Proof that type does not have decidable equality in Agda

Question

Can one create such function in Agda ?

ℕ→ℕ-undecidable : ¬ ( (f g :  ℕ → ℕ ) → Dec (f ≡ g)) 
ℕ→ℕ-undecidable = ?

I am particularly interested in proof using cubical Agda.

Answer 1

ℕ→ℕ-undecidable is not provable in Agda. If we postulate the law of excluded middle (LEM), it follows that equality on every set is decidable, contradicting ℕ→ℕ-undecidable. Since Agda is consistent with LEM, it follows that ℕ→ℕ-undecidable is not provable in base Agda. This holds the same for cubical and vanilla Agda.

Answer 2

The flag --injective-type-constructors makes Agda anti-classical. The proof of this can be seen here. See also this question on the Proof Assitants StackExchange.