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二人単貧民の定理を
Coqで証明する試み

Avatar for Katsuki Ohto Katsuki Ohto
August 31, 2020

 二人単貧民の定理を
Coqで証明する試み

単貧民をテーマに、Coqでゲームを扱うことについて検討する。

Avatar for Katsuki Ohto

Katsuki Ohto

August 31, 2020
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  1. ೋਓ୯ශຽͷఆࣜԽ ➤ ྫ ➤ 
 ઌख͕̏Λग़ͨ͠৔߹ɹɹɹɹɹ὎ {1} {2, 2, 4}

    3
 ޙख͕̐Λग़͢ɹɹɹɹɹɹɹɹ὎ {1} {2, 2} 4
 ઌख͸ग़ͤͳ͍ͷͰύεɹɹɹɹ὎ {1} {2, 2} 0
 ޙख͕̎Λग़͢ɹɹɹɹɹɹɹɹ὎ {1} {2} 2
 ઌख͸ύεɹɹɹɹɹɹɹɹɹɹ὎ {1} {2} 0
 ޙख͕̎Λग़ͯ͠ޙखউͪɹɹɹ὎ {1} {} 2 X = {1,3}, ¯ X = {2,2,4}, r = 0
  2. ೋਓ୯ශຽͷఆࣜԽ ➤ ྫ ➤ ઌख͕ Λग़͠
 ˠ ޙख͕ Λग़ͨ͠৔߹
 ˠ

    ઌख͕ Λग़ͯ͠ઌखউͪ ➤ ઌख͕ Λग़͠
 ˠ ޙख͕ Λग़ͨ͠৔߹
 ˠ ઌखύεɺޙख͕ Λग़͢
 ˠ ઌख͕ Λग़ͯ͠ઌखউͪ ➤ ὎Ҏ্ͷ݁Ռ͔Βɹઌखউͪɹͷہ໘ X = {1,3}, ¯ X = {2,2,4}, r = 0 1 2 3 1 4 2 3
  3. ೋਓ୯ශຽͷղ๏ ➤ ໦୩ɾখ໺ (2018) ➤ ҎԼͷΞϧΰϦζϜʹΑΓखࡳͷϚονϯά਺Λܭࢉ 1 ڧ͞ 3 2

    2 4 0 ख൪ଆ ඇख൪ଆ ࣗ෼ͷࡳ͔Βɺ
 ͦΕΑΓऑ͍૬खͷࡳʹ
 ϚονϯάลΛҾ͘ ख൪ଆ͸
 ৔ࡳ΋ର৅ʹ
  4. ೋਓ୯ශຽͷղ๏ ➤ ໦୩ɾখ໺ (2018) ➤ ҎԼͷΞϧΰϦζϜʹΑΓखࡳͷϚονϯά਺Λܭࢉ 1 ڧ͞ 3 2

    2 4 0 ख൪ଆ ඇख൪ଆ ૬खͷ
 ࠷ऑࡳʹ͸
 Ҿ͚ͳ͍ʂ ࣗ෼ͷࡳ͔Βɺ
 ͦΕΑΓऑ͍૬खͷࡳʹ
 ϚονϯάลΛҾ͘
  5. ೋਓ୯ශຽͷΞϧΰϦζϜͷূ໌ ➤ ূ໌͍ͨ͠ఆཧΛ࣍ͷ (a) (b) ʹ෼ׂ
 (a) ͳΒ͹ख൪ଆඞউ
 (b) ͳΒ͹ඇख൪ଆඞউ

    ➤ جૅεςοϓ
 खࡳ̍ຕͣͭʢ߹ܭ̎ຕʣͷͱ͖ (a) (b) ͕੒ཱ ➤ ؼೲεςοϓ
 खࡳ߹ܭ͕ ຕͷͱ͖ɺ
 ɾ(a)ہ໘ͳΒ͹ҰखͰউͪ or ૬खͷ(b)ہ໘ʹભҠ͢Δख͕͋Δ
 ɾ(b)ہ໘ͳΒ͹૬खͷ(a)ہ໘ʹඞͣભҠ͢Δ
 ͜ͱ͕ࣔͤͨͱ͢Δʢ͕͜͜೉͍͠ͷ͕ͩʣ μ(X, ¯ X− + {r}) > μ( ¯ X, X− ) μ(X, ¯ X− + {r}) ≤ μ( ¯ X, X− ) k
  6. ೋਓ୯ශຽͷΞϧΰϦζϜͷূ໌ ➤ ূ໌͍ͨ͠ఆཧΛ࣍ͷ (a) (b) ʹ෼ׂ
 (a) ͳΒ͹ख൪ଆඞউ
 (b) ͳΒ͹ඇख൪ଆඞউ

    ➤ ؼೲεςοϓ
 खࡳ߹ܭ͕ ຕͷͱ͖ɺ
 ɾ(i) (a)ہ໘Ͱग़ͤΔࡳ͕͋Ε͹ؼೲ๏ͷԾఆͷ(b)Λద༻
 ɾ(ii) (b)ہ໘Ͱग़ͤΔࡳ͕͋Ε͹ؼೲ๏ͷԾఆͷ(a)ɺ
 ɹɹɹग़ͤΔࡳ͕ͳ͚Ε͹ (i) Λద༻
 ɾ(i) (a)ہ໘Ͱग़ͤΔࡳ͕ͳ͚Ε͹ (ii) Λద༻
 Αͬͯखࡳ߹ܭ ຕͰ΋(a)(b)͕੒ཱ μ(X, ¯ X− + {r}) > μ( ¯ X, X− ) μ(X, ¯ X− + {r}) ≤ μ( ¯ X, X− ) k + 1 k + 1