104 lines
3.8 KiB
Org Mode
104 lines
3.8 KiB
Org Mode
* Reducible hints
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Lean automation can be configured using different commands and
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annotations. The =reducible= hint/annotation instructs automation
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which declarations can be freely unfolded. One of the main components
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of the Lean elaborator is a procedure for solving simultaneous
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higher-order unification constraints. Higher-order unification is a
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undecidable problem. Thus, the procedure implemented in Lean is
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clearly incomplete, that is, it may fail to find a solution for a set
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of constraints. One way to guide/help the procedure is to indicate
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which declarations can be unfolded. We should not confuse the
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=reducible= hint with whether a declaration is opaque or not. We say
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_opaqueness_ is part of the Lean logic, and is implemented inside of
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its trusted kernel. The =reducible= hint is just a way to
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control/guide Lean automation to fill missing gaps in our proofs and
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definitions. The Lean kernel ignores this annotation.
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The higher-order unification procedure has to perform case-analysis.
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The procedure is essentially implementing a backtracking search. This
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procedure has to decide whether a definition =C= should be unfolded or
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not. Here, we roughly divide this decision in two groups: _simple_
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and _complex_. We say an unfolding decision is _simple_ if the
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procedure does not have to consider an extra case (aka
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case-split). That is, it does not increase the search space. We say an
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unfolding decision is _complex_ if it produces at least one extra
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case, and consequently increases the search space.
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Users can mark whether a definition is =reducible= or =irreducible=.
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We write =reducible(C)= to denote that =C= was marked as reducible by the user,
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and =irreducible(C)= to denote that =C= was marked as irreducible by the user.
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Theorems are never unfolded. For a transparent definition =C=, the
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higher-order unification procedure uses the following decision tree.
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#+BEGIN_SRC
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if simple unfolding decision then
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if irreducible(C) then
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do not unfold
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else
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unfold
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end
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else -- complex unfolding decision
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if reducible(C) then
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unfold
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else if irreducible(C) then
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do not unfold
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else if C was defined in the current module then
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unfold
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else
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do not unfold
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end
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end
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#+END_SRC
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For an opaque definition =D=, the higher-order unification procedure uses the
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same decision tree if =D= was declared in the current module. Otherwise, it does
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not unfold =D=.
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#+END_SRC
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The following command declares a transparent definition =pr= and mark it as reducible.
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#+BEGIN_SRC lean
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definition pr1 [reducible] (A : Type) (a b : A) : A := a
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#+END_SRC
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The =reducible= mark is saved in the compiled .olean files. The user
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can temporarily change the =reducible= and =irreducible= marks using
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the following commands. The temporary modification is effective only in the
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current file, and is not saved in the produced .olean file.
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#+BEGIN_SRC lean
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definition id (A : Type) (a : A) : A := a
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definition pr2 (A : Type) (a b : A) : A := b
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-- mark pr2 as reducible
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reducible pr2
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-- ...
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-- mark id and pr2 as irreducible
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irreducible id pr2
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#+END_SRC
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The annotation =[persistent]= can be used to instruct Lean to make the
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modification permanent, and save it in the .olean file.
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#+BEGIN_SRC lean
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definition pr2 (A : Type) (a b : A) : A := b
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-- Mark pr2 as irreducible.
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-- The modification will affect modules that import this one.
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irreducible [persistent] pr2
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#+END_SRC
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The reducible and irreducible annotations can be removed using the modifier =[none]=.
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#+BEGIN_SRC lean
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definition pr2 (A : Type) (a b : A) : A := b
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-- temporarily remove any reducible and irreducible annotation from pr2
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reducible [none] pr2
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-- permanently remove any reducible and irreducible annotation from pr2
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reducible [persistent] [none] pr2
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#+END_SRC
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Finally, the command =irreducible= is syntax sugar for =reducible [off]=.
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The commands =reducible= and =reducible [on]= are equivalent.
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