CYC8CNBYCXCIC1CRCHCOCK K L ? K L DGAVBBC1N L BXC3AW?AMASA0AMDFAFA9A1 ? K L DGBF PAWBQALD9BZCFAWA7 K L AWBCBWAI?B5D6AGA8 N L C0CQCGD5CV K L AUDGAJAZB7CTBBD4AXD2AXAWB4CMBRB6BKLBSBNA2B0ARAWA7 AVA0B6BKC9A8 1. B4CMBRB6BKA8 LBPB6BKA7 2. CMBRB6BKA8 (2.1)BBA5ABBAB6BKA8 x 0 ,x 1 ,x 2 , ···. (2.2)CIAMB6BKA8 ? . (2.3)CGC4AMB6BKA8 ?, → . (2.4)B7BRB6BKA8 ), , ,(. B1A0K L AWBDDF(B8 N L AWBDDFCBDJA7) 1. K L AWAKBHCUAXAYD6AGA8 (1.1)BBA5ABBAB8BBA5AGBAADK L AWAK. (1.2)D8t 1 ,t 2 , ···,t m AD K L AWAKA6f m AD K L AWAVBB mBABJDIABBAB6BKA6BH f m (t 1 ,t 2 , ···,t m ) AD K L AWAKA7 1 2. K L AWBDDFBHCUAXAYD6AGA8 (2.1)D8t 1 ,t 2 , ···,t n ADLAWAKA6F n AD K L AWAVBB n BA?AMABBAB6BKA6BH F n (t 1 ,t 2 , ···,t n ) AD K L AW BDDFA7 BBBTBDDF (2.2)D8α 1 ,α 2 AD K L AWBDDFA6BH (?α 1 ), (α 1 →α 2 ) AD K L AWBDDFA7 (2.3)D8 α AD L AWBDDFA6x AD K L AWBBA5ABBAB6BKA6BH (?x)αAD K L AWAVBBBDDFA7 DAA8 K L AWAMDFB9ARDG N L AWAMDFB9ARAWAVBBBTB9ARA6B0B0A2CPDE B3D6AGAJA8BACXBLBCAXA8 ?BUB4B8BCDHA9 ?CABKDBCLBGBHA9 ? P BPBDDFBF K L BPAWASD7DDCE ?BXALBDDFA8 (α∨β)AD ((?α)→β)AWBXALA9 (α∧β)AD (?(α → (?β)))AWBXALA9 (α?β)AD ((α→β) ∧(β→α))AWBXALA9 (?x)αAD(?((?x)(?α)))AWBXALA7 2 K L C0CQCGCLC9C5D6 1. BDCDA8 (K1) α→(β→α) (K2) (α→(β→γ))→((α→β)→(α→γ)) (K3) (?α→?β)→(β→α) (K4) ?xα → α(x/t),D8tAZxBFαBPBUB4A7 (K5) α →?xα,D8xADBFαBPBUB4AKAHA7 (K6) ?x(α→β) → (?xα→?xβ) (K7) D8αDGK L AWAVBBBDCDA6BH (?x)αAUADK L AWAVBBBDCDA7 CYBPA8α, β, γADK L AWBDDFA6xADK L AWBBA5ABBAB6BKA6tADK L AWAKA7 2. BGBHA8 B5CCBGBH (M)A8 B4αBTα→β C8AVAUβ. D7CCCUCB C4CZ3.13 K L BDDFAWAVBBB5AIAQCK α 1 ,α 2 , ···,α n AHAD K L BPAW AVBBBKCRAQCKA4BKCRA5A6D6BICOBB α i (1 ≤ i ≤ n)AYCNBVAGCKA7BYBM AVA8 (1) α i DG K L AWAVBBBDCDA9BP (2) α i DGB4CTCHBBα j ,α k (1 ≤ j, k < i)B2B3(M)AVAUAWA7 ANDCA6AHα n AD K L AWAVBBCVAXCD,BVAD turnstileleft K L α n ,BPBXALADturnstileleft α n . 3 BZCECFCA C4C93.5 DAαADPAWAVBBCVAXCDA6α prime DGαBFK L BPAWAVBBASD7DD CEA6BHturnstileleft K L α prime C4C93.6 (1)D8turnstileleft K L α→β,D0turnstileleft K L α,BHA8 turnstileleft K L β . (2)D8turnstileleft K L (α→β),D0turnstileleft K L (β→γ),BHturnstileleft K L (α→γ). ANCSA4BPAW (1)D5BVAD(M), (2)D5BVAD (Tr). CA 3.14 D8AKtAZBBA5ABBAB6BKxBFαBPBUB4A6BHA8 turnstileleft α(x/t)→?xα . D7A8 B0AD(?α)(x/t)=?(α(x/t)),APB0 turnstileleft?x(?α)→(?α)(x/t K4) turnstileleft?x(?α)→?(α(x/t)) turnstileleft (?x(?α)→?(α(x/t))) → (α(x/t)→??x(?α)) (CSA4CVAXCD) turnstileleft α(x/t)→??x(?α)(M) BUA8 turnstileleft α(x/t)→?xα 4 C4C9 3.7 D8turnstileleft K L α,BHturnstileleft K L ?xα. BKA8B0turnstileleft K L α,BFAQBF K L BPBDDFAQCKA8 α 1 ,α 2 , ···,α n (= α) ADαAWAVBBBKCRA7 AGAZi (1 ≤ i ≤ n)BHCUBKCRA8 turnstileleft K L ?xα i (?) (1)ATi =1DCA6α 1 ADAVBBBDCDA6APB0?xα 1 AUADAVBBBDCDA6BF turnstileleft K L ?xα 1 . (2)DAi<kDCA6(?)AICFA6AGBKi = k DC(?)AUAICFA7 (2.1)D8α k D5ADBDCDA6B3(1)C8BKA7 (2.2)D8α k DGB4α l ,α j (1 ≤ l, j < k)B3(M)AVAUAWA6ADB2DA α j = α l →α k . BHA8 turnstileleft K L ?xα l ,(BHCUBWDA) turnstileleft K L ?xα j A1 (BHCUBWDA) turnstileleft K L ?x(α l →α k ). (BHCUBWDA) turnstileleft K L ?x(α l →α k ) → (?xα l →?xα k )(BDCDK5). turnstileleft K L ?xα l →?xα k ,(AXCD3.6) turnstileleft K L ?xα k .(AXCD3.6) BHCUBKAAA6(?)AICFA6APB0turnstileleft K L ?xα n ,BUturnstileleft K L ?xα. 5 CA 3.15(1) D8xADBFαBPBUB4AKAHA6BHA8 turnstileleft?x(α→β)→(α→?xβ) D7A8 B0turnstileleft P (p→(q→r)) → ((s→q)→(p→(s→r))). AW?x(α→β), ?xα, ?xβ, αB5ACA6BOCYBPAW p, q, r, sAVA8 turnstileleft K L (?x(α→β)→(?xα→?xβ)) → ((α→?xα)→(?x(α→β)→(α→?xβ))). turnstileleft K L ?x(α→β)→(?xα→?xβ), (K5) turnstileleft K L (α→?xα)→(?x(α→β)→(α→?xβ)). turnstileleft K L α→?xα. xADBFαBPBUB4AKAHA6 (K5) turnstileleft K L ?x(α→β)→(α→?xβ). 6 CA 3.15(2) D8xADBFαBPBUB4AKAHA6BHA8 turnstileleft (α→?xβ)→?x(α→β) D7A8 B0turnstileleft P (p→q)→((r→p)→(r→q)), B5ACAW?xβ, β, αASBOCYBPAW p, q, rAVA8 turnstileleft K L (?xβ→β) → ((α→?xβ)→(α→β)). turnstileleft K L ?xβ→β, turnstileleft K L (α→?xβ) → (α→β). turnstileleft K L A→B. ABVα→?xβ, B BVα→β xADBFABPBUB4AKAH turnstileleft K L ?x(A→B). (AXCD3.7) turnstileleft K L ?x(A→B) → (A→?xB). (1) turnstileleft K L A→?xB, turnstileleft K L (α→?xβ) →?x(α→β). 7 CA 3.16 D8turnstileleft α→β,BHA8 1. turnstileleft?xα→?xβ A1 2. turnstileleft?xα→?xβ. D7A8 1. (1) turnstileleft α→β (A4DA) (2) turnstileleft?x(α→β)(AXCD3.7) (3) turnstileleft?x(α→β)→(?xα→?xβ)(K6) (4) turnstileleft?xα→?xβ (M) 2. (1) turnstileleft α→β (2) turnstileleft (α→β)→(?β→?α)(CSA4CVAXCD) (3) turnstileleft?β→?α (M) (4) turnstileleft?x?β→?x?α (1) (5) turnstileleft (?x?β→?x?α)→(??x?α→??x?β) (6) turnstileleft??x?α→??x?β (M) BUA8 turnstileleft?xα →?xβ 8 D2CDCJC1CMCX C4CZ C4CZ3.14 DAΓDG K L AWAVBBBDDFBS(ADAVAXB5AI). K L BPBDDFAWAV BBB5AIAQCKα 1 ,α 2 , ···,α n AHADK L BPB4CZA3ΓAAAKα n AWAVBBBK CR,D6BICOBBα i (1 ≤ i ≤ n)CNBVAGCKA7BYBMAVA8 (i) α i ∈ Γ. (ii) α i DGAVBBBDCDA7 (iii) α i DGB4α j ,α k (1 ≤ j, k ≤ i)B3(M)AVAUA7 ANDCA6AUAHBFK L BPB4CZA3ΓC8AAAKα n ,BVADΓ turnstileleft K L α n BPΓ turnstileleft α n . CSD8 CSD8 (1) turnstileleft K L αAWAJATA7BYDGA8AZK L AWD4AVBBBDDFBSΓ, Γ turnstileleft K L α. CSD8 (2)DA Σ,αB5ACDG P BPBDDFBSB8BDDFA6 Σ ∪{α} AWBDDFBP AKAHAWCSA4ABBAB6BKAYBF p 0 ,p 1 , ···,p n BMBPA6C0 Σ B8α BPAW p 0 ,p 1 , ···,p n B5ACA6BOAD K L BPBDDFα 0 ,α 1 , ···,α n , AVAU K L AWBDDFBS Σ prime B8α prime . D8Σ turnstileleft P α,BHΣ prime turnstileleft K L α prime . CSD8 (3)D8Σ turnstileleft K L α,Σturnstileleft K L α→β,BHΣ turnstileleft K L β . CSD8 (4)D8Σ turnstileleft K L α→β,Σturnstileleft K L β→γ,BHΣ turnstileleft K L α→γ . CSD8 (5)D8Σ turnstileleft K L α,B0xDGAVBBADBF ΣAWD4BMBDDFBPBUB4AKAHAW AVBBBBA5ABBAB6BKA6BH Σ turnstileleft K L ?xα . 9 CSD8 (5) C0D7CC D7A8 B0Σ turnstileleft K L α,BHAQBF K L BPBDDFAQCKA8 α 1 ,α 2 , ···,α n (= α) ADBFCZA4 ΣAGAAAKαAWAVBBBKCRA7 AGAZi (1 ≤ i ≤ n)BHCUBKCRA8 Σ turnstileleft K L ?xα i (?) (1)ATi =1DCA6α 1 ADAVBBBDCDBPα 1 ∈ Σ. (1.1)D8α 1 ADAVBBBDCDA6BH?xα 1 AUADAVBBBDCDA6BFturnstileleft K L ?xα 1 , APB0Σ turnstileleft K L ?xα 1 . (1.2) D8 α 1 ∈ Σ, BH x ADBF α 1 BPBUB4AKAHA6B4 (K5) BLA8 turnstileleft K L α 1 →?xα 1 ,APB0Σ turnstileleft K L α 1 →?xα 1 . B6Σ turnstileleft K L α 1 ,BFA8Σ turnstileleft K L ?xα 1 . (2)DAi<kDCA6(?)AICFA6AGBKi = k DC(?)AUAICFA7 (2.1)D8α k D5ADBDCDBPα k ∈ Σ,B3(1)C8BKA7 (2.2) D8 α k DGB4 α l ,α j (1 ≤ l, j < k) B3 (M) AVAUAWA6ADB2 DAα j = α l →α k . B4BHCUBWDAAVA8 Σ turnstileleft K L ?xα l ,Σturnstileleft K L ?xα j ,BUA8 Σ turnstileleft K L ?x(α l →α k ). B6B4B7turnstileleft K L ?x(α l →α k ) → (?xα l →?xα k ) (BDCDK5). BFA8 Σ turnstileleft K L ?x(α l →α k ) → (?xα l →?xα k ) B4AOBO(2) BLA8 Σ turnstileleft K L ?xα l →?xα k ,Σturnstileleft K L ?xα k . BHCUBKAAA6(?)AICFA6APB0Σ turnstileleft K L ?xα n ,BUA8 Σ turnstileleft K L ?xα. DAA8D9CQAWBKCRBNDGAZAXCD3.7AWBKCRBXCJADARAWAPB9A7 10 CA 3.17 D8xADBFβ BPBUB4AKAHA6BHA8 {?x(α→β), ?β}turnstileleft?x?α D7A8 ?x(α→β)(CZA3) ?x(α→β)→(α→β)(K4) α→β (M) (α→β)→(?β→?α)(CSA4BQARDF) ?β→?α (M) ?β (CZA3) ?α (M) ?x?α (AOBO(4)) K L C0CWD0C4C9 C4C93.8 Σ,αturnstileleft K L β ATD0C5AT Σ turnstileleft K L α→β. BKCRB8 PBPASAZAXCDAWBKCRB4AGCBDJA6BNATC0A2 PAWBDDFA3B9 ADA2 K L AWBDDFA3BUC8A7 11 CA 3.15 C0D9CPD7CC BKCRA8 {α→?xβ, α}turnstileleftβ D7A8 α→?xβ (CZA3) α (CZA3) ?xβ (M) ?xβ→β (K4) β (M) B4ASAZAXCDAVA8 {α→?xβ}turnstileleftα→β . B4AOBO(4)AVA8 {α→?xβ) turnstileleft?x(α→β)(BSAXA8 xADBFα→?xβ BPBUB4AKAH) BEB4ASAZAXCDAVA8 turnstileleft (α→?xβ)→?x(α→β). CA 3.19 D8xADBFβ BPBUB4AKAHA6BKCRA8 turnstileleft K L ?x(α→β)→(?xα→β) D7A8 B4CE3.17AVA8 {?x(α→β), ?β}turnstileleft?x?α. APB0A8 {?x(α→β)}turnstileleft?β→?x?α. B0A8 turnstileleft (?β→?x?α)→(??x?α→β) BFA8 {?x(α→β)}turnstileleft(?β→?x?α)→(??x?α→β) B4AOBO(1)BLA8 {?x(α→β)}turnstileleft??x?α → β. BUA8 {?x(α→β)}turnstileleft?xα → β. BFA8 turnstileleft?x(α→β)→(?xα→β) 12 N L D4 K L C1C3C7CT D1C9 3.2 D8γ ADK L AWAVBBBDCDA6BHAZ K L AWD4AVB5AIBDDFBSΣ, Σ turnstileleft N L γ . D7A8 AZγ AWBEBGB8BDAOBHCUBKCRA7 (1) D8γ AD(K1)—(K6)BPAWCTAVA7DCA7 (1.1)D8γ AD(K1)—(K3)BPCTA7DCA6B4AXCD3.1C8BKA9 (1.2)ATγ AD(K4)DCA7B4(??)BLA8 ?xα turnstileleft N L α(x/t)(CYBPA8 tAZ xBF α BPBUB4), B4 (→ +) BLA8 ?turnstileleft N L ?xα→α(x/t). B4B7Σ DGB5AIBSA6BFC8B5AIAODEB3 (+)AVA8 Σ turnstileleft N L ?xα→α(x/t). (1.3)ATγ AD(K5)DCA6B4(?+)BLA8 α turnstileleft N L ?xα (CYBPA8 xAD BFαBPBUB4AKAH), APB0Σ turnstileleft N L α→?xα . (1.4) AT γ AD (K6) DCA6B4CE 3.6 C8BKA8 Σ turnstileleft N L ?x(α→β) → (?xα→?xβ). (2) D8 γ AD ?xγ prime DCA6CYBP γ prime AD K L AWAVBBBDCDA6B4BHCUBWDA AVA8 ?turnstileleft N L γ prime ,APB0B4 (?+)BLA8 ?turnstileleft N L ?xγ prime ,BFA8 Σ turnstileleft N L ?xγ prime . 13 K L ” ? ”N L DAΣ,αB5ACADK L AWB5AIBDDFBSB8BDDFA6D8Σ turnstileleft K L α,BHΣ turnstileleft N L α. D7A8 B4B7Σ turnstileleft K L α,BF K L BPAQBFB4ΣAAAKαAWBKCRAQCKA8 α 1 ,α 2 , ···,α n (= α) AGBKA8AZD4AX i (1 ≤ i ≤ n), Σ turnstileleft N L α i (?) AZiC6ANBHCUBKCRA7 (1) ATi =1DCA6α 1 AD K L AWBDCDBP α 1 ∈ Σ. (1.1)D8α 1 AD K L AWBDCDA6B4B1CD 3.2BLA8 Σ turnstileleft N L α 1 . (1.2)D8α 1 ∈ Σ,B4(+)BLA8 Σ turnstileleft N L α 1 . (2) DA (?) AZCNBV i<kAWA1B5BUD3DIi AICF(k>1), ACBK i = k DC(?)AUAICFA7 (2.1)D8α k ∈ ΣBPα k AD K L AWBDCDA6B3(1)C8BKA7 (2.2)D8α k DGB4α j ,α l (1 ≤ j,l < k)B3(M)AVAUA6ADB2DAα j AD α l →α k ,B4BHCUBWDAAVA8Σ turnstileleft N L α j ,Σturnstileleft N L α l ,BUA8Σ turnstileleft N L α l →α k . B4(→?)BLA8 Σ turnstileleft N L α k . BHCUBKABA6 (?)AICFA7APB0Σ turnstileleft N L α n ,BUA8 Σ turnstileleft N L α. 14 K L ” ? ”N L DAΣ,αB5ACN L BPAWB5AIBDDFBSB8BDDFA6D8Σ turnstileleft N L α,BHΣ turnstileleft K L α. D7A8 B4B7Σ turnstileleft N L α,BFN L BPAQBFBKCRAQCKA8 Σ 1 turnstileleft α 1 , Σ 2 turnstileleft α 2 , ···, Σ n turnstileleft α n DEAVA8 Σ n =Σ,α n = α . AGBKA8AZD4AX i (1 ≤ i ≤ n), Σ i turnstileleft K L α i (??) AZiC6ANBHCUBKCRA7 (1) AT i =1DCA6Σ 1 turnstileleft α 1 BNCWB4 (∈) AVAUA6APB0α 1 ∈ Σ 1 , BF Σ turnstileleft K L α 1 . (2) BWDA(??)AZCNBVi<kAWA1B5iAICFA6C7AF(??)ATi = k DCD1AMA7 (2.1)D8Σ k turnstileleft α k DGB3(∈)A0(??)A0(∨?)A0∨+)A0(∧?)A0 (∧+)A0(→?)A0(→ +)A0(??)BP(? +)AVAUAWA6B3D9BJAXCD 2.15C8BKA7 (2.2) D8 Σ k turnstileleft α k DGAZCTBBΣ i turnstileleft α i (1 ≤ i<k) B3 (+) AVAU AWA6BUA8 Σ k =Σ i ∪{γ}, α k = α i A7B4BHCUBWDAAVA8 Σ i turnstileleft K L α i ,AP B0Σ k turnstileleft K L α i ,BUΣ k turnstileleft K L α k . (2.3)D8Σ k turnstileleft α k DGAZΣ i turnstileleft α i B3 (??)AVAUAWA6BUA8 Σ k =Σ i , α i = ?xβ, α k = β(x/t), CYBPA8 t DG N L BPAWAKA6 t AZ x BF β BPBUB4A7B4BHCUBWDABLA8 Σ i turnstileleft K L α i , BUA8 Σ k turnstileleft K L ?xβ. B4BDCD (K4) BLA8 turnstileleft K L ?xβ → β(x/t), BF Σ k turnstileleft K L ?xβ → β(x/t), APB0 Σ k turnstileleft K L β(x/t),BUA8 Σ k turnstileleft K L α k . 15 (2.4)D8 Σ k turnstileleft α k DGAZΣ i turnstileleft α i B3 (?+)AVAUAWA6BUA8 Σ k =Σ i , α k = ?xα i ,CYBPA8BBA5ABBAB6BKxADBFΣ i AWD4BMBDDFBPBUB4AKAHA7 B4BHCUBWDAAVA8Σ i turnstileleft K L α i . B4D9C2AOBO(4)BLA8Σ i turnstileleft K L ?xα i ,BUA8 Σ k turnstileleft K L α k . (2.5) D8 Σ k turnstileleft α k DGAZ Σ i turnstileleft α i B3 (??) AVAUAWA6BUA8 Σ i = Γ∪{γ},Σ k =Γ∪{?xγ}, α k = α i ,CYBPA8ΓDGN L AWAVBBB5AIBD DFBSA6xADBFΓ∪{α i }AWD4AVBBBDDFBPBUB4AKAHA7B4BHCUBWDABLA8 Σ i turnstileleft K L α i ,BUA8Γ∪{γ}turnstileleft K L α i . B4ASBIAXCDBLA8Γ turnstileleft K L γ→α i . B4 B7xADBFΓAWD4BMBDDFBPBUB4AKAHA6BF Γ turnstileleft K L γ→α i . B0xADBFα i BPBUB4AKAHA6B4D9C2CE3.19BLA8 turnstileleft K L ?x(γ→α i )→(?xγ→α i ),BF Γ turnstileleft K L ?x(γ→α i )→(?xγ→α i ), APB0 Γ turnstileleft K L ?xγ→α i . BEB4ASBI AXCDBLA8 Γ∪{?xγ}turnstileleft K L α i ,BUA8 Σ k turnstileleft K L α k . (2.6)D8 Σ k turnstileleft α k DGAZΣ i turnstileleft α i B3 (?+)AVAUAWA6BUA8 Σ k =Σ i , α i = β(x/t), α k = ?xβ, CYBPA8 N L AWBBA5ABBAB6BKx ADBF β BP BUB4A7B4BHCUBWDABLA8 Σ i turnstileleft K L β(x/t). B4D9C2CE3.14BLA8 Σ i turnstileleft K L β(x/t)→?xβ,APB0Σ i turnstileleft K L ?xβ,BUΣ k turnstileleft K L α k . BHCUBKABA6 (??)AICFA7 K L D3 N L C0C2C6CS AZ N L (K L )BPAWB5AIBDDFBSΣB8BDDFα, Σ turnstileleft K L αATD0C5ATΣ turnstileleft N L α . 16