module Data.Term.Zipper

import Data.DPair
import Data.Fin.Occurs
import Data.Vect.Properties.Insert
import Data.Vect.Properties.Map
import Data.Term
import Syntax.PreorderReasoning

-- Definition ------------------------------------------------------------------

public export
data Zipper : Signature -> Nat -> Type where
  Top : Zipper sig n
  Op : sig.Operator (S k) -> Fin (S k) -> Vect k (Term sig n) -> Zipper sig n -> Zipper sig n

%name Zipper zip

-- Monoid ----------------------------------------------------------------------

export
(++) : Zipper sig n -> Zipper sig n -> Zipper sig n
Top ++ zip2 = zip2
Op op i ts zip1 ++ zip2 = Op op i ts (zip1 ++ zip2)

-- Properties

export
leftUnit : (zip : Zipper sig n) -> Top ++ zip = zip
leftUnit zip = Refl

export
rightUnit : (zip : Zipper sig n) -> zip ++ Top = zip
rightUnit Top = Refl
rightUnit (Op op i ts zip) = cong (Op op i ts) (rightUnit zip)

export
assoc : (zip1, zip2, zip3 : Zipper sig n) -> (zip1 ++ zip2) ++ zip3 = zip1 ++ (zip2 ++ zip3)
assoc Top zip2 zip3 = Refl
assoc (Op op i ts zip1) zip2 zip3 = cong (Op op i ts) (assoc zip1 zip2 zip3)

-- Action ----------------------------------------------------------------------

export
(+) : Zipper sig n -> Term sig n -> Term sig n
Top + t = t
Op op i ts zip + t = Op op (insertAt i (zip + t) ts)

-- Properties

export
actionAssoc :
  (zip1, zip2 : Zipper sig n) ->
  (t : Term sig n) ->
  (zip1 ++ zip2) + t = zip1 + (zip2 + t)
actionAssoc Top zip2 t = Refl
actionAssoc (Op op i ts zip1) zip2 t =
  cong (\t => Op op (insertAt i t ts)) $ actionAssoc zip1 zip2 t

-- Substitution ----------------------------------------------------------------

export
(<$>) : (Fin k -> Term sig n) -> Zipper sig k -> Zipper sig n
f <$> Top = Top
f <$> Op op i ts zip = Op op i (map (f <$>) ts) (f <$> zip)

-- Properties

export
actionHomo :
  (0 f : Fin k -> Term sig n) ->
  (zip : Zipper sig k) ->
  (0 t : Term sig k) ->
  f <$> zip + t = (f <$> zip) + (f <$> t)
actionHomo f Top t = Refl
actionHomo f (Op op i ts zip) t = cong (Op op) $ Calc $
  |~ map (f <$>) (insertAt i (zip + t) ts)
  ~~ insertAt i (f <$> zip + t) (map (f <$>) ts)           ...(mapInsertAt (f <$>) i (zip + t) ts)
  ~~ insertAt i ((f <$> zip) + (f <$> t)) (map (f <$>) ts) ...(cong (\t => insertAt i t (map (f <$>) ts)) $ actionHomo f zip t)

-- Cycles ----------------------------------------------------------------------

pivotAt : sig.Operator k -> Fin k -> Vect k (Term sig n) -> Zipper sig n
pivotAt op i ts =
  let op' = replace {p = sig.Operator} (snd prf) op in
  let i' = replace {p = Fin} (snd prf) i in
  let ts' = replace {p = flip Vect (Term sig n)} (snd prf) ts in
  Op op' i' (deleteAt i' ts') Top
  where
  prf : Exists (\j => k = S j)
  prf = indexIsSuc i

actionPivotAt :
  (op : sig.Operator k) ->
  (i : Fin k) ->
  (ts : Vect k (Term sig n)) ->
  pivotAt op i ts + index i ts = Op op ts
actionPivotAt op i@FZ ts = cong (Op op) $ insertAtIndex i ts
actionPivotAt op i@(FS _) ts = cong (Op op) $ insertAtIndex i ts

pivotNotTop : (zip : Zipper sig k) -> Not (zip ++ pivotAt op i ts = Top)
pivotNotTop (Op op' j us zip) prf impossible

export
noCycles : (zip : Zipper sig k) -> (t : Term sig k) -> zip + t = t -> zip = Top
noCycles Top t prf = Refl
noCycles (Op {k} op i ts zip') t@(Op {k = j} op' us) prf =
  void $ pivotNotTop zip' prf''
  where
  I : Fin j
  I = replace {p = Fin} (opInjectiveLen prf) i

  prf' : ((zip' ++ pivotAt op' I us) + index I us = index I us)
  prf' = Calc $
    |~ (zip' ++ pivotAt op' I us) + index I us
    ~~ zip' + (pivotAt op' I us + index I us)      ...(actionAssoc zip' (pivotAt op' I us) (index I us))
    ~~ zip' + Op op' us                            ...(cong (zip' +) $ actionPivotAt op' I us)
    ~~ index i (insertAt i (zip' + Op op' us) ts)  ..<(indexInsertAt i (zip' + Op op' us) ts)
    ~~ index i (rewrite opInjectiveLen prf in us)  ...(cong (index i) $ opInjectiveTs prf)
    ~~ index I us                                  ...(rewrite opInjectiveLen prf in Refl)

  prf'' : zip' ++ pivotAt op' I us = Top
  prf'' = noCycles (zip' ++ pivotAt op' I us) (assert_smaller t (index I us)) prf'