recursion - Assisting Agda's termination checker to make a recursive function -

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Text after "

Suppose that we want to define a function for some types A₀ ⋯ Aₙ and B. 

  f: A₀ ⋯ Aₙ → B f x₀ ⋯ xₙ = ... recursive call for F ...

Assume this definition is not structurally recursive.

Also assume that I have definitions for the following:

  1. A function g: A₀ ⋯ Aₙ → ℕ which For each case of f for a natural number

  2. f : There is a proof that the argument G passed arguments for the recurring call passed (in relation to the built _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _

    3. I will put these parts together to create a recursive function;

    This question is the follow up on the same topic, for which they have been fully answered. However, I think the type of function of the example is misfortune ℕ → ℕ , which makes it difficult for me to see how the general case spreads.

    Also, as a note, I was not expecting a big answer with explanation on the well-established induction and implementation of its implementation, as in @VITUS Has been provided.

    You have a function a → b and you know that b is a Well established relationship, you can build a good on-founded relationship a : 

      _ & lt; _: B → b → set f: a → b _≺_: a → a → set x ≺ y = fx & lt; Fiscal year - or more concisely _≺_ = _ & lt; On the '_' function - the function module comes 'on'

    certification that _≺ _ is well-only that is not difficult, but it is not From the standard library, we will use it: 

      Open Inverse-Image {A = A} - New Type {_ & lt; _ = _≺_} - Changing the name of the new relative (well-established to set up properly)

    inverted-image Export as well as established value value We can then use this value to get the actual value that is recursive for us: 

      ≺-well established = & lt; ⇒≺- Well established & lt;

    And so on, ≺-rec can now be used to implement with recursive functions> A As an argument type.

    ≺-rec then it is used almost: 

      g: a → cg = ≺-rap_ { - Layer Parameter -} _ {- A → set; In case of return, if fire dependence -} go {-help functions -} can go: ∀ X → → (∀ y → y ≺ x → c) → go to x rk = ... (RR RGR proof) . ..

    arg is the argument for the recurring event and the evidence is an evidence that the argument is actually small.

    When the return type depends on logic, the helper function looks almost like this: 

      go: ∀x → (∀ y → y ≺ x → cy) → C x

    While dealing with the function with many arguments, you have to bunch them in the same type so that relations can talk about them all. It usually means the pair or, if there is a dependency between logic, dependent pairs.


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