B ͕྆ํূ໌Ͱ͖ͨͱ͖ A ∧ B ͕ ূ໌͞ΕΔ 2 ೖྗɾ1 ग़ྗɻԾఆͷΩϟϯηϧෆཁ ∧-আڈنଇʢ∧Eʣղମ A ∧ B ∧−L A A ∧ B ∧−R B ∧EL ɿA ∧ B ͔Β A ΛऔΓग़͢ ∧ER ɿA ∧ B ͔Β B ΛऔΓग़͢ ϙΠϯτ ∧ ಋೖ 1 ͭɾআڈ 2 ͭʢࠨӈʣ ɻ ͲͪΒ͔Ұํ͚ͩऔΓग़ͤΔɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 5
B →+u A → B →Eʢআڈɾલఏ 2 ͭʣ A A → B →− B ¬ ͷఆٛʢུهʣ ¬A := A → ⊥ ⊥ɿنଇΛ࣋ͨͳ໋͍ఆ ∧ ͷنଇ ∧Iʢಋೖɾ2 ೖྗʣ A B ∧+ A ∧ B ∧EL ʢআڈɾࠨʣ A ∧ B ∧−L A ∧ER ʢআڈɾӈʣ A ∧ B ∧−R B ∨ ͷنଇ ∨IL ʢಋೖɾࠨʣ ∨IR ʢಋೖɾӈʣ A ∨+L A ∨ B B ∨+R A ∨ B ∨Eʢআڈɾ߹͚ʣ A ∨ B [u:A] . . . C [v:B] . . . C ∨−[u, v] C ߹ܭ 7 نଇʢ→2ɾ∧3ɾ∨2ʣ͜Ε͕࠷খ໋ཧͷશͯ ΩϟϯηϧΛ͏نଇɿ→Iʢ1 Ծఆʣͱ ∨Eʢ2 Ծఆʣ ɻ∧I ͱ ∨I Ωϟϯηϧͳ͠ɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 10
ূ໌थ [u : A ∧ B] ∧−R B [u : A ∧ B] ∧−L A ∧+ B ∧ A →+u A ∧ B → B ∧ A ٯࢉɿ࠷ޙ →I Ͱ [u] ΛΩϟϯηϧ B ∧ A ∧I Ͱߏ B, A ͲͪΒ [u : A ∧ B] ͔ΒऔΓग़͢ proof net [u:A ∧ B] Y ࣈ ∧ER B ∧EL A ∧I B ∧ A Y ࣈɿ[u] Λ 2 ճ͏ B, A ͷϫΠϠ͕ೖΕସΘΓ ∧I ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 15
. . B →+u A → B A →− B A → B Λ࡞ͬͯ ͙͢ A Ͱআڈ ˠ A ͔Β B Λಋ͚Α͔ͬͨ ∧-detour A B ∧+ A ∧ B ∧−L A A ∧ B Λ࡞ͬͯ ͙͢ A ͚ͩऔΓग़͢ ˠ ࠷ॳ͔Β A Λ͑Α͔ͬͨ ∨-detour A ∨+L A ∨ B [u:A] . . . C [v:B] . . . C ∨−[u, v] C A ∨ B Λ࡞ͬͯ ͙͢߹͚Ͱղମ ˠ A ͷূ໌Λ͑Α͔ͬͨ proof net Ͱͷݟ͑ํ detour ʹ ಋೖϊʔυͱআڈϊʔυ͕ͭͳ͕͍ͬͯΔ෦ɻͦͷ 2 ͭͷϊʔυΛফͯ͠ϫΠϠΛ݁͢Δ͚ͩͰ detour Λআڈ Ͱ͖Δɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 17
∧ B, A, B A → B ͷ෦ཧࣜɿA → B, A, B A ∨ B ͷ෦ཧࣜɿA ∨ B, A, B ࠶ؼతʹఆٛʢA ͷ෦ཧࣜͷ෦ཧࣜ ؚΉʣ ྫɿA ∧ B → B ∧ A ͷূ໌Ͱ֬ೝ ݁ A ∧ B → B ∧ Aɻ ূ໌ʹݱΕΔࣜɿ A ∧ Bʢ݁ͷ෦ࣜ ✓ʣ AʢA ∧ B, B ∧ A ͷ෦ࣜ ✓ʣ Bʢಉ্ ✓ʣ B ∧ Aʢ݁ͷ෦ࣜ ✓ʣ ʮಥવ C ͕ग़ͯ͘Δʯ͜ͱͳ͍ʂ ҙຯ ূ໌ʮ݁ͷ෦͚ͩʯͰߏ͞ΕΔ ˠʮͲ͔͜Β C ͕ग़͖ͯͨʁʯ͕ Մೳɻਖ਼Խͷࠜڌ͕ಁ໌ɻ ແໃ६ੑূ໌ɾܾఆՄೳੑʹԠ༻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 20
A ∧ (A → B) → B ͷ࠷֎݁߹ࢠ → ˠ ࠷ޙ →IɻԾఆ [u : A ∧ (A → B)] ΛՃɻ ˠ αϒΰʔϧɿ[u] ⊢ B Step 2ɿB Λग़͢ʹʁ B લఏʹͳ͍͕ A → B ͕͋Δ ˠ →E ͕͑ΔʢA ͱ A → B ͕ඞཁʣ Step 3ɿA ͱ A → B ΛऔΓग़͢ A ˡ [u] ʹ ∧EL A → B ˡ [u] ʹ ∧ER ʢ[u] Λ 2 ճ͏ ˠ Y ࣈϊʔυʣ ͨ͠ূ໌ [u : A ∧ (A → B)] ∧−L A [u : A ∧ (A → B)] ∧−R A → B →− B →+u A ∧ (A → B) → B [u] [u] Y ࣈʢॖʣ ∧EL ∧ER A A → B →E B ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 23
ඳ͘͜ͱɻ 1 A ∧ B → (B → C) → C ʢ∧E ͱ →E ͷΈ߹Θͤɻٯࢉ๏Ͱղ͜͏ʣ 2 (A → C) ∧ (B → C) → (A ∨ B → C) ʢ∧E Ͱ྆ํͷ → ΛऔΓग़͠ɼ∨E Ͱ߹͚ʣ 3 (A → C) ∨ (B → C) → (A ∧ B → C) ʢ∨E ͷԠ༻ɻ 2 ͱԿ͕ҧ͏ʁʣ ʣ A ∨ B → A ∧ B ʢূ໌Ͱ͖Δʁ Ͱ͖ͳ͍ʁͲ͜Ͱ٧·Δ͔͔֬ΊΑ͏ʣ ϯδʣ (A ∧ B) ∨ C → (A ∨ C) ∧ (B ∨ C) ʢଇɻ∨E ͱ ∧I ͷΈ߹Θͤʣ 4 ʹ͍ͭͯ A ∨ B → A ∧ B ূ໌Ͱ͖·ͤΜɻ ʮA ·ͨ Bʯ͔ΒʮA ͔ͭ Bʯཧతʹಋ͚ͳ͍ ٯࢉ๏Λࢼͯ͠ΈΔͱɼͲ͜Ͱߦ͖٧·Δ͔͕Θ͔Γ·͢ɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 25
→ C) → C [u : A ∧ B] ∧−R B [v : B → C] →− C →+v (B → C) → C →+u A ∧ B → (B → C) → C ٯࢉɿ→I×2 ˠ ∧ER + →E [u] ͔Β B ΛऔΓग़͠ [v : B → C] ʹ͢ ʢA Θͳ͍ʹҙʣ 2 ղɿ(A→C) ∧ (B→C) → (A ∨ B → C) [w : A ∨ B] [u : (A → C) ∧ (B → C)] ∧−L A → C [p : A] →− C [u : (A → C) ∧ (B → C)] ∧−R B → C [q : B] →− C ∨−[p, q] C →+w A ∨ B → C →+u (A → C) ∧ (B → C) → (A ∨ B → C) ٯࢉɿ→I×2 ˠ ∨Eʢ߹͚ʣ Case [p : A]ɿ∧EL Ͱ A → C ˠ →E Ͱ C Case [q : B]ɿ∧ER Ͱ B → C ˠ →E Ͱ C ʢ[u] Λ 2 ճ͏ ˠ proof net Ͱ Y ࣈʣ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 26