Converting Coq to Idris

Question

What would be some useful guidelines for converting Coq source to Idris (e.g. how similar are their type systems and what can be made of translating the proofs)? From what I gather, Idris' built-in library of tactics is minimal yet extendable, so I suppose with some extra work this should be possible.

Answer 1

I've recently translated a chunk of Software Foundations and did a partial port of {P|N|Z}Arith, some observations I've made in the process:

Generally using Idris tactics (in their Pruvloj/Elab.Reflection form) is not really recommended at the moment, this facility is somewhat fragile, and pretty hard to debug when something goes wrong. It's better to use the so-called "Agda style", relying on pattern matching where possible. Here are some rough equivalences for simpler Coq tactics:

• intros - just mention variables on the LHS
• reflexivity - Refl
• apply- calling the function directly
• simpl - simplification is done automatically by Idris
• unfold - also done automatically for you
• symmetry - sym
• congruence/f_equal - cong
• split - split in LHS
• rewrite - rewrite ... in
• rewrite <- - rewrite sym $ ... in
• rewrite in - to rewrite inside something you have as a parameter you can use the replace {P=\x=>...} equation term construct. Sadly Idris is not able to infer P most of the time, so this becomes a bit bulky. Another option is to extract the type into a lemma and use rewrites, however this won't always work (e.g., when replace makes a large chunk of a term disappear)
• destruct - if on a single variable, try splitting in LHS, otherwise use the with construct
• induction - split in LHS and use a recursive call in a rewrite or directly. Make sure that at least one of arguments in recursion is structurally smaller, or you'll fail totality and won't be able to use the result as a lemma. For more complex expressions you can also try SizeAccessible from Prelude.WellFounded.
• trivial - usually amounts to splitting in LHS as much as possible and solving with Refls
• assert - a lemma under where
• exists - dependent pair (x ** prf)
• case - either case .. of or with. Be careful with case however - don't use it in anything you will later want to prove things about, otherwise you'll get stuck on rewrite (see issue #4001). A workaround is to make top-level (currently you can't refer to lemmas under where/with, see issue #3991) pattern-matching helpers.
• revert - "unintroduce" a variable by making a lambda and later applying it to said variable

• inversion - manually define and use trivial lemmas about constructors:

  -- injectivity, used same as `cong`/`sym`
  FooInj : Foo a = Foo b -> a = b
  FooInj Refl = Refl
  
  -- disjointness, this sits in scope and is invoked when using `uninhabited`/`absurd`
  Uninhabited (Foo = Bar) where   
    uninhabited Refl impossible