\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#3})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]

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).

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

\[λx:T.~(t~x)\]

for $x$ an arbitrary variable name fresh in $t$.

Note

We deliberately do not define η-reduction:

\[λ x:T.~(t~x)~\not\triangleright_η~t\]

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:

\[f ~:~ ∀ x:\Type(2),~\Type(1)\]

then

\[λ x:\Type(1).~(f~x) ~:~ ∀ x:\Type(1),~\Type(1)\]

We could not allow

\[λ x:\Type(1).~(f~x) ~\triangleright_η~ f\]

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

\[E[Γ] ⊢ t~w_1 … w_m =_{βδιζη} t~w_1' … w_m'\]

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

\[E[Γ] ⊢ t~w_1 … w_m ≤_{βδιζη} t~w_1' … w_m'\]

and

\[E[Γ] ⊢ t~w_1' … w_m' ≤_{βδιζη} t~w_1 … w_m.\]

Furthermore, we consider

\[E[Γ] ⊢ c~v_1 … v_m =_{βδιζη} c'~v_1' … v_m'\]

convertible if

\[E[Γ] ⊢ v_i =_{βδιζη} v_i'\]

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

\[E[Γ] ⊢ c~v_1 … v_m : t~w_1 … w_m\]

and

\[E[Γ] ⊢ c'~v_1' … v_m' : t'~ w_1' … w_m '\]

and we have

\[E[Γ] ⊢ t~w_1 … w_m =_{βδιζη} t~w_1' … w_m'.\]

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

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` term
cofix-reduction	replaces a `cofix` with a beta-redex; recursive calls to the symbol are replaced with the `cofix` term
iota-reduction	match-, fix- and cofix-reduction together