Implicit arguments

Stream: lean4

[Dan Velleman]

This example works:

example (P : Nat → Prop) : (∀ {x : Nat}, P x) → ∀ {x : Nat}, P x := by
  intro h
  exact h

But this one doesn't:

example (P : Nat → Prop) : (∀ {x : Nat}, P x) → ∀ {x : Nat}, P x := by
  intro h
  have h2 : 0 = 0 := Eq.refl 0
  exact h

The exact h in the second example generates the error:

type mismatch
  h
has type
  P ?m.190 : Prop
but is expected to have type
  ∀ {x : Nat}, P x : Prop

In both examples, in the tactic state, both h and the goal are listed as ∀ {x : Nat}, P x, so it looks to a user like exact h should work. Examining the expressions for h and the goal, they are identical except that in the second example, the goal has mdata associated with it that says "noImplicitLambda". It is the introduction of h2 that causes that mdata to be attached to the goal, which explains why the first example works but the second doesn't.
Is this how it's supposed to work?

[Yury G. Kudryashov]

Here is an #mwe

structure TestProp (p : Nat → Prop) : Prop where
  add : ∀ ⦃m n⦄, p m → p n → p (m + n)

structure TestProp' (p : Nat → Prop) : Prop where
  add : ∀ {m} ⦃n⦄, p m → p n → p (m + n)

example : TestProp fun _ ↦ True where
  add := fun _ _ _ _ ↦ trivial

example : TestProp' fun _ ↦ True where
  add := fun _ _ ↦ trivial

Note that in the second example, I need two underscores, not three. IMHO, this is a bug. Occurred to me on Mathlib with the actual predicate implying s ⊆ t for some sets: ∀ {{x}} from the definition of ⊆ got autointroduced.