\[\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}\]

# Sections

Sections are naming scopes that permit creating section-local declarations that can be used by other declarations in the section. Declarations made with `Variable`, `Hypothesis`, `Context` (or the plural variants of the first two) and definitions made with `Let`, `Let Fixpoint` and `Let CoFixpoint` within sections are local to the section.

In proofs done within the section, section-local declarations are included in the local context of the initial goal of the proof. They are also accessible in definitions made with the `Definition` command.

## Using sections

Sections are opened by the `Section` command, and closed by `End`. Sections can be nested. When a section is closed, its local declarations are no longer available. Global declarations that refer to them will be adjusted so they're still usable outside the section as shown in this example.

Command Section ident

  Opens the section named `ident`. Section names do not need to be unique.

Command End ident

  Closes the section or module named `ident`. See Terminating an interactive module or module type definition for a description of its use with modules.

  After closing the section, the section-local declarations (variables and section-local definitions, see `Variable`) are discharged, meaning that they stop being visible and that all global objects defined in the section are generalized with respect to the variables and local definitions they each depended on in the section.

  Error There is nothing to end.
  Error Last block to end has name ident.

Note

Most commands, such as the Hint commands, `Notation` and option management commands that appear inside a section are canceled when the section is closed. In some cases, this behaviour can be tuned with locality attributes. See this table.

Command Let ident_decl def_body
Command Let Fixpoint fix_definition with fix_definition*
Command Let CoFixpoint cofix_definition with cofix_definition*

  These are similar to `Definition`, `Fixpoint` and `CoFixpoint`, except that the declared constant is local to the current section. When the section is closed, all persistent definitions and theorems within it that depend on the constant will be wrapped with a `term_let` with the same declaration.

  As for `Definition`, `Fixpoint` and `CoFixpoint`, if `term` is omitted, `type` is required and Coq enters proof mode. This can be used to define a term incrementally, in particular by relying on the `refine` tactic. See Entering and exiting proof mode.

Attribute clearbody

  When used with `Let` in a section, clears the body of the definition in the proof context of following proofs. The kernel will still use the body when checking.

Note

Terminating the proof for a `Let` with `Qed` produces an opaque side definition. `Let foo : T. Proof. tactics. Qed.` is equivalent to

Lemma foo_subproof : T. Proof. tactics. Qed. #[clearbody] Let foo := foo_subproof.

Command Context binder+

  Declare variables in the context of the current section, like `Variable`, but also allowing implicit variables, Implicit generalization, and let-binders.

  Context {A : Type} (a b : A). Context `{EqDec A}. Context (b' := b).

See also

Section Binders. Section Sections and contexts in chapter Typeclasses.

Example: Section-local declarations

Section s1.

Variables x y : nat.
  x is declared y is declared

The command `Let` introduces section-wide Let-in definitions. These definitions won't persist when the section is closed, and all persistent definitions which depend on `y'` will be prefixed with `let y' := y in`.

Let y' := y.
  y' is defined
Definition x' := S x.
  x' is defined
Definition x'' := x' + y'.
  x'' is defined

Print x'.
  x' = S x : nat x' uses section variable x.
Print x''.
  x'' = x' + y' : nat x'' uses section variables x y.
End s1.
Print x'.
  x' = fun x : nat => S x : nat -> nat Arguments x' x%nat_scope
Print x''.
  x'' = fun x y : nat => let y' := y in x' x + y' : nat -> nat -> nat Arguments x'' (x y)%nat_scope

Notice the difference between the value of `x'` and `x''` inside section `s1` and outside.

## Summary of locality attributes in a section

This table sums up the effect of locality attributes on the scope of vernacular commands in a `Section`, when outside the `Section` where they were entered. In the following table:

- a cross (❌) marks an unsupported attribute (compilation error);

- “not available” means that the command has no effect outside the `Section` it was entered;

- “available” means that the effects of the command persists outside the `Section`.

- For `Definition` (and `Lemma`, ...), `Canonical Structure`, `Coercion` and `Set` (and `Unset`), some locality attributes will be passed on to the `Module` containing the current `Section`, see the associated footnotes.

A similar table for `Module` can be found here.

`Command` | no attribute | `local` | `export` | `global`
--- | --- | --- | --- | ---
`Definition`, `Lemma`, `Axiom`, ... | available [1] | `local` in module [1] | ❌ | ❌
`Ltac` | `local` | not available | ❌ | ❌
`Ltac2` | `local` | not available | ❌ | ❌
`Notation (abbreviation)` | `local` | not available | ❌ | ❌
`Notation` | `local` | not available | ❌ | ❌
`Tactic Notation` | `local` | not available | ❌ | ❌
`Ltac2 Notation` | `local` | not available | ❌ | ❌
`Coercion` | `global` | not available | ❌ | `global` in module [2]
`Canonical Structure` | `global` | not available | ❌ | `global` in module [2]
`Hints` (and `Instance`) | `local` | not available | ❌ | ❌
`Set` or `Unset` a flag | available [3] | not available | `export` in module [3] | `global` in module [3]

[1] (1,2)

For `Definition`, `Lemma`, ... the default visibility is to be available outside the section and available with a short name when the current `Module` is imported (with `Import` or cmd:`Export`) outside the current `Module`. The `local` attribute make the corresponding identifiers available in the current `Module` but only with a fully qualified name outside the current `Module`.

[2] (1,2)

For `Coercion` and `Canonical Structure`, the `global` visibility, which is the default, makes them available outside the section, in the current `Module`, and outside the current `Module` when it is imported (with `Import` or cmd:`Export`).

[3] (1,2,3)

For `Set` and `Unset`, the `export` and `global` attributes both make the command's effects persist outside the current section, in the current `Module`. It will also persist outside the current `Module` with the `global` attribute, or with the `export` attribute, when the `Module` is imported (with `Import` or cmd:`Export`). The default behaviour (no attribute) is to make the setting persist outside the section in the current `Module`, but not outside the current `Module`.

## Typing rules used at the end of a section

From the original rules of the type system, one can show the admissibility of rules which change the local context of definition of objects in the global environment. We show here the admissible rules that are used in the discharge mechanism at the end of a section.

Abstraction. One can modify a global declaration by generalizing it over a previously assumed constant \(c\). For doing that, we need to modify the reference to the global declaration in the subsequent global environment and local context by explicitly applying this constant to the constant \(c\).

Below, if \(Γ\) is a context of the form \([y_1 :A_1 ;~…;~y_n :A_n]\), we write \(∀x:U,~\subst{Γ}{c}{x}\) to mean \([y_1 :∀ x:U,~\subst{A_1}{c}{x};~…;~y_n :∀ x:U,~\subst{A_n}{c}{x}]\) and \(\subst{E}{|Γ|}{|Γ|c}\) to mean the parallel substitution \(E\{y_1 /(y_1~c)\}…\{y_n/(y_n~c)\}\).

First abstracting property:

\[\frac{\WF{E;~c:U;~E′;~c′:=t:T;~E″}{Γ}} {\WF{E;~c:U;~E′;~c′:=λ x:U.~\subst{t}{c}{x}:∀x:U,~\subst{T}{c}{x};~\subst{E″}{c′}{(c′~c)}} {\subst{Γ}{c′}{(c′~c)}}}\]
\[\frac{\WF{E;~c:U;~E′;~c′:T;~E″}{Γ}} {\WF{E;~c:U;~E′;~c′:∀ x:U,~\subst{T}{c}{x};~\subst{E″}{c′}{(c′~c)}}{\subst{Γ}{c′}{(c′~c)}}}\]
\[\frac{\WF{E;~c:U;~E′;~\ind{p}{Γ_I}{Γ_C};~E″}{Γ}} {\WFTWOLINES{E;~c:U;~E′;~\ind{p+1}{∀ x:U,~\subst{Γ_I}{c}{x}}{∀ x:U,~\subst{Γ_C}{c}{x}};~ \subst{E″}{|Γ_I ;Γ_C |}{|Γ_I ;Γ_C | c}} {\subst{Γ}{|Γ_I ;Γ_C|}{|Γ_I ;Γ_C | c}}}\]

One can similarly modify a global declaration by generalizing it over a previously defined constant \(c\). Below, if \(Γ\) is a context of the form \([y_1 :A_1 ;~…;~y_n :A_n]\), we write \(\subst{Γ}{c}{u}\) to mean \([y_1 :\subst{A_1} {c}{u};~…;~y_n:\subst{A_n} {c}{u}]\).

Second abstracting property:

\[\frac{\WF{E;~c:=u:U;~E′;~c′:=t:T;~E″}{Γ}} {\WF{E;~c:=u:U;~E′;~c′:=(\letin{x}{u:U}{\subst{t}{c}{x}}):\subst{T}{c}{u};~E″}{Γ}}\]
\[\frac{\WF{E;~c:=u:U;~E′;~c′:T;~E″}{Γ}} {\WF{E;~c:=u:U;~E′;~c′:\subst{T}{c}{u};~E″}{Γ}}\]
\[\frac{\WF{E;~c:=u:U;~E′;~\ind{p}{Γ_I}{Γ_C};~E″}{Γ}} {\WF{E;~c:=u:U;~E′;~\ind{p}{\subst{Γ_I}{c}{u}}{\subst{Γ_C}{c}{u}};~E″}{Γ}}\]

Pruning the local context. If one abstracts or substitutes constants with the above rules then it may happen that some declared or defined constant does not occur any more in the subsequent global environment and in the local context. One can consequently derive the following property.

First pruning property:
  \[\frac{% \WF{E;~c:U;~E′}{Γ}% \hspace{3em}% c~\kw{does not occur in}~E′~\kw{and}~Γ% }{% \WF{E;E′}{Γ}% }\]

Second pruning property:
  \[\frac{% \WF{E;~c:=u:U;~E′}{Γ}% \hspace{3em}% c~\kw{does not occur in}~E′~\kw{and}~Γ% }{% \WF{E;E′}{Γ}% }\]

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