this improves support for structural recursion over inductive
*predicates* when there are reflexive arguments.
Consider
```lean
inductive F: Prop where
| base
| step (fn: Nat → F)
-- set_option trace.Meta.IndPredBelow.search true
set_option pp.proofs true
def F.asdf1 : (f : F) → True
| base => trivial
| step f => F.asdf1 (f 0)
termination_by structural f => f`
```
Previously the search for the right induction hypothesis would fail with
```
could not solve using backwards chaining x✝¹ : F
x✝ : x✝¹.below
f : Nat → F
a✝¹ : ∀ (a : Nat), (f a).below
a✝ : Nat → True
⊢ True
```
The backchaining process will try to use `a✝ : Nat → True`, but then has
no idea what to use for `Nat`.
There are three steps here to fix this.
1. We let-bind the function's type before the whole process. Now the
goal is
```
funType : F → Prop := fun x => True
x✝ : x✝¹.below
f : Nat → F
a✝¹ : ∀ (a : Nat), (f a).below
a✝ : ∀ (a : Nat), funType (f a)
⊢ funType (f 0)
```
2. Instead of using the general purpose backchaining proof search, which
is more
powerful than we need here (we need on recursive search and no
backtracking),
we have a custom search that looks for local assumptions that
provide evidence of `funType`, and extracts the arguments from that
“type” application to construct the recursive call.
Above, it will thus unify `f a =?= f 0`.
3. In order to make progress here, we also turn on use
`withoutProofIrrelevance`,
because else `isDefEq` is happy to say “they are equal” without actually
looking
at the terms and thus assigning `?a := 0`.
This idea of let-binding the function's motive may also be useful for
the other recursion compilers, as it may simplify the FunInd
construction. This is to be investigated.
fixes#4751
671ce7afd3