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Sunday, November 17, 2013

Induction

Linear Induction

$$\begin{aligned} &\phi[a] \\ &\forall \mu.(\phi[\mu] \implies \phi[s(\mu]) \\ \hline \\ &\forall \nu.\phi[\nu] \\ \end{aligned}$$

It's crucial that the signature consists of no other object constants or function constants !

Tree Induction

$$\begin{aligned} &\phi[a] \\ &\forall \mu.(\phi[\mu] \implies \phi[f(\mu]) \\ &\forall \mu.(\phi[\mu] \implies \phi[g(\mu]) \\ \hline \\ &\forall \nu.\phi[\nu] \\ \end{aligned}$$

Structural Induction

The most general form of induction !

$$\begin{aligned} &\phi[a] \\ &\phi[b] \\ &\forall \lambda. \forall \mu.((\phi[\lambda] \land \phi[\mu]) \implies \phi[c(\lambda, \mu)]) \\ \hline \\ &\forall \nu.\phi[\nu] \\ \end{aligned}$$

More at Introduction to Logic.

# posted by rot13(Unafba Pune) @ 8:11 AM

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