# Conversion rules¶

Coq has conversion rules that can be used to determine if two terms are equal by definition in CIC, or convertible. Conversion rules consist of reduction rules and expansion rules. Equality is determined by converting both terms to a normal form, then verifying they are syntactically equal (ignoring differences in the names of bound variables by alpha-conversion).

See also

Applying conversion rules, which describes tactics that apply these conversion rules.

Reductions convert terms to something that is incrementally
closer to its normal form. For example, zeta-reduction removes
`let ident := term`

constructs from a term by replacing
_{1} in term_{2}`ident`

with `term`

wherever _{1}`ident`

appears in `term`

.
The resulting term may be longer or shorter than the original._{2}

- Eval cbv zeta in let i := 1 in i + i.
- = 1 + 1 : nat

Expansions are reductions applied in the opposite direction,
for example expanding `2 + 2`

to `let i := 2 in i + i`

. While applying
reductions gives a unique result, the associated
expansion may not be unique. For example, `2 + 2`

could also be
expanded to `let i := 2 in i + 2`

. Reductions that have a unique inverse
expansion are also referred to as contractions.

The normal form is defined as the result of applying a particular set of conversion rules (beta-, delta-, iota- and zeta-reduction and eta-expansion) repeatedly until it's no longer possible to apply any of them.

Sometimes the result of a reduction tactic will be a simple value, for example reducing
`2*3+4`

with `cbv beta delta iota`

to `10`

, which requires applying several
reduction rules repeatedly. In other cases, it may yield an expression containing
variables, axioms or opaque contants that can't be reduced.

The useful conversion rules are shown below. All of them except for eta-expansion
can be applied with conversion tactics such as `cbv`

:

Conversion name

Description

beta-reduction

eliminates

`fun`

delta-reduction

replaces a defined variable or constant with its definition

zeta-reduction

eliminates

`let`

eta-expansion

replaces a term

`f`

of type`forall a : A, B`

with`fun x : A => f x`

match-reduction

eliminates

`match`

fix-reduction

replaces a

`fix`

with a beta-redex; recursive calls to the symbol are replaced with the`fix`

termcofix-reduction

replaces a

`cofix`

with a beta-redex; recursive calls to the symbol are replaced with the`cofix`

termiota-reduction

match-, fix- and cofix-reduction together

Applying conversion rules describes tactics that only apply conversion rules. (Other tactics may use conversion rules in addition to other changes to the proof state.)

## α-conversion¶

Two terms are α-convertible if they are syntactically
equal ignoring differences in the names of variables bound within the expression.
For example `forall x, x + 0 = x`

is α-convertible with `forall y, y + 0 = y`

.

## β-reduction¶

β-reduction reduces a beta-redex, which is
a term in the form `(fun x => t) u`

. (Beta-redex
is short for "beta-reducible expression", a term from lambda calculus.
See Beta reduction
for more background.)

Formally, in any global environment \(E\) and local context \(Γ\), the beta-reduction rule is:

- Beta
- \[\frac{% % }{% E[Γ] ⊢ ((λx:T.~t)~u)~\triangleright_β~\subst{t}{x}{u}% }\]

We say that \(\subst{t}{x}{u}\) is the *β-contraction* of
\(((λx:T.~t)~u)\) and, conversely, that \(((λ x:T.~t)~u)\) is the
*β-expansion* of \(\subst{t}{x}{u}\).

Terms of the *Calculus of Inductive Constructions*
enjoy some fundamental properties such as confluence,
strong normalization, subject reduction. These results are
theoretically of great importance but we will not detail them here and
refer the interested reader to [Coq85].

## δ-reduction¶

δ-reduction replaces variables defined in local contexts or constants defined in the global environment with their values. Unfolding means to replace a constant by its definition. Formally, this is:

- Delta-Local
- \[\frac{% \WFE{\Gamma}% \hspace{3em}% (x:=t:T) ∈ Γ% }{% E[Γ] ⊢ x~\triangleright_Δ~t% }\]

- Delta-Global
- \[\frac{% \WFE{\Gamma}% \hspace{3em}% (c:=t:T) ∈ E% }{% E[Γ] ⊢ c~\triangleright_δ~t% }\]

Delta-reduction only unfolds constants that are marked transparent. Opaque is the opposite of transparent; delta-reduction doesn't unfold opaque constants.

## ι-reduction¶

A specific conversion rule is associated with the inductive objects in the global environment. We shall give later on (see Section Well-formed inductive definitions) the precise rules but it just says that a destructor applied to an object built from a constructor behaves as expected. This reduction is called ι-reduction and is more precisely studied in [PM93a][Wer94].

## ζ-reduction¶

ζ-reduction removes let-in definitions in terms by replacing the defined variable by its value. One way this reduction differs from δ-reduction is that the declaration is removed from the term entirely. Formally, this is:

- Zeta
- \[\frac{% \WFE{\Gamma}% \hspace{3em}% \WTEG{u}{U}% \hspace{3em}% \WTE{\Gamma::(x:=u:U)}{t}{T}% }{% E[Γ] ⊢ \letin{x}{u:U}{t}~\triangleright_ζ~\subst{t}{x}{u}% }\]

## η-expansion¶

Another important concept is η-expansion. It is legal to identify any term \(t\) of functional type \(∀ x:T,~U\) with its so-called η-expansion

for \(x\) an arbitrary variable name fresh in \(t\).

Note

We deliberately do not define η-reduction:

This is because, in general, the type of \(t\) need not be convertible to the type of \(λ x:T.~(t~x)\). E.g., if we take \(f\) such that:

then

We could not allow

because the type of the reduced term \(∀ x:\Type(2),~\Type(1)\) would not be convertible to the type of the original term \(∀ x:\Type(1),~\Type(1)\).

## Examples¶

Example: Simple delta, fix, beta and match reductions

`+`

is a notation for`Nat.add`

, which is defined with a`Fixpoint`

.

- Print Nat.add.
- Nat.add = fix add (n m : nat) {struct n} : nat := match n with | 0 => m | S p => S (add p m) end : nat -> nat -> nat Arguments Nat.add (n m)%nat_scope
- Goal 1 + 1 = 2.
- 1 goal ============================ 1 + 1 = 2
- cbv delta.
- 1 goal ============================ (fix add (n m : nat) {struct n} : nat := match n with | 0 => m | S p => S (add p m) end) 1 1 = 2
- cbv fix.
- 1 goal ============================ (fun n m : nat => match n with | 0 => m | S p => S ((fix add (n0 m0 : nat) {struct n0} : nat := match n0 with | 0 => m0 | S p0 => S (add p0 m0) end) p m) end) 1 1 = 2
- cbv beta.
- 1 goal ============================ match 1 with | 0 => 1 | S p => S ((fix add (n m : nat) {struct n} : nat := match n with | 0 => m | S p0 => S (add p0 m) end) p 1) end = 2
- cbv match.
- 1 goal ============================ S ((fix add (n m : nat) {struct n} : nat := match n with | 0 => m | S p => S (add p m) end) 0 1) = 2
The term can be fully reduced with

`cbv`

:

- Goal 1 + 1 = 2.
- 1 goal ============================ 1 + 1 = 2
- cbv.
- 1 goal ============================ 2 = 2

## Proof Irrelevance¶

It is legal to identify any two terms whose common type is a strict
proposition \(A : \SProp\). Terms in a strict propositions are
therefore called *irrelevant*.

## Convertibility¶

Let us write \(E[Γ] ⊢ t \triangleright u\) for the contextual closure of the relation \(t\) reduces to \(u\) in the global environment \(E\) and local context \(Γ\) with one of the previous reductions β, δ, ι or ζ.

We say that two terms \(t_1\) and \(t_2\) are
*βδιζη-convertible*, or simply convertible, or
definitionally equal, in the
global environment \(E\) and local context \(Γ\) iff there
exist terms \(u_1\) and \(u_2\) such that \(E[Γ] ⊢ t_1 \triangleright
… \triangleright u_1\) and \(E[Γ] ⊢ t_2 \triangleright … \triangleright u_2\) and either \(u_1\) and
\(u_2\) are identical up to irrelevant subterms, or they are convertible up to η-expansion,
i.e. \(u_1\) is \(λ x:T.~u_1'\) and \(u_2 x\) is
recursively convertible to \(u_1'\), or, symmetrically,
\(u_2\) is \(λx:T.~u_2'\)
and \(u_1 x\) is recursively convertible to \(u_2'\). We then write
\(E[Γ] ⊢ t_1 =_{βδιζη} t_2\).

Apart from this we consider two instances of polymorphic and cumulative (see Chapter Polymorphic Universes) inductive types (see below) convertible

if we have subtypings (see below) in both directions, i.e.,

and

Furthermore, we consider

convertible if

and we have that \(c\) and \(c'\) are the same constructors of different instances of the same inductive types (differing only in universe levels) such that

and

and we have

The convertibility relation allows introducing a new typing rule which says that two convertible well-formed types have the same inhabitants.