desi-ivanov / agda-permutations

Formally verified soundness and completeness of permutations generation, in Agda

agda-permutations

Formally verified permutations generation.

Permutation definition

data Permutation : List A → List A → Set where
  base : Permutation {A} [] []
  swap : (x y : A) → (l : List A) → Permutation (x ∷ y ∷ l) (y ∷ x ∷ l)
  skip :  ∀{ l l' }  → (x : A) → Permutation l l' → Permutation (x ∷ l) (x ∷ l')
  trans : ∀ {l l' l''} → Permutation {A} l l' → Permutation {A} l' l'' → Permutation {A} l l''  

Permutations generation

distr : A → List A → List (List A)
distr a [] = (a ∷ []) ∷ []
distr a (x ∷ xs) = (a ∷ x ∷ xs) ∷ map (x ∷_) (distr a xs)

perm : List A → List (List A)
perm [] = [] ∷ []
perm (x ∷ xs) = concatMap (distr x) (perm xs)

Soundness and completeness

Available in Soundness.agda and Completeness.agda

sound : ∀{xs ys : List A} 
  → ys ∈ perm xs
    -----------------
  → Permutation xs ys

complete : ∀{xs ys : List A} 
  → Permutation xs ys 
    -----------------
  → ys ∈ perm xs 

Credits

Co-authored by @iwilare