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wadler
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2 changed files with 71 additions and 152 deletions
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@ -410,7 +410,7 @@ the argument on which we induct becomes *smaller*.
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Generalising on an auxiliary argument, which becomes larger as the argument on
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which we recurse or induct becomes smaller, is a common trick. It belongs in
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you sling of arrows, ready to slay the right problem.
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your quiver of arrows, ready to slay the right problem.
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Having defined shunt be generalisation, it is now easy to respecialise to
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give a more efficient definition of reverse.
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@ -14,11 +14,12 @@ module Typed where
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; refl; sym; trans; cong; cong₂; _≢_)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Data.List using (List; []; _∷_; [_]; _++_; map; foldr; filter)
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open import Data.List using (List; []; _∷_; _++_; map; foldr; filter)
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open import Data.List.Any using (Any; here; there)
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open import Data.Nat using (ℕ; zero; suc; _+_; _∸_; _≤_; _⊔_; _≟_)
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open import Data.Nat.Properties using (≤-refl; ≤-trans; m≤m⊔n; n≤m⊔n; 1+n≰n)
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open import Data.Product using (_×_; proj₁; proj₂; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
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open import Data.Product using (_×_; proj₁; proj₂; ∃; ∃-syntax)
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renaming (_,_ to ⟨_,_⟩)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Unit using (⊤; tt)
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open import Function using (_∘_)
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@ -27,6 +28,7 @@ open import Function.Equivalence using (_⇔_; equivalence)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Nullary.Negation using (contraposition; ¬?)
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open import Relation.Nullary.Product using (_×-dec_)
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open import Collections using (_↔_)
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\end{code}
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@ -216,58 +218,20 @@ erase-lemma (⊢L · ⊢M) = cong₂ _·_ (erase-lemma ⊢L) (er
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### Lists as sets
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\begin{code}
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infix 4 _∈_
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infix 4 _⊆_
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infixl 5 _∪_
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infixl 5 _\\_
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_∈_ : Id → List Id → Set
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x ∈ xs = Any (x ≡_) xs
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_⊆_ : List Id → List Id → Set
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xs ⊆ ys = ∀ {w} → w ∈ xs → w ∈ ys
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_∪_ : List Id → List Id → List Id
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xs ∪ ys = xs ++ ys
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_\\_ : List Id → Id → List Id
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xs \\ x = filter (¬? ∘ (x ≟_)) xs
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open Collections.CollectionDec (Id) (_≟_)
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\end{code}
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### Properties of sets
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\begin{code}
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⊆∷ : ∀ {y xs ys} → xs ⊆ ys → xs ⊆ y ∷ ys
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⊆∷ xs⊆ ∈xs = there (xs⊆ ∈xs)
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∷⊆∷ : ∀ {x xs ys} → xs ⊆ ys → (x ∷ xs) ⊆ (x ∷ ys)
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∷⊆∷ xs⊆ (here refl) = here refl
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∷⊆∷ xs⊆ (there ∈xs) = there (xs⊆ ∈xs)
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[]⊆ : ∀ {x xs} → [ x ] ⊆ xs → x ∈ xs
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[]⊆ ⊆xs = ⊆xs (here refl)
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⊆[] : ∀ {x xs} → x ∈ xs → [ x ] ⊆ xs
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⊆[] x∈ (here refl) = x∈
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⊆[] x∈ (there ())
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bind : ∀ {x xs} → xs \\ x ⊆ xs
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bind {x} {[]} ()
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bind {x} {y ∷ ys} with x ≟ y
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... | yes refl = ⊆∷ (bind {x} {ys})
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... | no _ = ∷⊆∷ (bind {x} {ys})
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left : ∀ {xs ys} → xs ⊆ xs ∪ ys
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left (here refl) = here refl
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left (there x∈) = there (left x∈)
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right : ∀ {xs ys} → ys ⊆ xs ∪ ys
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right {[]} y∈ = y∈
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right {x ∷ xs} y∈ = there (right {xs} y∈)
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prev : ∀ {z y xs} → y ≢ z → z ∈ y ∷ xs → z ∈ xs
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prev y≢z (here z≡y) = ⊥-elim (y≢z (sym z≡y))
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prev _ (there z∈) = z∈
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-- ⊆∷ : ∀ {y xs ys} → xs ⊆ ys → xs ⊆ y ∷ ys
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-- ∷⊆∷ : ∀ {x xs ys} → xs ⊆ ys → (x ∷ xs) ⊆ (x ∷ ys)
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-- []⊆ : ∀ {x xs} → [ x ] ⊆ xs → x ∈ xs
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-- ⊆[] : ∀ {x xs} → x ∈ xs → [ x ] ⊆ xs
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-- bind : ∀ {x xs} → xs \\ x ⊆ xs
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-- left : ∀ {xs ys} → xs ⊆ xs ∪ ys
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-- right : ∀ {xs ys} → ys ⊆ xs ∪ ys
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-- prev : ∀ {z y xs} → y ≢ z → z ∈ y ∷ xs → z ∈ xs
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\end{code}
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### Free variables
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@ -287,7 +251,7 @@ fresh : List Id → Id
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fresh = foldr _⊔_ 0 ∘ map suc
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⊔-lemma : ∀ {x xs} → x ∈ xs → suc x ≤ fresh xs
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⊔-lemma {x} {.x ∷ xs} (here refl) = m≤m⊔n (suc x) (fresh xs)
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⊔-lemma {x} {.x ∷ xs} here = m≤m⊔n (suc x) (fresh xs)
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⊔-lemma {x} {y ∷ xs} (there x∈) = ≤-trans (⊔-lemma {x} {xs} x∈)
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(n≤m⊔n (suc y) (fresh xs))
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@ -302,9 +266,9 @@ fresh-lemma x∈ refl = 1+n≰n (⊔-lemma x∈)
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∅ x = ⌊ x ⌋
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_,_↦_ : (Id → Term) → Id → Term → (Id → Term)
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(ρ , x ↦ M) y with x ≟ y
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(ρ , x ↦ M) w with w ≟ x
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... | yes _ = M
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... | no _ = ρ y
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... | no _ = ρ w
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\end{code}
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### Substitution
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@ -408,85 +372,47 @@ dom ε = []
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dom (Γ , x ⦂ A) = x ∷ dom Γ
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dom-lemma : ∀ {Γ y B} → Γ ∋ y ⦂ B → y ∈ dom Γ
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dom-lemma Z = here refl
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dom-lemma Z = here
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dom-lemma (S x≢y ⊢y) = there (dom-lemma ⊢y)
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f : ∀ {x y} → y ∈ [ x ] → y ≡ x
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f (here y≡x) = y≡x
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f (there ())
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g : ∀ {w xs ys} → w ∈ xs ∪ ys → w ∈ xs ⊎ w ∈ ys
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g {_} {[]} {ys} w∈ = inj₂ w∈
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g {_} {x ∷ xs} {ys} (here px) = inj₁ (here px)
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g {_} {x ∷ xs} {ys} (there w∈) with g w∈
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... | inj₁ ∈xs = inj₁ (there ∈xs)
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... | inj₂ ∈ys = inj₂ ∈ys
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k : ∀ {w x xs} → w ∈ xs \\ x → x ≢ w
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k {w} {x} {[]} ()
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k {w} {x} {x′ ∷ xs′} w∈ with x ≟ x′
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k {w} {x} {x′ ∷ xs′} w∈ | yes refl = k {w} {x} {xs′} w∈
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k {w} {x} {x′ ∷ xs′} (here w≡x′) | no x≢x′ = λ x≡w → x≢x′ (trans x≡w w≡x′)
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k {w} {x} {x′ ∷ xs′} (there w∈) | no x≢x′ = k {w} {x} {xs′} w∈
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h : ∀ {x xs ys} → xs ⊆ x ∷ ys → xs \\ x ⊆ ys
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h {x} {xs} {ys} xs⊆ {w} w∈ with xs⊆ (bind w∈)
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... | here w≡x = ⊥-elim (k {w} {x} {xs} w∈ (sym w≡x))
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... | there w∈′ = w∈′
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free-lemma : ∀ {Γ M A} → Γ ⊢ M ⦂ A → free M ⊆ dom Γ
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free-lemma ⌊ ⊢x ⌋ w∈ rewrite f w∈ = dom-lemma ⊢x
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free-lemma {Γ} (ƛ_ {x = x} {N = N} ⊢N) = h ih
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where
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ih : free N ⊆ x ∷ dom Γ
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ih = free-lemma ⊢N
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free-lemma (⊢L · ⊢M) w∈ with g w∈
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... | inj₁ ∈L = free-lemma ⊢L ∈L
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... | inj₂ ∈M = free-lemma ⊢M ∈M
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free-lemma ⌊ ⊢x ⌋ w∈ with w∈
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... | here = dom-lemma ⊢x
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... | there ()
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free-lemma {Γ} (ƛ_ {x = x} {N = N} ⊢N) = proj₂ lemma-\\-∷ (free-lemma ⊢N)
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free-lemma (⊢L · ⊢M) w∈ with proj₂ lemma-⊎-∪ w∈
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... | inj₁ ∈L = free-lemma ⊢L ∈L
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... | inj₂ ∈M = free-lemma ⊢M ∈M
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\end{code}
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Wow! A lot of work to prove stuff that is obvious. Gulp!
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### Renaming
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\begin{code}
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∷drop : ∀ {v vs ys} → v ∷ vs ⊆ ys → vs ⊆ ys
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∷drop ⊆ys ∈vs = ⊆ys (there ∈vs)
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i : ∀ {w x xs} → w ∈ xs → x ≢ w → w ∈ xs \\ x
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i {w} {x} {.w ∷ xs} (here refl) x≢w with x ≟ w
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... | yes refl = ⊥-elim (x≢w refl)
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... | no _ = here refl
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i {w} {x} {y ∷ xs} (there w∈) x≢w with x ≟ y
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... | yes refl = (i {w} {x} {xs} w∈ x≢w)
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... | no _ = there (i {w} {x} {xs} w∈ x≢w)
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j : ∀ {x xs ys} → xs \\ x ⊆ ys → xs ⊆ x ∷ ys
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j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
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... | yes refl = here refl
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... | no x≢w = there (⊆ys (i w∈ x≢w))
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⊢rename : ∀ {Γ Δ xs} → (∀ {x A} → x ∈ xs → Γ ∋ x ⦂ A → Δ ∋ x ⦂ A) →
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(∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ M ⦂ A)
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⊢rename ⊢σ ⊆xs (⌊ ⊢x ⌋) = ⌊ ⊢σ ∈xs ⊢x ⌋
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⊢rename ⊢σ ⊆xs (⌊ ⊢x ⌋) = ⌊ ⊢σ ∈xs ⊢x ⌋
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where
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∈xs = []⊆ ⊆xs
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∈xs = proj₂ lemma-[_] ⊆xs
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⊢rename {Γ} {Δ} {xs} ⊢σ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N)
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= ƛ (⊢rename {Γ′} {Δ′} {xs′} ⊢σ′ ⊆xs′ ⊢N) -- ⊆xs : free N \\ x ⊆ xs
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= ƛ (⊢rename {Γ′} {Δ′} {xs′} ⊢σ′ ⊆xs′ ⊢N)
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where
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Γ′ = Γ , x ⦂ A
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Δ′ = Δ , x ⦂ A
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xs′ = x ∷ xs
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⊢σ′ : ∀ {y B} → y ∈ xs′ → Γ′ ∋ y ⦂ B → Δ′ ∋ y ⦂ B
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⊢σ′ ∈xs′ Z = Z
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⊢σ′ ∈xs′ (S x≢y k) with ∈xs′
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... | here refl = ⊥-elim (x≢y refl)
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... | there ∈xs = S x≢y (⊢σ ∈xs k)
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⊢σ′ ∈xs′ Z = Z
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⊢σ′ ∈xs′ (S x≢y ⊢y) with ∈xs′
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... | here = ⊥-elim (x≢y refl)
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... | there ∈xs = S x≢y (⊢σ ∈xs ⊢y)
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⊆xs′ : free N ⊆ xs′
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⊆xs′ = j ⊆xs
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⊢rename {xs = xs} ⊢σ {L · M} ⊆xs (⊢L · ⊢M) = ⊢rename ⊢σ (⊆xs ∘ left) ⊢L · ⊢rename ⊢σ (⊆xs ∘ right) ⊢M
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⊆xs′ = proj₁ lemma-\\-∷ ⊆xs
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⊢rename {xs = xs} ⊢σ {L · M} ⊆xs (⊢L · ⊢M)
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= ⊢rename ⊢σ L⊆ ⊢L · ⊢rename ⊢σ M⊆ ⊢M
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where
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L⊆ = trans-⊆ (proj₁ lemma-⊎-∪ ∘ inj₁) ⊆xs
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M⊆ = trans-⊆ (proj₁ lemma-⊎-∪ ∘ inj₂) ⊆xs
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\end{code}
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@ -494,20 +420,20 @@ j {x} {xs} {ys} ⊆ys {w} w∈ with x ≟ w
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\begin{code}
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lemma₁ : ∀ {y ys} → [ y ] ⊆ y ∷ ys
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lemma₁ (here refl) = here refl
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lemma₁ (there ())
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lemma₁ = proj₁ lemma-[_] here
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lemma₂ : ∀ {z x xs} → x ≢ z → z ∈ x ∷ xs → z ∈ xs
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lemma₂ x≢z (here refl) = ⊥-elim (x≢z refl)
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lemma₂ _ (there z∈xs) = z∈xs
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lemma₂ : ∀ {w x xs} → x ≢ w → w ∈ x ∷ xs → w ∈ xs
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lemma₂ x≢ here = ⊥-elim (x≢ refl)
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lemma₂ _ (there w∈) = w∈
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⊢subst : ∀ {Γ Δ xs ys ρ} →
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(∀ {x} → x ∈ xs → free (ρ x) ⊆ ys) →
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(∀ {x A} → x ∈ xs → Γ ∋ x ⦂ A → Δ ⊢ ρ x ⦂ A) →
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(∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ subst ys ρ M ⦂ A)
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⊢subst Σ ⊢ρ ⊆xs ⌊ ⊢x ⌋ = ⊢ρ (⊆xs (here refl)) ⊢x
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⊢subst Σ ⊢ρ ⊆xs ⌊ ⊢x ⌋
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= ⊢ρ (⊆xs here) ⊢x
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⊢subst {Γ} {Δ} {xs} {ys} {ρ} Σ ⊢ρ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N)
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= ƛ_ {x = y} {A = A} (⊢subst {Γ′} {Δ′} {xs′} {ys′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N)
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= ƛ_ {x = y} {A = A} (⊢subst {Γ′} {Δ′} {xs′} {ys′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N)
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where
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y = fresh ys
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Γ′ = Γ , x ⦂ A
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@ -516,43 +442,35 @@ lemma₂ _ (there z∈xs) = z∈xs
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ys′ = y ∷ ys
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ρ′ = ρ , x ↦ ⌊ y ⌋
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Σ′ : ∀ {z} → z ∈ xs′ → free (ρ′ z) ⊆ ys′
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Σ′ {z} (here refl) with x ≟ z
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Σ′ : ∀ {w} → w ∈ xs′ → free (ρ′ w) ⊆ ys′
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Σ′ {w} here with w ≟ x
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... | yes refl = lemma₁
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... | no x≢z = ⊥-elim (x≢z refl)
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Σ′ {z} (there x∈) with x ≟ z
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... | no w≢ = ⊥-elim (w≢ refl)
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Σ′ {w} (there w∈) with w ≟ x
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... | yes refl = lemma₁
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... | no _ = there ∘ (Σ x∈)
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... | no _ = there ∘ (Σ w∈)
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⊆xs′ : free N ⊆ xs′
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⊆xs′ = j ⊆xs
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{-
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free (ƛ x ⦂ A ⇒ N) ⊆ xs
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= def'n
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free N \\ x ⊆ xs
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= adjoint
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free N ⊆ x ∷ xs
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-}
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⊆xs′ = proj₁ lemma-\\-∷ ⊆xs
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⊢σ : ∀ {z C} → z ∈ ys → Δ ∋ z ⦂ C → Δ′ ∋ z ⦂ C
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⊢σ z∈ ⊢z = S (fresh-lemma z∈) ⊢z
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⊢σ : ∀ {w C} → w ∈ ys → Δ ∋ w ⦂ C → Δ′ ∋ w ⦂ C
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⊢σ w∈ ⊢w = S (fresh-lemma w∈) ⊢w
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⊢ρ′ : ∀ {z C} → z ∈ xs′ → Γ′ ∋ z ⦂ C → Δ′ ⊢ ρ′ z ⦂ C
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⊢ρ′ : ∀ {w C} → w ∈ xs′ → Γ′ ∋ w ⦂ C → Δ′ ⊢ ρ′ w ⦂ C
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⊢ρ′ _ Z with x ≟ x
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... | yes _ = ⌊ Z ⌋
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... | no x≢x = ⊥-elim (x≢x refl)
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⊢ρ′ {z} z∈′ (S x≢z ⊢z) with x ≟ z
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... | yes x≡z = ⊥-elim (x≢z x≡z)
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... | no _ = ⊢rename {Δ} {Δ′} {ys} ⊢σ (Σ z∈) (⊢ρ z∈ ⊢z)
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⊢ρ′ {w} w∈′ (S x≢w ⊢w) with w ≟ x
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... | yes refl = ⊥-elim (x≢w refl)
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... | no _ = ⊢rename {Δ} {Δ′} {ys} ⊢σ (Σ w∈) (⊢ρ w∈ ⊢w)
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where
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z∈ = lemma₂ x≢z z∈′
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w∈ = lemma₂ x≢w w∈′
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⊢subst {xs = xs} Σ ⊢ρ {L · M} ⊆xs (⊢L · ⊢M) = ⊢subst Σ ⊢ρ L⊆xs ⊢L · ⊢subst Σ ⊢ρ M⊆xs ⊢M
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⊢subst {xs = xs} Σ ⊢ρ {L · M} ⊆xs (⊢L · ⊢M)
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= ⊢subst Σ ⊢ρ L⊆ ⊢L · ⊢subst Σ ⊢ρ M⊆ ⊢M
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where
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L⊆xs : free L ⊆ xs
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L⊆xs = ⊆xs ∘ left
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M⊆xs : free M ⊆ xs
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M⊆xs = ⊆xs ∘ right
|
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L⊆ = trans-⊆ lemma-∪₁ ⊆xs
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M⊆ = trans-⊆ lemma-∪₂ ⊆xs
|
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⊢substitution : ∀ {Γ x A N B M} →
|
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Γ , x ⦂ A ⊢ N ⦂ B →
|
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|
|
@ -568,19 +486,20 @@ lemma₂ _ (there z∈xs) = z∈xs
|
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ρ = ∅ , x ↦ M
|
||||
|
||||
Σ : ∀ {w} → w ∈ xs → free (ρ w) ⊆ ys
|
||||
Σ {w} w∈ y∈ with x ≟ w
|
||||
... | yes _ = left y∈
|
||||
... | no x≢w with y∈
|
||||
... | here refl = right (i {w} {x} {free N} w∈ x≢w)
|
||||
... | there ()
|
||||
Σ {w} w∈ y∈ with w ≟ x
|
||||
... | yes _ = lemma-∪₁ y∈
|
||||
... | no w≢ with y∈
|
||||
... | here = lemma-∪₂
|
||||
(proj₂ lemma-\\-∈-≢ ⟨ w∈ , w≢ ⟩)
|
||||
... | there ()
|
||||
|
||||
⊢ρ : ∀ {z C} → z ∈ xs → Γ′ ∋ z ⦂ C → Γ ⊢ ρ z ⦂ C
|
||||
⊢ρ {.x} z∈ Z with x ≟ x
|
||||
... | yes _ = ⊢M
|
||||
... | no x≢x = ⊥-elim (x≢x refl)
|
||||
⊢ρ {z} z∈ (S x≢z ⊢z) with x ≟ z
|
||||
... | yes x≡z = ⊥-elim (x≢z x≡z)
|
||||
... | no _ = ⌊ ⊢z ⌋
|
||||
⊢ρ {z} z∈ (S x≢z ⊢z) with z ≟ x
|
||||
... | yes refl = ⊥-elim (x≢z refl)
|
||||
... | no _ = ⌊ ⊢z ⌋
|
||||
|
||||
⊆xs : free N ⊆ xs
|
||||
⊆xs x∈ = x∈
|
||||
|
|
|
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Reference in a new issue