Proving to Agda that we're talking about the same thing

Question

I'm trying to prove a contradiction, but I run into an issue trying to prove to Agda that the sigma domain type returned by the <>-wt-inv is the same sigma as seen earlier in the proof. I expect that the uniq-type proof should help me there, but I can't fit them together.

I hope the comments in the code below give enough context.

-- given a type for (f ⟨⟩), we can derive that f is a function type
-- and we can prove that the context yields σ 
⟨⟩-wt-inv : ∀ {n m f τ} {K : Ktx n m} → K ⊢ (f ⟨⟩) ∶ τ → 
            ∃ λ σ → K Δ↝ σ × K ⊢ f ∶ (σ ⇒ τ)
⟨⟩-wt-inv (_⟨_⟩ {τ = σ} K⊢f∶σ⇒τ KΔ↝σ) = σ , (KΔ↝σ , K⊢f∶σ⇒τ)

uniq-type : ∀ {n m} {K : Ktx n m} {t τ τ'} → K ⊢ t ∶ τ → K ⊢ t ∶ τ' → τ ≡ τ'

-- excerpt from the typeof decision procedure
typeof : ∀ {n m} (K : Ktx n m) t → Dec (HasType K t)
typeof (Γ , Δ) (f ⟨⟩)   with typeof (Γ , Δ) f
typeof (Γ , Δ) (f ⟨⟩) | yes (σ ⇒ τ , _)     with (Δ-resolve (Γ , Δ) σ)
typeof (Γ , Δ) (f ⟨⟩) | yes (σ ⇒ τ , f∶φ) | no KΔ↝̸σ = 
  -- I'm trying to derive a contraction based on the fact that we've proven that
  -- K Δ↝̸ σ, but assuming a type for (f ⟨⟩) will yield an instance of K Δ↝ σ' (see ⟨⟩-wt-inv)
  -- the problem is that I don't know how to make agda see that σ' ≡ σ
  -- such that the following typechecks.
  -- (while agda will now complain that the σ in the wt-inv is not the
     same one as used in the KΔ↝̸σ instance, which is sensible)
  -- I think I have to use the uniq-type prove on f somewhere...
  no $ KΔ↝̸σ ∘ proj₁ ∘ proj₂ ⟨⟩-wt-inv ∘ proj₂

Any help is appreciated

score 3 · Accepted Answer · answered Apr 07 '15 at 20:10

Saizan on the #agda channel was kind enough to point me in the right direction: using the function subst to use an equality proof to "substitute" σ for σ' in the proof that I had to obtain an instance of KΔ↝σ from KΔ↝σ':

typeof (Γ , Δ) (f ⟨⟩) | yes (σ ⇒ τ , f∶φ) | no KΔ↝̸σ = 
  no $ KΔ↝̸σ ∘ helper
    where
      helper : (HasType (Γ , Δ) (f ⟨⟩)) → (Γ , Δ) Δ↝ σ
      helper p with (⟨⟩-wt-inv ∘ proj₂) p
      helper p | (σ' , KΔ↝σ' , f∶φ') = subst (λ s → (Γ , Δ) Δ↝ s) σ'≡σ KΔ↝σ' 
        where
          σ'≡σ : σ' ≡ σ
          σ'≡σ = ≡⇒dom $ uniq-type f∶φ' f∶φ

Proving to Agda that we're talking about the same thing

1 Answers1