福井大学 情報・メディア学科 学士 研究発表

Transcript

1. 
   ࣗಈతɾର࿩తͳ ఆཧূ໌ثΛ༻͍ͨ ۠ؒԋࢉϥΠϒϥϦͷݕূ 
   ଔۀݚڀൃද ෱Ҫେֶ ޻ֶ෦ ৘ใɾϝσΟΞ޻ֶՊ ੴҪݚڀࣨ ൃදऀ
   ༅ஐਔ ֶ੶൪߸  1
2. എܠ ۠ؒԋࢉ ܭࢉػ্Ͱߴ৴པʹ਺஋ܭࢉΛ͢Δํ๏ ۠ؒԋࢉϥΠϒϥϦ #PPTU $
   LW #PPTUվྑ
   */5-"# ."5-"#  LWͷ࣮૷ྫ ۠ؒಉ࢜ͷֻ͚ࢉͷΞϧΰϦζϜͷൈਮ ∀ ", $ ∈ &(() " ≤ 0 → " > 0 → $ ≤ 0 → $ ≤ 0 → "×$ ≈ [1("×$), ∆("×$)] 2
3. എܠ ۠ؒԋࢉ ܭࢉػ্Ͱߴ৴པʹ਺஋ܭࢉΛ͢Δํ๏ ۠ؒԋࢉϥΠϒϥϦ #PPTU $
   LW #PPTUվྑ
   */5-"# ."5-"#  LWͷ࣮૷ྫ ۠ؒಉ࢜ͷֻ͚ࢉͷΞϧΰϦζϜͷൈਮ ∀ ", $ ∈ &(() " ≤ 0 → " > 0 → $ ≤ 0 → $ ≤ 0 → "×$ ≈ [1("×$), ∆("×$)] 3 όάͷࠞೖՄೳੑ ϗϯτʹߴ৴པʁ
4. Ϟνϕʔγϣϯ ˠ ҆৺ͯ۠ؒ͠ԋࢉϥΠϒϥϦΛ࢖͍͍ͨ ߴ৴པͳ਺஋ܭࢉ ໨త ۠ؒԋࢉϥΠϒϥϦͷ࣮૷ͷਖ਼͠͞ͷݕূ 4
5. ఏҊख๏ 8IZʹΑΓࣗಈత
   ର࿩తͳఆཧূ໌ث Λ૊Έ߹ΘͤͨϓϩάϥϜݕূ 5 ಛ௕ • ࢓༷෇͖ ϓϩάϥϜ͔Βݕূ৚݅ͷࣗಈੜ੒ • ॊೈͳূ໌ ·ͣࣗಈূ໌ ˠ ࢒ΓΛػցࢧԉ෇ͷର࿩ূ໌ • ূ໌ࡁΈϓϩάϥϜͷࣗಈΤϯίʔυ ˠ ূ໌ͨ͠ϓϩάϥϜΛͦͷ··࣮ߦͰ͖Δʂ
6. ۠ؒԋࢉ ܭࢉػ্Ͱ ߴ৴པʹ਺஋ܭࢉΛ͢Δํ๏ දهͱఆٛ
   ! = ! , ! ∶= % ∈ ℝ ! ≤ % ≤ ! } 6 •Լݶ ! ͱ ্ݶ ! ͸ුಈখ਺఺਺ Լݶ͸Լ޲͖ *
   ্ݶ͸্޲͖ ∆ ʹؙΊΔ ˠ ଘࡏ͢ΔͰ͋Ζ͏ਅͷղΛؚΈͳ͕Βܭࢉ •࣮਺ͷԋࢉಉ༷ʹ
   ࢛ଇԋࢉΛఆٛՄೳ Ճࢉͷఆٛͱ࣮૷ ! + - ≔ % + / % ∈ ! ∧ / ∈ -} ≈ [* ! + - , ∆( ! + - )] ࣮૷ͷܭࢉྫ
   2,4 + −3,5 = [2 − 3, 4 + 5] = [−1,9]
7. ఆཧূ໌ث ࿦ཧֶʹج͖ͮఆཧͷଥ౰ੑΛ൑ఆ ఆཧͷྫ ∀", $ ∈ ℕ, " + $ = $ + " ަ׵๏ଇ 7 4.5ιϧόʔ ࣗಈఆཧূ໌ث • ιϧόʔ͕ࣗಈతʹݕূ • ۙ೥ͷϚγϯεϖοΫͷ޲্ˠ ੑೳ͕ඈ༂తʹ޲্ • ໰୊
   ܭࢉ͕ݱ࣮త࣌ؒͰఀࢭ͠ͳ͍৔߹͕͋Δ ఆཧূ໌ࢧԉܥ $PR
   ର࿩ఆཧূ໌ث • ਓͷखʹΑΔର࿩ূ໌ • 4.5ιϧόʔͰূ໌Ͱ͖ͳ͍৔߹ʹ$PRΛ࢖͏
8. ॳظ৚݅ Λຬͨ͢ ϓϩάϥϜ Λ࣮ߦͯ͠ఀࢭ ࣄޙ৚݅ Λຬͨ͢ ԋ៷త ϓϩάϥϜݕূ 8 ࢓༷෇͖ϓϩάϥϜ͕ਖ਼͍͜͠ͱΛ ԋ៷తͳਪ࿦ )PBSF࿦ཧ ʹΑΓূ໌ ॳظ৚݅ɺࣄޙ৚݅
   ࿦ཧࣜ ϓϩάϥϜ
   ஞ࣍ݴޠͷจ ϓϩάϥϜ͕ਖ਼͍͠
9. 8IZ
   ϓϥοτϑΥʔϜ ࢓༷෇͖ ϓϩάϥϜ ݕূ৚݅ͷ ܭࢉ ূ໌ثʹΑΔ ଥ౰ੑ൑ఆ 8IZݴޠ 9 ԋ៷త ϓϩάϥϜݕূͷͨΊʹ։ൃ • ࣮਺΍ුಈখ਺఺਺ͳͲΛѻ͑Δඪ४ϥΠϒϥϦ • ෳ਺ͷఆཧূ໌ث 4.5ιϧόʔ
   $PR Λར༻Մೳ • (6*΋࢖͑Δʂ ϓϩάϥϜͷਖ਼͠͞ͷূ໌Λ ΄΅ࣗಈԽʂ
10. ྫ
    Ճࢉԋࢉͷݕূ ࣮૷ͷΈ 10 type interval = { inf: double; sup: double; } let add (X Y: interval) : interval = { inf = add_down X.inf Y.inf; sup = add_up X.sup Y.sup; } ! + # ≈ [&(! + #), ∆(! + Y)] ࣮૷
11. ྫ
    Ճࢉԋࢉͷݕূ ࢓༷෇͖ 11 type interval = { inf: double; sup: double; } invariant { inf ≤ sup } let add (X Y: interval) : interval = ensures { forall x y: real. (in x X ∧ in y Y) -> in (x + y) result } { inf = add_down X.inf Y.inf; sup = add_up X.sup Y.sup; } ᶃ ৗʹʮԼݶ ≤ ্ݶʯ ᶄʮ# ∈ % ∧ & ∈ ' → # + & ∈ % + 'ʯ
12. ྫ
    Ճࢉԋࢉͷݕূ ࢓༷෇͖ ೚ҙͷ۠ؒ 9
    : ʹର͠
    ࣮૷ͷܭࢉΛͨ͠ͱ͖ ݁Ռͷ۠ؒ͸ ࣄޙ৚݅ᶄ Λຬ͔ͨ͢ʁ ·ͨ
    ۠ؒ஋͸ৗʹ ৚݅ᶃ Λຬ͔ͨ͢ʁ 12 type interval = { inf: double; sup: double; } invariant { inf ≤ sup } let add (X Y: interval) : interval = ensures { forall x y: real. (in x X ∧ in y Y) -> in (x + y) result } { inf = add_down X.inf Y.inf; sup = add_up X.sup Y.sup; } ᶃ ৗʹʮԼݶ ≤ ্ݶʯ ᶄʮ# ∈ % ∧ & ∈ ' → # + & ∈ % + 'ʯ ࢓༷
13. Ճࢉԋࢉͷݕূ 4.5ιϧόʔ "MU&SHP Ͱূ໌Ͱ͖ͨ 13 4.5ιϧόʔʹΑΔՃࢉͷݕূ݁Ռ 8IZ (6*
14. ৐ࢉԋࢉͷݕূ 4.5ιϧόʔ Ͱ͸Ұ ෦ͷ৔߹Ͱ͔ࣗ͠ಈূ ໌Ͱ͖ͣ 14 4.5ιϧόʔʹΑΔ෼ׂޙͷݕ ূ৚݅ͷݕূ݁Ռ 8IZ (6*
15. ৐ࢉԋࢉͷݕূ 4.5ιϧόʔ Ͱ͸Ұ ෦ͷΈͰ͔ࣗ͠ಈূ໌ Ͱ͖ͣ ˠ $PR
    ʹΑΔର࿩ূ ໌ 15 8IZ
    ͕৐ࢉͷݕূϓϩάϥϜ͔Β ࣗಈܭࢉͨ͠ $PR
    ༻ͷݕূ৚݅ $PR
    *%&
16. ৐ࢉԋࢉͷݕূ 4.5ιϧόʔ Ͱ͸Ұ ෦ͷΈͰ͔ࣗ͠ಈূ໌ Ͱ͖ͣ ˠ $PR
    ʹΑΔର࿩ূ ໌ ˠ ӈΛূ໌͢Ε͹Α ͍ ্෦෼͸લఏͰԼ ͕ূ໌͢΂͖ఆཧ 16 ੔ཧޙͷ $PR
    ༻ͷݕূ৚݅ $PR
    *%&
17. ݕূ݁Ռ  ۠ؒಉ࢜ͷ࢛ଇԋࢉͷ࣮૷Λݕূͨ͠ 17 ݕূ݁Ռ • ୺఺Λ࣮਺ real ͱͨ۠ؒ͠ ˠ શͯূ໌ • ୺఺Λුಈখ਺఺਺ double ͱͨ۠ؒ͠ ˠ Ճࢉͱݮࢉ͸ূ໌ ৐ࢉ͸Ұ෦
    আࢉ͸ূ໌Ͱ͖ͣ
18. 18   )  !  $    + 2 2 0 0 − 2 2 0 0 × 22 13 9 0 ÷ 8 0 8 0 :
    "% $$#&'( ࢛ଇԋࢉͷݕূ݁Ռ real୺఺
19. 19   )  !  $    + 2 2 - 0 − 2 2 - 0 × 30 16 - 14 ÷ 12 0 - 12 ࢛ଇԋࢉͷݕূ݁Ռ double୺఺ :
    "% $$#&'( ؙΊ΍ಛघͳ஋ ແݶେͳͲ ʹΑΓ ݕূ৚͕͔݅ͳΓෳࡶʹͳΓɺର࿩ূ໌͸׬੒Ͱ͖ͳ͔ͬͨ
20. ۠ؒԋࢉͷݕূ  ۠ؒಉ࢜ͷ࢛ଇԋࢉͷ࣮૷Λ༻͍ͨ ϓϩάϥϜΛݕূΛͨ͠ 20 ݕূ݁Ռ • ۠ؒͱ੔਺ͷੵ real
    double ˠ ূ໌Ͱ͖ͨ • ྦྷ৐ real
    double
    ˠ ূ໌Ͱ͖ͨ • ฏํࠜ real
    ˠ ূ໌Ͱ͖ͨ • ฏํࠜ double) ˠ ূ໌Ͱ͖ͳ͔ͬͨ
21. ࢛ଇԋࢉͷ࣮૷Λ༻͍ͨ ϓϩάϥϜͷݕূ݁Ռ real୺఺ 21 !                 21 21 0 0 
    10 10 0 0   11 10 1 0
22. ࢛ଇԋࢉͷ࣮૷Λ༻͍ͨ ϓϩάϥϜͷݕূ݁Ռ double୺఺ 22 !                 21 21 - 0 
    10 10 - 0   14 10 - 4
23. ·ͱΊ ఆཧূ໌ثΛ༻͍ͨ۠ؒԋࢉϥΠϒϥϦͷ ݕূͷ֓ཁͱݕূ݁ՌΛࣔͨ͠ 23 ࠓճͷݕূʹΑΓূ໌Ͱ͖ͨ΋ͷ • ୺఺͕࣮਺ͷ۠ؒ • ۠ؒಉ࢜ͷ࢛ଇԋࢉ • ۠ؒಉ࢜ͷ࢛ଇԋࢉΛ༻͍ͨ਺஋ܭࢉϓϩάϥϜ • ୺఺͕ුಈখ਺఺਺۠ؒ • ۠ؒಉ࢜ͷՃࢉͱݮࢉ • ۠ؒಉ࢜ͷ৐ࢉͷ৔߹෼͚ͷҰ෦ • ۠ؒಉ࢜ͷ࢛ଇԋࢉΛ༻͍ͨ਺஋ܭࢉϓϩάϥϜ
24. ࠓޙͷ՝୊ •ුಈখ਺఺਺୺఺ͷ࢛ଇԋࢉͷূ໌ ݕূ৚݅ͷෳࡶԽ ˠ ର࿩ূ໌͸͔ͳΓ೉͍͠ ߟ͑ΒΕΔղܾҊ • ࣮਺୺఺Λิॿఆཧʹ༻͍ͨ4.5ιϧόʔʹΑΔূ໌ •࢒Γͷ۠ؒԋࢉϥΠϒϥϦͷ࣮૷ͷݕূ • ॳ౳ؔ਺ ࡾ֯ؔ਺ͳͲ • ුಈখ਺఺਺ ˠ จࣈྻ ͷม׵ •ݕূࡁΈϓϩάϥϜ ˠ 0$BNM ͷΤϯίʔυ • 8IZʹ͸ුಈখ਺఺਺༻ͷΤϯίʔυυϥΠόʔ͸·ͩͳ͍ 24
25. ࢀߟจݙ • കଜߊ޿ 4.5
    ιϧόɾ4.5
    ιϧόͷٕज़ͱԠ༻ ίϯϐϡʔλιϑτ΢ΣΞ
    
    
    QQ
     • 3BNPO
    &
    .PPSF
    FU
    BM Introduction to Interval Analysis 4*".
    1SFTT
     • :WFT
    #FSUPU BOE
    1JFSSF
    $BTUFSBO Interactive theorem proving and program develop-ment - Coq’art: The calculus of inductive constructions 4QSJOHFS7FSMBH
     25