷఆࣜԽ (࠶ܝ)
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    (T)
    

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47. Subspaceؔ܎
    ໢ཏͷఆࣜԽ (࠶ܝ)
    ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace
    ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔
    ▸ ύλʔϯ ͕ ܕ TΛ໢ཏ͍ͯ͠Δͱ͸ҎԼΛຬͨ͢͜ͱͰ͋Δɻ
    (T)
    p1
    , p2
    , . . .
    (T) ⪯ (p1
    ) ∣ (p2
    ) ∣ . . .
    ͜͜·ͰͰ”໢ཏ”ͷ͕ࣜಡΊͨʂ(͸ͣʂ)
    

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48. Subspaceؔ܎
    SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    ▸ ໢ཏ͍ͯ͠Δ͜ͱΛSubspaceؔ܎Λ࢖ͬͯҎԼͷΑ͏ʹදͤͨ
    (T) ⪯ (p1
    ) ∣ (p2
    ) ∣ . . .
    ▸ ࣍͸͜ͷSubspaceؔ܎ͷ۩ମతͳϧʔϧΛ·ͨผͷ֓೦Λ࢖ͬͯఆ͍ٛͯ͘͠…. (ΊΜͲ͍)
    s1
    ⪯ s2
    ⇔ s1
    ⊖ s2
    ≐
    ΞϧΰϦζϜΛཧղ͢Δͷʹඞཁͳఆٛ͸
    ͜ΕͰ࠷ޙͳͷͰ͕Μ͹Ζ͏…
    

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49. ⊖Λ࢖ͬͨSubspace
    ؔ܎ͷఆٛ
    

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50. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    ▸ Subspaceؔ܎Λ௚઀తʹఆ͍ٛͯ͘͜͠ͱΛͤͣʹ৽ͨʹ㱿 (Subtraction) Λಋ
    ೖ͢Δ͜ͱͰSubspaceؔ܎Λఆٛ͢Δ
    ▸ ௚઀ఆٛ͠Α͏ͱ͕ͨ͠ɺ“We tried but failed.” Β͍͠ɻ
    ▸ ௚؍తʹ͸ू߹ͷsubtractionͱಉ͕ͩ͡ (ry
    

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51. ▸ Subspaceؔ܎ΛSubtractionΛ༻͍ͯఆٛ͢Δ
    ▸ Space ʹ͍ͭͯɺ ͕ ҎԼΛຬͨ࣌͢(͔ͭͦͷ࣌ͷΈ) ͕ ͷ
    SubspaceͰ͋Δͱ͍͏
    ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    ఆٛ (Subspaceؔ܎)
    s1
    , s2
    s1
    s2
    s1
    s2
    s1
    ⪯ s2
    ⇔ s1
    ⊖ s2
    ≐
    

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52. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    Subtractionͷϧʔϧ
    A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
    

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53. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    Subtractionͷϧʔϧ
    A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
    empty spaceʹؔ͢Δϧʔϧ
    

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54. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    Subtractionͷϧʔϧ
    A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
    subtypeʹؔ͢Δϧʔϧ
    

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55. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ
    (T) ⪯ (p1
    ) ∣ (p2
    ) ∣ . . .
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    “໢ཏ͍ͯ͠Δ”
    ͜͜·ͰΘ͔Ε͹ϝΠϯͷϩδοΫ͸ಡΊΔʂ
    

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56. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    

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57. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    

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58. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    

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59. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    

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60. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    

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61. Decompose
    

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62. Decompose
    Decompose
    ▸ enum E { case a, b } ͕͋ͬͨͱͯ͠ ͷܭࢉΛͲ͏ਐΊΔ
    ͔ʁ
    ▸ ͜ͷΑ͏ͳܭࢉΛਐΊΔͨΊʹܕEʹରͯ͠ Decompose (෼ղ) ͱ͍͏ૢ࡞Λఆ
    ٛ͢Δඞཁ͕͋Δ
    (E) ⊖ (a) ∣ (b)
    

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63. Decompose
    Decompose
    ▸ ۩ମతʹͲ͏ఆٛ͞ΕΔ͔͸ͦͷϓϩάϥϛϯάݴޠʹΑΔ͕ɺSwiftʹ͓͍ͯ͸
    enumΛ֤caseͷprojectionͷunion spaceʹ෼ղ͢Δ͜ͱ
    ▸ (E) = (a) ∣ (b) = (a) ∣ (b)
    

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64. Decompose
    Decompose·ͰΘ͔Ε͹͍͍͍͚ͩͨΔ
    (E) ⊖ (a) ∣ (b) ≐
    (E) ⪯ (a) ∣ (b)
    ໢ཏ͍ͯ͠Δͱ͸
    subtractionΛ࢖ͬͯ
    

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65. Decompose
    Decompose·ͰΘ͔Ε͹͍͍͍͚ͩͨΔ
    (E) ⊖ (a) ∣ (b)
    DecomposeΛ࢖ͬͯ
    (E) = (a) ∣ (b)
    (a) ∣ (b) ⊖ (a) ∣ (b)
    ͋ͱ͸Α͠ͳʹܭࢉΛਐΊΕ͹
    (a) ∣ (b) ⊖ (a) ∣ (b) ≐
    

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66. Swiftͷ࣮૷ΛಡΉ
    

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67. Swiftͷ࣮૷ΛಡΉ
    Swiftͷ࣮૷ΛಡΉ
    ▸ ͓ർΕ༷Ͱͨ͠ɻ͜͜·ͰΘ͔Ε͹ίʔυΛಡΉͷ͸೉͘͠ͳ͍Ͱ͢ɻ
    ▸ ͜͜·Ͱղઆͨ͜͠ͱ͕΄΅ͦͷ··࣮૷͞Ε͍ͯΔ
    

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68. Swiftͷ࣮૷ΛಡΉ
    Swiftʹ͓͚ΔSpace
    Bool͸ಛผѻ͍ɻ
    ↓͜ΜͳΠϝʔδ
    

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69. Swiftͷ࣮૷ΛಡΉ
    ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ
    

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70. Swiftͷ࣮૷ΛಡΉ
    ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ
    ͢΂ͯͷCase(Pattern)ʹ͍ͭͯ
    

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71. Swiftͷ࣮૷ΛಡΉ
    ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ
    ProjectionΛٻΊͯ
    

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72. Swiftͷ࣮૷ΛಡΉ
    ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ
    spacesʹ٧ΊΔ
    

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73. Swiftͷ࣮૷ΛಡΉ
    ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ
    (T) ⪯ (p1
    ) ∣ (p2
    ) ∣ . . .
    

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74. Swiftͷ࣮૷ΛಡΉ
    ໢ཏੑνΣοΫ
    (T) ⊖ (p1
    ) ∣ (p2
    ) ∣ . . . ≐
    

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75. ▸ ͜͜·ͰͷղઆͲ͓ΓͳΒ ʹΑͬͯఆٛ͞Ε͍ͯΔ͸͕ͣͩɺEmpty sapce
    ޲͚ʹOptimze͞Ε͍ͯΔͨΊʹཧ࿦௨ΓͰ͸ͳ͍
    Swiftͷ࣮૷ΛಡΉ
    Subspaceͷఆٛ
    ⊖
    

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76. Swiftͷ࣮૷ΛಡΉ
    Subspaceͷఆٛ
    

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77. Swiftͷ࣮૷ΛಡΉ
    Subtractionͷఆٛ
    

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78. Swiftͷ࣮૷ΛಡΉ
    Subtractionͷఆٛ
    

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79. ৑௕ੑνΣοΫ
    

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80. ৑௕ੑνΣοΫ
    ৑௕ੑ(Redundancy)νΣοΫ
    ▸ ৑௕ͳέʔεͷνΣοΫ΋Spaceͷ࢓૊Έʹ৐͔͍ͬͬͯΔ
    ▸ ݩͷPaperʹ͸ॻ͔Ε͍ͯͳ͍͕ɺΞϧΰϦζϜ͸ίʔυ͔ΒಡΈऔΕΔ
    

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81. ৑௕ੑνΣοΫ
    ৑௕ੑνΣοΫͷΞϧΰϦζϜ
    ▸ ໢ཏੑνΣοΫͷաఔͰ֤έʔεͷunion spaceΛܭࢉ͍ͯ͘͠
    (T) ⪯ (p1
    ) ∣ (p2
    ) ∣ . . .
    ͜ͷ෦෼
    ▸ ܭࢉͷաఔͰɺ৽ͨʹ௥Ճ͢Δέʔε͕ͦ͜·Ͱͷunion spaceͷsubspaceʹ
    ͳ͍ͬͯͳ͍͔ΛνΣοΫ͢Δ
    (p1
    ) ∣ (p2
    )| . . . |(pn
    ) ⪯ (pn
    + 1)
    

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82. ৑௕ੑνΣοΫ
    ৑௕ੑνΣοΫͷΞϧΰϦζϜ
    

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83. ৑௕ੑνΣοΫ
    ৑௕ੑνΣοΫͷΞϧΰϦζϜ
    (a) ∣ (b) ⪯ (c)
    (a) ∣ (b) ∣ (c) ⪯ (a)
    (a) ⪯ (b)
    ੒ཱ͠ͳ͍ͷͰOK
    ੒ཱ͢ΔͷͰ৑௕
    

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84. ৑௕ੑνΣοΫ
    ࣮૷
    

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85. where۟
    

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86. Where۟
    Where۟
    ▸ ܕ΍ܕߏஙࢠಉ࢜ͷؔ܎ʹج͍ͮͯSubspace(Subtraction)͕ఆٛ͞ΕΔ
    ▸ ஋ʹ௚઀ݴٴ͢ΔΑ͏ͳ΋ͷ͸͜ͷΞϧΰϦζϜͰ͸ѻ͑ͳ͍
    ▸ 3.4 Limitations ʹ “Guards in pattern clauses pose a theoretical difficulty, thus
    are not handled by the algorithm.” ͱॻ͔Ε͍ͯΔ௨Γ
    

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87. Where۟
    Where۟
    

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88. Int
    

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89. Int
    IntͷΑ͏ʹҰҙʹDecompose͕ఆٛͰ͖ͳ͍ܕͷ৔߹
    ▸ Intͷ৔߹Decompose͕ҰҙʹఆٛͰ͖ͳ͍ → ѻ͑ͳ͍
    ▸ ٯʹBool͸Ұҙʹ෼ղ͕ఆٛͰ͖Δ͕Ώ͑ʹಛผѻ͍Ͱ͖Δ
    

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90. ·ͱΊ
    

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91. switchͷ໢ཏੑ/৑௕ੑνΣοΫ
    ·ͱΊ
    ▸ swiftc͕͍͔ʹͯ͠switchͷ໢ཏੑνΣοΫΛ͍ͯ͠Δ͔Λɺഎܠʹ͋ΔSpaceͱ
    ͍͏֓೦Λݩʹͨ͠ύλʔϯϚονͷ໢ཏੑνΣοΫͷཧ࿦ͱͦͷ࣮૷ʹ͍ͭ
    ͯɺlib/Sema/TypeCheckSwitchStmt.cppͱιʔείʔυதͰ঺հ͞Ε͍ͯΔ࿦
    จΛݩʹ঺հͨ͠ɻ
    

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92. Appendix:
    Intersection
    

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93. Intersection
    Intersectionͷఆٛ
    A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
    

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94. Intersection
    Intersectionͷఆٛ
    A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
    

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