Definition of `Induction.WellFounded.WfRec`

[jamesmckinna]

Currently, the definition:

WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
WfRec _<_ P x = ∀ y → y < x → P y

suffers from two (unfortunate) defects:

• the use of the name RecStruct, when this type is nothing more or less than that of PredicateTransformer (or: what you will...) which only really takes on the role of being a structure-for-recursion once a given such T : PredicateTransformer is accompanied by a proof that T P ⊆′ P for suitable P...; ADDRESSED by Predicate transformers: Reconciling Induction.RecStruct with Relation.Unary.PredicateTransformer.PT #2140
• the explicit quantification over y in WfRec, which leads, infectively, to a profusion of (almost-?)always-inferrable arguments when working with subsequent Accessibility and WellFoundedness proofs. ADDRESSED by Changes explicit argument y to implicit in Induction.WellFounded.WfRec #2084

The first of these is, indeed, unfortunate but ignorable.

The second is, frankly, a complete pain, but likewise (almost) completely avoidable.

PROPOSAL: to make the breaking change for v2.0 that the definition be changed to:

WfRec _<_ P x = ∀ {y} → y < x → P y

together with all of its downstream consequences... including the in-progress #2077 #2082 etc.

[jamesmckinna]

A separate issue exposed by this proposal: in Data.Vec.Relation.Binary.Lex.Strict, the following lemma proposed in #602 and added in #1672 and its immediate corollary (ditto. for List in #1648)

  xs≮[] : ∀ {n} (xs : Vec A n) → ¬ xs < []
  xs≮[] _ (base ())

  ¬[]<[] : ¬ [] < []
  ¬[]<[] = xs≮[] []

should, likewise, make the xs argument implicit. FIXED by #2085

[JacquesCarette]

Both of these explicit proposals make sense to me.

[MatthewDaggitt]

the explicit quantification over y in WfRec, which leads, infectively, to a profusion of (almost-?)always-inferrable arguments when working with subsequent Accessibility and WellFoundedness proofs. FIXED by #2084

I'm not sure I agree with this. What happens when A is a record type and < is a strict partial order over it that only inspects some of the fields? Then you're not going to be able to infer the values of the missing fields right?