{-# OPTIONS --without-K --safe #-} open import Level using (0ℓ) open import Axiom.Extensionality.Propositional -- Various properties of domain-level operations -- -- This modulke covers properties of domain-level operations including insertion, -- composition, truncation and truncation offsets. -- -- This module is one of the root sources that use functional extensionality. module Mint.Semantics.Properties.Domain (fext : Extensionality 0ℓ 0ℓ) where open import Data.Nat.Properties as Nₚ open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡×≡⇒≡) open import Lib open import Mint.Statics.Syntax open import Mint.Semantics.Domain open import Mint.Semantics.Properties.NoFunExt.Domain public vone-stable : ins vone 1 ≡ vone vone-stable = fext λ { zero → refl ; (suc n) → refl } vone-∥ : ∀ n → (vone ∥ n) ≡ vone vone-∥ n = fext λ m → refl ins-ø : ∀ n κ κ′ → (ins κ n ø κ′) ≡ ins (κ ø κ′ ∥ n) (O κ′ n) ins-ø n κ κ′ = fext λ { zero → refl ; (suc m) → refl } ∥-+ : ∀ (κ : UMoT) n m → κ ∥ n + m ≡ κ ∥ n ∥ m ∥-+ κ n m = fext (λ i → cong κ (+-assoc n m i)) ø-∥ : ∀ κ κ′ n → (κ ø κ′) ∥ n ≡ (κ ∥ n ø (κ′ ∥ O κ n)) ø-∥ κ κ′ zero = refl ø-∥ κ κ′ (suc n) rewrite ∥-+ κ′ (κ 0) (O (κ ∥ 1) n) | ø-∥ (κ ∥ 1) (κ′ ∥ κ 0) n = refl ø-idx : ∀ κ κ′ n → (κ ø κ′) n ≡ O (κ′ ∥ O κ n) (κ n) ø-idx κ κ′ zero = refl ø-idx κ κ′ (suc n) = trans (ø-idx (κ ∥ 1) (κ′ ∥ κ 0) n) (cong (λ k → M-O k (κ (suc n))) (fext λ m → cong κ′ (sym (+-assoc (κ 0) (O (κ ∥ 1) n) m)))) -- monoidal structor of vone and ø vone-ø : ∀ κ → (vone ø κ) ≡ κ vone-ø κ = fext helper where helper : ∀ n → (vone ø κ) n ≡ κ n helper n rewrite ø-idx vone κ n | O-vone n | +-identityʳ n = +-identityʳ (κ n) ø-vone : ∀ κ → (κ ø vone) ≡ κ ø-vone κ = fext helper where helper : ∀ n → (κ ø vone) n ≡ κ n helper n rewrite ø-idx κ vone n = O-vone (κ n) ins-vone-ø : ∀ n κ → (ins vone n ø κ) ≡ ins (κ ∥ n) (O κ n) ins-vone-ø n κ rewrite ins-ø n vone κ | vone-ø (κ ∥ n) = refl ins-1-ø-ins-n : ∀ κ κ′ n → (ins κ 1 ø ins κ′ n) ≡ ins (κ ø κ′) n ins-1-ø-ins-n κ κ′ n rewrite ins-ø 1 κ (ins κ′ n) | +-identityʳ n = refl ins-1-ø-ins-vone : ∀ κ n → (ins κ 1 ø ins vone n) ≡ ins κ n ins-1-ø-ins-vone κ n rewrite ins-1-ø-ins-n κ vone n | ø-vone κ = refl ø-assoc : ∀ κ κ′ κ″ → (κ ø κ′ ø κ″) ≡ (κ ø (κ′ ø κ″)) ø-assoc κ κ′ κ″ = fext (helper κ κ′ κ″) where helper : ∀ κ κ′ κ″ n → (κ ø κ′ ø κ″) n ≡ (κ ø (κ′ ø κ″)) n helper κ κ′ κ″ zero = sym (O-ø κ′ κ″ (κ 0)) helper κ κ′ κ″ (suc n) rewrite ø-∥ κ′ κ″ (κ 0) = helper (κ ∥ 1) (κ′ ∥ κ 0) (κ″ ∥ O κ′ (κ 0)) n -- distributivity of O over UMoT application to an Envs O-ρ-[] : ∀ (ρ : Envs) (κ : UMoT) n → O (ρ [ κ ]) n ≡ O κ (O ρ n) O-ρ-[] ρ κ zero = refl O-ρ-[] ρ κ (suc n) rewrite O-+ κ (proj₁ (ρ 0)) (O (ρ ∥ 1) n) | sym (O-ρ-[] (ρ ∥ 1) (κ ∥ proj₁ (ρ 0)) n) = cong (O κ (proj₁ (ρ 0)) +_) (cong (λ k → M-O k n) (fext (λ m → cong (λ k → M-O k (proj₁ (ρ (suc m)))) (∥-+ κ (proj₁ (ρ 0)) (O (ρ ∥ 1) m))))) -- composition of UMoT application mutual D-comp : ∀ (a : D) κ κ′ → a [ κ ] [ κ′ ] ≡ a [ κ ø κ′ ] D-comp N κ κ′ = refl D-comp (□ A) κ κ′ rewrite sym (ins-ø 1 κ (ins κ′ 1)) = cong □ (D-comp A (ins κ 1) (ins κ′ 1)) D-comp (Π A t ρ) κ κ′ rewrite D-comp A κ κ′ | ρ-comp ρ κ κ′ = refl D-comp (U i) κ κ′ = refl D-comp ze κ κ′ = refl D-comp (su a) κ κ′ = cong su (D-comp a κ κ′) D-comp (Λ t ρ) κ κ′ rewrite ρ-comp ρ κ κ′ = refl D-comp (box a) κ κ′ rewrite sym (ins-ø 1 κ (ins κ′ 1)) = cong box (D-comp a (ins κ 1) (ins κ′ 1)) D-comp (↑ A c) κ κ′ = cong₂ ↑ (D-comp A κ κ′) (Dn-comp c κ κ′) Dn-comp : ∀ (c : Dn) κ κ′ → c [ κ ] [ κ′ ] ≡ c [ κ ø κ′ ] Dn-comp (l x) κ κ′ = refl Dn-comp (rec T a t ρ c) κ κ′ rewrite D-comp a κ κ′ | ρ-comp ρ κ κ′ | Dn-comp c κ κ′ = refl Dn-comp (c $ d) κ κ′ = cong₂ _$_ (Dn-comp c κ κ′) (Df-comp d κ κ′) Dn-comp (unbox n c) κ κ′ rewrite O-ø κ κ′ n | ø-∥ κ κ′ n = cong (unbox (O κ′ (O κ n))) (Dn-comp c (κ ∥ n) (κ′ ∥ O κ n)) Df-comp : ∀ (d : Df) κ κ′ → d [ κ ] [ κ′ ] ≡ d [ κ ø κ′ ] Df-comp (↓ A a) κ κ′ = cong₂ ↓ (D-comp A κ κ′) (D-comp a κ κ′) ρ-comp : ∀ (ρ : Envs) κ κ′ → ρ [ κ ] [ κ′ ] ≡ ρ [ κ ø κ′ ] ρ-comp ρ κ κ′ = fext λ n → ≡×≡⇒≡ (helper n , helper′ n) where helper : ∀ n → proj₁ ((ρ [ κ ] [ κ′ ]) n) ≡ proj₁ ((ρ [ κ ø κ′ ]) n) helper n rewrite ø-∥ κ κ′ (O ρ n) | O-ρ-[] ρ κ n = sym (O-ø (κ ∥ O ρ n) (κ′ ∥ O κ (O ρ n)) (proj₁ (ρ n))) helper′ : ∀ n → proj₂ ((ρ [ κ ] [ κ′ ]) n) ≡ proj₂ ((ρ [ κ ø κ′ ]) n) helper′ n rewrite ø-∥ κ κ′ (O ρ n) | O-ρ-[] ρ κ n = fext λ m → D-comp (proj₂ (ρ n) m) (κ ∥ O ρ n) (κ′ ∥ O κ (O ρ n)) -- distributivity of truncation over UMoT application to an Env ρ-∥-[] : ∀ (ρ : Envs) (κ : UMoT) n → (ρ [ κ ]) ∥ n ≡ ρ ∥ n [ κ ∥ O ρ n ] ρ-∥-[] ρ κ n = fext λ m → ≡×≡⇒≡ (helper m , helper′ m) where helper : ∀ m → proj₁ (((ρ [ κ ]) ∥ n) m) ≡ proj₁ ((ρ ∥ n [ κ ∥ O ρ n ]) m) helper m rewrite O-+ (toUMoT ρ) n m = cong (λ k → M-O k (proj₁ (ρ (n + m)))) (fext λ i → cong κ (+-assoc (O ρ n) (O (ρ ∥ n) m) i)) helper′ : ∀ m → proj₂ (((ρ [ κ ]) ∥ n) m) ≡ proj₂ ((ρ ∥ n [ κ ∥ O ρ n ]) m) helper′ m rewrite O-+ (toUMoT ρ) n m = fext λ i → cong (mtran (proj₂ (ρ (n + m)) i)) (fext λ i → cong κ (+-assoc (O ρ n) (O (ρ ∥ n) m) i)) -- identity of UMoT application mutual D-ap-vone : ∀ (a : D) → a [ vone ] ≡ a D-ap-vone N = refl D-ap-vone (□ A) rewrite vone-stable = cong □ (D-ap-vone A) D-ap-vone (Π A t ρ) rewrite D-ap-vone A | ρ-ap-vone ρ = refl D-ap-vone (U i) = refl D-ap-vone ze = refl D-ap-vone (su a) = cong su (D-ap-vone a) D-ap-vone (Λ t ρ) = cong (Λ t) (ρ-ap-vone ρ) D-ap-vone (box a) rewrite vone-stable = cong box (D-ap-vone a) D-ap-vone (↑ A c) = cong₂ ↑ (D-ap-vone A) (Dn-ap-vone c) Dn-ap-vone : ∀ (c : Dn) → c [ vone ] ≡ c Dn-ap-vone (l x) = refl Dn-ap-vone (rec T a t ρ c) rewrite D-ap-vone a | ρ-ap-vone ρ | Dn-ap-vone c = refl Dn-ap-vone (c $ d) = cong₂ _$_ (Dn-ap-vone c) (Df-ap-vone d) Dn-ap-vone (unbox k c) rewrite O-vone k = cong (unbox k) (Dn-ap-vone c) Df-ap-vone : ∀ (d : Df) → d [ vone ] ≡ d Df-ap-vone (↓ A a) = cong₂ ↓ (D-ap-vone A) (D-ap-vone a) ρ-ap-vone : ∀ (ρ : Envs) → ρ [ vone ] ≡ ρ ρ-ap-vone ρ = fext helper where helper : ∀ n → (ρ [ vone ]) n ≡ ρ n helper n = ≡×≡⇒≡ (O-vone (proj₁ (ρ n)) , fext λ m → D-ap-vone (proj₂ (ρ n) m)) -- monotonicity of various operations ↦-mon : ∀ ρ a (κ : UMoT) → (ρ ↦ a) [ κ ] ≡ ρ [ κ ] ↦ a [ κ ] ↦-mon ρ a κ = fext λ { 0 → ≡×≡⇒≡ (refl , (fext λ { 0 → refl ; (suc m) → refl })) ; (suc n) → refl } ext1-mon-ins : ∀ ρ κ k → ext ρ 1 [ ins κ k ] ≡ ext (ρ [ κ ]) k ext1-mon-ins ρ κ k = fext λ { 0 → ≡×≡⇒≡ (+-identityʳ _ , refl) ; (suc n) → refl } ext1-mon : ∀ ρ n → ext ρ 1 [ ins vone n ] ≡ ext ρ n ext1-mon ρ n rewrite ext1-mon-ins ρ vone n | ρ-ap-vone ρ = refl ext-mon : ∀ ρ k (κ : UMoT) → ext ρ k [ κ ] ≡ ext (ρ [ κ ∥ k ]) (O κ k) ext-mon ρ k κ = fext λ { 0 → refl ; (suc n) → ≡×≡⇒≡ ( cong (λ κ′ → O κ′ (proj₁ (ρ n))) (fext λ m → cong κ (+-assoc k (O ρ n) m)) , fext λ m → cong (proj₂ (ρ n) m [_]) (fext λ l → cong κ (+-assoc k (O ρ n) l))) } drop-mon : ∀ ρ (κ : UMoT) → drop ρ [ κ ] ≡ drop (ρ [ κ ]) drop-mon ρ κ = fext λ { 0 → refl ; (suc n) → refl } drop-↦ : ∀ ρ a → drop (ρ ↦ a) ≡ ρ drop-↦ ρ a = fext λ { 0 → refl ; (suc n) → refl } drop-same : ∀ ρ → drop ρ ↦ lookup ρ 0 ≡ ρ drop-same ρ = fext λ { zero → ≡×≡⇒≡ (refl , fext λ { zero → refl ; (suc x) → refl }) ; (suc n) → refl } D-ins-ins : ∀ (a : D) κ n → a [ ins κ 1 ] [ ins vone n ] ≡ a [ ins κ n ] D-ins-ins a κ n = trans (D-comp a (ins κ 1) (ins vone n)) (cong (a [_]) (ins-1-ø-ins-vone κ n)) D-ins-ins′ : ∀ (a : D) κ κ′ n → a [ ins κ 1 ] [ ins κ′ n ] ≡ a [ ins (κ ø κ′) n ] D-ins-ins′ a κ κ′ n = trans (D-comp a (ins κ 1) (ins κ′ n)) (cong (a [_]) (ins-1-ø-ins-n κ κ′ n))