2436587:9;<>=@?ACBDBEGFH=I.JA=KELNM
◦
OQPR$SUTWV X4Y[Z\]Z^`_4TZX4abZ\
cd)egf:h"i8jlkmjlnlhmd4do$ijpo.qQohmjlir
• p
sq
sp 1s p 0s t.ulvxw *zy {0y|{.{0)ww*
• A
sB
sA 1s A 0s t.u>v~}{0 w*(xtwtw>y {0yQ{.{0)w
• Γ
s∆
sΠ
sΓ 1s Γ 0s t.u|v~w
")Kw*(}{0 w*u8{0
Γ
∆
*W){0*Γ, ∆
s w,w ")wp)t*{0g>) {t*t )t*,p) )}{0 ,w
A
*z){0A
u• Γ . ∆
v*)g• Γ, A . ∆, A
v)[~{0=
*g)({0N) }{0 w). g)t),(x {0•
*)g8*Q 0¡~¢[£g¤¥¦w)@~b zCy z{0w§ K0t)z*t*(*g)8{0g {0*(w*}ww*z{0g*([~{0*u)()}{0 wA
*Cy {0.wpw*g). A
wm¨*u•
©ª .w*zpx} )t*vΓ . A, ∆
Γ, ¬A . ∆ ( ¬
) Γ, A . ∆ Γ . ¬A, ∆ ( ¬
) Γ . A, ∆ Γ, B . ∆
Γ, A → B . ∆ (→
) Γ, A . B, ∆
Γ . A → B, ∆ (→
) Γ, A . ∆ Γ, B . ∆
Γ, A ∨ B . ∆ (∨
) Γ . A, B, ∆ Γ . A ∨ B, ∆ (∨
) Γ, A, B . ∆
Γ, A ∧ B . ∆ (∧
) Γ . A, ∆ Γ . B, ∆ Γ . A ∧ B, ∆ (∧
)
«¬>g®¯g°±¯G² ³ ´®µ·¶l¸x¹º°»°§¼½)¾ ¿ {0 zw*C}{0 ,w*v
• p → (q → p)
• (p → (q → r)) → ((p → q) → (p → r))
• (p → q) → ((p → ¬q) → ¬p)
• (p ∧ q) ∨ (p → ¬q)
!${0g )w§,}{0 w0 >¨*CyÀ0Cy {0.wUv
• (p ∨ q) ∧ (p → ¬q)
«¬>g®¯g°±¯Á ³ ´®[~¶>¸Qg ¼¼¹º»±gÂÃH¸·½®°Ä¼½¾ '~{0g8*w *¿y {0yQ{.{0)ww*
p 1 , . . . , p n , q
s{0
n ≥ 0
*CÅ zy|{. }Æ.uÀÇW{0) Å y w§,}{0 wv[(· · · (p 1 ∧ p 2 ) · · · ∧ p n ) → q] → [p 1 → (p 2 → · · · (p n → q) · · ·)]
È )ww(*
s
}{0)t{0U
n
s w§À WCwmÉ (Cy )Àt*{0) ·Ê 'À{008twtw w§Kw w§,}{0 ,wt "*)s
g)w>K Kzw){0 , w.)*·Ê
«¬>g®¯g°±¯ÌË ³ÎÍ>°ÏGл°§Ñ¯¹x¼°±µ·ÒÓÃÆÌ»&Ô¹x¬l°±µ·Ï̾ Õ4t) tK)U{0g >w*y|{.wH, *
)x w§H){0{0pÉ[~{0w>t0({0)U>É{0 K))Cw*}ww*z* *z
y )H*[~{0*Hw§}{0
Γ, p . p, ∆
s {0p
* )Hw Hy {0y|{.{0)ww K)y|{0Kg *x w§tw§0K *z*)gzy {0.w*u){0 (t*w§{0Åy {t
ª
xNy)v
Ö uz!${0g ),{0K*g)w§U}{0
Γ, A ` A, ∆
0xG Hy ))w*)w w*C}ww*{0g(*[~{0*Cy|{0 Kg C)}{0 w{0).u+xuz *x ){0C*g)y {00w0x y Q>Éy|{0 W}ww*)
*[~{0*(yQ{0 K( (*}{0 w*{0g)*u
«¬>g®¯g°±¯ ³8¹)°±º»°ÄÂÂÏÌÒQ¼0¾ !${0g )yQ{0 {0)
Γ
sΓ 0s ∆
s ∆ 0s w*)g Γ . ∆
Γ . ∆
*Cy {0.w
s
w{0 Cw*g)
Γ, Γ 0 . ∆, ∆ 0 *z 0wgwgu
£ÆÕÅyQ{0 K K{0) yÀ C *t )t* Cw§À (* *zy .u
«¬>g®¯g°±¯ ³pµÀ®0ÏGÂUÒÆµÀ®0ÏG¹)»±g¯µQÒ*µ·ÒƯ.¼°Ä¸·g¼Uðĵ·ÒƯ.¼°Ä¸·)¾
•
¥@£0¥·*{0)w Wy {0yQ{.{0)wws
{0w§)g{0p)w y {0yQ{.{0)ww.u
•
)¥£0¡¦ !*"±¢¦*C)x#{0)t{0Å w Kxpu•
)%$.&U¦'·0(&£0¥¦)*!+"Ä¢0¦*z) t*{0# {0)t{0$ tw§)*(x# {0)t*u '-,~ g)s
{0@)v
•
)¥£0¡¦.!*.Ä¢¦,*C)t*{0# {0)t{0$w Kx|u•
)%$.&U¦'·0(&£0¥¦) !*"±¢¦*C)x#{0)t{0Åtw§)K*zt*{0# {0)t*u±w§){0{0b(w KwQ*w§@u~yw(}{0 z){0 Uwt*{0# {0)tv
(p ∨ ¬q ∨ r) ∧ (q ∨ r)
u¿ {0 g)y|{0 C{0}{0 w
A
s w>@x)}{0 ){0 Uwt*{0#{0)tA ∧ C)}{0
){0 Uwx# {0)t
A ∨ ww)w*C {0z}{0 ,w*
A
sA ∧ A ∨ {0g(*.wg*u
«¬>g®¯g°±¯0/ '~{0
A 0 = p → q
s yQ{0 z{0n ≤ 0
sA n+1 = A n → q
u!${0g )w *)g
. A n , p
*zy {0.w>zwn
*CyÀ u«¬>g®¯g°±¯21 3t 4 Çww*b}{0)t{0)ÅgtwtwgÅ*b}{0 * ){0 Uw*t*{0#{0)t4
x#{0)tU@)*wm¨@ tt*,1y|{0 (tKÀ0g)}{0 w{0)*.u