# Allgemeine Gleichheitssubstitution/Relation und Gleichheit/Ableitbar/Aufgabe

Zur Navigation springen Zur Suche springen

Es seien ${\displaystyle {}s_{1},\ldots ,s_{n},t_{1},\ldots ,t_{n}}$ Terme einer prädikatenlogischen Sprache ${\displaystyle {}L^{S}}$ und seien ${\displaystyle {}x_{1},\ldots ,x_{n}}$ verschiedene Variablen.

1. Es sei ${\displaystyle {}R}$ ein ${\displaystyle {}k}$-stelliges Relationssymbol und ${\displaystyle {}r_{1},\ldots ,r_{k}}$ seien Terme. Zeige die Ableitbarkeit
${\displaystyle \vdash s_{1}=t_{1}\wedge \ldots \wedge s_{n}=t_{n}\rightarrow {\left((Rr_{1}\ldots r_{k}){\frac {s_{1},\ldots ,s_{n}}{x_{1},\ldots ,x_{n}}}\rightarrow (Rr_{1}\ldots r_{k}){\frac {t_{1},\ldots ,t_{n}}{x_{1},\ldots ,x_{n}}}\right)}.}$
2. Es seien ${\displaystyle {}r_{1}}$ und ${\displaystyle {}r_{2}}$ Terme. Zeige die Ableitbarkeit
${\displaystyle \vdash s_{1}=t_{1}\wedge \ldots \wedge s_{n}=t_{n}\rightarrow {\left(r_{1}{\frac {s_{1},\ldots ,s_{n}}{x_{1},\ldots ,x_{n}}}=r_{2}{\frac {s_{1},\ldots ,s_{n}}{x_{1},\ldots ,x_{n}}}\rightarrow r_{1}{\frac {t_{1},\ldots ,t_{n}}{x_{1},\ldots ,x_{n}}}=r_{2}{\frac {t_{1},\ldots ,t_{n}}{x_{1},\ldots ,x_{n}}}\right)}.}$