diff options
| -rw-r--r-- | Bidir.agda | 20 | ||||
| -rw-r--r-- | Precond.agda | 21 |
2 files changed, 18 insertions, 23 deletions
@@ -4,20 +4,15 @@ module Bidir (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open import Data.Nat using (ℕ) open import Data.Fin using (Fin) -open import Data.Fin.Props using (_≟_) open import Data.Maybe using (Maybe ; nothing ; just ; maybe′) open import Data.List using (List) -open import Data.List.Any using (here ; there) open import Data.List.All using (All) -open Data.List.Any.Membership-≡ using (_∉_) -open import Data.Vec using (Vec ; [] ; _∷_ ; toList ; fromList ; map ; tabulate) renaming (lookup to lookupVec) +open import Data.Vec using (Vec ; [] ; _∷_ ; toList ; map ; tabulate) renaming (lookup to lookupVec) open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate ; map-cong ; map-∘) open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂) -open import Data.Empty using (⊥-elim) open import Function using (id ; _∘_ ; flip) -open import Relation.Nullary using (yes ; no) open import Relation.Binary.Core using (refl) -open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; _≗_ ; trans ; cong₂) +open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans ; cong₂) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) import FreeTheorems @@ -45,17 +40,6 @@ lemma-lookupM-assoc i is x xs .h refl | just h | ._ | same pl = pl lemma-lookupM-assoc i is x xs ._ refl | just h' | ._ | new _ = lemma-lookupM-insert i x h' lemma-lookupM-assoc i is x xs h () | just h' | ._ | wrong _ _ _ -lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing -lemma-∉-lookupM-assoc i [] [] .empty refl i∉is = lemma-lookupM-empty i -lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs' -lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ] -lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = begin - lookupM i h - ≡⟨ sym (lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' h ph) ⟩ - lookupM i h' - ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩ - nothing ∎ - _in-domain-of_ : {n : ℕ} {A : Set} → (is : List (Fin n)) → (FinMapMaybe n A) → Set _in-domain-of_ is h = All (λ i → ∃ λ x → lookupM i h ≡ just x) is diff --git a/Precond.agda b/Precond.agda index f3df111..f1b5e82 100644 --- a/Precond.agda +++ b/Precond.agda @@ -5,21 +5,21 @@ module Precond (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open import Data.Nat using (ℕ) open import Data.Fin using (Fin) open import Data.List using (List ; [] ; _∷_) +open import Data.Maybe using (nothing ; just) open import Data.Vec using (Vec ; [] ; _∷_ ; map ; lookup ; toList) -import Data.List.Any +open import Data.List.Any using (here ; there) open Data.List.Any.Membership-≡ using (_∉_) open import Data.Maybe using (just) open import Data.Product using (∃ ; _,_) open import Function using (flip ; _∘_) -open import Relation.Binary.PropositionalEquality using (refl ; cong) +open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) -open import FinMap using (FinMap ; FinMapMaybe ; union ; fromFunc ; empty ; insert) +open import FinMap using (FinMap ; FinMapMaybe ; lookupM ; union ; fromFunc ; empty ; insert ; lemma-lookupM-empty) import CheckInsert -open CheckInsert Carrier deq using (checkInsert ; lemma-checkInsert-new) +open CheckInsert Carrier deq using (checkInsert ; lemma-checkInsert-new ; lemma-lookupM-checkInsert-other) open import BFF using (fmap ; _>>=_) import Bidir -open Bidir Carrier deq using (lemma-∉-lookupM-assoc) open BFF.VecBFF Carrier deq using (get-type ; assoc ; enumerate ; denumerate ; bff) @@ -33,6 +33,17 @@ data All-different {A : Set} : List A → Set where different-[] : All-different [] different-∷ : {x : A} {xs : List A} → x ∉ xs → All-different xs → All-different (x ∷ xs) +lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing +lemma-∉-lookupM-assoc i [] [] .empty refl i∉is = lemma-lookupM-empty i +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs' +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ] +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = begin + lookupM i h + ≡⟨ sym (lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' h ph) ⟩ + lookupM i h' + ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩ + nothing ∎ + different-assoc : {m n : ℕ} → (u : Vec (Fin n) m) → (v : Vec Carrier m) → All-different (toList u) → ∃ λ h → assoc u v ≡ just h different-assoc [] [] p = empty , refl different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) with different-assoc us vs diff-us |