. B →+u A → B Ծఆ [u : A] ΛΩϟϯηϧ A → B ࡞Δ →-আڈنଇʢ→Eʣղମ A A → B →− B proof net ͷه๏ʢ࠶ܝʣ ໋ʢࣜʣ ˠ ϫΠϠʢઢʣ ਪنଇ ˠ ϊʔυʢശʣ ΩϟϯηϧԾఆ ˠ ઢϫΠϠ ະ༻Ծఆ ˠ ͿΒΓΜϫΠϠ ॖʢෳ༻ʣˠ Y ࣈϊʔυ ࠓͷඪ ∧ʢ͔ͭʣͱ ∨ʢ·ͨʣͷ ਪنଇΛֶͼɺproof net ඳ͘ ˠ ϰΝΠΦϦϯͷᶅᶆݭ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 6
B ͕྆ํূ໌Ͱ͖ͨͱ͖ A ∧ B ͕ ূ໌͞ΕΔ ˠ 2 ೖྗɾ1 ग़ྗɻԾఆͷΩϟϯηϧ ෆཁ ∧I = ߏࢠɺ∧EL,R = ղମࢠ ∧-আڈنଇʢ∧Eʣղମ A ∧ B ∧−L A A ∧ B ∧−R B ∧EL ɿA ∧ B ͔Β A ΛऔΓग़͢ ∧ER ɿA ∧ B ͔Β B ΛऔΓग़͢ → ͱͷൺֱ ಋೖ আڈ → 1 ͭʢΩϟϯηϧ͋Γʣ 1 ͭ ∧ 1 ͭʢΩϟϯηϧͳ͠ʣ 2 ͭʢL/R ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 8
∧ B 2 ຊͷϫΠϠ͕߹ྲྀ 1 ຊ͕ग़ྗ ∧EL ͷϊʔυ A ∧ B ∧EL A 1 ຊೖྗɾ1 ຊग़ྗ ʢB ࣺͯΒΕΔʣ ∧ER ͷϊʔυ A ∧ B ∧ER B 1 ຊೖྗɾ1 ຊग़ྗ ʢA ࣺͯΒΕΔʣ detourʢແବͳճΓಓʣͷྫͱফڈ ∧I ͷޙʹ ∧EL ˠʮA ∧ B Λ࡞͙ͬͯ͢ A ͚ͩऔΓग़͢ʯ= ࠷ॳ͔Β A Λ͑ Α͔ͬͨɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 9
: A ∧ B] ∧−R B [u : A ∧ B] ∧−L A ∧+ B ∧ A →+u A ∧ B → B ∧ A ٯࢉɿ࠷ޙ →I Ͱ u ΛΩϟϯηϧ B ∧ A Λ࡞Δʹ Bʢӈʣͱ Aʢࠨʣ͕ ඞཁ ͲͪΒ [u : A ∧ B] ͔Β ∧E ͰऔΓग़͢ proof net [u:A ∧ B] Y ࣈʢॖʣ ∧ER B ∧EL A ∧I B ∧ A Y ࣈϊʔυɿ[u : A ∧ B] Λ 2 ճ͏ B ͱ A ͷϫΠϠ͕ʮೖΕସΘͬͯʯ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 10
net ඳ͘͜ͱɻ 1 A ∧ B → A ʢ∧ ͷআڈͷ࿅शʣ 2 A ∧ B → B ∧ A ʢަଇɻূ໌͚ͩ࠶֬ೝʣ 3 A → B → A ∧ B ʢ→ ͱ ∧ ͷΈ߹Θͤʣ ϯδʣ (A ∧ B) ∧ C → A ∧ (B ∧ C) ʢ݁߹ଇʣ ώϯτɿٯࢉͷखॱ Step 1ɿূ໌໋͍ͨ͠ͷʮ࠷ޙͷ݁߹ࢠʯΛಛఆ Step 2ɿͦͷલͷܗΛ֬ఆͯ͠ԾఆΛܾΊΔ Step 3ɿআڈنଇͰԾఆ͔Β݁ΛΈཱͯΔ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 11
A ∧ B] ∧−L A →+u A ∧ B → A ٯࢉɿ࠷ޙ →I Ͱ u ΛΩϟϯηϧ [u : A ∧ B] ͔Β ∧EL Ͱ A ΛऔΓग़͢ proof net [u:A ∧ B] ∧EL A ઢ 1 ຊͷΈɺY ࣈϊʔυɾऑԽͳ͠ ˠ ࠷γϯϓϧͳྫ қɿ˒ˑˑ ∧EL ͷҰൃద༻ɻB ͷใΛʮࣺͯΔʯ͜ͱͰ A ΛऔΓग़͢ɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 12
: A ∨ B] [u : A] ∨+R B ∨ A [v : B] ∨+L B ∨ A ∨−[u, v] B ∨ A →+w A ∨ B → B ∨ A ٯࢉɿ࠷ޙ →I Ͱ w ΛΩϟϯηϧ ∨E Ͱ߹͚ɿ ࢬ 1ɿ[u : A] ͔Β ∨IR Ͱ B ∨ A ࢬ 2ɿ[v : B] ͔Β ∨IL Ͱ B ∨ A proof net [w:A ∨ B] [u:A] ∨IR B ∨ A [v:B] ∨IL B ∨ A ∨E[u, v] B ∨ A 2 ࢬ͕ ∨E Ͱ߹ྲྀ ˠ ߹͚ͷߏ͕ՄࢹԽ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 18
net ඳ͘͜ͱɻ 1 A → A ∨ B ʢ∨IL ͷ࿅शʣ 2 A ∨ A → A ʢ∨E ͷ࿅शɻ߹͚ͷ྆ࢬ͕ಉ͡ʹͳΔʣ 3 A ∨ B → B ∨ A ʢަଇɻূ໌͚ͩ࠶֬ೝʣ ϯδʣ (A ∨ B) ∨ C → A ∨ (B ∨ C) ʢ݁߹ଇʣ ∨-E Λ͏ͱ͖ͷखॱ Step 1ɿA ∨ B ͱ͍͏લఏΛಛఆ͢Δ Step 2ɿ[u : A] ΛԾఆͯ݁͠ C Λূ໌͢ΔࢬΛ࡞Δ Step 3ɿ[v : B] ΛԾఆͯ݁͠ C Λূ໌͢ΔࢬΛ࡞Δ Step 4ɿ∨-E Ͱ 3 ຊΛ߹ྲྀͤ͞ [u]ɾ[v] ΛΩϟϯηϧ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 19
A] ∨+L A ∨ B →+u A → A ∨ B ٯࢉɿ࠷ޙ →I Ͱ u ΛΩϟϯηϧ [u : A] ͔Β ∨IL Ͱ A ∨ B Λߏ ʢB ʮຒΊࠐ·ΕΔ͚ͩʯͰΘ ͳ͍ʣ proof net [u:A] ∨IL A ∨ B ઢ 1 ຊͷΈɺY ࣈϊʔυɾऑԽͳ͠ ˠ ∧EL ͷ proof net ͱಉ͡ܗʂ ∧E ͱ ∨I ͲͪΒʮ1 ຊˠ 1 ຊʯ қɿ˒ˑˑ ∨IL ͷҰൃద༻ɻB ΛʮຒΊࠐΉʯ͜ͱͰ A ∨ B Λ࡞Δɻ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 20
A ͷ proof net [u:A ∧ B] Y ࣈ ∧ER B ∧EL A ∧I B ∧ A Y ࣈʢॖʣ ˠ ∧E ʷ 2 ˠ ∧IʢϫΠϠ͕ ަࠩʣ A ∨ B → B ∨ A ͷ proof net [w:A ∨ B] [u:A] ∨IR B ∨ A [v:B] ∨IL B ∨ A ∨E[u, v] B ∨ A ∨I ʷ 2 ˠ ∨Eʢ2 ࢬ͕߹͚Ͱ߹ྲྀʣ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 25
ඳ͘͜ͱɻ 1 A ∧ B → A ʢ∧E ͷجຊʣ 2 A → A ∨ B ʢ∨I ͷجຊʣ 3 A ∧ B → B ∧ A ʢ∧ ަଇɻY ࣈϊʔυΛඳ͜͏ʣ 4 A ∨ B → B ∨ A ʢ∨ ަଇɻ∨E ͷ߹͚Λඳ͜͏ʣ 5 A → B → A ∧ B ʢ→ ͱ ∧ ͷΈ߹Θͤʣ ϯδʣ A ∧ (B ∨ C) → (A ∧ B) ∨ (A ∧ C) ʢଇʣ ࣍ճʢ6/2ʣͷ༧ఆ ໋ཧͷ͖Ε͍ͳূ໌ͱਖ਼نԽ ՊֶֶՊֶ࢙ (ԋश) ཧֶʢલظʣ ୲ɿా෦ढ़հ 27