diff options
Diffstat (limited to 'FinMap.agda')
| -rw-r--r-- | FinMap.agda | 60 |
1 files changed, 30 insertions, 30 deletions
diff --git a/FinMap.agda b/FinMap.agda index 8322b79..57d3ecf 100644 --- a/FinMap.agda +++ b/FinMap.agda @@ -18,8 +18,8 @@ open import Function.Equality using (module Π) open import Function.Surjection using (module Surjection) open import Relation.Nullary using (yes ; no) open import Relation.Nullary.Negation using (contradiction) -open import Relation.Binary.Core using (_≡_ ; refl ; _≢_ ; Decidable) -open import Relation.Binary.PropositionalEquality as P using (cong ; sym ; _≗_ ; trans ; cong₂) +open import Relation.Binary.Core using (Decidable) +open import Relation.Binary.PropositionalEquality as P using (_≡_ ; _≢_ ; _≗_) open P.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import Generic using (just-injective) @@ -77,79 +77,79 @@ delete-many = flip (foldr (const _) delete) lemma-insert-same : {n : ℕ} {A : Set} → (m : FinMapMaybe n A) → (f : Fin n) → {a : A} → lookupM f m ≡ just a → m ≡ insert f a m lemma-insert-same [] () p -lemma-insert-same {suc n} (x ∷ xs) zero p = cong (flip _∷_ xs) p -lemma-insert-same (x ∷ xs) (suc i) p = cong (_∷_ x) (lemma-insert-same xs i p) +lemma-insert-same {suc n} (x ∷ xs) zero p = P.cong (flip _∷_ xs) p +lemma-insert-same (x ∷ xs) (suc i) p = P.cong (_∷_ x) (lemma-insert-same xs i p) lemma-lookupM-empty : {A : Set} {n : ℕ} → (i : Fin n) → lookupM {A} i empty ≡ nothing -lemma-lookupM-empty zero = refl +lemma-lookupM-empty zero = P.refl lemma-lookupM-empty (suc i) = lemma-lookupM-empty i lemma-lookupM-restrict : {A : Set} {n m : ℕ} → (i : Fin n) → (f : Fin n → A) → (is : Vec (Fin n) m) → {a : A} → lookupM i (restrict f is) ≡ just a → f i ≡ a -lemma-lookupM-restrict i f [] p = contradiction (trans (sym p) (lemma-lookupM-empty i)) (λ ()) +lemma-lookupM-restrict i f [] p = contradiction (P.trans (P.sym p) (lemma-lookupM-empty i)) (λ ()) lemma-lookupM-restrict i f (i' ∷ is) p with i ≟ i' -lemma-lookupM-restrict i f (.i ∷ is) {a} p | yes refl = just-injective (begin +lemma-lookupM-restrict i f (.i ∷ is) {a} p | yes P.refl = just-injective (begin just (f i) - ≡⟨ sym (lookup∘update i (restrict f is) (just (f i))) ⟩ + ≡⟨ P.sym (lookup∘update i (restrict f is) (just (f i))) ⟩ lookupM i (insert i (f i) (restrict f is)) ≡⟨ p ⟩ just a ∎) lemma-lookupM-restrict i f (i' ∷ is) {a} p | no i≢i' = lemma-lookupM-restrict i f is (begin lookupM i (restrict f is) - ≡⟨ sym (lookup∘update′ i≢i' (restrict f is) (just (f i'))) ⟩ + ≡⟨ P.sym (lookup∘update′ i≢i' (restrict f is) (just (f i'))) ⟩ lookupM i (insert i' (f i') (restrict f is)) ≡⟨ p ⟩ just a ∎) lemma-lookupM-restrict-∈ : {A : Set} {n m : ℕ} → (i : Fin n) → (f : Fin n → A) → (js : Vec (Fin n) m) → i ∈ js → lookupM i (restrict f js) ≡ just (f i) lemma-lookupM-restrict-∈ i f [] () lemma-lookupM-restrict-∈ i f (j ∷ js) p with i ≟ j -lemma-lookupM-restrict-∈ i f (.i ∷ js) p | yes refl = lookup∘update i (restrict f js) (just (f i)) +lemma-lookupM-restrict-∈ i f (.i ∷ js) p | yes P.refl = lookup∘update i (restrict f js) (just (f i)) lemma-lookupM-restrict-∈ i f (j ∷ js) (Any.here i≡j) | no i≢j = contradiction i≡j i≢j lemma-lookupM-restrict-∈ i f (j ∷ js) (Any.there p) | no i≢j = - trans (lookup∘update′ i≢j (restrict f js) (just (f j))) - (lemma-lookupM-restrict-∈ i f js p) + P.trans (lookup∘update′ i≢j (restrict f js) (just (f j))) + (lemma-lookupM-restrict-∈ i f js p) lemma-lookupM-restrict-∉ : {A : Set} {n m : ℕ} → (i : Fin n) → (f : Fin n → A) → (js : Vec (Fin n) m) → i ∉ js → lookupM i (restrict f js) ≡ nothing lemma-lookupM-restrict-∉ i f [] i∉[] = lemma-lookupM-empty i lemma-lookupM-restrict-∉ i f (j ∷ js) i∉jjs = - trans (lookup∘update′ (All.head i∉jjs) (restrict f js) (just (f j))) - (lemma-lookupM-restrict-∉ i f js (All.tail i∉jjs)) + P.trans (lookup∘update′ (All.head i∉jjs) (restrict f js) (just (f j))) + (lemma-lookupM-restrict-∉ i f js (All.tail i∉jjs)) lemma-tabulate-∘ : {n : ℕ} {A : Set} → {f g : Fin n → A} → f ≗ g → tabulate f ≡ tabulate g -lemma-tabulate-∘ {zero} {_} {f} {g} f≗g = refl -lemma-tabulate-∘ {suc n} {_} {f} {g} f≗g = cong₂ _∷_ (f≗g zero) (lemma-tabulate-∘ (f≗g ∘ suc)) +lemma-tabulate-∘ {zero} {_} {f} {g} f≗g = P.refl +lemma-tabulate-∘ {suc n} {_} {f} {g} f≗g = P.cong₂ _∷_ (f≗g zero) (lemma-tabulate-∘ (f≗g ∘ suc)) lemma-lookupM-fromFunc : {n : ℕ} {A : Set} → (f : Fin n → A) → flip lookupM (fromFunc f) ≗ Maybe.just ∘ f -lemma-lookupM-fromFunc f zero = refl +lemma-lookupM-fromFunc f zero = P.refl lemma-lookupM-fromFunc f (suc i) = lemma-lookupM-fromFunc (f ∘ suc) i lemma-lookupM-delete : {n : ℕ} {A : Set} {i j : Fin n} → (f : FinMapMaybe n A) → i ≢ j → lookupM i (delete j f) ≡ lookupM i f -lemma-lookupM-delete {i = zero} {j = zero} (_ ∷ _) p = contradiction refl p -lemma-lookupM-delete {i = zero} {j = suc j} (_ ∷ _) p = refl -lemma-lookupM-delete {i = suc i} {j = zero} (x ∷ xs) p = refl -lemma-lookupM-delete {i = suc i} {j = suc j} (x ∷ xs) p = lemma-lookupM-delete xs (p ∘ cong suc) +lemma-lookupM-delete {i = zero} {j = zero} (_ ∷ _) p = contradiction P.refl p +lemma-lookupM-delete {i = zero} {j = suc j} (_ ∷ _) p = P.refl +lemma-lookupM-delete {i = suc i} {j = zero} (x ∷ xs) p = P.refl +lemma-lookupM-delete {i = suc i} {j = suc j} (x ∷ xs) p = lemma-lookupM-delete xs (p ∘ P.cong suc) lemma-lookupM-delete-many : {n m : ℕ} {A : Set} (h : FinMapMaybe n A) → (i : Fin n) → (js : Vec (Fin n) m) → i ∉ js → lookupM i (delete-many js h) ≡ lookupM i h -lemma-lookupM-delete-many {n} h i [] i∉[] = refl +lemma-lookupM-delete-many {n} h i [] i∉[] = P.refl lemma-lookupM-delete-many {n} h i (j ∷ js) i∉jjs = - trans (lemma-lookupM-delete (delete-many js h) (All.head i∉jjs)) - (lemma-lookupM-delete-many h i js (All.tail i∉jjs)) + P.trans (lemma-lookupM-delete (delete-many js h) (All.head i∉jjs)) + (lemma-lookupM-delete-many h i js (All.tail i∉jjs)) lemma-reshape-id : {n : ℕ} {A : Set} → (m : FinMapMaybe n A) → reshape m n ≡ m -lemma-reshape-id [] = refl -lemma-reshape-id (x ∷ xs) = cong (_∷_ x) (lemma-reshape-id xs) +lemma-reshape-id [] = P.refl +lemma-reshape-id (x ∷ xs) = P.cong (_∷_ x) (lemma-reshape-id xs) lemma-disjoint-union : {n m : ℕ} {A : Set} → (f : Fin n → A) → (t : Vec (Fin n) m) → union (restrict f t) (delete-many t (fromFunc f)) ≡ fromFunc f lemma-disjoint-union {n} f t = lemma-tabulate-∘ inner where inner : (x : Fin n) → maybe′ just (lookupM x (delete-many t (fromFunc f))) (lookupM x (restrict f t)) ≡ just (f x) inner x with is-∈ _≟_ x t - inner x | yes-∈ x∈t = cong (maybe′ just (lookupM x (delete-many t (fromFunc f)))) (lemma-lookupM-restrict-∈ x f t x∈t) + inner x | yes-∈ x∈t = P.cong (maybe′ just (lookupM x (delete-many t (fromFunc f)))) (lemma-lookupM-restrict-∈ x f t x∈t) inner x | no-∉ x∉t = begin maybe′ just (lookupM x (delete-many t (fromFunc f))) (lookupM x (restrict f t)) - ≡⟨ cong₂ (maybe′ just) (lemma-lookupM-delete-many (fromFunc f) x t x∉t) (lemma-lookupM-restrict-∉ x f t x∉t) ⟩ + ≡⟨ P.cong₂ (maybe′ just) (lemma-lookupM-delete-many (fromFunc f) x t x∉t) (lemma-lookupM-restrict-∉ x f t x∉t) ⟩ maybe′ just (lookupM x (fromFunc f)) nothing ≡⟨ lemma-lookupM-fromFunc f x ⟩ just (f x) ∎ lemma-exchange-maps : {n m : ℕ} → {A : Set} → {h h′ : FinMapMaybe n A} → {P : Fin n → Set} → (∀ j → P j → lookupM j h ≡ lookupM j h′) → {is : Vec (Fin n) m} → All P (toList is) → mapV (flip lookupM h) is ≡ mapV (flip lookupM h′) is -lemma-exchange-maps h≈h′ {[]} All.[] = refl -lemma-exchange-maps h≈h′ {i ∷ is} (pi All.∷ pis) = cong₂ _∷_ (h≈h′ i pi) (lemma-exchange-maps h≈h′ pis) +lemma-exchange-maps h≈h′ {[]} All.[] = P.refl +lemma-exchange-maps h≈h′ {i ∷ is} (pi All.∷ pis) = P.cong₂ _∷_ (h≈h′ i pi) (lemma-exchange-maps h≈h′ pis) |