Some definitions and adjusted proofs
This commit is contained in:
JKF
@@ -11,7 +11,7 @@ open import Data.Empty using (⊥; ⊥-elim)
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open import univTypes.SubjectReduction.Specification
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open import univTypes.SubjectReduction.Confluence using (↪ₚ*-trans; ξ*-λ; ξ*-·₁; ξ*-·₂; ξ*-∀₁; ξ*-∀₂; ξ*-subst₁; ξ*-subst₂; ξ*-subst₃)
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open import univTypes.SubjectReduction.Confluence using (ξ*-∃₁; ξ*-∃₂; ξ*-,₁; ξ*-,₂; ξ*-≡₁; ξ*-≡₂; ξ*-proj₁; ξ*-proj₂)
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open import univTypes.SubjectReduction.Confluence using (ξ*-∃₁; ξ*-∃₂; ξ*-,₁; ξ*-,₂; ξ*-≡₁; ξ*-≡₂; ξ*-proj₁; ξ*-proj₂; ξ*-Setn; ξ*-Setω; ξ*-lsuc; ξ*-⊔₁; ξ*-⊔₂)
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open import univTypes.SubjectReduction.Confluence using (confluence; _↪ₚ_; _↪ₚ*_; ↪ₚ-refl; ↪ₚ-ren; ↪ₚ*→↪*; ↪*→↪ₚ*); open _↪ₚ*_; open _↪ₚ_
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open import univTypes.SubjectReduction.Composition using (swap-0↦-extᵣ-fusion; swap-0↦-extₛ-fusion)
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open import univTypes.Util.Fin
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@@ -126,13 +126,35 @@ open import univTypes.Util.Fin
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a = ↪*-trans ξ-t ξ-1
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b = ↪*-trans a ξ-2
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-- New
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ξₚ*-Setn : ∀ {n} {e e' : Term n}
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→ e ↪ₚ* e'
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---------------
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→ `Setn e ↪ₚ* `Setn e'
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ξₚ*-Setn e↪ₚ*e' = {! ↪*→↪ₚ* ? !}
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ξₚ*-Setn e↪ₚ*e' = ↪*→↪ₚ* (ξ*-Setn (↪ₚ*→↪* e↪ₚ*e'))
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ξₚ*-Setω : ∀ {n} {e e' : Term n}
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→ e ↪ₚ* e'
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---------------
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→ `Setω e ↪ₚ* `Setω e'
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ξₚ*-Setω e↪ₚ*e' = ↪*→↪ₚ* (ξ*-Setω (↪ₚ*→↪* e↪ₚ*e'))
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ξₚ*-lsuc : ∀ {n} {e e' : Term n}
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→ e ↪ₚ* e'
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---------------
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→ `lsuc e ↪ₚ* `lsuc e'
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ξₚ*-lsuc e↪ₚ*e' = ↪*→↪ₚ* (ξ*-lsuc (↪ₚ*→↪* e↪ₚ*e'))
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ξₚ*-⊔ : ∀ {n} {e₁ e₂ e₁' e₂' : Term n}
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→ e₁ ↪ₚ* e₁'
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→ e₂ ↪ₚ* e₂'
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---------------------
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→ e₁ `⊔ e₂ ↪ₚ* e₁' `⊔ e₂'
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ξₚ*-⊔ e₁↪ₚ*e₁' e₂↪ₚ*e₂' = ↪ₚ*-trans
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(↪*→↪ₚ* (ξ*-⊔₁ (↪ₚ*→↪* e₁↪ₚ*e₁')))
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(↪*→↪ₚ* (ξ*-⊔₂ (↪ₚ*→↪* e₂↪ₚ*e₂')))
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-- ↪*→↪ₚ* (ξ*-Setn (↪ₚ*→↪* e↪ₚ*e'))
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--------------------------------------------------------------------------------
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-- "Substituting two convertible terms into another term, yields again convertible terms."
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@@ -182,10 +204,10 @@ _≈σ_ : ∀ {m n} (σ₁ σ₂ : Sub m n) → Set
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= ξₚ*-subst (↪ₚ*σ-sub {e = t} (↪ₚ*σ-ext σ↪σ')) (↪ₚ*σ-sub {e = e₁} σ↪σ') (↪ₚ*σ-sub {e = e₂} σ↪σ')
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↪ₚ*σ-sub {m} {n} {`Level} {σ} {σ'} σ↪σ' = ↪ₚ*-refl
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↪ₚ*σ-sub {m} {n} {`lzero} {σ} {σ'} σ↪σ' = ↪ₚ*-refl
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↪ₚ*σ-sub {m} {n} {`Setω x} {σ} {σ'} σ↪σ' = {! ξₚ*-Setω (↪ₚ*σ-sub {e = e} (σ↪σ') !}
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↪ₚ*σ-sub {m} {n} {`Setn x} {σ} {σ'} σ↪σ' = {! !}
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↪ₚ*σ-sub {m} {n} {`lsuc x} {σ} {σ'} σ↪σ' = {! !}
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↪ₚ*σ-sub {m} {n} {l `⊔ r} {σ} {σ'} σ↪σ' = {! ξₚ*-⊔ (↪ₚ*σ-sub {e = l} σ↪σ') (↪ₚ*σ-sub {e = r} σ↪σ') !}
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↪ₚ*σ-sub {m} {n} {`Setω e} {σ} {σ'} σ↪σ' = ξₚ*-Setω (↪ₚ*σ-sub {e = e} σ↪σ')
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↪ₚ*σ-sub {m} {n} {`Setn e} {σ} {σ'} σ↪σ' = ξₚ*-Setn (↪ₚ*σ-sub {e = e} σ↪σ')
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↪ₚ*σ-sub {m} {n} {`lsuc e} {σ} {σ'} σ↪σ' = ξₚ*-lsuc (↪ₚ*σ-sub {e = e} σ↪σ')
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↪ₚ*σ-sub {m} {n} {l `⊔ r} {σ} {σ'} σ↪σ' = ξₚ*-⊔ (↪ₚ*σ-sub {e = l} σ↪σ') (↪ₚ*σ-sub {e = r} σ↪σ')
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↪*σ-sub : ∀ {m n} {e : Term m} {σ σ' : Sub m n} →
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σ ↪*σ σ' →
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@@ -231,8 +253,17 @@ ext-≈σ ≈ₛ (suc x) with ≈ₛ x
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| mk-≈ c e₂↪*c e₂'↪*c
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= mk-≈ (`subst a b c) (↪ₚ*→↪* (ξₚ*-subst (↪*→↪ₚ* t↪*a) (↪*→↪ₚ* e₁↪*b) (↪*→↪ₚ* e₂↪*c)))
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(↪ₚ*→↪* (ξₚ*-subst (↪*→↪ₚ* t'↪*a) (↪*→↪ₚ* e₁'↪*b) (↪*→↪ₚ* e₂'↪*c)))
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≈σ-sub {zero} ≈σ' = {! !}
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≈σ-sub {suc e} ≈σ' = {! !}
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≈σ-sub {e = `Level} ≈σ' = mk-≈ `Level ↪*-refl ↪*-refl
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≈σ-sub {e = `lzero} ≈σ' = mk-≈ `lzero ↪*-refl ↪*-refl
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≈σ-sub {e = `Setω e} ≈σ' with ≈σ-sub {e = e} ≈σ'
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... | mk-≈ e' e↪*e' e₁↪*e' = mk-≈ (`Setω e') (ξ*-Setω e↪*e') (ξ*-Setω e₁↪*e')
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≈σ-sub {e = `Setn e} ≈σ' with ≈σ-sub {e = e} ≈σ'
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... | mk-≈ e' e↪*e' e₁↪*e' = mk-≈ (`Setn e') (ξ*-Setn e↪*e') (ξ*-Setn e₁↪*e')
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≈σ-sub {e = `lsuc e} ≈σ' with ≈σ-sub {e = e} ≈σ'
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... | mk-≈ e' e↪*e' e₁↪*e' = mk-≈ (`lsuc e') (ξ*-lsuc e↪*e') (ξ*-lsuc e₁↪*e')
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≈σ-sub {e = l `⊔ r} ≈σ' with ≈σ-sub {e = l} ≈σ' | ≈σ-sub {e = r} ≈σ'
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... | mk-≈ l' l↪*l' l₂↪*l' | mk-≈ r' r↪*r' r₂↪*r'
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= mk-≈ (l' `⊔ r') (↪*-trans (ξ*-⊔₁ l↪*l') (ξ*-⊔₂ r↪*r')) ((↪*-trans (ξ*-⊔₁ l₂↪*l') (ξ*-⊔₂ r₂↪*r')))
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≈→≈σ : ∀ {n} {e e' : Term n}
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→ e ≈ e'
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@@ -288,11 +319,11 @@ ext-≈σ ≈ₛ (suc x) with ≈ₛ x
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↪-ren ρ (ξ-subst₂ x) = ξ-subst₂ (↪-ren ρ x)
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↪-ren ρ (ξ-subst₃ x) = ξ-subst₃ (↪-ren ρ x)
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-- New
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↪-ren ρ (ξ-⊔₁ x) = {! !}
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↪-ren ρ (ξ-⊔₂ x) = {! !}
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↪-ren ρ (ξ-lsuc x) = {! !}
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↪-ren ρ (ξ-Setn x) = {! !}
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↪-ren ρ (ξ-Setω x) = {! !}
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↪-ren ρ (ξ-⊔₁ x) = ξ-⊔₁ (↪-ren ρ x)
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↪-ren ρ (ξ-⊔₂ x) = ξ-⊔₂ (↪-ren ρ x)
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↪-ren ρ (ξ-lsuc x) = ξ-lsuc (↪-ren ρ x)
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↪-ren ρ (ξ-Setn x) = ξ-Setn (↪-ren ρ x)
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↪-ren ρ (ξ-Setω x) = ξ-Setω (↪-ren ρ x)
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-- Renaming preserves many reduction steps
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@@ -335,12 +366,12 @@ ext-≈σ ≈ₛ (suc x) with ≈ₛ x
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↪-sub σ (ξ-subst₁ x) = ξ-subst₁ (↪-sub (extₛ σ) x)
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↪-sub σ (ξ-subst₂ x) = ξ-subst₂ (↪-sub σ x)
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-- New
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↪-sub σ (ξ-⊔₁ x) = {! !}
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↪-sub σ (ξ-⊔₂ x) = {! !}
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↪-sub σ (ξ-lsuc x) = {! !}
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↪-sub σ (ξ-Setn x) = {! !}
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↪-sub σ (ξ-subst₃ x) = {! !}
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↪-sub σ (ξ-Setω x) = {! !}
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↪-sub σ (ξ-⊔₁ x) = ξ-⊔₁ (↪-sub σ x)
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↪-sub σ (ξ-⊔₂ x) = ξ-⊔₂ (↪-sub σ x)
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↪-sub σ (ξ-lsuc x) = ξ-lsuc (↪-sub σ x)
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↪-sub σ (ξ-Setn x) = ξ-Setn (↪-sub σ x)
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↪-sub σ (ξ-subst₃ x) = ξ-subst₃ (↪-sub σ x)
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↪-sub σ (ξ-Setω x) = ξ-Setω (↪-sub σ x)
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-- Substitution preserves many reduction steps
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@@ -396,12 +427,12 @@ _≈ᶜ_ : ∀ {n} → Context n → Context n → Set
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≈-Γ-⊢ Γ₁≈Γ₂ (⊢-subst t'⊢t ⊢u₁ ⊢u₂ ⊢≡ ⊢t[u₁])
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= ⊢-subst (≈-Γ-⊢ (≈-ext Γ₁≈Γ₂ ≈-refl) t'⊢t) (≈-Γ-⊢ Γ₁≈Γ₂ ⊢u₁) (≈-Γ-⊢ Γ₁≈Γ₂ ⊢u₂) (≈-Γ-⊢ Γ₁≈Γ₂ ⊢≡) (≈-Γ-⊢ Γ₁≈Γ₂ ⊢t[u₁])
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-- New
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-Setn = {! !}
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-Setω = {! !}
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-lzero = {! !}
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≈-Γ-⊢ Γ₁≈Γ₂ (⊢-lsuc ⊢x) = {! !}
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≈-Γ-⊢ Γ₁≈Γ₂ (⊢-⊔ ⊢x ⊢x₁) = {! !}
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-Level = {! !}
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-Setn = ⊢-Setn
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-Setω = ⊢-Setω
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-lzero = ⊢-lzero
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≈-Γ-⊢ Γ₁≈Γ₂ (⊢-lsuc ⊢x) = ⊢-lsuc (≈-Γ-⊢ Γ₁≈Γ₂ ⊢x)
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≈-Γ-⊢ Γ₁≈Γ₂ (⊢-⊔ ⊢x ⊢x₁) = ⊢-⊔ (≈-Γ-⊢ Γ₁≈Γ₂ ⊢x) (≈-Γ-⊢ Γ₁≈Γ₂ ⊢x₁)
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≈-Γ-⊢ Γ₁≈Γ₂ ⊢-Level = ⊢-Level
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≈-Γ-⊢₁ : ∀ {n} {Γ : Context n} {t₁ t₂ : Term n} {e t : Term (suc n)}
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→ t₁ ≈ t₂
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@@ -424,9 +455,9 @@ _≈ᶜ_ : ∀ {n} → Context n → Context n → Set
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(≈-Γ-⊢₁ t₁≈t₂ t₁⊢u₂)
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(≈-Γ-⊢₁ t₁≈t₂ t₁⊢≡)
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(≈-Γ-⊢₁ t₁≈t₂ t₁⊢t[u₁])
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≈-Γ-⊢₁ t₁≈t₂ ⊢-Setn = {! !}
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≈-Γ-⊢₁ t₁≈t₂ ⊢-Setω = {! !}
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≈-Γ-⊢₁ t₁≈t₂ ⊢-lzero = {! !}
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≈-Γ-⊢₁ t₁≈t₂ (⊢-lsuc ⊢x) = {! !}
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≈-Γ-⊢₁ t₁≈t₂ (⊢-⊔ ⊢x ⊢x₁) = {! !}
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≈-Γ-⊢₁ t₁≈t₂ ⊢-Level = {! !}
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≈-Γ-⊢₁ t₁≈t₂ ⊢-Setn = ⊢-Setn
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≈-Γ-⊢₁ t₁≈t₂ ⊢-Setω = ⊢-Setω
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≈-Γ-⊢₁ t₁≈t₂ ⊢-lzero = ⊢-lzero
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≈-Γ-⊢₁ t₁≈t₂ (⊢-lsuc ⊢x) = ⊢-lsuc (≈-Γ-⊢₁ t₁≈t₂ ⊢x)
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≈-Γ-⊢₁ t₁≈t₂ (⊢-⊔ ⊢x ⊢x₁) = ⊢-⊔ (≈-Γ-⊢₁ t₁≈t₂ ⊢x) (≈-Γ-⊢₁ t₁≈t₂ ⊢x₁)
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≈-Γ-⊢₁ t₁≈t₂ ⊢-Level = ⊢-Level
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@@ -41,27 +41,28 @@ invert-≈-∀ (mk-≈ x ∀₁↪*x ∀₂↪*x) with ↪*-∀-shape ∀₁↪*
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invert-⊢λ' : ∀ {n} {Γ : Context n} {e : Term (suc n)} {t : Term n} →
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Γ ⊢ `λ e ⦂ t →
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∃[ t₁ ] ∃[ t₂ ]
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∃[ t₁ ] ∃[ t₂ ] ∃[ l ]
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t ≈ (∀[x⦂ t₁ ] t₂) ×
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Γ ⊢ t₁ ⦂ `Set ×
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Γ ⊢ t₁ ⦂ (`Setn l) ×
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Γ , t₁ ⊢ e ⦂ t₂
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invert-⊢λ' (⊢-λ ⊢λ ⊢e) = ⟨ _ , ⟨ _ , ⟨ ≈-refl , ⟨ ⊢λ , ⊢e ⟩ ⟩ ⟩ ⟩
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invert-⊢λ' (⊢-λ ⊢λ ⊢e) = ⟨ _ , ⟨ _ , ⟨ _ , ⟨ ≈-refl , ⟨ ⊢λ , ⊢e ⟩ ⟩ ⟩ ⟩ ⟩
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invert-⊢λ' (⊢-≈ t₁≈t ⊢λ⦂t₁) with invert-⊢λ' ⊢λ⦂t₁
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... | ⟨ X , ⟨ E , ⟨ t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩
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= ⟨ X , ⟨ E , ⟨ ≈-trans (≈-sym t₁≈t) t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩
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... | ⟨ X , ⟨ E , ⟨ L , ⟨ t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩
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= ⟨ X , ⟨ E , ⟨ _ , ⟨ ≈-trans (≈-sym t₁≈t) t₁≈∀[X]E , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩
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invert-⊢λ : ∀ {n} {Γ : Context n} {e : Term (suc n)} {t₁ : Term n} {t₂ : Term (suc n)}
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→ Γ ⊢ `λ e ⦂ ∀[x⦂ t₁ ] t₂
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→ ∃[ t₁' ] ∃[ t₂' ]
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→ ∃[ t₁' ] ∃[ t₂' ] ∃[ l ]
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t₁ ≈ t₁' ×
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t₂ ≈ t₂' ×
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Γ ⊢ t₁' ⦂ `Set ×
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Γ ⊢ t₁' ⦂ (`Setn l) ×
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Γ , t₁' ⊢ e ⦂ t₂'
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invert-⊢λ (⊢-λ ⊢λ ⊢λ₁) = ⟨ _ , ⟨ _ , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ ⊢λ , ⊢λ₁ ⟩ ⟩ ⟩ ⟩ ⟩
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invert-⊢λ (⊢-λ ⊢λ ⊢λ₁) = ⟨ _ , ⟨ _ , ⟨ _ , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ mk-≈ _ ↪*-refl ↪*-refl , ⟨ ⊢λ , ⊢λ₁ ⟩ ⟩ ⟩ ⟩ ⟩ ⟩
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invert-⊢λ (⊢-≈ t≈∀ ⊢λ) with invert-⊢λ' ⊢λ
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... | ⟨ X , ⟨ E , ⟨ t≈∀' , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ with invert-≈-∀ (≈-trans (≈-sym t≈∀) t≈∀')
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... | ⟨ X , ⟨ E , ⟨ L , ⟨ t≈∀' , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩ with invert-≈-∀ (≈-trans (≈-sym t≈∀) t≈∀')
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... | ⟨ ≈₁ , ≈₂ ⟩
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= ⟨ X , ⟨ E , ⟨ ≈₁ , ⟨ ≈₂ , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩
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= ⟨ X , ⟨ E , ⟨ L , ⟨ ≈₁ , ⟨ ≈₂ , ⟨ X⦂Set , Γ,X⊢e⦂E ⟩ ⟩ ⟩ ⟩ ⟩ ⟩
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-- Inversion for pairs
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@@ -27,12 +27,7 @@ subject-reduction : ∀ {n} {Γ : Context n} {e e' t : Term n}
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→ e ↪ e'
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----------
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→ Γ ⊢ e' ⦂ t
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subject-reduction (⊢-≈ t≈t₁ ⊢e) β-λ = ⊢-≈ t≈t₁ (subject-reduction ⊢e β-λ)
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subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-λ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-λ e↪e'))
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subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-·₁ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-·₁ e↪e'))
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subject-reduction (⊢-≈ t≈t₁ ⊢e) (ξ-·₂ e↪e') = ⊢-≈ t≈t₁ (subject-reduction ⊢e (ξ-·₂ e↪e'))
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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
|
||||
|
||||
@@ -58,7 +58,6 @@ sub-type {a = a} _ = a
|
||||
→ Γ₁ ⊢ e ⦂ t
|
||||
------------------------
|
||||
→ Γ₂ ⊢ ren ρ e ⦂ ren ρ t
|
||||
⊢ren ⊢ρ ⊢-Set = ⊢-Set
|
||||
⊢ren ⊢ρ (⊢-refl ⊢-e) = ⊢-refl (⊢ren ⊢ρ ⊢-e)
|
||||
⊢ren ⊢ρ (⊢-` x refl) = ⊢-` _ (⊢ρ x)
|
||||
⊢ren ⊢ρ (⊢-≡ ⊢l ⊢r) = ⊢-≡ (⊢ren ⊢ρ ⊢l) (⊢ren ⊢ρ ⊢r)
|
||||
@@ -76,6 +75,14 @@ sub-type {a = a} _ = a
|
||||
= ⊢-≈ (≡→≈ (swap-0↦-extᵣ-fusion ρ (term ⊢u₂) t* ))
|
||||
(⊢-subst (⊢ren (⊢extᵣ ⊢ρ) t'⊢t) (⊢ren ⊢ρ ⊢u₁) (⊢ren ⊢ρ ⊢u₂) (⊢ren ⊢ρ ⊢≡)
|
||||
(⊢-≈ (≡→≈ (sym (swap-0↦-extᵣ-fusion ρ (term ⊢u₁) (t*)))) (⊢ren ⊢ρ ⊢t[u₁])))
|
||||
-- New
|
||||
⊢ren ⊢ρ ⊢-Setn = ⊢-Setn
|
||||
⊢ren ⊢ρ ⊢-Setω = ⊢-Setω
|
||||
⊢ren ⊢ρ ⊢-lzero = ⊢-lzero
|
||||
⊢ren ⊢ρ (⊢-lsuc ⊢e) = ⊢-lsuc (⊢ren ⊢ρ ⊢e)
|
||||
⊢ren ⊢ρ (⊢-⊔ ⊢l ⊢r) = ⊢-⊔ (⊢ren ⊢ρ ⊢l) (⊢ren ⊢ρ ⊢r)
|
||||
⊢ren ⊢ρ ⊢-Level = ⊢-Level
|
||||
|
||||
|
||||
⊢wk : ∀ {n} {Γ : Context n} {t : Term n}
|
||||
→ wk ⦂ Γ ⇒ᵣ (Γ , t)
|
||||
@@ -99,7 +106,6 @@ _⦂_⇒ₛ_ : ∀ {m n} → Sub m n → Context m → Context n → Set
|
||||
→ Γ₁ ⊢ e ⦂ t
|
||||
------------------------
|
||||
→ Γ₂ ⊢ sub σ e ⦂ sub σ t
|
||||
⊢sub ⊢σ ⊢-Set = ⊢-Set
|
||||
⊢sub ⊢σ (⊢-` x refl) = ⊢σ x
|
||||
⊢sub ⊢σ (⊢-refl e) = ⊢-refl (⊢sub ⊢σ e)
|
||||
⊢sub {σ = σ} ⊢σ (⊢-· ⊢l ⊢r) = ⊢-≈ (≡→≈ (swap-0↦-extₛ-fusion σ (term ⊢r) (∀-type ⊢l))) (⊢-· (⊢sub ⊢σ ⊢l) (⊢sub ⊢σ ⊢r))
|
||||
@@ -117,6 +123,13 @@ _⦂_⇒ₛ_ : ∀ {m n} → Sub m n → Context m → Context n → Set
|
||||
= ⊢-≈ (≡→≈ (swap-0↦-extₛ-fusion σ (term ⊢u₂) t* ))
|
||||
(⊢-subst (⊢sub (⊢extₛ ⊢σ) t'⊢t) (⊢sub ⊢σ ⊢u₁) (⊢sub ⊢σ ⊢u₂) (⊢sub ⊢σ ⊢≡)
|
||||
(⊢-≈ (≡→≈ (sym (swap-0↦-extₛ-fusion σ (term ⊢u₁) (t*)))) (⊢sub ⊢σ ⊢t[u₁])))
|
||||
-- New
|
||||
⊢sub ⊢σ ⊢-Setn = ⊢-Setn
|
||||
⊢sub ⊢σ ⊢-Setω = ⊢-Setω
|
||||
⊢sub ⊢σ ⊢-lzero = ⊢-lzero
|
||||
⊢sub ⊢σ ⊢-Level = ⊢-Level
|
||||
⊢sub ⊢σ (⊢-lsuc ⊢e) = ⊢-lsuc (⊢sub ⊢σ ⊢e)
|
||||
⊢sub ⊢σ (⊢-⊔ ⊢l ⊢r) = ⊢-⊔ (⊢sub ⊢σ ⊢l) (⊢sub ⊢σ ⊢r)
|
||||
|
||||
⊢sub₁ : ∀ {n} {Γ : Context n} {e : Term n} {t}
|
||||
→ Γ ⊢ e ⦂ t
|
||||
|
||||
Reference in New Issue
Block a user