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。

1. Θ͍Θ͍swiftc #8 @ukitaka switchͷexhaustiveness/redundancy νΣοΫ ཧ࿦ͱ࣮૷
4. switchͷ໢ཏੑ/৑௕ੑνΣοΫ ࠓ೔ͷ࿩ ▸ swiftc͕͍͔ʹͯ͠switchͷ໢ཏੑνΣοΫΛ͍ͯ͠Δ͔Λɺഎܠʹ͋ΔSpaceͱ ͍͏֓೦Λݩʹͨ͠ύλʔϯϚονͷ໢ཏੑνΣοΫͷཧ࿦ͱͦͷ࣮૷ʹ͍ͭ ͯɺlib/Sema/TypeCheckSwitchStmt.cppͱιʔείʔυதͰ঺հ͞Ε͍ͯΔ࿦ จΛݩʹ঺հͯ͠Έ͍ͨͱࢥ͍·͢ɻ
5. switchͷ໢ཏੑ/৑௕ੑνΣοΫ ࠓ೔ͷ࿩
6. switchΛ஌Δ
7. switch ໢ཏੑνΣοΫ
8. switch ଍Γͳ͍caseͷ௥Ճ
9. switch ΋͏গ͠ෳࡶͳέʔεͰ΋Մ
10. switch ଍Γͳ͍caseͷ௥Ճ
11. switch Unreachableͳέʔε͸লུՄೳ
12. switch where۟Λ࢖͏ͱ࣮ࡍʹ͸໢ཏ͞Ε͍ͯͯ΋Τϥʔʹ
13. switch IntͳͲ͸໢ཏ͞Ε͍ͯͯ΋Τϥʔʹ
14. switch ৑௕ͳέʔε͕͋ΔͱwarningʹͳΔ
15. switch SwitchͷεϖοΫ ▸ έʔε͕໢ཏ͞Ε͍ͯΔ͜ͱΛݕࠪͰ͖ɺ໢ཏ͞Ε͍ͯͳ͍৔߹͸Τϥʔʹ͢Δ ▸ ͦͷࡍ(ิ׬͢ΔͨΊʹ)Ͳͷέʔε͕଍Γͳ͍͔Λ೺ѲͰ͖͍ͯΔඞཁ͕͋Δ ▸ ॏෳ͢Δέʔε΋ݕग़Ͱ͖Δඞཁ͕͋Δ → ͜ΕΛ࣮ݱ͍ͯ͠Δͷ͕Space (Engine)ͱݺ͹ΕΔ࢓૊Έ
16. Space EngineΛ༻ ͍ͨ໢ཏੑνΣοΫ
17. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ TypeCheckSwitchStmt.cpp
18. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ TypeCheckSwitchStmt.cpp
19. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ TypeCheckSwitchStmt.cpp
20. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ TypeCheckSwitchStmt.cpp
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 )
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/
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
24. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ ͦ΋ͦ΋SpaceҎલͷ࣮૷ ▸ ݕࠪ͢Δର৅ͷܕ (Subject type) ͔Β֤έʔεΛ औΓআ͍ͯߦͬͯ࢒Γ͕ͳ͍͔ݟΔɺΈ͍ͨͳ࣮ ૷ ▸ ࣮͸SpaceΛ༻͍࣮ͨ૷ʹ͍ۙ͜ͱΛ͍ͯ͠Δ ͕ɺ͍͔ͭ͘໰୊͕͋ͬͨͨΊScala (Dotty)Ͱͷ ࣮੷ͷ͋ΔํࣜΛվΊͯ࠾༻
25. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ ݁Ռ
26. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ (ࢀߟ) SR-483
27. Space EngineΛ༻͍ͨ໢ཏੑνΣοΫ (ࢀߟ) SR-2293
28. SpaceΛ༻͍ͨ “໢ཏ”ͷఆࣜԽ
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
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
31. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SwiftͰͷΠϝʔδ: (Bool) = {true, false} (Never) =
32. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SwiftͰͷΠϝʔδ: (Bool)|(Never) = {true, false}
33. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SwiftͰͷΠϝʔδ: (Optional . some, (Bool)) = {.some(true), . some(false)} Optional<Bool>.someͷྫ Type constructor
34. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T) ·ͩఆ͍ٛͯ͠ͳ͍͕ɺ㲈 ू߹ͷแؚؔ܎
35. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T)
36. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T)
37. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T) (Bool?) = {.some(true), . some(false), . none}
38. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T) (Optional . none) = {.none}
39. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T) s1 |s2 |s3 s1 s2 s3
40. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ࢖ͬͯ”໢ཏੑνΣοΫ”Λݴ͍׵͑Δ ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T) (Bool?) ⪯ s1 |s2 |s3
41. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ ΑΓ௚؍తʹ͸ (Bool?) ⪯ s1 |s2 |s3 ֤case͕Χόʔ͍ͯ͠Δൣғͷ΄͏͕ େ͖͍ΑɺΈ͍ͨͳΠϝʔδ “େ͖͍” Λఆٛ͢Δํ๏ → Subspaceؔ܎
42. Subspaceؔ܎
43. Subspaceؔ܎ ఆٛ (Subspaceؔ܎) ▸ ΛSpace্ͷೋ߲ؔ܎ Subspaceؔ܎ ͱ͢Δ ⪯ s1 ⪯ s2 ▸ ௚؍తʹ͸ू߹ͷแؚؔ܎͕ͩɺཁૉʹ͸௚઀ݴٴͤͣʹ ܕ΍ܕߏஙࢠ (case) ͷؔ܎Λ࢖ͬͯఆ͍ٛͯ͘͠ɻৄࡉͳϧʔϧ͸ޙड़ɻ
44. Subspaceؔ܎ ఆٛ (projection) ▸ Λύλʔϯͱ͢Δɻ͜ͷͱ͖ ͕Χόʔ͢ΔSpaceΛ ͱද͢ ▸ ύλʔϯ = case ͱࢥͬͯࠩ͠ࢧ͑ͳ͍ (ਖ਼֬Ͱ͸ͳ͍) p (p) p ͬͪ͜͸type constructor ͬͪ͜͸pattern
45. Subspaceؔ܎ SwiftͰͷΠϝʔδ: (.some(true)) = {.some(true)} (.some(_)) = {.some(true), some(false)}
46. Subspaceؔ܎ ໢ཏͷఆࣜԽ (࠶ܝ) ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ (T)
47. Subspaceؔ܎ ໢ཏͷఆࣜԽ (࠶ܝ) ▸ Ϛονͤ͞Δର৅ͷܕTʹ͍ͭͯɺ ͕͢΂ͯͷcase͕Χόʔ͍ͯ͠Δspace ͷunion spaceͷsubspaceʹͳ͍ͬͯΔ͔ ▸ ύλʔϯ ͕ ܕ TΛ໢ཏ͍ͯ͠Δͱ͸ҎԼΛຬͨ͢͜ͱͰ͋Δɻ (T) p1 , p2 , . . . (T) ⪯ (p1 ) ∣ (p2 ) ∣ . . . ͜͜·ͰͰ”໢ཏ”ͷ͕ࣜಡΊͨʂ(͸ͣʂ)
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…
49. ⊖Λ࢖ͬͨSubspace ؔ܎ͷఆٛ
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
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 ≐
52. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ Subtractionͷϧʔϧ A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
53. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ Subtractionͷϧʔϧ A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻ empty spaceʹؔ͢Δϧʔϧ
54. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ Subtractionͷϧʔϧ A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻ subtypeʹؔ͢Δϧʔϧ
55. SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ SpaceΛ༻͍ͨ”໢ཏ”ͷఆࣜԽ (T) ⪯ (p1 ) ∣ (p2 ) ∣ . . . (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐ “໢ཏ͍ͯ͠Δ” ͜͜·ͰΘ͔Ε͹ϝΠϯͷϩδοΫ͸ಡΊΔʂ
56. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐
57. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐
58. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐
59. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐
60. ⊖Λ࢖ͬͨSubspaceؔ܎ͷఆٛ (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐
61. Decompose
62. Decompose Decompose ▸ enum E { case a, b } ͕͋ͬͨͱͯ͠ ͷܭࢉΛͲ͏ਐΊΔ ͔ʁ ▸ ͜ͷΑ͏ͳܭࢉΛਐΊΔͨΊʹܕEʹରͯ͠ Decompose (෼ղ) ͱ͍͏ૢ࡞Λఆ ٛ͢Δඞཁ͕͋Δ (E) ⊖ (a) ∣ (b)
63. Decompose Decompose ▸ ۩ମతʹͲ͏ఆٛ͞ΕΔ͔͸ͦͷϓϩάϥϛϯάݴޠʹΑΔ͕ɺSwiftʹ͓͍ͯ͸ enumΛ֤caseͷprojectionͷunion spaceʹ෼ղ͢Δ͜ͱ ▸ (E) = (a) ∣ (b) = (a) ∣ (b)
64. Decompose Decompose·ͰΘ͔Ε͹͍͍͍͚ͩͨΔ (E) ⊖ (a) ∣ (b) ≐ (E) ⪯ (a) ∣ (b) ໢ཏ͍ͯ͠Δͱ͸ subtractionΛ࢖ͬͯ
65. Decompose Decompose·ͰΘ͔Ε͹͍͍͍͚ͩͨΔ (E) ⊖ (a) ∣ (b) DecomposeΛ࢖ͬͯ (E) = (a) ∣ (b) (a) ∣ (b) ⊖ (a) ∣ (b) ͋ͱ͸Α͠ͳʹܭࢉΛਐΊΕ͹ (a) ∣ (b) ⊖ (a) ∣ (b) ≐
66. Swiftͷ࣮૷ΛಡΉ
67. Swiftͷ࣮૷ΛಡΉ Swiftͷ࣮૷ΛಡΉ ▸ ͓ർΕ༷Ͱͨ͠ɻ͜͜·ͰΘ͔Ε͹ίʔυΛಡΉͷ͸೉͘͠ͳ͍Ͱ͢ɻ ▸ ͜͜·Ͱղઆͨ͜͠ͱ͕΄΅ͦͷ··࣮૷͞Ε͍ͯΔ
68. Swiftͷ࣮૷ΛಡΉ Swiftʹ͓͚ΔSpace Bool͸ಛผѻ͍ɻ ↓͜ΜͳΠϝʔδ
69. Swiftͷ࣮૷ΛಡΉ ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ
70. Swiftͷ࣮૷ΛಡΉ ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ ͢΂ͯͷCase(Pattern)ʹ͍ͭͯ
71. Swiftͷ࣮૷ΛಡΉ ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ ProjectionΛٻΊͯ
72. Swiftͷ࣮૷ΛಡΉ ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ spacesʹ٧ΊΔ
73. Swiftͷ࣮૷ΛಡΉ ໢ཏੑνΣοΫͷΤϯτϦϙΠϯτ (T) ⪯ (p1 ) ∣ (p2 ) ∣ . . .
74. Swiftͷ࣮૷ΛಡΉ ໢ཏੑνΣοΫ (T) ⊖ (p1 ) ∣ (p2 ) ∣ . . . ≐
75. ▸ ͜͜·ͰͷղઆͲ͓ΓͳΒ ʹΑͬͯఆٛ͞Ε͍ͯΔ͸͕ͣͩɺEmpty sapce ޲͚ʹOptimze͞Ε͍ͯΔͨΊʹཧ࿦௨ΓͰ͸ͳ͍ Swiftͷ࣮૷ΛಡΉ Subspaceͷఆٛ ⊖
76. Swiftͷ࣮૷ΛಡΉ Subspaceͷఆٛ
77. Swiftͷ࣮૷ΛಡΉ Subtractionͷఆٛ
78. Swiftͷ࣮૷ΛಡΉ Subtractionͷఆٛ
79. ৑௕ੑνΣοΫ
80. ৑௕ੑνΣοΫ ৑௕ੑ(Redundancy)νΣοΫ ▸ ৑௕ͳέʔεͷνΣοΫ΋Spaceͷ࢓૊Έʹ৐͔͍ͬͬͯΔ ▸ ݩͷPaperʹ͸ॻ͔Ε͍ͯͳ͍͕ɺΞϧΰϦζϜ͸ίʔυ͔ΒಡΈऔΕΔ
81. ৑௕ੑνΣοΫ ৑௕ੑνΣοΫͷΞϧΰϦζϜ ▸ ໢ཏੑνΣοΫͷաఔͰ֤έʔεͷunion spaceΛܭࢉ͍ͯ͘͠ (T) ⪯ (p1 ) ∣ (p2 ) ∣ . . . ͜ͷ෦෼ ▸ ܭࢉͷաఔͰɺ৽ͨʹ௥Ճ͢Δέʔε͕ͦ͜·Ͱͷunion spaceͷsubspaceʹ ͳ͍ͬͯͳ͍͔ΛνΣοΫ͢Δ (p1 ) ∣ (p2 )| . . . |(pn ) ⪯ (pn + 1)
82. ৑௕ੑνΣοΫ ৑௕ੑνΣοΫͷΞϧΰϦζϜ
83. ৑௕ੑνΣοΫ ৑௕ੑνΣοΫͷΞϧΰϦζϜ (a) ∣ (b) ⪯ (c) (a) ∣ (b) ∣ (c) ⪯ (a) (a) ⪯ (b) ੒ཱ͠ͳ͍ͷͰOK ੒ཱ͢ΔͷͰ৑௕
84. ৑௕ੑνΣοΫ ࣮૷
85. where۟
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
87. Where۟ Where۟
88. Int
89. Int IntͷΑ͏ʹҰҙʹDecompose͕ఆٛͰ͖ͳ͍ܕͷ৔߹ ▸ Intͷ৔߹Decompose͕ҰҙʹఆٛͰ͖ͳ͍ → ѻ͑ͳ͍ ▸ ٯʹBool͸Ұҙʹ෼ղ͕ఆٛͰ͖Δ͕Ώ͑ʹಛผѻ͍Ͱ͖Δ
90. ·ͱΊ
91. switchͷ໢ཏੑ/৑௕ੑνΣοΫ ·ͱΊ ▸ swiftc͕͍͔ʹͯ͠switchͷ໢ཏੑνΣοΫΛ͍ͯ͠Δ͔Λɺഎܠʹ͋ΔSpaceͱ ͍͏֓೦Λݩʹͨ͠ύλʔϯϚονͷ໢ཏੑνΣοΫͷཧ࿦ͱͦͷ࣮૷ʹ͍ͭ ͯɺlib/Sema/TypeCheckSwitchStmt.cppͱιʔείʔυதͰ঺հ͞Ε͍ͯΔ࿦ จΛݩʹ঺հͨ͠ɻ
92. Appendix: Intersection
93. Intersection Intersectionͷఆٛ A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻
94. Intersection Intersectionͷఆٛ A generic algorithm for checking exhaustivity of pattern matching, F. Liu, Scala 16’, 2017 ΑΓҾ༻