Normal form (natural deduction)

In mathematical logic and proof theory, a derivation in normal form in the context of natural deduction refers to a proof which contains no detours — steps in which a formula is first introduced and then immediately eliminated.

The concept of normalization in natural deduction was introduced by Dag Prawitz in the 1960s as part of a general effort to analyze the structure of proofs and eliminate unnecessary reasoning steps.^[1] The associated normalization theorem establishes that every derivation in natural deduction can be transformed into normal form.

Natural deduction is a system of formal logic that uses introduction and elimination rules for each logical connective. Introduction rules describe how to construct a formula of a particular form, while elimination rules describe how to infer information from such formulas. A derivation is in normal form if it contains no formula which is both:

• the conclusion of an introduction rule, and
• the major premise of an elimination rule.

A derivation containing such a structure is said to include a detour. Normalization involves transforming a derivation to remove all such detours, thereby producing a proof that directly reflects the logical dependencies of the conclusion on the assumptions.

Another definition of normal derivation in classical logic is:^[2]

A derivation in NK is normal if all major premisses of E-rules are assumptions.

The normalization theorem for natural deduction states that:

Every derivation in natural deduction can be converted into a derivation in normal form.

This result was first proved by Dag Prawitz in 1965.^[1] The normalization process typically involves identifying and eliminating maximal formulas — formulas introduced and immediately eliminated—through a sequence of local reduction steps.

Normalization has several important consequences:

• It implies the subformula property: any formula occurring in the proof is a subformula of the assumptions or conclusion.
• It guarantees consistency of the system: there is no derivation of a contradiction from no assumptions.
• It supports constructive content in logic: proofs correspond to explicit constructions or computations.

A derivation of ${\displaystyle A\rightarrow A}$ that includes a detour:

1. [A]    (assumption)
2. A → A  (→ introduction, discharging 1)
3. [A]    (assumption)
4. A      (→ elimination on 2 and 3)

This introduces and then immediately eliminates an implication. A normal derivation is:

1. [A]
2. A → A  (→ introduction)

A derivation of ${\displaystyle A,B\vdash A}$ that includes a detour:

${\displaystyle {\frac {{\frac {A\quad B}{A\land B}}[\land {\text{I}}]}{A}}[\land {\text{E}}]\quad \Rightarrow \quad A}$

The elimination is unnecessary if ${\displaystyle A}$ is already available.

Normalization is central to several areas of logic and computer science:

• In proof theory, it ensures that logical systems have desirable meta-properties such as consistency and the subformula property.
• In type theory, it underlies the soundness and completeness of type-checking algorithms.
• In proof assistants (e.g. Coq, Agda), normalization is used to verify that formal proofs are constructive and terminating.
• In functional programming, the normalization process corresponds to evaluation strategies for typed lambda calculi.

• Natural deduction
• Curry–Howard correspondence
• Cut-elimination theorem
• Sequent calculus

• Prawitz, Dag (1965). Natural Deduction: A Proof-Theoretical Study (Thesis). Stockholm Studies in Philosophy. Vol. 3. Stockholm: Almqvist & Wiksell.
• Prawitz, Dag (2006) [1965]. Natural Deduction: A Proof-Theoretical Study (Reprint of the 1965 thesis ed.). Mineola, New York: Dover Publications. ISBN 9780486446554. OCLC 61296001.
• Sørensen, Morten Heine; Urzyczyn, Paweł (2006) [1998]. Lectures on the Curry–Howard isomorphism. Studies in Logic and the Foundations of Mathematics. Vol. 149. Elsevier Science. CiteSeerX 10.1.1.17.7385. ISBN 978-0-444-52077-7.
• Troelstra, A. S.; Schwichtenberg, H. (2000). Basic Proof Theory. Cambridge University Press. ISBN 9780521779111.
• von Plato, Jan (2013). Elements of logical reasoning (1 ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-03659-8.