Ein pragmatisierter Kalkul des naturlichen Schlieβens nebst Metatheorie



44    1 Zum grammatischen Rahmen

., θ*k>, <ξo, ., ξk>, r∏ζΔoπ]] = [θ*k, ξk, r∏ζ[<θ*o, ., θ*k>, <ξo, ., ξk>, ʌo]ɔ] =
r∏ζ[<θ*o, ., θ*k>, <ξo, ., ξk>, Δo,' = [<θ*o, ., θ*k}, <ξo, ., ξk>, ∏ζΔ I = [<θ*0, .,
θ*k}, <ξo, ., ξk>, Δ].

Angenommen ξ ζ fur alle i k+1. Dann ist [θ*k, ξk, [<θ*o, ., θ*k), <ξo, ., ξk>, Δ]]
= [θ*
k, ξk, [<θ*o, ., θ*k>, <ξo, ., ξk-1>, r∏ζΔoπ]] = [θ*k, ξk, r∏ζ[<θ*o, ., θ*k>, <ξo, .,
ξ
k>, Δ ] ] = r∏ζ[θ*k, ξk, [<θ*o, ., θ*k>, <ξo, ., ξk>, Δ H . Mit I.V. gilt [θ*k, ξk, [<θ*o,
., θ*
k>, <ξo, ., ξk>, Δo]] = [<θ*o, ., θ*k>, <ξo, ., ξk>, Δo]. Also [θ*k, ξk, [<θ*o, .,
θ*
k-1>, <ξo, ., ξk-1>, Δ]] = r∏ζ[<θ*o, ., θ*k>, <ξo, ., ξk>, Δ ] = [<θ*o, ., θ*k>, <ξo, ., ξk>,
r∏ζΔoπ] = [<θ*o, ., θ*k>, <ξo, ., ξk>, Δ]. ■

Theorem 1-29. Mehrfache Substitution von geschlossenen Termen fur paarweise verschiedene
Variablen in Termen und Formeln (b)

Wenn k N{o}, {θ*o, ., θ*k} GTERM und {ξo, ., ξ} VAR, wobei ξi ≠ ξ∙ fur alle i, j
< k+1 mit i j, dann:

(i) Wenn θ TERM, dann

[<θ*o, ., θ*k-1>, <ξo, ., ξk-1>, [θ*k, ξk, θ]] = [<θ*o, ., θ*ft>, <ξo, ., ξfc>, θ], und
(ii) Wenn Δ
FORM, dann

[<θ*o, ., θ*k-1>, <ξo, ., ξk-1>, [θ*k, ξk, Δ]] = [<θ*o, ., θ*k>, <ξo, ., ξk>, Δ].

Beweis: Seien k N\{o}, {θ*o, ., θ*k} GTERM und {ξo, ., ξk} VAR, wobei ξi
ξ
j fur alle i, j k+1 mit i j. Zu (i): Sei θ TERM. Der Beweis wird mittels Induktion
uber
k gefuhrt. Sei k = 1. Dann ist mit Theorem 1-25-(i) und Theorem 1-28-(i) [θ*o, ξo,
[θ*
1, ξ1, θ]] = [θ*1, ξ1, [θ*o, ξo, θ]] = [<θ*o, θ*1>, <ξo, ξ1>, θ]. Sei nun 1 < k. Durch Anwen-
dung von I.V., Theorem 1-25-(i), I.V., Theorem 1-28-(i), I.V. und Theorem 1-28-(i) in
dieser Reihenfolge ergibt sich: [
<θ*o, ., θ*k-1>, <ξo, ., ξk-1>, [θ*k, ξk, θ]] = [<θ*o, ., θ*k-
2
>, <ξo, ., ξk-2>, [θ*k-1, ξk-1, [θ*k, ξk, θ]]] = [<θ*o, ., θ*k-2>, <ξo, ., ξk-2>, [θ*k, ξk, [θ*k-1, ξk-1,
θ]]] = [
<θ*o, ., θ*k-2, θ*k>, <ξo, ., ξk-2, ξk>, [θ*k-1, ξk-1, θ]] = [θ*k, ξk, [<θ*o, ., θ*k-2>, <ξo,
., ξ
k-2>, [θ*k-1, ξk-1, θ]]] = [θ*k, ξk, [<θ*o, ., θ*k-1>, <ξo, ., ξk-1>, θ]] = [<θ*o, ., θ*k>, <ξo,
., ξ
k>, θ].

(ii) folgt auf analogem Wege aus Theorem 1-25-(ii) und Theorem 1-28-(ii). ■



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