diff options
| -rw-r--r-- | BFF.agda | 27 | ||||
| -rw-r--r-- | Bidir.agda | 18 | ||||
| -rw-r--r-- | FreeTheorems.agda | 21 | ||||
| -rw-r--r-- | Generic.agda | 10 | ||||
| -rw-r--r-- | GetTypes.agda | 27 | ||||
| -rw-r--r-- | Precond.agda | 14 |
6 files changed, 93 insertions, 24 deletions
@@ -11,15 +11,19 @@ open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_) open import Data.List using (List ; [] ; _∷_ ; map ; length) open import Data.Vec using (Vec ; toList ; fromList ; tabulate ; allFin) renaming (lookup to lookupV ; map to mapV ; [] to []V ; _∷_ to _∷V_) open import Function using (id ; _∘_ ; flip) -open import Relation.Binary using (DecSetoid ; module DecSetoid) +open import Function.Equality using (_⟶_ ; _⟨$⟩_) +open import Function.Injection using (module Injection) renaming (Injection to _↪_ ; id to id↪) +open import Relation.Binary using (Setoid ; DecSetoid ; module DecSetoid) +open import Relation.Binary.PropositionalEquality using (cong) renaming (setoid to EqSetoid) +open Injection using (to) open import FinMap -open import Generic using (mapMV) +open import Generic using (mapMV ; ≡-to-Π) import CheckInsert -import GetTypes +open import GetTypes using (VecVec-to-PartialVecVec) -module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where - open GetTypes.VecVec public using (Get) +module PartialVecBFF (A : DecSetoid ℓ₀ ℓ₀) where + open GetTypes.PartialVecVec public using (Get) open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq) open CheckInsert A @@ -33,7 +37,8 @@ module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where denumerate : {n : ℕ} → Vec Carrier n → Fin n → Carrier denumerate = flip lookupV - bff : (G : Get) → ({n : ℕ} → Vec Carrier n → Vec Carrier (Get.getlen G n) → Maybe (Vec Carrier n)) + + bff : (G : Get) → ({i : Setoid.Carrier (Get.I G)} → Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i) → Vec Carrier ((Get.gl₂ G) ⟨$⟩ i) → Maybe (Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i))) bff G s v = let s′ = enumerate s t′ = Get.get G s′ g = fromFunc (denumerate s) @@ -41,3 +46,13 @@ module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where h = assoc t′ v h′ = (flip union g′) <$> h in h′ >>= flip mapMV s′ ∘ flip lookupV + +module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where + open GetTypes.VecVec public using (Get) + open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq) + open CheckInsert A + + open PartialVecBFF A public using (assoc ; enumerate ; denumerate) + + bff : (G : Get) → ({n : ℕ} → Vec Carrier n → Vec Carrier (Get.getlen G n) → Maybe (Vec Carrier n)) + bff G = PartialVecBFF.bff A (VecVec-to-PartialVecVec G) @@ -18,6 +18,9 @@ open import Data.Vec.Equality using () renaming (module Equality to VecEq) open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate ; map-cong ; map-∘) open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂) open import Function using (id ; _∘_ ; flip) +open import Function.Equality using (_⟶_ ; _⟨$⟩_) +open import Function.Injection using (module Injection) renaming (Injection to _↪_) +open Injection using (to) open import Relation.Binary.Core using (refl ; _≡_) open import Relation.Binary.Indexed using (_at_) renaming (Setoid to ISetoid) open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans ; cong₂ ; decSetoid ; module ≡-Reasoning) renaming (setoid to EqSetoid) @@ -25,13 +28,13 @@ open import Relation.Binary using (Setoid ; module Setoid ; module DecSetoid) import Relation.Binary.EqReasoning as EqR import GetTypes -open GetTypes.VecVec using (Get ; module Get) +open GetTypes.PartialVecVec using (Get ; module Get) open import Generic using (mapMV ; mapMV-cong ; mapMV-purity ; sequenceV ; sequence-map ; VecISetoid) open import FinMap import CheckInsert open CheckInsert A import BFF -open BFF.VecBFF A using (assoc ; enumerate ; denumerate ; bff) +open BFF.PartialVecBFF A using (assoc ; enumerate ; denumerate ; bff) open Setoid using () renaming (_≈_ to _∋_≈_) open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq) @@ -125,7 +128,7 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin as ∎) where open ≡-Reasoning -theorem-1 : (G : Get) → {m : ℕ} → (s : Vec Carrier m) → bff G s (Get.get G s) ≡ just s +theorem-1 : (G : Get) → {i : Setoid.Carrier (Get.I G)} → (s : Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i)) → bff G s (Get.get G s) ≡ just s theorem-1 G s = begin bff G s (get s) ≡⟨ cong (bff G s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ @@ -189,9 +192,8 @@ lemma-mapM-successful (x ∷ xs) () | just y | nothing | _ lemma-mapM-successful (x ∷ xs) p | just y | just ys | [ p′ ] with lemma-mapM-successful xs p′ lemma-mapM-successful (x ∷ xs) p | just y | just ys | [ p′ ] | w , pw = y ∷ w , cong (_∷_ (just y)) pw - -lemma-get-mapMV : {A B : Set} {f : A → Maybe B} {n : ℕ} {v : Vec A n} {r : Vec B n} → mapMV f v ≡ just r → (get : Get) → Get.get get <$> mapMV f v ≡ mapMV f (Get.get get v) -lemma-get-mapMV {f = f} {v = v} p G = let w , pw = lemma-mapM-successful v p in begin +lemma-get-mapMV : {A B : Set} {f : A → Maybe B} → (G : Get) → {i : Setoid.Carrier (Get.I G)} {v : Vec A (to (Get.gl₁ G) ⟨$⟩ i)} {r : Vec B (to (Get.gl₁ G) ⟨$⟩ i)} → mapMV f v ≡ just r → Get.get G <$> mapMV f v ≡ mapMV f (Get.get G v) +lemma-get-mapMV {f = f} G {v = v} p = let w , pw = lemma-mapM-successful v p in begin get <$> mapMV f v ≡⟨ cong (_<$>_ get) (sym (sequence-map f v)) ⟩ get <$> (sequenceV (map f v)) @@ -219,7 +221,7 @@ sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecE sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecEq.∷-cong xs≈ys) | nothing | nothing | nothing = Setoid.refl (MaybeSetoid (VecISetoid S at _)) sequence-cong {S} (nothing VecEq.∷-cong xs≈ys) = Setoid.refl (MaybeSetoid (VecISetoid S at _)) -theorem-2 : (G : Get) → {m : ℕ} → (v : Vec Carrier (Get.getlen G m)) → (s u : Vec Carrier m) → bff G s v ≡ just u → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v +theorem-2 : (G : Get) → {i : Setoid.Carrier (Get.I G)} → (v : Vec Carrier (Get.gl₂ G ⟨$⟩ i)) → (s u : Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i)) → bff G s v ≡ just u → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v theorem-2 G v s u p with (lemma->>=-just ((flip union (delete-many (Get.get G (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (Get.get G (enumerate s)) v)) p) theorem-2 G v s u p | h′ , ph′ with (lemma-<$>-just (assoc (Get.get G (enumerate s)) v) ph′) theorem-2 G v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (VecISetoid A.setoid at _) ⟩ @@ -228,7 +230,7 @@ theorem-2 G v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (V get <$> (bff G s v) ≡⟨ cong (_<$>_ get ∘ flip _>>=_ h′↦r ∘ _<$>_ h↦h′) ph ⟩ get <$> mapMV (flip lookupM (h↦h′ h)) s′ - ≡⟨ lemma-get-mapMV (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) G ⟩ + ≡⟨ lemma-get-mapMV G (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) ⟩ mapMV (flip lookupM (h↦h′ h)) (get s′) ≡⟨ sym (sequence-map (flip lookupM (h↦h′ h)) (get s′)) ⟩ sequenceV (map (flip lookupM (h↦h′ h)) (get s′)) diff --git a/FreeTheorems.agda b/FreeTheorems.agda index 2695491..2181163 100644 --- a/FreeTheorems.agda +++ b/FreeTheorems.agda @@ -1,10 +1,17 @@ module FreeTheorems where +open import Level using () renaming (zero to ℓ₀) open import Data.Nat using (ℕ) open import Data.List using (List ; map) open import Data.Vec using (Vec) renaming (map to mapV) open import Function using (_∘_) -open import Relation.Binary.PropositionalEquality using (_≗_) +open import Function.Equality using (_⟶_ ; _⟨$⟩_) +open import Function.Injection using (module Injection) renaming (Injection to _↪_) +open import Relation.Binary.PropositionalEquality using (_≗_ ; cong) renaming (setoid to EqSetoid) +open import Relation.Binary using (Setoid) +open Injection using (to) + +open import Generic using (≡-to-Π) import GetTypes @@ -31,3 +38,15 @@ module VecVec where assume-get : {getlen : ℕ → ℕ} → (get : get-type getlen) → Get assume-get {getlen} get = record { getlen = getlen; get = get; free-theorem = free-theorem get } + +module PartialVecVec where + get-type : {I : Setoid ℓ₀ ℓ₀} → (I ↪ (EqSetoid ℕ)) → (I ⟶ (EqSetoid ℕ)) → Set₁ + get-type {I} gl₁ gl₂ = {A : Set} {i : Setoid.Carrier I} → Vec A (to gl₁ ⟨$⟩ i) → Vec A (gl₂ ⟨$⟩ i) + + open GetTypes.PartialVecVec public + + postulate + free-theorem : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ (EqSetoid ℕ)) → (gl₂ : I ⟶ (EqSetoid ℕ)) (get : get-type gl₁ gl₂) → {α β : Set} → (f : α → β) → {i : Setoid.Carrier I} → get {_} {i} ∘ mapV f ≗ mapV f ∘ get + + assume-get : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ (EqSetoid ℕ)) → (gl₂ : I ⟶ (EqSetoid ℕ)) (get : get-type gl₁ gl₂) → Get + assume-get {I} gl₁ gl₂ get = record { I = I; gl₁ = gl₁; gl₂ = gl₂; get = get; free-theorem = free-theorem gl₁ gl₂ get } diff --git a/Generic.agda b/Generic.agda index 81292ff..a734ec2 100644 --- a/Generic.agda +++ b/Generic.agda @@ -9,16 +9,20 @@ open import Data.Product using (_×_ ; _,_) open import Data.Vec using (Vec ; toList ; fromList ; map) renaming ([] to []V ; _∷_ to _∷V_) open import Data.Vec.Equality using () renaming (module Equality to VecEq) open import Function using (_∘_ ; id) +open import Function.Equality using (_⟶_) open import Level using () renaming (zero to ℓ₀) open import Relation.Binary using (Setoid ; module Setoid) open import Relation.Binary.Core using (_≡_ ; refl) open import Relation.Binary.Indexed using (_at_) renaming (Setoid to ISetoid) -open import Relation.Binary.PropositionalEquality using (_≗_ ; cong ; subst ; trans) renaming (setoid to PropEq) +open import Relation.Binary.PropositionalEquality using (_≗_ ; cong ; subst ; trans) renaming (setoid to EqSetoid) open Setoid using () renaming (_≈_ to _∋_≈_) open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_) open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_) +≡-to-Π : {A B : Set} → (A → B) → EqSetoid A ⟶ EqSetoid B +≡-to-Π f = record { _⟨$⟩_ = f; cong = cong f } + just-injective : {A : Set} → {x y : A} → Maybe.just x ≡ Maybe.just y → x ≡ y just-injective refl = refl @@ -40,11 +44,11 @@ mapMV-purity : {A B : Set} {n : ℕ} → (f : A → B) → (v : Vec A n) → map mapMV-purity f []V = refl mapMV-purity f (x ∷V xs) rewrite mapMV-purity f xs = refl -maybeEq-from-≡ : {A : Set} {a b : Maybe A} → a ≡ b → MaybeEq (PropEq A) ∋ a ≈ b +maybeEq-from-≡ : {A : Set} {a b : Maybe A} → a ≡ b → MaybeEq (EqSetoid A) ∋ a ≈ b maybeEq-from-≡ {a = just x} {b = .(just x)} refl = just refl maybeEq-from-≡ {a = nothing} {b = .nothing} refl = nothing -maybeEq-to-≡ : {A : Set} {a b : Maybe A} → MaybeEq (PropEq A) ∋ a ≈ b → a ≡ b +maybeEq-to-≡ : {A : Set} {a b : Maybe A} → MaybeEq (EqSetoid A) ∋ a ≈ b → a ≡ b maybeEq-to-≡ (just refl) = refl maybeEq-to-≡ nothing = refl diff --git a/GetTypes.agda b/GetTypes.agda index 99675f9..a52ec24 100644 --- a/GetTypes.agda +++ b/GetTypes.agda @@ -1,10 +1,17 @@ module GetTypes where +open import Level using () renaming (zero to ℓ₀) open import Data.Nat using (ℕ) open import Data.List using (List ; map) open import Data.Vec using (Vec) renaming (map to mapV) open import Function using (_∘_) -open import Relation.Binary.PropositionalEquality using (_≗_) +open import Function.Equality using (_⟶_ ; _⟨$⟩_) +open import Function.Injection using (module Injection) renaming (Injection to _↪_ ; id to id↪) +open import Relation.Binary.PropositionalEquality using (_≗_) renaming (setoid to EqSetoid) +open import Relation.Binary using (Setoid) +open Injection using (to) + +open import Generic using (≡-to-Π) module ListList where record Get : Set₁ where @@ -18,3 +25,21 @@ module VecVec where getlen : ℕ → ℕ get : {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n) free-theorem : {α β : Set} (f : α → β) {n : ℕ} → get {_} {n} ∘ mapV f ≗ mapV f ∘ get + +module PartialVecVec where + record Get : Set₁ where + field + I : Setoid ℓ₀ ℓ₀ + gl₁ : I ↪ EqSetoid ℕ + gl₂ : I ⟶ EqSetoid ℕ + get : {A : Set} {i : Setoid.Carrier I} → Vec A (to gl₁ ⟨$⟩ i) → Vec A (gl₂ ⟨$⟩ i) + free-theorem : {α β : Set} → (f : α → β) → {i : Setoid.Carrier I} → get {_} {i} ∘ mapV f ≗ mapV f ∘ get + +VecVec-to-PartialVecVec : VecVec.Get → PartialVecVec.Get +VecVec-to-PartialVecVec G = record + { I = EqSetoid ℕ + ; gl₁ = id↪ + ; gl₂ = ≡-to-Π getlen + ; get = get + ; free-theorem = free-theorem + } where open VecVec.Get G diff --git a/Precond.agda b/Precond.agda index 2be3f3a..787f010 100644 --- a/Precond.agda +++ b/Precond.agda @@ -6,7 +6,7 @@ open import Data.Nat using (ℕ) open import Data.Fin using (Fin ; zero ; suc) open import Data.Fin.Props using (_≟_) open import Data.List using (List ; [] ; _∷_) -import Level +open import Level using () renaming (zero to ℓ₀) import Category.Monad import Category.Functor open import Data.Maybe using (Maybe ; nothing ; just ; maybe′) @@ -20,7 +20,11 @@ open Data.List.Any.Membership-≡ using (_∉_) open import Data.Maybe using (just) open import Data.Product using (∃ ; _,_ ; proj₂) open import Function using (flip ; _∘_ ; id) -open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid) +open import Function.Equality using (_⟶_ ; _⟨$⟩_) +open import Function.Injection using (module Injection) renaming (Injection to _↪_) +open Injection using (to) +open import Relation.Binary using (Setoid) +open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid) renaming (setoid to EqSetoid) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import Relation.Nullary using (yes ; no) @@ -31,8 +35,8 @@ open CheckInsert (decSetoid deq) using (checkInsert ; lemma-checkInsert-new ; le import BFF open import Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain ; lemma-just-sequence) import GetTypes -open GetTypes.VecVec using (Get ; module Get) -open BFF.VecBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff) +open GetTypes.PartialVecVec using (Get ; module Get) +open BFF.PartialVecBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff) lemma-lookup-map-just : {n : ℕ} (f : Fin n) {A : Set} (v : Vec A n) → lookup f (map Maybe.just v) ≡ Maybe.just (lookup f v) lemma-lookup-map-just zero (x ∷ xs) = refl @@ -70,7 +74,7 @@ lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} ((x , px) Da maybe′ just (lookupM i (delete-many is (map just g))) (lookup i h) ∎ inner f | no f≢i = cong (flip (maybe′ just) (lookup f h)) (lemma-lookupM-delete (delete-many is (map just g)) f≢i) -assoc-enough : (G : Get) → {m : ℕ} → (s : Vec Carrier m) → (v : Vec Carrier (Get.getlen G m)) → ∃ (λ h → assoc (Get.get G (enumerate s)) v ≡ just h) → ∃ λ u → bff G s v ≡ just u +assoc-enough : (G : Get) → {i : Setoid.Carrier (Get.I G)} → (s : Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i)) → (v : Vec Carrier (Get.gl₂ G ⟨$⟩ i)) → ∃ (λ h → assoc (Get.get G (enumerate s)) v ≡ just h) → ∃ λ u → bff G s v ≡ just u assoc-enough G s v (h , p) = let w , pw = lemma-union-delete-fromFunc (lemma-assoc-domain (get s′) v h p) in _ , (begin bff G s v ≡⟨ cong (flip _>>=_ (flip mapMV s′ ∘ flip lookupM) ∘ _<$>_ (flip union g′)) p ⟩ |