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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/SubjectReduction.agda
+7 -22
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@@ -27,12 +27,7 @@ subject-reduction : ∀ {n} {Γ : Context n} {e e' t : Term n}
→ e ↪ e'
----------
→ Γ ⊢ e' ⦂ t
subject-reduction (⊢-≈ t≈t₁ ⊢e) β-λ = ⊢-≈ t≈t₁ (subject-reduction ⊢e β-λ)
subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-λ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-λ e↪e'))
subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-·₁ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-·₁ e↪e'))
subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-·₂ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-·₂ e↪e'))
subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-∀₁ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-∀₁ e↪e'))
subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-∀₂ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-∀₂ e↪e'))
subject-reduction (⊢-≈ t≈t ⊢e) e↪e' = ⊢-≈ t≈t (subject-reduction ⊢e e↪e')
subject-reduction (⊢-λ ⊢e₁ ⊢e₂) (ξ-λ e₁↪e₁') = ⊢-λ ⊢e₁ (subject-reduction ⊢e₂ e₁↪e₁')
subject-reduction (⊢-· ⊢e₁ ⊢e₂) (ξ-·₁ e₁↪e₁') = ⊢-· (subject-reduction ⊢e₁ e₁↪e₁') ⊢e₂
subject-reduction (⊢-· ⊢e₁ ⊢e₂) (ξ-·₂ e₂↪e₂') =
@@ -41,26 +36,11 @@ subject-reduction (⊢-∀ ⊢t ⊢e) (ξ-∀₁ t↪t') =
⊢-∀ (subject-reduction ⊢t t↪t') (≈-Γ-⊢₁ (↪→≈ t↪t') ⊢e)
subject-reduction (⊢-∀ ⊢e₁ ⊢e₂) (ξ-∀₂ e₁↪e₁') = ⊢-∀ ⊢e₁ (subject-reduction ⊢e₂ e₁↪e₁')
subject-reduction (⊢-· ⊢e₁ ⊢e₂) β-λ with invert-⊢λ ⊢e₁
... | ⟨ t₁' , ⟨ t₂' , ⟨ t₁≈t₁' , ⟨ t₂≈t₂' , ⟨ ⊢t₁' , Γ,t₁'⊢e₁⦂t₂' ⟩ ⟩ ⟩ ⟩ ⟩
... | ⟨ t₁' , ⟨ t₂' , ⟨ lvl , ⟨ t₁≈t₁' , ⟨ t₂≈t₂' , ⟨ ⊢t₁' , Γ,t₁'⊢e₁⦂t₂' ⟩ ⟩ ⟩ ⟩ ⟩ ⟩
= ⊢sub (⊢sub₁ (⊢-≈ t₁≈t₁' ⊢e₂)) ⊢e₁⦂t₂
where
⊢e₁⦂t₂ = ⊢-≈ (≈-sym t₂≈t₂') Γ,t₁'⊢e₁⦂t₂'
subject-reduction (⊢-` x x₁) ()
subject-reduction ⊢-Set ()
subject-reduction (⊢-≈ x ⊢e) β-proj₁ = ⊢-≈ x (subject-reduction ⊢e β-proj₁)
subject-reduction (⊢-≈ x ⊢e) β-proj₂ = ⊢-≈ x (subject-reduction ⊢e β-proj₂)
subject-reduction (⊢-≈ x ⊢e) β-subst = ⊢-≈ x (subject-reduction ⊢e β-subst)
subject-reduction (⊢-≈ x ⊢e) (ξ-∃₁ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-∃₁ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-∃₂ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-∃₂ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-proj₁ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-proj₁ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-proj₂ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-proj₂ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-,₁ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-,₁ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-,₂ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-,₂ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-≡₁ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-≡₁ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-≡₂ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-≡₂ e↪e'))
subject-reduction (⊢-≈ t₁≈t ⊢e) (ξ-subst₁ t₂↪t') = ⊢-≈ t₁≈t (subject-reduction ⊢e (ξ-subst₁ t₂↪t'))
subject-reduction (⊢-≈ x ⊢e) (ξ-subst₂ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-subst₂ e↪e'))
subject-reduction (⊢-≈ x ⊢e) (ξ-subst₃ e↪e') = ⊢-≈ x (subject-reduction ⊢e (ξ-subst₃ e↪e'))
subject-reduction (⊢-∃ ⊢t₁ t₁⊢t₂) (ξ-∃₁ t₁↪t₁')
= ⊢-∃ (subject-reduction ⊢t₁ t₁↪t₁') (≈-Γ-⊢₁ (↪→≈ t₁↪t₁') t₁⊢t₂)
subject-reduction (⊢-∃ ⊢e ⊢e₁) (ξ-∃₂ e↪e') = ⊢-∃ ⊢e (subject-reduction ⊢e₁ e↪e')
@@ -88,3 +68,8 @@ subject-reduction (⊢-subst {u₁ = u₁} {u₂ = u₂} ⊢t ⊢u₁ ⊢u₂
= ⊢-≈ (≈-sym (≈-sub (0↦ u₂) (↪→≈ t↪t'))) (⊢-subst (subject-reduction ⊢t t↪t') ⊢u₁ ⊢u₂ ⊢≈ (⊢-≈ (≈-sub (0↦ u₁) (↪→≈ t↪t')) ⊢t[u₁]))
subject-reduction (⊢-subst ⊢e ⊢e₁ ⊢e₂ ⊢e₃ ⊢e₄) (ξ-subst₂ e↪e') = ⊢-subst ⊢e ⊢e₁ ⊢e₂ (subject-reduction ⊢e₃ e↪e') ⊢e₄
subject-reduction (⊢-subst ⊢e ⊢e₁ ⊢e₂ ⊢e₃ ⊢e₄) (ξ-subst₃ e↪e') = ⊢-subst ⊢e ⊢e₁ ⊢e₂ ⊢e₃ (subject-reduction ⊢e₄ e↪e')
subject-reduction (⊢-lsuc a) (ξ-lsuc l↪l') = ⊢-lsuc (subject-reduction a l↪l')
subject-reduction (⊢-⊔ a a₁) (ξ-⊔₁ l↪l') = ⊢-⊔ (subject-reduction a l↪l') a₁
subject-reduction (⊢-⊔ a a₁) (ξ-⊔₂ l↪l') = ⊢-⊔ a (subject-reduction a₁ l↪l')
subject-reduction ⊢-Setn (ξ-Setn l↪l') = ⊢-≈ (mk-≈ (`Setn `lsuc _) ↪*-refl (↪*-step (ξ-Setn (ξ-lsuc l↪l')) ↪*-refl) ) ⊢-Setn
subject-reduction ⊢-Setω (ξ-Setω l↪l') = ⊢-≈ (mk-≈ (`Setω `lsuc _) ↪*-refl (↪*-step (ξ-Setω (ξ-lsuc l↪l')) ↪*-refl) ) ⊢-Setω