I'm trying to define the following quantity partn:
variable pi : nat -> Prop
variable (Hdecp : ∀ p, decidable (pi p))
definition partn (n : nat) : nat := ∏ p ∈ (prime_factors n), (if pi p then p^(mult p n) else 1)
but get the error
error: failed to synthesize placeholder
pi : ℕ → Prop,
n p : ℕ
⊢ decidable (pi p)
How can I help Lean recognize that (pi p) is indeed decidable thanks to Hdecp?
edit: The elaborator can actually infer the instance completely on its own, as long it's available in the definition's context:
variable (Hdecp : ∀ p, decidable (pi p))
include Hdecp
definition partn (n : nat) : nat := ∏ p ∈ (prime_factors n), (if pi p then p^(mult p n) else 1)
original answer (still useful if the instance has more complex hypotheses):
If you want to avoid the explicit call to ite, you can locally introduce the decidable instance:
definition partn (n : nat) : nat := ∏ p ∈ (prime_factors n),
have decidable (pi p), from Hdecp p,
if pi p then p^(mult p n) else 1
I found a solution:
definition partn (n : nat) : nat := ∏ p ∈ (prime_factors n), (@ite (pi p) (Hdecp p) nat (p^(mult p n)) 1)
which allows me to explicitly use Hdecp in my if-the-else