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Some definitions and adjusted proofs

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JKF
2026-04-25 21:57:10 +02:00
parent 5cbf679c0c
commit e8ee5b200c
4 changed files with 96 additions and 66 deletions
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SubjectReduction/Inversion.agda
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@@ -41,27 +41,28 @@ invert-≈-∀ (mk-≈ x ∀₁↪*x ∀₂↪*x) with ↪*-∀-shape ∀₁↪*
invert-⊢λ' : ∀ {n} {Γ : Context n} {e : Term (suc n)} {t : Term n} →
Γ ⊢ `λ e ⦂ t →
∃[ t₁ ] ∃[ t₂ ]
∃[ t₁ ] ∃[ t₂ ] ∃[ l ]
t ≈ (∀[x⦂ t₁ ] t₂) ×
Γ ⊢ t₁ ⦂ `Set ×
Γ ⊢ t₁ ⦂ (`Setn l) ×
Γ , t₁ ⊢ e ⦂ t₂
invert-⊢λ' (⊢-λ ⊢λ ⊢e) = ⟨ _ , ⟨ _ , ⟨ ≈-refl , ⟨ ⊢λ , ⊢e ⟩ ⟩ ⟩ ⟩
invert-⊢λ' (⊢-λ ⊢λ ⊢e) = ⟨ _ , ⟨ _ , ⟨ _ , ⟨ ≈-refl , ⟨ ⊢λ , ⊢e ⟩ ⟩ ⟩ ⟩ ⟩
invert-⊢λ' (⊢-≈ t₁≈t ⊢λ⦂t₁) with invert-⊢λ' ⊢λ⦂t₁
... | ⟨ X , ⟨ E , ⟨ t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩
= ⟨ X , ⟨ E , ⟨ ≈-trans (≈-sym t₁≈t) t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩
... | ⟨ X , ⟨ E , ⟨ L , ⟨ t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩
= ⟨ X , ⟨ E , ⟨ _ , ⟨ ≈-trans (≈-sym t₁≈t) t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩
invert-⊢λ : ∀ {n} {Γ : Context n} {e : Term (suc n)} {t₁ : Term n} {t₂ : Term (suc n)}
→ Γ ⊢ `λ e ⦂ ∀[x⦂ t₁ ] t₂
→ ∃[ t₁' ] ∃[ t₂' ]
→ ∃[ t₁' ] ∃[ t₂' ] ∃[ l ]
t₁ ≈ t₁' ×
t₂ ≈ t₂' ×
Γ ⊢ t₁' ⦂ `Set ×
Γ ⊢ t₁' ⦂ (`Setn l) ×
Γ , t₁' ⊢ e ⦂ t₂'
invert-⊢λ (⊢-λ ⊢λ ⊢λ₁) = ⟨ _ , ⟨ _ , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ ⊢λ , ⊢λ₁ ⟩ ⟩ ⟩ ⟩ ⟩
invert-⊢λ (⊢-λ ⊢λ ⊢λ₁) = ⟨ _ , ⟨ _ , ⟨ _ , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ ⊢λ , ⊢λ₁ ⟩ ⟩ ⟩ ⟩ ⟩ ⟩
invert-⊢λ (⊢-≈ t≈∀ ⊢λ) with invert-⊢λ' ⊢λ
... | ⟨ X , ⟨ E , ⟨ t≈∀' , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ with invert-≈-∀ (≈-trans (≈-sym t≈∀) t≈∀')
... | ⟨ X , ⟨ E , ⟨ L , ⟨ t≈∀' , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩ with invert-≈-∀ (≈-trans (≈-sym t≈∀) t≈∀')
... | ⟨ ≈₁ , ≈₂ ⟩
= ⟨ X , ⟨ E , ⟨ ≈₁ , ⟨ ≈₂ , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩
= ⟨ X , ⟨ E , ⟨ L , ⟨ ≈₁ , ⟨ ≈₂ , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩ ⟩
-- Inversion for pairs