module Quox.Whnf.Coercion

import Quox.Whnf.Interface
import Quox.Whnf.ComputeElimType
import Quox.Whnf.TypeCase

%default total



private
coeScoped : {s : Nat} -> DScopeTerm d n -> Dim d -> Dim d -> Loc ->
            ScopeTermN s d n -> ScopeTermN s d n
coeScoped ty p q loc (S names (N body)) =
  S names $ N $ E $ Coe ty p q body loc
coeScoped ty p q loc (S names (Y body)) =
  SY names $ E $ Coe (weakDS s ty) p q body loc
where
  weakDS : (by : Nat) -> DScopeTerm d n -> DScopeTerm d (by + n)
  weakDS by (S names (Y body)) = S names $ Y $ weakT by body
  weakDS by (S names (N body)) = S names $ N $ weakT by body


parameters {auto _ : CanWhnf Term Interface.isRedexT}
           {auto _ : CanWhnf Elim Interface.isRedexE}
           (defs : Definitions) (ctx : WhnfContext d n) (sg : SQty)
  ||| reduce a function application `App (Coe ty p q val) s loc`
  export covering
  piCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) ->
          (val, s : Term d n) -> Loc ->
          Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
  piCoe sty@(S [< i] ty) p q val s loc = do
    -- (coe [i ⇒ π.(x : A) → B] @p @q t) s ⇝
    -- coe [i ⇒ B[𝒔‹i›/x] @p @q ((t ∷ (π.(x : A) → B)‹p/i›) 𝒔‹p›)
    --   where 𝒔‹j› ≔ coe [i ⇒ A] @q @j s
    --
    -- type-case is used to expose A,B if the type is neutral
    let ctx1 = extendDim i ctx
    Element ty tynf <- whnf defs ctx1 SZero $ getTerm ty
    (arg, res) <- tycasePi defs ctx1 ty
    let s0   = CoeY i arg q p s s.loc
        body = E $ App (Ann val (ty // one p) val.loc) (E s0) loc
        s1   = CoeY i (arg // (BV 0 i.loc ::: shift 2)) (weakD 1 q) (BV 0 i.loc)
                    (s // shift 1) s.loc
    whnf defs ctx sg $ CoeY i (sub1 res s1) p q body loc

  ||| reduce a pair elimination `CasePair pi (Coe ty p q val) ret body loc`
  export covering
  sigCoe : (qty : Qty) ->
           (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
           (ret : ScopeTerm d n) -> (body : ScopeTermN 2 d n) -> Loc ->
           Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
  sigCoe qty sty@(S [< i] ty) p q val ret body loc = do
    -- caseπ (coe [i ⇒ (x : A) × B] @p @q s) return z ⇒ C of { (a, b) ⇒ e }
    --   ⇝
    -- caseπ s ∷ ((x : A) × B)‹p/i› return z ⇒ C
    -- of { (a, b) ⇒
    --   e[(coe [i ⇒ A] @p @q a)/a,
    --     (coe [i ⇒ B[(coe [j ⇒ A‹j/i›] @p @i a)/x]] @p @q b)/b] }
    --
    -- type-case is used to expose A,B if the type is neutral
    let ctx1 = extendDim i ctx
    Element ty tynf <- whnf defs ctx1 SZero $ getTerm ty
    (tfst, tsnd) <- tycaseSig defs ctx1 ty
    let [< x, y] = body.names
        a' = CoeY i (weakT 2 tfst) p q (BVT 1 x.loc) x.loc
        tsnd' = tsnd.term //
          (CoeY i (weakT 2 $ tfst // (B VZ tsnd.loc ::: shift 2))
            (weakD 1 p) (B VZ i.loc) (BVT 1 tsnd.loc) y.loc ::: shift 2)
        b' = CoeY i tsnd' p q (BVT 0 y.loc) y.loc
    whnf defs ctx sg $ CasePair qty (Ann val (ty // one p) val.loc) ret
      (SY body.names $ body.term // (a' ::: b' ::: shift 2)) loc

  ||| reduce a pair projection `Fst (Coe ty p q val) loc`
  export covering
  fstCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
           Loc -> Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
  fstCoe sty@(S [< i] ty) p q val loc = do
    -- fst (coe (𝑖 ⇒ (x : A) × B) @p @q s)
    --   ⇝
    -- coe (𝑖 ⇒ A) @p @q (fst (s ∷ (x : A‹p/𝑖›) × B‹p/𝑖›))
    --
    -- type-case is used to expose A,B if the type is neutral
    let ctx1 = extendDim i ctx
    Element ty tynf <- whnf defs ctx1 SZero $ getTerm ty
    (tfst, _) <- tycaseSig defs ctx1 ty
    whnf defs ctx sg $
      Coe (SY [< i] tfst) p q
          (E (Fst (Ann val (ty // one p) val.loc) val.loc)) loc

  ||| reduce a pair projection `Snd (Coe ty p q val) loc`
  export covering
  sndCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
           Loc -> Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
  sndCoe sty@(S [< i] ty) p q val loc = do
    -- snd (coe (𝑖 ⇒ (x : A) × B) @p @q s)
    --   ⇝
    -- coe (𝑖 ⇒ B[coe (𝑗 ⇒ A‹𝑗/𝑖›) @p @𝑖 (fst (s ∷ (x : A) × B))/x]) @p @q
    --     (snd (s ∷ (x : A‹p/𝑖›) × B‹p/𝑖›))
    --
    -- type-case is used to expose A,B if the type is neutral
    let ctx1 = extendDim i ctx
    Element ty tynf <- whnf defs ctx1 SZero $ getTerm ty
    (tfst, tsnd) <- tycaseSig defs ctx1 ty
    whnf defs ctx sg $
      Coe (SY [< i] $ sub1 tsnd $
            Coe (SY [< !(fresh i)] $ tfst // (BV 0 i.loc ::: shift 2))
                (weakD 1 p) (BV 0 loc)
                (E (Fst (Ann (dweakT 1 val) ty val.loc) val.loc)) loc)
          p q
          (E (Snd (Ann val (ty // one p) val.loc) val.loc))
          loc

  ||| reduce a dimension application `DApp (Coe ty p q val) r loc`
  export covering
  eqCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
          (r : Dim d) -> Loc ->
          Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
  eqCoe sty@(S [< j] ty) p q val r loc = do
    -- (coe [j ⇒ Eq [i ⇒ A] L R] @p @q eq) @r
    --   ⇝
    -- comp [j ⇒ A‹r/i›] @p @q ((eq ∷ (Eq [i ⇒ A] L R)‹p/j›) @r)
    --   @r { 0 j ⇒ L; 1 j ⇒ R }
    let ctx1 = extendDim j ctx
    Element ty tynf <- whnf defs ctx1 SZero $ getTerm ty
    (a0, a1, a, s, t) <- tycaseEq defs ctx1 ty
    let a'   = dsub1 a (weakD 1 r)
        val' = E $ DApp (Ann val (ty // one p) val.loc) r loc
    whnf defs ctx sg $ CompH j a' p q val' r j s j t loc

  ||| reduce a pair elimination `CaseBox pi (Coe ty p q val) ret body`
  export covering
  boxCoe : (qty : Qty) ->
           (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
           (ret : ScopeTerm d n) -> (body : ScopeTerm d n) -> Loc ->
           Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
  boxCoe qty sty@(S [< i] ty) p q val ret body loc = do
    -- caseπ (coe [i ⇒ [ρ. A]] @p @q s) return z ⇒ C of { [a] ⇒ e }
    --   ⇝
    -- caseπ s ∷ [ρ. A]‹p/i› return z ⇒ C of { [a] ⇒ e[(coe [i ⇒ A] p q a)/a] }
    let ctx1 = extendDim i ctx
    Element ty tynf <- whnf defs ctx1 SZero $ getTerm ty
    ta <- tycaseBOX defs ctx1 ty
    let xloc = body.name.loc
    let a' = CoeY i (weakT 1 ta) p q (BVT 0 xloc) xloc
    whnf defs ctx sg $ CaseBox qty (Ann val (ty // one p) val.loc) ret
      (SY body.names $ body.term // (a' ::: shift 1)) loc


-- new params block to call the above functions at different `n`
parameters {auto _ : CanWhnf Term Interface.isRedexT}
           {auto _ : CanWhnf Elim Interface.isRedexE}
           (defs : Definitions) (ctx : WhnfContext d n) (sg : SQty)
  ||| pushes a coercion inside a whnf-ed term
  export covering
  pushCoe : BindName ->
            (ty : Term (S d) n) -> (p, q : Dim d) -> (s : Term d n) -> Loc ->
            (0 pc : So (canPushCoe sg ty s)) =>
            Eff Whnf (NonRedex Elim d n defs ctx sg)
  pushCoe i ty p q s loc =
    case ty of
      -- (coe ★ᵢ @_ @_ s) ⇝ (s ∷ ★ᵢ)
      TYPE l tyLoc =>
        whnf defs ctx sg $ Ann s (TYPE l tyLoc) loc

      -- (coe IOState @_ @_ s) ⇝ (s ∷ IOState)
      IOState tyLoc =>
        whnf defs ctx sg $ Ann s (IOState tyLoc) loc

      -- η expand, then simplify the Coe/App in the body
      --
      -- (coe (𝑖 ⇒ π.(x : A) → B) @p @q s)
      --   ⇝
      -- (λ y ⇒ (coe (𝑖 ⇒ π.(x : A) → B) @p @q s) y) ∷ (π.(x : A) → B)‹q/𝑖›
      --   ⇝     ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
      -- (λ y ⇒ ⋯) ∷ (π.(x : A) → B)‹q/𝑖›   -- see `piCoe`
      --
      -- do the piCoe step here because otherwise equality checking keeps
      -- doing the η forever
      Pi {arg, res = S [< x] _, _} => do
        let ctx' = extendTy x (arg // one p) ctx
        body <- piCoe defs ctx' sg
          (weakDS 1 $ SY [< i] ty) p q (weakT 1 s) (BVT 0 loc) loc
        whnf defs ctx sg $
          Ann (LamY x (E body.fst) loc) (ty // one q) loc

      -- no η!!!
      -- push into a pair constructor, otherwise still stuck
      --
      --   s̃‹𝑘› ≔ coe (𝑗 ⇒ A‹𝑗/𝑖›) @p @𝑘 s
      -- -----------------------------------------------
      --   (coe (𝑖 ⇒ (x : A) × B) @p @q (s, t))
      --     ⇝
      --   (s̃‹q›, coe (𝑖 ⇒ B[s̃‹𝑖›/x]) @p @q t)
      --     ∷ ((x : A) × B)‹q/𝑖›
      Sig tfst tsnd tyLoc => do
        let Pair fst snd sLoc = s
            fst' = CoeY i tfst p q fst fst.loc
            fstInSnd =
              CoeY !(fresh i)
                   (tfst // (BV 0 loc ::: shift 2))
                   (weakD 1 p) (BV 0 loc) (dweakT 1 fst) fst.loc
            snd' = CoeY i (sub1 tsnd fstInSnd) p q snd snd.loc
        whnf defs ctx sg $
          Ann (Pair (E fst') (E snd') sLoc) (ty // one q) loc

      -- (coe {𝐚̄} @_ @_ s) ⇝ (s ∷ {𝐚̄})
      Enum cases tyLoc =>
        whnf defs ctx sg $ Ann s (Enum cases tyLoc) loc

      -- η expand/simplify, same as for Π
      --
      -- (coe (𝑖 ⇒ Eq (𝑗 ⇒ A) l r) @p @q s)
      --   ⇝
      -- (δ 𝑘 ⇒ (coe (𝑖 ⇒ Eq (𝑗 ⇒ A) l r) @p @q s) @𝑘) ∷ (Eq (𝑗 ⇒ A) l r)‹q/𝑖›
      --   ⇝     ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
      -- (δ 𝑘 ⇒ ⋯) ∷ (Eq (𝑗 ⇒ A) l r)‹q/𝑖›   -- see `eqCoe`
      --
      -- do the eqCoe step here because otherwise equality checking keeps
      -- doing the η forever
      Eq {ty = S [< j] _, _} => do
        let ctx' = extendDim j ctx
        body <- eqCoe defs ctx' sg
          (dweakDS 1 $ S [< i] $ Y ty) (weakD 1 p) (weakD 1 q)
          (dweakT 1 s) (BV 0 loc) loc
        whnf defs ctx sg $
          Ann (DLamY i (E body.fst) loc) (ty // one q) loc

      -- (coe ℕ @_ @_ s) ⇝ (s ∷ ℕ)
      NAT tyLoc =>
        whnf defs ctx sg $ Ann s (NAT tyLoc) loc

      -- (coe String @_ @_ s) ⇝ (s ∷ String)
      STRING tyLoc =>
        whnf defs ctx sg $ Ann s (STRING tyLoc) loc

      -- η expand/simplify
      --
      -- (coe (𝑖 ⇒ [π.A]) @p @q s)
      --   ⇝
      -- [case coe (𝑖 ⇒ [π.A]) @p @q s return A‹q/𝑖› of {[x] ⇒ x}]
      --   ⇝
      -- [case1 s ∷ [π.A]‹p/𝑖› ⋯] ∷ [π.A]‹q/𝑖›   -- see `boxCoe`
      --
      -- do the eqCoe step here because otherwise equality checking keeps
      -- doing the η forever
      BOX qty inner tyLoc => do
        body <- boxCoe defs ctx sg qty
          (SY [< i] ty) p q s
          (SN $ inner // one q)
          (SY [< !(mnb "inner" loc)] (BVT 0 loc)) loc
        whnf defs ctx sg $ Ann (Box (E body.fst) loc) (ty // one q) loc