1. (Naturau Deduction System, N-system) 1.1,,,,, n- R t 1,..., t n Rt 1... t n atomic formula : x, y, z, u, v, w,... : f, g, h,... : c, d,... : t, s,
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- よしお はにうだ
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1 1 (Naturau Deduction System, N-system) 11,,,,, n- R t 1,, t n Rt 1 t n atomic formula : x, y, z, u, v, w, : f, g, h, : c, d, : t, s, r, : P, Q, : R, : A, B, C, D, 12 A A A A N-system 1 1 N-system N-system Hilbert-style system Gentzen-system 1
2 (Introduction rules) A B ( I) A B (Elimination rules) A B A ( E r) A B B ( E l) [A] B A B ( I) A A B ( I r) ϕ(x) xϕ(x) ϕ(t) xϕ(x) ( I) ( I) B A B ( I l) A B B A B xϕ(x) ϕ[t/x] xϕ(x) ψ A [A] C C ( E) ( E) [A] C ( E) [ϕ(x)] ψ ( E) 10 A A ( E) [A] ( I) A (minimal logic) P (P Q) P Q, P Q A ( ) ( ) ex falso sequitur quodlibet 2 A A A A ( E) 10 ( ) 11 2 material implication 2
3 13 N-system N-system Ni 1 [A (B C)] 1 [A] 3 E [A B]2 [A] 3 E B C B E C A C I 3 (A B) (A C) I 2 (A (B C)) ((A B) (A C)) I 1 (A (B C)) ((A B) (A C)) deduction derivation I [A] B A B ( I) (A (B C)) ((A B) (A C)) (A (B C)) ((A B) (A C)) ((A B) (A C)) A B A C A C A C 1 (1) (A B) (B A) (2) (A (B C)) ((A B) C) (3) A A (4) (A B) C (A C) (B C) (5) (A C) (B C) (A B) C (4) (5) (A B) C (A C) (B C) 3
4 [(A C) (B C)] D (A B) C [(A B) C] D (A C) (B C) (A B) C (A C) (B C) 2( cancellation [A] 1 A A I 1 N-system N-system N-system (scheme) A, B A B A B A A B A, B active A, B Γ D A D B A B D -I 3 Γ A B Γ A B -I Γ B Γ {A} A B 3 Γ 4
5 A Γ Γ {A} Γ A 2 A A A A [A] A A Γ = {A} Γ A 4 3 [A] B A I 1 A (B A) I 1 -I A A A B A I A (B A) I A B A B vacuous discharge 2 4 (1) -E sequent calculus style (2) 3 B A [A (A B)] 2 [A] 1 E A B [A] 1 E B A B I 1 (A (A B)) (A B) I 2 -I 1 discharge -I sequent style 4 {A} A 5
6 A (A B)) A (A B) A A A (A B), A A B A A A (A B), A B (A (A B)) (A B) I (A (A B)) (A B) I E 14 1 constant 2 3 t 1,, t n n- f n f n (t 1 t n ) t free variable F V (t) 1 t = a F V (a) = φ, 2 t = x F V (x) = {x}, 3 t = f n (t 1,, t n ) F V (f n (t 1,, t n )) = F V (t 1 ) F V (t n ) A F V (A) 1 F V ( ) = φ, 2 F V (P (t 1 t n )) = F V (t 1 ) F V (t n ), 3 F V (A B) = F V (A B) = F V (A B) = F V (A) F V (B), 4 F V ( xa) = F V ( xa) = F V (A) {x} x t x t [t /x] t x t t[t /x] [t/x] 1 a[t/x] = a, 2 y x y[t/x] = y, y = x y[t/x] = t, 6
7 3 f n (t 1,, t n )[t/x] = f n (t 1 [t/x],, t n [t/x]) [t/x] 1 [t/x] =, 2 (P n (t 1,, t n ))[t/x] = P n (t 1 [t/x],, t n [t/x]), 3 =,, (A B)[t/x] = A[t/x] B[t/x], 4 y x ( ya)[t/x] = ya(t/x), y = x ( ya)[t/x] = ya 5 y x ( ya)[t/x] = ya(t/x), y = x ( ya)[t/x] = ya [ ] t A x A x t t A y x(y < x) y x y x x(x < x) 5 1z xp (y, x) y 2f(x 0, x 1 ) x 1 P (x 0, x 3 ) x 0 3z P (x, y) xq(x, w) x 1 xa y A[y/x] 2 xa a A[a/x] A(y/x) A(t/x) xa I xa I ( I t 6 ) I 5 x(y < x) y x rename z(y < z) 6 t A x 7
8 y A(y/x) xa [A(y/x)] xa I A(y/x) A(y/x) A(y/x) [ x ya(x, y)] 1 E ya(x, y) E A(x, y) xa(x, y) I y xa(x, y) I x ya(x, y) y xa(x, y) I 1 I A(x, y) [ x ya(x, y)] x E xa A(t/x) E t 7 E [A(y/x)] n xa C D C E n y C xa [A(y/x)] C [A(y/x)] 7 t xa x x y(x = y) y(y = y) 8
9 x F V (A) x(a B(x)) (A xb(x)) [ x(a B(x))] 1 E A B(x) [A] 2 E B(x) xb(x) I A xb(x) I 2 x(a B(x)) (A xb(x)) I 1 x F V (B) x(a(x) B) ( xa(x) B) [ x(a(x) B)] 3 E A(x) B [A(x)] 1 [ xa(x)] 2 B E B 1 xa(x) B I 2 x(a(x) B) ( xa(x) B) I 3 E 3 (1) x(a(x) B(x)) xa(x) xb(x), (2) x(a(x) B(x)) ( xa(x) xb(x)), (3) xa(x) A(x), (4) x(a(x) B) xa(x) B, x F V (B) x(x = x) x = x E y(x = y) I occurrences (RAA) 9
10 [ A] A (RAA) ( I) (RAA) ( I) P P (RAA) P P P P [A] 1 A A I [ (A A)] 2 E A I 1 A A I [ (A A)] 2 E A A RAA 2 (A B) (B A) [B] 1 A B I 1 (A B) (B A) I [ ((A B) (B A))] 2 E B A B I 1 (A B) (B A) I [ ((A B) (B A))] 2 E (A B) (B A) RAA 2 x A(x) xa(x) 4 [A(x)] 3 [ xa(x)] 2 xa(x) I E A(x) I 3 [ x A(x)] 1 x A(x) I E xa(x) RAA 2 x A(x) xa(x) I 1 10
11 (1) xa(x) x A(x), (2) A A, (3) A B ( A B), (4) A ( A B), (5) ((A B) (A B)) A 11
12 G-system, Sequent Calculus) G-system cancellation -I explicit Γ D A D B A B Γ A B Γ A B G-system Γ A B Γ A B G right rules 8 Γ A B Γ, A B R A, Γ B Γ A B R Γ A Γ A B R 1 Γ B Γ A B R 2 G N-system -E A, Γ C B, C A B, Γ, C 8 R Γ A B Γ, A B R L G1 R Γ, A Γ, B Γ, A B p17 (3) 12
13 [A] A B C C [B] C ( E) G A, B, Γ C A B, Γ C L Γ A B, C A B, Γ, C A B A E r A B B E l A B B A E A B C A B C C A B resouce A B D 1 D 2 A B A B A B C [A, B] C E C A B L A B A B A B I C [A, B] C E 13
14 A B A, B C C A B C convert ( A B) Inversion principle A B - A, B A, B C A B C - A B A B C C B C A B A A B A C [B] C E A B [A] B A B I C A [B] C E 14
15 Ạ Ḅ C L G1 G1 Ax A A (structural rule) LW Γ A, Γ RW Γ Γ, A LC A, A, Γ A, Γ RC Γ, A, A Γ, A LEx Γ, A, B, Λ Γ, B, A, Λ REx Γ, A A, Λ Θ Γ, Λ, Θ Γ, A, B, Λ Γ, B, A, Λ Γ, A Γ, B R Γ, A B L A, B, Γ A B, Γ R A, Γ, B Γ, A B L Γ, A B, Γ A B, Γ, R Γ, A Γ, A B Γ, B Γ, A B L A, Γ B, Γ A B, Γ 15
16 R A, Γ Γ, A L Γ, A A, Γ LK 9 (1) G- A A A A A A (2) G- (A (A B)) (A B) (A (A B)) (A B) sequent A A B, A B A A A B, A B A (A B), A B A (A B) A B R (A (A B)) (A B) R sequent A A B B B, A B LW LW weakning( Thinning vacuous discharge 5 A A A, B A LW A B A R A (B A) R 9 16
17 (3) G G1 G1 G1 G1 G1 G1 G1c G1i 10 R Γ A B Γ, A B R Γ Θ, A Ξ, B Γ, Θ, Ξ, A B Γ Delta shared context A Γ B Γ Γ, A Γ, B Γ, A B (4) G1i Γ Γ ϕ Γ Γ - Γ Γ ϕ weakening 10 LK,LJ 17
18 ϕ, Γ Γ ϕ R -I Γ ϕ ϕ, Γ L -E Γ sequent Γ sequent Γ sequent falsifiable A, Γ Γ, A sequent Γ A A G1c ϕ(t), Γ xϕ, Γ L Γ, xϕ(x) Γ, ϕ(a) R L t R a Γ, a eigenvaliable ϕ(a), Γ xϕ(x), Γ L Γ, ϕ(t) Γ, xϕ(x) R 18
19 L a Γ, a eigenvariable R t G1c A A A, A L A, A A R A A, A REx A A, A A R RC A A G1c F (a) F (a) F (a) xf (x) R xf (x), F (a) R xf (x), y G(y) R y F (y) xf (x) L y F (y) xf (x) R 1 F (a) F (a) F (a), G(a) F (a) LW G(a) G(a) F (a), G(a) G(a) LW R F (a), G(a) F (a) G(a) F (a), G(a) x(f (x) G(x)) R F (a), xg(x) x(f (x) G(x)) L xf (x), xg(x) x(f (x) G(x)) L xf (x) xg(x) x(f (x) G(x)) L xf (x) xg(x) x(f (x) G(x)) R 2 (1) A B ( A B) (2) ( A B) A B (3) (A B) A B (4) (A B) A B (5) ( A B) ( B A) (6) ( A B) (B A) (7) xf (x) y F (y) (8) yf (y) x F (x) (9) x(a B(x)) (A xb(x)) (10) (A xb(x)) x(a B(x)) (11) x(a(x) B(x)) ( xa(x) xb(x)) 19
20 G1i G1c sequent Γ G1i A A A A A L A, A A L A A, A A L LC A A (A A) R G1i (1) A B A B (2) A B A (3) A A B (4) A B (A B) (5) (A B) A B (6) A B (A B) (7) (A C) (B C) (A B) C (8) A B B A (9) xf (x) y F (y) (10) x F (x) xf (x) (11) x(f (x) G(x)) xf (x) xg(x) (12) (A B), A B (13) B B, (A B) A B (14) A A (15) A A G3ip( ) G3 P, Γ P ( P ) A, B, Γ C A B, Γ C L Γ A Γ B Γ A B R A, Γ C B, Γ C A B, Γ C L Γ A Γ A B R 1 Γ B Γ A B R 2 20
21 A B, Γ A B, Γ C A B, Γ C A, Γ B Γ A B R, Γ C L the root first proof search G1 (1) weakning A A P, Γ P P (2) Γ multiset Γ {A, B, C} {A, C, B} multiset {A, B, C} {A, A, B, C} Γ (3) R Γ A B Γ, A B Γ R Γ A Γ B Γ A B Shared context A, B A A, B B R A, B A B A B A B R A (B A B) R 5 (1) (A B) (B A) (2) (A (B C)) ((A B) C) 21
22 (3) A A (4) (A B) C (A C) (B C) (5) (A C) (B C) (A B) C 5 G3ip A A (3) A ((A ) ) A, A A, A L A, A A (A ) R A ((A ) ) R 6 (1) (A (B C)) ((A B) (A C)) (2) (A B C) ((A C) (B C)) (3) ( A B) (A B) (4) (P P ) (5) (A (A B)) (A B) admissibility G3ip Γ D D, C Γ, C -free system G3ip cut-free G3ip cut cut cut G3ip cut admissible G3ip+cut G3ip cut G3ip conservative extension 22
23 cut admissibility weight 1 A w(a) 1 w( ) = 0 2 P w(p ) = 1 3 w(a B) = w(a) + w(b) + 1,, w( A) = w(a ) = w(a) + w( ) + 1 = w(a) height L 0 1 C contextγ sequent C, Γ C [ ] C w(c) 1 C = P C = P C = 11 C, Γ C L C = P C =,, Γ L, Γ R w(c) n C C, Γ C w(d) n + 1 D D, Γ D D = A B w(a) n w(b) n A B, Γ A B A, B, Γ A A B, Γ A L A B, Γ A B A, B, Γ B A B, Γ B L R A, B, Γ A A, B, Γ B B, Γ = Γ A, Γ = Γ A, Γ A B, Γ B context 11 23
24 D = A B w(a) n w(b) n A, Γ A A, Γ A B R B, Γ B 1 B, Γ A B R 2 L A B, Γ A B D = A B A, A B, Γ A B, A, Γ B A, A B, Γ B A B, Γ A B R sequent QED n Γ C sequent Γ C G3ip n 4 Height-preserving weakening D n D, Γ C n Γ C [ ] Base case: n = 0 Γ C L C Γ Γ D, Γ C L Induction base: n height-preserving weakening n Induction step: L Γ = A B, Γ A, B, Γ C A B, Γ C L A, B, Γ C n D, A, B, Γ C n L n + 1 D, A B, Γ C QED 24
25 5: Inversion lemma (i) (ii) (iii) n A B, Γ C n A, B, Γ C, n A B, Γ C n A, Γ C n B, Γ C, n A B, Γ C n B, Γ C [ ] n (i) A B, Γ C L A B A B context A B, Γ C C C, A B, Γ C A B, Γ context A, B, Γ C L n (i) n+1 A B, Γ C A B A B L A, B, Γ C n A B A B, Γ C A B, Γ C n n A, B, Γ C n A, BΓ C A, B, Γ C n + 1 (ii) (i) A B, Γ C A, Γ C B, Γ C A B L A, Γ C B, Γ C n A B A B, Γ C A B, Γ C n (1) n A, Γ C (2) n B, Γ C (3) n A, Γ C (4) n B, Γ C (1) (3) A, Γ C (2) (4) B, Γ C n + 1 (iii) A B, Γ C A B B, Γ C n A B A B, Γ C A B, Γ C n 25
26 n B, Γ C n B, Γ C B, Γ C n + 1 QED A B, Γ A B, Γ C A B, Γ C inversion (*) n A B, Γ C n A B, Γ A Inversion lemma (i) n A B, Γ C n A, B, Γ C A B, Γ C ( A, B, Γ C n (*) A B, Γ C A B, Γ A (*) L R cut 12 G3ip inconsistent, L R Contraction contraction 6 Height-preserving contraction n D, Γ C n D, D, Γ C 12 cut 26
27 [ ] n = 0 D, D, Γ C L C D, Γ C L n contraction contraction formula (1) contraction formula D contraction D, D, Γ C D, D, Γ C n n D, Γ C n+1 D, Γ C (2) D D D = A B L n A, B, A B, Γ C 5 n A, B, A, B, A B, Γ C n A, B, Γ C L n+1 A B, Γ C D = A B D = A B n A B, A B, Γ A n B, A B, Γ C sequent n A B, Γ A 5 sequent n B, B, Γ C n B, Γ C n+1 A B, Γ C QED D = A B cut L cut cut cut L L cut L cut, Γ C P, Γ P P, C 27
28 P, C C = P C L cut P, Γ, C L cut formula 7: -Height cut cut-height cut 8 cut Γ D D, C Γ, C G3ip (1) cut L (2) cut cut formula (3) cut cut formula (4) cut cut formula cut L 1 Γ D L 11 cut formula D Γ cut Γ, C D, C weakening cut 12 Γ Γ, C L 2 D, C L 21 C Γ, C 22 C = D Γ C weakening cut Γ, C (cut ) 23 Γ, C L 24 D = cut Γ, C Γ, C 28
29 Γ 1 L Γ 13 cut-height cut 3 cut formula D D (R- sequent n, m, k, 31 L Γ = A B, Γ cut-height n m cut A, B, Γ D A B, Γ D L D, C A B, Γ, C L, L A, B, Γ D D, C A, B, Γ, C A B, Γ, C L cut-height n + m 32 L Γ = A B, Γ cut-height max(n, m) k A, Γ D B, Γ D A B, Γ L D D, C A B, Γ, C n + k m + k cut A, Γ D D, C B, Γ D D, C A, Γ, C B, Γ, C A B, Γ L, C 13 sequent 29
30 33 Γ = A B, Γ cut-height max(n, m)+1+k A B, Γ A B, Γ D A B, Γ D D, C A B, Γ, C m + k cut A B, Γ A A B, Γ, A W k B, Γ D D, C B, Γ, C A B, Γ, C cut-height 4 cut formula D cut-height cut 41 L = A B, D, A, B, C Γ D D, A B, C L Γ, A B, C cut-height n + m + 1 n + m Γ D D, A, B, C Γ, A, B, C Γ, A B, C L 42 L = A B, cut-height n + max(m, k) + 1 D, A, C D, B, C Γ D D, A B, C Γ, A B, C L n + m n + k cut Γ D D, A, C Γ D D, B, C Γ, A, C Γ, B, C Γ, A B, L C 43 cut-height n + max(m, k)
31 Γ D D, A B, A D, B, C D, A B, C Γ, A B, C n + m n + k cut Γ D D, A B, A Γ D D, B, C Γ, A B, A Γ, B, C Γ, A B, C 44 R C = A B cut-height n+max(m, k)+ 1 D, A D, B R Γ D D, A B Γ, A B n + m n + k cut Γ D D, A Γ D D, B Γ, A Γ, B R Γ, A B 45 R C = A B cut-height n + m + 1 n + k + 1 cut D, A Γ D D, A B R 1 Γ, A B D, B Γ D D, A B R 2 Γ, A B cut-height n + m n + k Γ D D, A Γ, A Γ, A B R 1 Γ D D, B Γ, B Γ, A B R 2 46 R C = A B cut-height n + m + 1 D, A, B Γ D D, A B R Γ, A B cut-height n + m cut 31
32 Γ D D, A, B Γ, A, B Γ, A B R cut-height 5 cut formula 51 D = A B cut-height max(n, m) k + 1 cut Γ A Γ B A, B, C R Γ A B A B, C L Γ, C height n + k m + max(n, k) + 1 cut Γ A A, B, C Γ B Γ, B, C Γ, Γ, C Γ, C Ctr contraction cut-height cut-height cut-formula cut cut-height 52 D = A B cut-height n max(m, k) + 1 Γ A Γ A B R A, C B, C 1 L A B, C Γ, C cut-height Γ B Γ A B R A, C B, C 2 L A B, C Γ, C cut-height n + m n + k 32
33 Γ A A, C Γ, C Γ B B, C Γ, C cut-height cut 53 D = A B n max(m, k) + 1 A, Γ B Γ A B R A B A B, C A B, C Γ, C A, Γ B Γ A B R A B, A Γ, A Γ, C A, Γ B B, C A, Γ, C cut n+1+m, n + k, max(n + 1, m) max(n, k) + 1 cut cut cut cut cut QED cut cut cut (a) (subformula property) 9 Γ C G3ip Γ, C G3ip cut 9 G3ip consistency 33
34 10 G3ip A B A B [ ] sequent R QED (b) 10 disjunction property Underivability results:g3ip sequent A A P P sequent sequent A A A P 11 sequent G3ip (i) P P (ii) (iii) (iv) P P ((P Q) P ) P (P Q R) ((P Q) (P R)) [ ] (i) P P disjunction property P P P P P R inversion P (ii): P P P P (i) root P P P L P P P (P ) R root P P 34
35 (iii): ((P Q) P ) P (Peirce s Law) 2 (P Q) P P Q P P (P Q) P P ((P Q) P ) P R sequent R P, (P Q) P P Q P, P Q P, (P Q) P Q (P Q) P P Q R P, P Q (P Q) P P Q P P Q R (P Q) P P Q ((P Q) P ) P (iv): (P Q R) ((P Q) (P R)) sequent R (P Q R) ((P Q) (P R)) sequent R 1 (P Q R) (P Q) (P Q R) ((P Q) (P R)) R 1 R R P, Q Q P, R Q L P, P Q R P P, Q R Q P, P Q R Q P Q R P Q R P Q R (P Q) (P R) R 1 P, R Q R P Q R P Q R P P Q R P Q R P Q P Q R P Q P Q R (P Q) (P R) R 1 35
36 R 2 sequent QED (c) A B ( A B) sequent (i) (ii) (iii) ( A B) A B A B A B (A B) A B (d) G3ip cut contraction 12 G3ip sequent Γ C [ ] sequent sequent sequent sequent L sequent sequent L endsequent G3cp: (1) A A (3) A A (4)RAA 36
37 Ạ A G-system sequent (multi succedent ) sequent A, Γ, B Γ, A B A, Γ B Γ A B A A, A, A multisuccedent R A (A G3cp Logical axioms: P, Γ, P Logical rules: G3cp A, B, Γ A B, Γ L Γ, A Γ, B Γ, A B R 37
38 A, Γ B, Γ A B, Γ L Γ, A, B Γ, A B R Γ, A B, Γ A B, Γ A, Γ, B Γ, A B R, Γ L G3cp G3cp invertible inversion 13 G3cp invertible invertible sequent R Γ, A Γ, B R Γ, A B n Γ, A B n Γ, A n Γ, B R invertible G3cp L, L invertible G3ip sequent C R R Γ, A B L A B Γ, A Γ, B L n inversion n+1 Γ, A B A B Γ, A B Γ, A B n n Γ, A n Γ, B n Γ, A n Γ, B 38
39 Γ, A Γ, B n + 1 A B R Γ, A Γ, B n QED G3cp weakning contraction Γ A, Γ LW Γ Γ, A RW A, A, Γ A, Γ LC Γ, A, A Γ, A RC G3cp cut rule, Γ, D D, Γ Γ, Γ, admissible G3ip 14 Height-preserving contraction: n C, C, Γ n C, Γ n Γ, C, C n Γ, C [ ] n = 0 L contraction L n contraction (1)contract contraction rule (2)contract 6 L L 6 R n Γ, A B, A n Γ, A B, B 13 invertibility Γ, A, A n Γ, B, B n contraction n Γ, A n Γ, B R n+1 Γ, A B n R R R vdash n A, Γ, A B, B height-preserving invertibility n A, A, Γ, B, B 39
40 n A, Γ, B R contraction A B, Γ, A B, A B, Γ A B, A B, Γ inversion n Γ, A, A n B, B, Γ n Γ, A n B, Γ A B, Γ n + 1 QED G3cp cut G3ip 15 cut Γ, D D, Γ Γ, Γ, G3cp cut G3cp 16 G3cp sequent Γ Γ, G3i G3c G3i A(t/x), xa, Γ C xa, Γ C L Γ A(y/x) Γ xa R A(y/x), Γ C Γ A(t/x) xa, Γ C L Γ A R R y Γ xa L y xa, Γ, C G3c 40
41 A(t/x), xa, Γ xa, Γ L Γ, A(y/x) Γ, xa R A(y/x), Γ Γ, A, A(t/x) xa, Γ L Γ, A R y Γ, xa L y xa, Γ, G3i G3c 16 t A(t/x) xa xa G3i G3c 17 G3i Γ C G3c Γ Γ, C (Γ, sequent G3i G3c G3i G3i xa cut R 18 G3i xa t A(t/x) existence property G3i x A xa 19 sequent x A xa G3i [ ] root R x A x A xa x A xa x A xa R x A x A A(t/x) x A A(t/x) x A xa R x A xa R R 41
42 R x A, A(y/x), A(z/x) x A, A(y/x) A(z/x) R x A, A(y/x) x A R, A(y/x) x A, A(y/x) x A A(y/x) R x A x A R xa x A xa x A xa R R x A, A(y/x), A(z/x), sequents G3ip sequent Γ sequent G3ip topsequent,, P 1,, P m Q 1,, Q n,,, G3ip Γ sequent 0 m, n 0 20 G3ip sequent Γ topsequent G3ip root-first C C (1)sequent C G3ip (2) tree topsequents L (3) sequent C sequent 42
43 21 regular sequence P 1,, P m Q 1,, Q n,,, sequent P i Q j sequent regular sequent trace formula 1 P 1 P m Q 1 Q n, m, n > 0 2 Q 1 Q n, m = 0, n > 0 3 (P 1 P m ), m > 0, n = 0 4, m, n = 0 regular sequence P i, Q j C L topsequent regular sequent tree trace formula trace formula C 1,, C n C (C 1 C n ) C 1 C n C C 1 C n C i P 1 P m Q 1 Q n trace formula P 1 P m Q 1 Q n C 1 C n 22 C regular sequence trace formulas C (C 1 C n ) sequent 23 valuation 0, 1 v(p ) = 0 v(p ) = 1 43
44 v( ) = 0 v(a B) = min(v(a), v(b)) v(a B) = max(v(a), v(b)) v(a B) = max(1 v(a), v(b)) v(a B) = 1 iff v(a) v(b) multiset Γ (Γ), (Γ) C Γ v (Γ) = min(v(c)), v (Γ) = max(v(c)) 24 sequent Γ v (Γ) > v ( ) v sequent Γ Γ v v (Γ) v ( ) 25 v min(v(a), v(b)) v(c) iff v(a) v(b C) 26 min(v(a B), v(a)) v(b) G3cp 27 Soundness sequent Γ G3cp Γ sequent L v (P, Γ) v (, P ) v (, Γ) v ( ) A, B, Γ A B, Γ L v (A, B, Γ) v( ) v (A B, Γ) = v (A, B, Γ) v (A B, Γ) v( ) 25,26 R QED 44
45 28 Completeness Γ G3cp Γ G3cp root first L regular sequent Γ regular sequents regular sequents Γ 1 1, Γ n n n > 0) regular sequent trace formula C i 22 C C 1 C n C (Γ) ( ) Γ v(c) = 1 C C 1 C n v(c) v(c 1 C n ) C i v v(c i ) = 1 C i 1 C i (P 1 P m ) P j 1 C i P 1 P m Q i Q r Q i Q r P j 1 Q k 0 C i 0 regular sequent QED 45
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寄せられた質問 : 演習問題について この講義の範囲に含まれる適切な演習問題が載っている参考書がありますか? できれば解答や解説が付いているものがあると良いのですが 第 回の授業の中で 演習問題に取り組む方法を説明しますこの授業は 回だけ行うもので 書籍の1 冊分に比べると少ない分量しかカバーしていません 回の講義の概観 : 1 完全性と不完全性 命題論理 命題論理 ( 真理値 ) ( 公理と推論規則
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数学ノート 2018-
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2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac