我试图证明一个矛盾,但是遇到一个问题,试图向Agda证明所返回的sigma域类型<>-wt-inv
与该证明前面所见的sigma是相同的。我希望uniq型证明可以帮助我,但我无法将它们放在一起。
我希望下面代码中的注释能提供足够的上下文。
-- 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
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#agda频道上的Saizan足以将我指向正确的方向:使用该函数subst
使用相等证明将σ替换为σ,以证明我必须从KΔ↝获得KΔ↝σ的实例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
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