changes
This commit is contained in:
JKF
@@ -32,7 +32,12 @@ ext-dist-∘ᵣᵣ ρ₁ ρ₂ = fun-ext eq
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fusion-∘ᵣᵣ : ∀ {n₁ n₂ n₃} (ρ₁ : Ren n₂ n₃) (ρ₂ : Ren n₁ n₂) (e : Term n₁)
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→ ren ρ₁ (ren ρ₂ e) ≡ ren (ρ₁ ∘ᵣᵣ ρ₂) e
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fusion-∘ᵣᵣ ρ₁ ρ₂ (` x) = refl
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fusion-∘ᵣᵣ ρ₁ ρ₂ `Set = refl
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fusion-∘ᵣᵣ ρ₁ ρ₂ (`Setω e) = cong `Setω_ (fusion-∘ᵣᵣ ρ₁ ρ₂ e)
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fusion-∘ᵣᵣ ρ₁ ρ₂ (`Setn e) = cong `Setn_ (fusion-∘ᵣᵣ ρ₁ ρ₂ e)
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fusion-∘ᵣᵣ ρ₁ ρ₂ `Level = refl
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fusion-∘ᵣᵣ ρ₁ ρ₂ `lzero = refl
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fusion-∘ᵣᵣ ρ₁ ρ₂ (`lsuc e) = cong `lsuc (fusion-∘ᵣᵣ ρ₁ ρ₂ e)
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fusion-∘ᵣᵣ ρ₁ ρ₂ (l `⊔ r) rewrite fusion-∘ᵣᵣ ρ₁ ρ₂ l | fusion-∘ᵣᵣ ρ₁ ρ₂ r = refl
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fusion-∘ᵣᵣ ρ₁ ρ₂ `refl = refl
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fusion-∘ᵣᵣ ρ₁ ρ₂ (∀[x⦂ x ] e)
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rewrite fusion-∘ᵣᵣ ρ₁ ρ₂ x | (fusion-∘ᵣᵣ (extᵣ ρ₁) (extᵣ ρ₂) e) | ext-dist-∘ᵣᵣ ρ₁ ρ₂ = refl
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@@ -70,7 +75,12 @@ ext-dist-∘ₛᵣ _ _ = fun-ext λ{zero → refl; (suc x) → refl }
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fusion-∘ₛᵣ : ∀ {n₁ n₂ n₃} (σ₁ : Sub n₂ n₃) (ρ₂ : Ren n₁ n₂) (e : Term n₁)
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→ sub σ₁ (ren ρ₂ e) ≡ sub (σ₁ ∘ₛᵣ ρ₂) e
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fusion-∘ₛᵣ σ ρ (` x) = refl
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fusion-∘ₛᵣ σ ρ `Set = refl
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fusion-∘ₛᵣ σ ρ (`Setω e) rewrite fusion-∘ₛᵣ σ ρ e = refl
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fusion-∘ₛᵣ σ ρ (`Setn e) rewrite fusion-∘ₛᵣ σ ρ e = refl
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fusion-∘ₛᵣ σ ρ `Level = refl
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fusion-∘ₛᵣ σ ρ `lzero = refl
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fusion-∘ₛᵣ σ ρ (`lsuc e) rewrite fusion-∘ₛᵣ σ ρ e = refl
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fusion-∘ₛᵣ σ ρ (l `⊔ r) rewrite fusion-∘ₛᵣ σ ρ l | fusion-∘ₛᵣ σ ρ r = refl
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fusion-∘ₛᵣ σ ρ `refl = refl
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fusion-∘ₛᵣ σ ρ (`λ e) rewrite fusion-∘ₛᵣ (extₛ σ) (extᵣ ρ) e | ext-dist-∘ₛᵣ σ ρ = refl
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fusion-∘ₛᵣ σ ρ (l · r) rewrite fusion-∘ₛᵣ σ ρ l | fusion-∘ₛᵣ σ ρ r = refl
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@@ -123,7 +133,13 @@ ext-dist-∘ᵣₛ ρ σ = fun-ext (fun-ext-aux₁ ρ σ)
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fusion-∘ᵣₛ : ∀ {n₁ n₂ n₃} (ρ₁ : Ren n₂ n₃) (σ₂ : Sub n₁ n₂) (e : Term n₁)
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→ ren ρ₁ (sub σ₂ e) ≡ sub (ρ₁ ∘ᵣₛ σ₂) e
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fusion-∘ᵣₛ ρ σ (` x) = refl
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fusion-∘ᵣₛ ρ σ `Set = refl
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fusion-∘ᵣₛ ρ σ (`Setn e) rewrite fusion-∘ᵣₛ ρ σ e = refl
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fusion-∘ᵣₛ ρ σ (`Setω e) rewrite fusion-∘ᵣₛ ρ σ e = refl
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fusion-∘ᵣₛ ρ σ `Level = refl
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fusion-∘ᵣₛ ρ σ `lzero = refl
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fusion-∘ᵣₛ ρ σ (`lsuc e) rewrite fusion-∘ᵣₛ ρ σ e = refl
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fusion-∘ᵣₛ ρ σ (l `⊔ r) rewrite fusion-∘ᵣₛ ρ σ l
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| fusion-∘ᵣₛ ρ σ r = refl
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fusion-∘ᵣₛ ρ σ `refl = refl
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fusion-∘ᵣₛ ρ σ (`λ e) rewrite fusion-∘ᵣₛ (extᵣ ρ) (extₛ σ) e
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| ext-dist-∘ᵣₛ ρ σ = refl
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@@ -181,7 +197,12 @@ ext-dist-∘ₛₛ σ₁ σ₂ = fun-ext (fun-ext-aux₂ σ₁ σ₂)
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fusion-∘ₛₛ : ∀ {n₁ n₂ n₃} (σ₁ : Sub n₂ n₃) (σ₂ : Sub n₁ n₂) (e : Term n₁)
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→ sub σ₁ (sub σ₂ e) ≡ sub (σ₁ ∘ₛₛ σ₂) e
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fusion-∘ₛₛ σ₁ σ₂ (` x) = refl
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fusion-∘ₛₛ σ₁ σ₂ `Set = refl
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fusion-∘ₛₛ σ₁ σ₂ (`Setn e) rewrite fusion-∘ₛₛ σ₁ σ₂ e = refl
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fusion-∘ₛₛ σ₁ σ₂ (`Setω e) rewrite fusion-∘ₛₛ σ₁ σ₂ e = refl
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fusion-∘ₛₛ σ₁ σ₂ `Level = refl
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fusion-∘ₛₛ σ₁ σ₂ `lzero = refl
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fusion-∘ₛₛ σ₁ σ₂ (`lsuc e) rewrite fusion-∘ₛₛ σ₁ σ₂ e = refl
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fusion-∘ₛₛ σ₁ σ₂ (l `⊔ r) rewrite fusion-∘ₛₛ σ₁ σ₂ l | fusion-∘ₛₛ σ₁ σ₂ r = refl
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fusion-∘ₛₛ σ₁ σ₂ `refl = refl
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fusion-∘ₛₛ σ₁ σ₂ (`λ e) rewrite fusion-∘ₛₛ (extₛ σ₁) (extₛ σ₂) e | ext-dist-∘ₛₛ σ₁ σ₂ = refl
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fusion-∘ₛₛ σ₁ σ₂ (l · r) rewrite fusion-∘ₛₛ σ₁ σ₂ l | fusion-∘ₛₛ σ₁ σ₂ r = refl
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@@ -210,7 +231,12 @@ extₛ-idₛ {n} = fun-ext λ{zero → refl; (suc x) → refl}
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sub-id : ∀ {n} (e : Term n)
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→ sub idₛ e ≡ e
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sub-id (` x) = refl
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sub-id `Set = refl
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sub-id (`Setn e) = cong (`Setn_) (trans (cong (λ a → sub a e) refl) (sub-id e))
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sub-id (`Setω e) = cong (`Setω_) (trans (cong (λ a → sub a e) refl) (sub-id e))
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sub-id (`lsuc e) = cong (`lsuc) (trans (cong (λ a → sub a e) refl) (sub-id e))
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sub-id `Level = refl
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sub-id `lzero = refl
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sub-id (l `⊔ r) rewrite sub-id l | sub-id r = refl
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sub-id `refl = refl
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sub-id (`λ e) = cong (λ a → `λ a) (trans (cong (λ a → sub a e) extₛ-idₛ) (sub-id e))
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sub-id (`proj₁ e) = cong (λ a → `proj₁ a) (trans (cong (λ a → sub a e) refl) (sub-id e))
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@@ -128,6 +128,41 @@ open import univTypes.Util.Fin using (Fin; zero; suc)
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→ `subst t e₁ e₂ ↪* `subst t e₁ e₂'
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ξ*-subst₃ ↪*-refl = ↪*-refl
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ξ*-subst₃ (↪*-step ↪ ↪*) = ↪*-step (ξ-subst₃ ↪) (ξ*-subst₃ ↪* )
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-- New
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ξ*-lsuc : ∀ {n} {x x' : Term n}
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→ x ↪* x'
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-------------------------------
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→ `lsuc x ↪* `lsuc x'
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ξ*-lsuc ↪*-refl = ↪*-refl
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ξ*-lsuc (↪*-step ↪ ↪*) = ↪*-step (ξ-lsuc ↪) (ξ*-lsuc ↪*)
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ξ*-Setω : ∀ {n} {x x' : Term n}
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→ x ↪* x'
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-------------------------------
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→ `Setω x ↪* `Setω x'
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ξ*-Setω ↪*-refl = ↪*-refl
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ξ*-Setω (↪*-step ↪ ↪*) = ↪*-step (ξ-Setω ↪) (ξ*-Setω ↪*)
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ξ*-Setn : ∀ {n} {x x' : Term n}
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→ x ↪* x'
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-------------------------------
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→ `Setn x ↪* `Setn x'
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ξ*-Setn ↪*-refl = ↪*-refl
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ξ*-Setn (↪*-step ↪ ↪*) = ↪*-step (ξ-Setn ↪) (ξ*-Setn ↪*)
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ξ*-⊔₁ : ∀ {n} {l l' r : Term n}
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→ l ↪* l'
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-------------------------------
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→ l `⊔ r ↪* l' `⊔ r
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ξ*-⊔₁ ↪*-refl = ↪*-refl
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ξ*-⊔₁ (↪*-step x l↪l') = ↪*-step (ξ-⊔₁ x) (ξ*-⊔₁ l↪l')
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ξ*-⊔₂ : ∀ {n} {l r r' : Term n}
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→ r ↪* r'
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-------------------------------
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→ l `⊔ r ↪* l `⊔ r'
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ξ*-⊔₂ ↪*-refl = ↪*-refl
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ξ*-⊔₂ (↪*-step x r↪r') = ↪*-step (ξ-⊔₂ x) (ξ*-⊔₂ r↪r')
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--------------------------------------------------------------------------------
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-- Parallel Reduction
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@@ -182,10 +217,26 @@ data _↪ₚ_ : ∀ {n} → Term n → Term n → Set where
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β-subst : ∀ {n} {t : Term (suc n)} {e e' : Term n}
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→ e ↪ₚ e'
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→ `subst t `refl e ↪ₚ e'
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ξ-Set : ∀ {n} →
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`Set ↪ₚ (`Set {n = n})
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ξ-refl : ∀ {n} →
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`refl ↪ₚ (`refl {n = n})
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-- New:
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ξ-Setω : ∀ {n} {x x' : Term n} →
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x ↪ₚ x' →
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`Setω x ↪ₚ (`Setω x')
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ξ-Setn : ∀ {n} {x x' : Term n} →
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x ↪ₚ x' →
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`Setn x ↪ₚ (`Setn x')
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ξ-Level : ∀ {n} →
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`Level ↪ₚ `Level {n = n}
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ξ-lzero : ∀ {n} →
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`lzero ↪ₚ `lzero {n = n}
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ξ-lsuc : ∀ {n} {x x' : Term n} →
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x ↪ₚ x' →
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`lsuc x ↪ₚ `lsuc x'
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ξ-⊔ : ∀ {n} {e₁ e₁' e₂ e₂' : Term n} →
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e₁ ↪ₚ e₁' →
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e₂ ↪ₚ e₂' →
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e₁ `⊔ e₂ ↪ₚ e₁' `⊔ e₂'
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↪ₚ₁ : ∀ {n} {fst snd : Term n} → (fst ↪ₚ snd) → Term n
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↪ₚ₁ {n} {fst} {snd} _ = fst
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@@ -245,7 +296,6 @@ module ↪ₚ*-Reasoning where
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↪ₚ-refl {n} {` x} = ξ-`
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↪ₚ-refl {n} {`proj₁ t} = ξ-proj₁ ↪ₚ-refl
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↪ₚ-refl {n} {`proj₂ t} = ξ-proj₂ ↪ₚ-refl
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↪ₚ-refl {n} {`Set} = ξ-Set
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↪ₚ-refl {n} {`λ t} = ξ-λ ↪ₚ-refl
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↪ₚ-refl {n} {t · t₁} = ξ-· ↪ₚ-refl ↪ₚ-refl
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↪ₚ-refl {n} {∀[x⦂ t ] t₁} = ξ-∀ ↪ₚ-refl ↪ₚ-refl
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@@ -253,6 +303,12 @@ module ↪ₚ*-Reasoning where
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↪ₚ-refl {n} {t `, t₁} = ξ-, ↪ₚ-refl ↪ₚ-refl
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↪ₚ-refl {n} {t `≡ t₁} = ξ-≡ ↪ₚ-refl ↪ₚ-refl
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↪ₚ-refl {n} {`subst t t₁ t₂} = ξ-subst ↪ₚ-refl ↪ₚ-refl ↪ₚ-refl
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↪ₚ-refl {n} {`Setω x} = ξ-Setω ↪ₚ-refl
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↪ₚ-refl {n} {`Setn x} = ξ-Setn ↪ₚ-refl
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↪ₚ-refl {n} {`Level} = ξ-Level
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↪ₚ-refl {n} {`lzero} = ξ-lzero
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↪ₚ-refl {n} {`lsuc x} = ξ-lsuc ↪ₚ-refl
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↪ₚ-refl {n} {x `⊔ x₁} = ξ-⊔ ↪ₚ-refl ↪ₚ-refl
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--------------------------------------------------------------------------------
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@@ -281,6 +337,12 @@ module ↪ₚ*-Reasoning where
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↪→↪ₚ (ξ-subst₃ t↪t') = ξ-subst ↪ₚ-refl ↪ₚ-refl(↪→↪ₚ t↪t')
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↪→↪ₚ β-proj₁ = β-proj₁ ↪ₚ-refl
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↪→↪ₚ β-proj₂ = β-proj₂ ↪ₚ-refl
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-- New
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↪→↪ₚ (ξ-⊔₁ x) = ξ-⊔ (↪→↪ₚ x) ↪ₚ-refl
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↪→↪ₚ (ξ-⊔₂ x) = ξ-⊔ ↪ₚ-refl (↪→↪ₚ x)
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↪→↪ₚ (ξ-Setn x) = ξ-Setn (↪→↪ₚ x)
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↪→↪ₚ (ξ-Setω x) = ξ-Setω (↪→↪ₚ x)
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↪→↪ₚ (ξ-lsuc x) = ξ-lsuc (↪→↪ₚ x)
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↪*→↪ₚ* : ∀ {n} {t t' : Term n}
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→ t ↪* t'
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@@ -295,7 +357,6 @@ module ↪ₚ*-Reasoning where
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→ t ↪ₚ t'
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→ t ↪* t'
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↪ₚ→↪* ξ-` = ↪*-refl
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↪ₚ→↪* ξ-Set = ↪*-refl
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↪ₚ→↪* (ξ-λ ↪₁) = ξ*-λ (↪ₚ→↪* ↪₁)
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↪ₚ→↪* (ξ-∀ ↪₁ ↪₂) = b
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where
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@@ -336,6 +397,15 @@ module ↪ₚ*-Reasoning where
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e = ↪*-trans c d
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↪ₚ→↪* (β-proj₁ x↪x') = ↪*-step β-proj₁ (↪ₚ→↪* x↪x')
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↪ₚ→↪* (β-proj₂ x↪x') = ↪*-step β-proj₂ (↪ₚ→↪* x↪x')
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↪ₚ→↪* (ξ-Setω x↪x') = ξ*-Setω (↪ₚ→↪* x↪x')
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↪ₚ→↪* (ξ-Setn x↪x') = ξ*-Setn (↪ₚ→↪* x↪x')
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↪ₚ→↪* ξ-Level = ↪*-refl
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↪ₚ→↪* ξ-lzero = ↪*-refl
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↪ₚ→↪* (ξ-lsuc x↪x') = ξ*-lsuc (↪ₚ→↪* x↪x')
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↪ₚ→↪* (ξ-⊔ l↪l' r↪r') = ↪*-trans a b
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where
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a = ξ*-⊔₁ (↪ₚ→↪* l↪l')
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b = ξ*-⊔₂ (↪ₚ→↪* r↪r')
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↪ₚ*→↪* : ∀ {n} {t t' : Term n}
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→ t ↪ₚ* t'
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@@ -361,7 +431,6 @@ module ↪ₚ*-Reasoning where
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↪ₚ-ren ρ (ξ-λ ↪) = ξ-λ (↪ₚ-ren (extᵣ ρ) ↪)
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↪ₚ-ren ρ (ξ-∀ ↪₁ ↪₂) = ξ-∀ (↪ₚ-ren ρ ↪₁) (↪ₚ-ren (extᵣ ρ) ↪₂)
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↪ₚ-ren ρ (ξ-· ↪₁ ↪₂) = ξ-· (↪ₚ-ren ρ ↪₁) (↪ₚ-ren ρ ↪₂)
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↪ₚ-ren ρ ξ-Set = ξ-Set
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↪ₚ-ren ρ (ξ-proj₁ t↪t') = ξ-proj₁ (↪ₚ-ren ρ t↪t')
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↪ₚ-ren ρ (ξ-proj₂ t↪t') = ξ-proj₂ (↪ₚ-ren ρ t↪t')
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↪ₚ-ren ρ (ξ-∃ t↪t' t↪t'') = ξ-∃ (↪ₚ-ren ρ t↪t') (↪ₚ-ren (extᵣ ρ) t↪t'')
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@@ -372,8 +441,14 @@ module ↪ₚ*-Reasoning where
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↪ₚ-ren ρ ξ-refl = ξ-refl
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↪ₚ-ren ρ (β-proj₁ x↪x') = β-proj₁ (↪ₚ-ren ρ x↪x')
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↪ₚ-ren ρ (β-proj₂ x↪x') = β-proj₂ (↪ₚ-ren ρ x↪x')
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↪ₚ-ren ρ (ξ-Setω x) = ξ-Setω (↪ₚ-ren ρ x)
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↪ₚ-ren ρ (ξ-Setn x) = ξ-Setn (↪ₚ-ren ρ x)
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↪ₚ-ren ρ ξ-Level = ξ-Level
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↪ₚ-ren ρ ξ-lzero = ξ-lzero
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↪ₚ-ren ρ (ξ-lsuc x) = ξ-lsuc (↪ₚ-ren ρ x)
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↪ₚ-ren ρ (ξ-⊔ l r) = ξ-⊔ (↪ₚ-ren ρ l) (↪ₚ-ren ρ r)
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-------{! !}-----------------------------------------------------------------------
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------------------------------------------------------------------------------
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-- A substitution σ₁ reduces to σ₂, if each term of σ₁ reduces to the
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-- corresponding term of σ₂.
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@@ -402,7 +477,6 @@ _↪ₚσ_ : ∀ {m n} (σ₁ σ₂ : Sub m n) → Set
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↪ₚσ-sub ↪σ (ξ-· l↪ₚl' r↪ₚr') = ξ-· (↪ₚσ-sub ↪σ l↪ₚl') (↪ₚσ-sub ↪σ r↪ₚr')
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↪ₚσ-sub ↪σ (ξ-≡ l↪ₚl' r↪ₚr') = ξ-≡ (↪ₚσ-sub ↪σ l↪ₚl') (↪ₚσ-sub ↪σ r↪ₚr')
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↪ₚσ-sub ↪σ (ξ-, l↪ₚl' r↪ₚr') = ξ-, (↪ₚσ-sub ↪σ l↪ₚl') (↪ₚσ-sub ↪σ r↪ₚr')
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↪ₚσ-sub ↪σ ξ-Set = ξ-Set
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↪ₚσ-sub {σ' = σ'} ↪σ (β-λ ₁↪ₚ ₂↪ₚ)
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= ↪ₚ-≡ (β-λ (↪ₚσ-sub (↪ₚσ-ext ↪σ) ₁↪ₚ) (↪ₚσ-sub ↪σ ₂↪ₚ)) (swap-0↦-extₛ-fusion σ' (↪ₚ₂ ₂↪ₚ) (↪ₚ₂ ₁↪ₚ))
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↪ₚσ-sub ↪σ (ξ-proj₁ e↪ₚe') = ξ-proj₁ (↪ₚσ-sub ↪σ e↪ₚe')
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@@ -411,6 +485,12 @@ _↪ₚσ_ : ∀ {m n} (σ₁ σ₂ : Sub m n) → Set
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↪ₚσ-sub ↪σ (β-subst e↪e') = β-subst (↪ₚσ-sub ↪σ e↪e')
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↪ₚσ-sub ↪σ (β-proj₁ x) = β-proj₁ (↪ₚσ-sub ↪σ x)
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↪ₚσ-sub ↪σ (β-proj₂ x) = β-proj₂ (↪ₚσ-sub ↪σ x)
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↪ₚσ-sub ↪σ (ξ-Setω x) = ξ-Setω (↪ₚσ-sub ↪σ x)
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↪ₚσ-sub ↪σ (ξ-Setn x) = ξ-Setn (↪ₚσ-sub ↪σ x)
|
||||
↪ₚσ-sub ↪σ ξ-Level = ξ-Level
|
||||
↪ₚσ-sub ↪σ ξ-lzero = ξ-lzero
|
||||
↪ₚσ-sub ↪σ (ξ-lsuc x) = ξ-lsuc (↪ₚσ-sub ↪σ x)
|
||||
↪ₚσ-sub ↪σ (ξ-⊔ x x₁) = ξ-⊔ (↪ₚσ-sub ↪σ x) (↪ₚσ-sub ↪σ x₁)
|
||||
|
||||
↪ₚσ-0↦ : ∀ {n} {e₁ e₂ : Term n} →
|
||||
e₁ ↪ₚ e₂ →
|
||||
@@ -435,7 +515,6 @@ _⁺ : ∀ {n} → Term n → Term n
|
||||
(e₁ `≡ e₂) ⁺ = (e₁ ⁺) `≡ (e₂ ⁺)
|
||||
(∀[x⦂ t₁ ] t₂) ⁺ = ∀[x⦂ (t₁ ⁺) ] (t₂ ⁺)
|
||||
(∃[x⦂ t₁ ] t₂) ⁺ = ∃[x⦂ (t₁ ⁺) ] (t₂ ⁺)
|
||||
`Set ⁺ = `Set
|
||||
`refl ⁺ = `refl
|
||||
`proj₁ (e `, _) ⁺ = (e ⁺)
|
||||
`proj₂ (_ `, e) ⁺ = (e ⁺)
|
||||
@@ -443,6 +522,12 @@ _⁺ : ∀ {n} → Term n → Term n
|
||||
`proj₂ e ⁺ = `proj₂ (e ⁺)
|
||||
(`subst t `refl e) ⁺ = (e ⁺)
|
||||
(`subst t e₁ e₂) ⁺ = `subst (t ⁺) (e₁ ⁺) (e₂ ⁺)
|
||||
(`Setω x) ⁺ = `Setω (x ⁺)
|
||||
(`Setn x) ⁺ = `Setn (x ⁺)
|
||||
`Level ⁺ = `Level
|
||||
`lzero ⁺ = `lzero
|
||||
`lsuc x ⁺ = `lsuc (x ⁺)
|
||||
(x `⊔ x₁) ⁺ = (x ⁺) `⊔ (x₁ ⁺)
|
||||
|
||||
-- Note, in part 4, this proof might have *many* cases. That's to be
|
||||
-- expected, and most of them are actually very similar.
|
||||
@@ -452,7 +537,6 @@ par-triangle : ∀ {n} {e e' : Term n}
|
||||
-------
|
||||
→ e' ↪ₚ (e ⁺)
|
||||
par-triangle ξ-` = ξ-`
|
||||
par-triangle ξ-Set = ξ-Set
|
||||
par-triangle ξ-refl = ξ-refl
|
||||
par-triangle (β-λ ₁↪ₚ ₂↪ₚ) = ↪ₚσ-[] (par-triangle ₁↪ₚ) (par-triangle ₂↪ₚ)
|
||||
par-triangle (ξ-λ x) = ξ-λ (par-triangle x)
|
||||
@@ -462,7 +546,6 @@ par-triangle (ξ-· (ξ-∀ ll lr) r) = ξ-· (ξ-∀ (par-triangle ll) (par-tri
|
||||
par-triangle (ξ-· ξ-` r) = ξ-· ξ-` (par-triangle r)
|
||||
par-triangle (ξ-· (ξ-λ l) r) = β-λ (par-triangle l) (par-triangle r)
|
||||
par-triangle (ξ-· (ξ-· ll lr) r) = ξ-· (par-triangle (ξ-· ll lr)) (par-triangle r)
|
||||
par-triangle (ξ-· ξ-Set r) = ξ-· ξ-Set (par-triangle r)
|
||||
par-triangle (ξ-· p@(β-proj₁ l) r) = ξ-· (par-triangle p) (par-triangle r)
|
||||
par-triangle (ξ-· p@(β-proj₂ l) r) = ξ-· (par-triangle p) (par-triangle r)
|
||||
par-triangle (ξ-· (ξ-proj₁ l) r) = ξ-· (par-triangle (ξ-proj₁ l)) (par-triangle r)
|
||||
@@ -489,7 +572,6 @@ par-triangle (ξ-proj₁ (ξ-≡ e e₁)) = ξ-proj₁ (ξ-≡ (par-triangle e)
|
||||
par-triangle (ξ-proj₁ (ξ-, e e₁)) = β-proj₁ (par-triangle e)
|
||||
par-triangle (ξ-proj₁ (ξ-subst e e₁ e₂)) = ξ-proj₁ (par-triangle (ξ-subst e e₁ e₂))
|
||||
par-triangle (ξ-proj₁ (β-subst e)) = ξ-proj₁ (par-triangle e)
|
||||
par-triangle (ξ-proj₁ ξ-Set) = ξ-proj₁ ξ-Set
|
||||
par-triangle (ξ-proj₁ ξ-refl) = ξ-proj₁ ξ-refl
|
||||
par-triangle (ξ-proj₂ ξ-`) = ξ-proj₂ ξ-`
|
||||
par-triangle (ξ-proj₂ (β-proj₁ e)) = ξ-proj₂ (par-triangle e)
|
||||
@@ -505,7 +587,6 @@ par-triangle (ξ-proj₂ (ξ-≡ e e₁)) = ξ-proj₂ (ξ-≡ (par-triangle e)
|
||||
par-triangle (ξ-proj₂ (ξ-, e e₁)) = β-proj₂ (par-triangle e₁)
|
||||
par-triangle (ξ-proj₂ (ξ-subst e e₁ e₂)) = ξ-proj₂ (par-triangle (ξ-subst e e₁ e₂))
|
||||
par-triangle (ξ-proj₂ (β-subst e)) = ξ-proj₂ (par-triangle e)
|
||||
par-triangle (ξ-proj₂ ξ-Set) = ξ-proj₂ ξ-Set
|
||||
par-triangle (ξ-proj₂ ξ-refl) = ξ-proj₂ ξ-refl
|
||||
par-triangle (ξ-∃ e e₁) = ξ-∃ (par-triangle e) (par-triangle e₁)
|
||||
par-triangle (ξ-≡ e e₁) = ξ-≡ (par-triangle e) (par-triangle e₁)
|
||||
@@ -525,8 +606,31 @@ par-triangle (ξ-subst e (ξ-≡ e₁ e₃) e₂) = ξ-subst (par-triangle e) (
|
||||
par-triangle (ξ-subst e (ξ-, e₁ e₃) e₂) = ξ-subst (par-triangle e) (ξ-, (par-triangle e₁) (par-triangle e₃)) (par-triangle e₂)
|
||||
par-triangle (ξ-subst e (ξ-subst e₁ e₃ e₄) e₂) = ξ-subst (par-triangle e) (par-triangle (ξ-subst e₁ e₃ e₄)) (par-triangle e₂)
|
||||
par-triangle (ξ-subst e (β-subst e₁) e₂) = ξ-subst (par-triangle e) (par-triangle e₁) (par-triangle e₂)
|
||||
par-triangle (ξ-subst e ξ-Set e₂) = ξ-subst (par-triangle e) ξ-Set (par-triangle e₂)
|
||||
par-triangle (ξ-subst e ξ-refl e₂) = β-subst (par-triangle e₂)
|
||||
par-triangle (ξ-proj₁ (ξ-Setω x)) = ξ-proj₁ (ξ-Setω (par-triangle x))
|
||||
par-triangle (ξ-proj₁ (ξ-Setn x)) = ξ-proj₁ (ξ-Setn (par-triangle x))
|
||||
par-triangle (ξ-proj₁ ξ-Level) = ξ-proj₁ ξ-Level
|
||||
par-triangle (ξ-proj₁ ξ-lzero) = ξ-proj₁ (par-triangle ξ-lzero)
|
||||
par-triangle (ξ-proj₁ (ξ-lsuc x)) = ξ-proj₁ (ξ-lsuc (par-triangle x))
|
||||
par-triangle (ξ-proj₁ (ξ-⊔ x x₁)) = ξ-proj₁ (ξ-⊔ (par-triangle x) (par-triangle x₁))
|
||||
par-triangle (ξ-proj₂ (ξ-Setω x)) = ξ-proj₂ (ξ-Setω (par-triangle x))
|
||||
par-triangle (ξ-proj₂ (ξ-Setn x)) = ξ-proj₂ (ξ-Setn (par-triangle x))
|
||||
par-triangle (ξ-proj₂ ξ-Level) = ξ-proj₂ ξ-Level
|
||||
par-triangle (ξ-proj₂ ξ-lzero) = ξ-proj₂ (par-triangle ξ-lzero)
|
||||
par-triangle (ξ-proj₂ (ξ-lsuc x)) = ξ-proj₂ (ξ-lsuc (par-triangle x))
|
||||
par-triangle (ξ-proj₂ (ξ-⊔ x x₁)) = ξ-proj₂ (ξ-⊔ (par-triangle x) (par-triangle x₁))
|
||||
par-triangle (ξ-· x x₁) = {! !}
|
||||
par-triangle (ξ-subst x x₁ x₂) = {! !}
|
||||
par-triangle (ξ-Setω x) = ξ-Setω (par-triangle x)
|
||||
par-triangle (ξ-Setn x) = ξ-Setn (par-triangle x)
|
||||
par-triangle ξ-Level = ξ-Level
|
||||
par-triangle ξ-lzero = ξ-lzero
|
||||
par-triangle (ξ-lsuc x) = ξ-lsuc (par-triangle x)
|
||||
par-triangle (ξ-⊔ x x₁) = ξ-⊔ (par-triangle x) (par-triangle x₁)
|
||||
-- par-triangle (ξ-proj₁ ξ-Set) = ξ-proj₁ ξ-Set
|
||||
-- par-triangle (ξ-proj₂ ξ-Set) = ξ-proj₂ ξ-Set
|
||||
-- par-triangle (ξ-· ξ-Set r) = ξ-· ξ-Set (par-triangle r)
|
||||
-- par-triangle (ξ-subst e ξ-Set e₂) = ξ-subst (par-triangle e) ξ-Set (par-triangle e₂)
|
||||
|
||||
1→* : ∀ {m} {e e' : Term m} → e ↪ₚ e' → (e ↪ₚ* e')
|
||||
1→* e↪ₚe' = ↪ₚ*-step e↪ₚe' ↪ₚ*-refl
|
||||
|
||||
@@ -37,8 +37,8 @@ data Term : ℕ → Set where
|
||||
`Setω_ : ∀ {n} (l : Term n) → Term n
|
||||
`Setn_ : ∀ {n} (l : Term n) → Term n
|
||||
`Level : ∀ {n} → Term n
|
||||
`lzero : ∀ {n} → Term n
|
||||
`lsuc : ∀ {n} → Term n → Term n
|
||||
`lzero : ∀ {n} → Term n
|
||||
`lsuc : ∀ {n} → Term n → Term n
|
||||
_`⊔_ : ∀ {n} → Term n → Term n → Term n
|
||||
|
||||
-- Extension
|
||||
@@ -149,10 +149,10 @@ mutual
|
||||
val-Level : ∀ {n}
|
||||
→ Value {n = n} `Level
|
||||
|
||||
val-l : ∀ {n v}
|
||||
val-l : ∀ {n}
|
||||
→ Value {n = n} (`lzero)
|
||||
|
||||
val-lsuc : ∀ {n v l}
|
||||
val-lsuc : ∀ {n l}
|
||||
→ Value l
|
||||
→ Value {n = n} (`lsuc l)
|
||||
|
||||
@@ -289,7 +289,7 @@ data _↪_ : ∀ {n} → Term n → Term n → Set where
|
||||
→ e₂ ↪ e₂'
|
||||
→ `subst t e₁ e₂ ↪ `subst t e₁ e₂'
|
||||
|
||||
-- Extension:
|
||||
-- New
|
||||
ξ-⊔₁ : ∀ {n} {t₁ t₁' : Term n} {t₂ : Term n}
|
||||
→ t₁ ↪ t₁'
|
||||
-------------------------------
|
||||
@@ -300,6 +300,10 @@ data _↪_ : ∀ {n} → Term n → Term n → Set where
|
||||
-------------------------------
|
||||
→ (t₁ `⊔ t₂) ↪ (t₁ `⊔ t₂')
|
||||
|
||||
ξ-lsuc : ∀ {n} {l l' : Term n}
|
||||
→ l ↪ l'
|
||||
→ `lsuc l ↪ `lsuc l'
|
||||
|
||||
ξ-Setn : ∀ {n} {l l' : Term n}
|
||||
→ l ↪ l'
|
||||
→ `Setn l ↪ `Setn l'
|
||||
@@ -467,3 +471,4 @@ data _⊢_⦂_ : ∀ {n} → Context n → Term n → Term n → Set where
|
||||
|
||||
cong₃ : ∀ {A B C D : Set} (f : A → B → C → D) {x y u v a b} → x ≡ y → u ≡ v → a ≡ b → f x u a ≡ f y v b
|
||||
cong₃ f refl refl refl = refl
|
||||
|
||||
|
||||
Reference in New Issue
Block a user