Merge PR #12782: Trying to rephrase complex sentences to make them ea…

…sier to read.

Reviewed-by: jfehrle
Ack-by: Mbodin
Ack-by: corwin-of-amber

doc/sphinx/language/extensions/evars.rst

@@ -13,13 +13,13 @@ Existential variables
    | ?[ ?@ident ]
    | ?@ident {? @%{ {+; @ident := @term } %} }
 
-|Coq| terms can include existential variables which represents unknown
-subterms to eventually be replaced by actual subterms.
+|Coq| terms can include existential variables that represent unknown
+subterms that are eventually replaced with actual subterms.
 
-Existential variables are generated in place of unsolvable implicit
+Existential variables are generated in place of unsolved implicit
 arguments or “_” placeholders when using commands such as ``Check`` (see
 Section :ref:`requests-to-the-environment`) or when using tactics such as
-:tacn:`refine`, as well as in place of unsolvable instances when using
+:tacn:`refine`, as well as in place of unsolved instances when using
 tactics such that :tacn:`eapply`. An existential
 variable is defined in a context, which is the context of variables of
 the placeholder which generated the existential variable, and a type,
@@ -43,22 +43,18 @@ existential variable is represented by “?” followed by an identifier.
    Check identity _ (fun x => _).
 
 In the general case, when an existential variable :n:`?@ident` appears
-outside of its context of definition, its instance, written under the
-form :n:`{ {*; @ident := @term} }` is appending to its name, indicating
+outside its context of definition, its instance, written in the
+form :n:`{ {*; @ident := @term} }`, is appended to its name, indicating
 how the variables of its defining context are instantiated.
-The variables of the context of the existential variables which are
-instantiated by themselves are not written, unless the :flag:`Printing Existential Instances` flag
-is on (see Section :ref:`explicit-display-existentials`), and this is why an
-existential variable used in the same context as its context of definition is written with no instance.
+Only the variables that are defined in another context are displayed:
+this is why an existential variable used in the same context as its
+context of definition is written with no instance.
+This behaviour may be changed: see :ref:`explicit-display-existentials`.
 
 .. coqtop:: all
 
    Check (fun x y => _) 0 1.
 
-   Set Printing Existential Instances.
-
-   Check (fun x y => _) 0 1.
-
 Existential variables can be named by the user upon creation using
 the syntax :n:`?[@ident]`. This is useful when the existential
 variable needs to be explicitly handled later in the script (e.g.
@@ -88,6 +84,14 @@ Explicit displaying of existential instances for pretty-printing
    context of an existential variable is instantiated at each of the
    occurrences of the existential variable.
 
+.. coqtop:: all
+
+   Check (fun x y => _) 0 1.
+
+   Set Printing Existential Instances.
+
+   Check (fun x y => _) 0 1.
+
 .. _tactics-in-terms:
 
 Solving existential variables using tactics

doc/sphinx/language/extensions/implicit-arguments.rst

@@ -70,7 +70,7 @@ is said *contextual* if it can be inferred only from the knowledge of
 the type of the context of the current expression. For instance, the
 only argument of::
 
-  nil : forall A:Set, list A`
+  nil : forall A:Set, list A
 
 is contextual. Similarly, both arguments of a term of type::
 
@@ -539,7 +539,7 @@ with free variables into a closed statement where these variables are
 quantified explicitly.  Use the :cmd:`Generalizable` command to designate
 which variables should be generalized.
 
-It is activated for a binder by prefixing a \`, and for terms by
+It is activated within a binder by prefixing it with \`, and for terms by
 surrounding it with \`{ }, or \`[ ] or \`( ).
 
 Terms surrounded by \`{ } introduce their free variables as maximally

doc/sphinx/language/extensions/match.rst

@@ -94,7 +94,7 @@ The expression :n:`let ( {*, @ident__i } ) := @term__0 in @term__1`
 performs case analysis on :n:`@term__0` whose type must be an
 inductive type with exactly one constructor.  The number of variables
 :n:`@ident__i` must correspond to the number of arguments of this
-contrustor.  Then, in :n:`@term__1`, these variables are bound to the
+constructor.  Then, in :n:`@term__1`, these variables are bound to the
 arguments of the constructor in :n:`@term__0`.  For instance, the
 definition