# Ground Instantiation plus Ground Resolution as a Proof System

We introduce this proof system as an intermediate step between the naive enumeration algorithm based on Herbrand's theorem and resolution for first-order logic.

Our goal is to avoid explicit enumeration and apply proof system where we do as little search (guessing) as possible, while still being complete.

## Definition

The proof system has two rules: ground instantiation and ground resolution.

#### Ground Instantiation Rule

where is a clause and is a ground substitution.

Ground substitution is of the form where each is a ground term.

#### Ground Resolution Rule

If is a ground atom and are ground claues, then

Note that this is propositional resolution where propositional variables have “long names” (they are ground atoms).

## Completeness

The proof is based on Herbrand's Expansion Theorem (see also the proof of Compactness for First-Order Logic).

Suppose a set of clauses is contradictory. By Herbrand's Expansion Theorem and Compactness Theorem for Propositional Formulas, some finite subset is contradictory. Then there exists a derivation of empty clause from viewed as set of propositional formulas, using propositional resolution. In other words, there exists a derivation of empty clause from using ground resolution rule. Each element of can be obtained from an element of using instantiation rule. This means that there exists a proof tree whose leaves are followed by a single application of instantiation rule, and inner nodes contain ground resolution steps.