lean2/doc/lean/reducible.org

* Reducible hints

Lean automation can be configured using different commands and
annotations. The =reducible= hint/annotation instructs automation
which declarations can be freely unfolded. One of the main components
of the Lean elaborator is a procedure for solving simultaneous
higher-order unification constraints. Higher-order unification is a
undecidable problem. Thus, the procedure implemented in Lean is
clearly incomplete, that is, it may fail to find a solution for a set
of constraints. One way to guide/help the procedure is to indicate
which declarations can be unfolded. We should not confuse the
=reducible= hint with whether a declaration is opaque or not.  We say
_opaqueness_ is part of the Lean logic, and is implemented inside of
its trusted kernel. The =reducible= hint is just a way to
control/guide Lean automation to fill missing gaps in our proofs and
definitions. The Lean kernel ignores this annotation.

The higher-order unification procedure has to perform case-analysis.
The procedure is essentially implementing a backtracking search.  This
procedure has to decide whether a definition =C= should be unfolded or
not.  Here, we roughly divide this decision in two groups: _simple_
and _complex_.  We say an unfolding decision is _simple_ if the
procedure does not have to consider an extra case (aka
case-split). That is, it does not increase the search space.  We say an
unfolding decision is _complex_ if it produces at least one extra
case, and consequently increases the search space.

Users can mark whether a definition is =reducible= or =irreducible=.
We write =reducible(C)= to denote that =C= was marked as reducible by the user,
and =irreducible(C)= to denote that =C= was marked as irreducible by the user.

Theorems are never unfolded. For a transparent definition =C=, the
higher-order unification procedure uses the following decision tree.

#+BEGIN_SRC
if simple unfolding decision then
  if irreducible(C) then
     do not unfold
  else
     unfold
  end
else -- complex unfolding decision
  if reducible(C) then
     unfold
  else
     do not unfold
  end
end
#+END_SRC

For an opaque definition =D=, the higher-order unification procedure uses the
same decision tree if =D= was declared in the current module. Otherwise, it does
not unfold =D=.

The following command declares a transparent definition =pr= and mark it as reducible.

#+BEGIN_SRC lean
definition pr1 [reducible] (A : Type) (a b : A) : A := a
#+END_SRC

The =reducible= mark is saved in the compiled .olean files.  The user
can temporarily change the =reducible= and =irreducible= marks using
the following commands. The temporary modification is effective only in the
current file, and is not saved in the produced .olean file.

#+BEGIN_SRC lean
  definition id (A : Type) (a : A) : A := a
  definition pr2 (A : Type) (a b : A) : A := b
  -- mark pr2 as reducible
  reducible pr2
  -- ...
  -- mark id and pr2 as irreducible
  irreducible id pr2
#+END_SRC

The annotation =[persistent]= can be used to instruct Lean to make the
modification permanent, and save it in the .olean file.

#+BEGIN_SRC lean
  definition pr2 (A : Type) (a b : A) : A := b
  -- Mark pr2 as irreducible.
  -- The modification will affect modules that import this one.
  irreducible [persistent] pr2
#+END_SRC

The reducible and irreducible annotations can be removed using the modifier =[none]=.

#+BEGIN_SRC lean
  definition pr2 (A : Type) (a b : A) : A := b

  -- temporarily remove any reducible and irreducible annotation from pr2
  reducible [none] pr2

  -- permanently remove any reducible and irreducible annotation from pr2
  reducible [persistent] [none] pr2
#+END_SRC

Finally, the command =irreducible= is syntax sugar for =reducible [off]=.
The commands =reducible= and =reducible [on]= are equivalent.