YES 3.606 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:



↳ HASKELL
  ↳ LR

mainModule List
  ((elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int) :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (\vv1 ->
case vv1 of
  (x,i)->  if p x then i : [] else []
  _-> []
) (zip xs (enumFrom 0))


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : _) = Just a



Lambda Reductions:
The following Lambda expression
\vv1→
case vv1 of
 (x,i) → if p x then i : [] else []
 _ → []

is transformed to
findIndices0 p vv1 = 
case vv1 of
 (x,i) → if p x then i : [] else []
 _ → []

The following Lambda expression
\ab→(a,b)

is transformed to
zip0 a b = (a,b)



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR

mainModule List
  ((elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int) :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 = 
case vv1 of
  (x,i)->  if p x then i : [] else []
  _-> []


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : _) = Just a



Case Reductions:
The following Case expression
case vv1 of
 (x,i) → if p x then i : [] else []
 _ → []

is transformed to
findIndices00 p (x,i) = if p x then i : [] else []
findIndices00 p _ = []



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR

mainModule List
  ((elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int) :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 = findIndices00 p vv1

  
findIndices00 p (x,i) =  if p x then i : [] else []
findIndices00 p _ = []


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : _) = Just a



If Reductions:
The following If expression
if p x then i : [] else []

is transformed to
findIndices000 i True = i : []
findIndices000 i False = []



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR

mainModule List
  ((elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int) :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 = findIndices00 p vv1

  
findIndices00 p (x,i) = findIndices000 i (p x)
findIndices00 p _ = []

  
findIndices000 i True = i : []
findIndices000 i False = []


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : _) = Just a



Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR

mainModule List
  ((elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int) :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 = findIndices00 p vv1

  
findIndices00 p (x,i) = findIndices000 i (p x)
findIndices00 p vw = []

  
findIndices000 i True = i : []
findIndices000 i False = []


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : vx) = Just a



Cond Reductions:
The following Function with conditions
undefined 
 | False
 = undefined

is transformed to
undefined  = undefined1

undefined0 True = undefined

undefined1  = undefined0 False



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed

mainModule List
  ((elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int) :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0))

  
findIndices0 p vv1 = findIndices00 p vv1

  
findIndices00 p (x,i) = findIndices000 i (p x)
findIndices00 p vw = []

  
findIndices000 i True = i : []
findIndices000 i False = []


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : vx) = Just a



Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow

mainModule List
  (elemIndex :: Eq a => [a]  ->  [[a]]  ->  Maybe Int)

module List where
  import qualified Maybe
import qualified Prelude
  elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int
elemIndex x = findIndex (== x)

  findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int
findIndex p = Maybe.listToMaybe . findIndices p

  findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int]
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom (Pos Zero)))

  
findIndices0 p vv1 = findIndices00 p vv1

  
findIndices00 p (x,i) = findIndices000 i (p x)
findIndices00 p vw = []

  
findIndices000 i True = i : []
findIndices000 i False = []


module Maybe where
  import qualified List
import qualified Prelude
  listToMaybe :: [a]  ->  Maybe a
listToMaybe [] = Nothing
listToMaybe (a : vx) = Just a



Haskell To QDPs


↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                                ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_primPlusNat(Succ(yv5400), Succ(yv4001000)) → new_primPlusNat(yv5400, yv4001000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                                ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_primMulNat(Succ(yv30100), Succ(yv400100)) → new_primMulNat(yv30100, Succ(yv400100))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                                ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_primEqNat(Succ(yv3000), Succ(yv40000)) → new_primEqNat(yv3000, yv40000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                                ↳ QDPSizeChangeProof
                              ↳ QDP
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_esEs1(Right(yv300), Right(yv4000), bf, app(ty_Maybe, baa)) → new_esEs3(yv300, yv4000, baa)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, bd, app(app(ty_Either, cg), da)) → new_esEs1(yv302, yv4002, cg, da)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), app(ty_Maybe, fg), bd, be) → new_esEs3(yv300, yv4000, fg)
new_esEs1(Left(yv300), Left(yv4000), app(app(app(ty_@3, ga), gb), gc), bg) → new_esEs0(yv300, yv4000, ga, gb, gc)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), bh, app(app(ty_@2, bah), bba)) → new_esEs2(yv301, yv4001, bah, bba)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), app(app(ty_Either, fb), fc), bd, be) → new_esEs1(yv300, yv4000, fb, fc)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, app(app(ty_@2, ec), ed), be) → new_esEs2(yv301, yv4001, ec, ed)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), app(ty_Maybe, bcc), ca) → new_esEs3(yv300, yv4000, bcc)
new_esEs1(Right(yv300), Right(yv4000), bf, app(app(app(ty_@3, hb), hc), hd)) → new_esEs0(yv300, yv4000, hb, hc, hd)
new_esEs(:(yv310, yv311), :(yv4010, yv4011), app(app(ty_@2, bh), ca)) → new_esEs2(yv310, yv4010, bh, ca)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), bh, app(ty_Maybe, bbb)) → new_esEs3(yv301, yv4001, bbb)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), app(app(ty_@2, bca), bcb), ca) → new_esEs2(yv300, yv4000, bca, bcb)
new_esEs1(Left(yv300), Left(yv4000), app(app(ty_Either, gd), ge), bg) → new_esEs1(yv300, yv4000, gd, ge)
new_esEs1(Right(yv300), Right(yv4000), bf, app(app(ty_Either, he), hf)) → new_esEs1(yv300, yv4000, he, hf)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, app(app(app(ty_@3, df), dg), dh), be) → new_esEs0(yv301, yv4001, df, dg, dh)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, bd, app(ty_[], cc)) → new_esEs(yv302, yv4002, cc)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), app(app(ty_@2, fd), ff), bd, be) → new_esEs2(yv300, yv4000, fd, ff)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, app(app(ty_Either, ea), eb), be) → new_esEs1(yv301, yv4001, ea, eb)
new_esEs1(Left(yv300), Left(yv4000), app(app(ty_@2, gf), gg), bg) → new_esEs2(yv300, yv4000, gf, gg)
new_esEs3(Just(yv300), Just(yv4000), app(app(ty_@2, bdb), bdc)) → new_esEs2(yv300, yv4000, bdb, bdc)
new_esEs3(Just(yv300), Just(yv4000), app(app(app(ty_@3, bce), bcf), bcg)) → new_esEs0(yv300, yv4000, bce, bcf, bcg)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, app(ty_Maybe, ee), be) → new_esEs3(yv301, yv4001, ee)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, bd, app(ty_Maybe, dd)) → new_esEs3(yv302, yv4002, dd)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), bh, app(app(ty_Either, baf), bag)) → new_esEs1(yv301, yv4001, baf, bag)
new_esEs3(Just(yv300), Just(yv4000), app(ty_[], bcd)) → new_esEs(yv300, yv4000, bcd)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, bd, app(app(ty_@2, db), dc)) → new_esEs2(yv302, yv4002, db, dc)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), app(ty_[], ef), bd, be) → new_esEs(yv300, yv4000, ef)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), bh, app(app(app(ty_@3, bac), bad), bae)) → new_esEs0(yv301, yv4001, bac, bad, bae)
new_esEs(:(yv310, yv311), :(yv4010, yv4011), ba) → new_esEs(yv311, yv4011, ba)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), bh, app(ty_[], bab)) → new_esEs(yv301, yv4001, bab)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, bd, app(app(app(ty_@3, cd), ce), cf)) → new_esEs0(yv302, yv4002, cd, ce, cf)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), bc, app(ty_[], de), be) → new_esEs(yv301, yv4001, de)
new_esEs1(Right(yv300), Right(yv4000), bf, app(app(ty_@2, hg), hh)) → new_esEs2(yv300, yv4000, hg, hh)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), app(app(app(ty_@3, bbd), bbe), bbf), ca) → new_esEs0(yv300, yv4000, bbd, bbe, bbf)
new_esEs3(Just(yv300), Just(yv4000), app(ty_Maybe, bdd)) → new_esEs3(yv300, yv4000, bdd)
new_esEs(:(yv310, yv311), :(yv4010, yv4011), app(app(app(ty_@3, bc), bd), be)) → new_esEs0(yv310, yv4010, bc, bd, be)
new_esEs(:(yv310, yv311), :(yv4010, yv4011), app(ty_Maybe, cb)) → new_esEs3(yv310, yv4010, cb)
new_esEs1(Left(yv300), Left(yv4000), app(ty_Maybe, gh), bg) → new_esEs3(yv300, yv4000, gh)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), app(ty_[], bbc), ca) → new_esEs(yv300, yv4000, bbc)
new_esEs(:(yv310, yv311), :(yv4010, yv4011), app(app(ty_Either, bf), bg)) → new_esEs1(yv310, yv4010, bf, bg)
new_esEs1(Right(yv300), Right(yv4000), bf, app(ty_[], ha)) → new_esEs(yv300, yv4000, ha)
new_esEs1(Left(yv300), Left(yv4000), app(ty_[], fh), bg) → new_esEs(yv300, yv4000, fh)
new_esEs3(Just(yv300), Just(yv4000), app(app(ty_Either, bch), bda)) → new_esEs1(yv300, yv4000, bch, bda)
new_esEs2(@2(yv300, yv301), @2(yv4000, yv4001), app(app(ty_Either, bbg), bbh), ca) → new_esEs1(yv300, yv4000, bbg, bbh)
new_esEs(:(yv310, yv311), :(yv4010, yv4011), app(ty_[], bb)) → new_esEs(yv310, yv4010, bb)
new_esEs0(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), app(app(app(ty_@3, eg), eh), fa), bd, be) → new_esEs0(yv300, yv4000, eg, eh, fa)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                                ↳ UsableRulesProof
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_listToMaybe(yv35, False, :(yv41110, yv41111), ba) → new_listToMaybe(new_primPlusNat0(yv35, Zero), new_esEs4([], yv41110, ba), yv41111, ba)

The TRS R consists of the following rules:

new_esEs23(yv310, yv4010, app(app(ty_@2, bb), bc)) → new_esEs5(yv310, yv4010, bb, bc)
new_esEs23(yv310, yv4010, ty_Integer) → new_esEs13(yv310, yv4010)
new_esEs6(yv301, yv4001, ty_Int) → new_esEs12(yv301, yv4001)
new_esEs18(Char(yv300), Char(yv4000)) → new_primEqNat0(yv300, yv4000)
new_primPlusNat1(Succ(yv5400), Succ(yv4001000)) → Succ(Succ(new_primPlusNat1(yv5400, yv4001000)))
new_primEqInt(Neg(Succ(yv3000)), Pos(yv4000)) → False
new_primEqInt(Pos(Succ(yv3000)), Neg(yv4000)) → False
new_esEs6(yv301, yv4001, ty_Float) → new_esEs8(yv301, yv4001)
new_esEs17(Right(yv300), Right(yv4000), bbh, app(ty_Ratio, bdh)) → new_esEs16(yv300, yv4000, bdh)
new_esEs22(yv300, yv4000, ty_Ordering) → new_esEs15(yv300, yv4000)
new_esEs17(Left(yv300), Left(yv4000), app(ty_Ratio, bcf), bca) → new_esEs16(yv300, yv4000, bcf)
new_esEs19(Just(yv300), Just(yv4000), app(ty_[], bad)) → new_esEs4(yv300, yv4000, bad)
new_esEs17(Left(yv300), Left(yv4000), app(ty_Maybe, bdc), bca) → new_esEs19(yv300, yv4000, bdc)
new_esEs20(yv302, yv4002, ty_Integer) → new_esEs13(yv302, yv4002)
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Double) → new_esEs10(yv300, yv4000)
new_primEqInt(Neg(Zero), Pos(Succ(yv40000))) → False
new_primEqInt(Pos(Zero), Neg(Succ(yv40000))) → False
new_esEs20(yv302, yv4002, ty_Char) → new_esEs18(yv302, yv4002)
new_esEs15(EQ, EQ) → True
new_esEs23(yv310, yv4010, ty_Bool) → new_esEs11(yv310, yv4010)
new_esEs17(Left(yv300), Left(yv4000), ty_Int, bca) → new_esEs12(yv300, yv4000)
new_esEs7(yv300, yv4000, ty_Bool) → new_esEs11(yv300, yv4000)
new_esEs21(yv301, yv4001, app(ty_Maybe, gh)) → new_esEs19(yv301, yv4001, gh)
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Int) → new_esEs12(yv300, yv4000)
new_primMulNat0(Zero, Zero) → Zero
new_esEs21(yv301, yv4001, ty_Int) → new_esEs12(yv301, yv4001)
new_esEs21(yv301, yv4001, ty_Char) → new_esEs18(yv301, yv4001)
new_esEs24(yv301, yv4001, ty_Integer) → new_esEs13(yv301, yv4001)
new_esEs17(Right(yv300), Left(yv4000), bbh, bca) → False
new_esEs17(Left(yv300), Right(yv4000), bbh, bca) → False
new_esEs21(yv301, yv4001, app(app(app(ty_@3, fh), ga), gb)) → new_esEs14(yv301, yv4001, fh, ga, gb)
new_esEs17(Left(yv300), Left(yv4000), ty_Float, bca) → new_esEs8(yv300, yv4000)
new_esEs21(yv301, yv4001, app(app(ty_@2, gf), gg)) → new_esEs5(yv301, yv4001, gf, gg)
new_esEs17(Left(yv300), Left(yv4000), app(app(app(ty_@3, bcc), bcd), bce), bca) → new_esEs14(yv300, yv4000, bcc, bcd, bce)
new_esEs21(yv301, yv4001, ty_Double) → new_esEs10(yv301, yv4001)
new_esEs19(Just(yv300), Just(yv4000), app(app(ty_Either, bba), bbb)) → new_esEs17(yv300, yv4000, bba, bbb)
new_esEs20(yv302, yv4002, app(ty_Ratio, eh)) → new_esEs16(yv302, yv4002, eh)
new_primPlusNat0(Zero, yv400100) → Succ(yv400100)
new_esEs19(Just(yv300), Just(yv4000), ty_@0) → new_esEs9(yv300, yv4000)
new_esEs5(@2(yv300, yv301), @2(yv4000, yv4001), bb, bc) → new_asAs(new_esEs7(yv300, yv4000, bb), new_esEs6(yv301, yv4001, bc))
new_esEs19(Just(yv300), Just(yv4000), ty_Bool) → new_esEs11(yv300, yv4000)
new_esEs21(yv301, yv4001, ty_Integer) → new_esEs13(yv301, yv4001)
new_esEs7(yv300, yv4000, app(app(ty_@2, df), dg)) → new_esEs5(yv300, yv4000, df, dg)
new_sr(Pos(yv3010), Neg(yv40010)) → Neg(new_primMulNat0(yv3010, yv40010))
new_sr(Neg(yv3010), Pos(yv40010)) → Neg(new_primMulNat0(yv3010, yv40010))
new_esEs23(yv310, yv4010, app(ty_Maybe, bac)) → new_esEs19(yv310, yv4010, bac)
new_esEs23(yv310, yv4010, ty_@0) → new_esEs9(yv310, yv4010)
new_esEs19(Just(yv300), Just(yv4000), ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs6(yv301, yv4001, app(ty_Maybe, ce)) → new_esEs19(yv301, yv4001, ce)
new_esEs4(:(yv310, yv311), :(yv4010, yv4011), ba) → new_asAs(new_esEs23(yv310, yv4010, ba), new_esEs4(yv311, yv4011, ba))
new_esEs22(yv300, yv4000, ty_Bool) → new_esEs11(yv300, yv4000)
new_esEs20(yv302, yv4002, ty_@0) → new_esEs9(yv302, yv4002)
new_esEs19(Nothing, Just(yv4000), bac) → False
new_esEs19(Just(yv300), Nothing, bac) → False
new_esEs22(yv300, yv4000, app(app(ty_@2, hh), baa)) → new_esEs5(yv300, yv4000, hh, baa)
new_esEs7(yv300, yv4000, ty_Ordering) → new_esEs15(yv300, yv4000)
new_esEs22(yv300, yv4000, ty_Char) → new_esEs18(yv300, yv4000)
new_esEs20(yv302, yv4002, ty_Float) → new_esEs8(yv302, yv4002)
new_esEs6(yv301, yv4001, ty_Bool) → new_esEs11(yv301, yv4001)
new_esEs19(Just(yv300), Just(yv4000), app(app(app(ty_@3, bae), baf), bag)) → new_esEs14(yv300, yv4000, bae, baf, bag)
new_esEs23(yv310, yv4010, ty_Ordering) → new_esEs15(yv310, yv4010)
new_esEs19(Nothing, Nothing, bac) → True
new_esEs7(yv300, yv4000, app(ty_[], cf)) → new_esEs4(yv300, yv4000, cf)
new_esEs17(Left(yv300), Left(yv4000), app(ty_[], bcb), bca) → new_esEs4(yv300, yv4000, bcb)
new_primEqNat0(Zero, Succ(yv40000)) → False
new_primEqNat0(Succ(yv3000), Zero) → False
new_esEs15(GT, LT) → False
new_esEs15(LT, GT) → False
new_esEs19(Just(yv300), Just(yv4000), ty_Float) → new_esEs8(yv300, yv4000)
new_primEqInt(Pos(Zero), Pos(Zero)) → True
new_esEs25(yv300, yv4000, ty_Int) → new_esEs12(yv300, yv4000)
new_esEs15(LT, LT) → True
new_esEs20(yv302, yv4002, app(ty_[], ed)) → new_esEs4(yv302, yv4002, ed)
new_esEs21(yv301, yv4001, app(app(ty_Either, gd), ge)) → new_esEs17(yv301, yv4001, gd, ge)
new_esEs7(yv300, yv4000, ty_Int) → new_esEs12(yv300, yv4000)
new_esEs6(yv301, yv4001, app(ty_[], bd)) → new_esEs4(yv301, yv4001, bd)
new_esEs17(Left(yv300), Left(yv4000), app(app(ty_Either, bcg), bch), bca) → new_esEs17(yv300, yv4000, bcg, bch)
new_esEs7(yv300, yv4000, ty_@0) → new_esEs9(yv300, yv4000)
new_esEs19(Just(yv300), Just(yv4000), ty_Ordering) → new_esEs15(yv300, yv4000)
new_esEs11(True, False) → False
new_esEs11(False, True) → False
new_esEs12(yv30, yv400) → new_primEqInt(yv30, yv400)
new_esEs15(EQ, LT) → False
new_esEs15(LT, EQ) → False
new_esEs17(Right(yv300), Right(yv4000), bbh, app(app(app(ty_@3, bde), bdf), bdg)) → new_esEs14(yv300, yv4000, bde, bdf, bdg)
new_esEs20(yv302, yv4002, app(app(ty_@2, fc), fd)) → new_esEs5(yv302, yv4002, fc, fd)
new_esEs6(yv301, yv4001, app(app(ty_@2, cc), cd)) → new_esEs5(yv301, yv4001, cc, cd)
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Float) → new_esEs8(yv300, yv4000)
new_esEs7(yv300, yv4000, app(ty_Maybe, dh)) → new_esEs19(yv300, yv4000, dh)
new_esEs19(Just(yv300), Just(yv4000), ty_Int) → new_esEs12(yv300, yv4000)
new_esEs23(yv310, yv4010, app(ty_Ratio, bbg)) → new_esEs16(yv310, yv4010, bbg)
new_esEs15(GT, EQ) → False
new_esEs15(EQ, GT) → False
new_esEs22(yv300, yv4000, app(ty_Ratio, he)) → new_esEs16(yv300, yv4000, he)
new_esEs21(yv301, yv4001, ty_Bool) → new_esEs11(yv301, yv4001)
new_esEs10(Double(yv300, yv301), Double(yv4000, yv4001)) → new_esEs12(new_sr(yv300, yv4000), new_sr(yv301, yv4001))
new_esEs7(yv300, yv4000, ty_Float) → new_esEs8(yv300, yv4000)
new_esEs17(Left(yv300), Left(yv4000), app(app(ty_@2, bda), bdb), bca) → new_esEs5(yv300, yv4000, bda, bdb)
new_esEs24(yv301, yv4001, ty_Int) → new_esEs12(yv301, yv4001)
new_esEs23(yv310, yv4010, ty_Char) → new_esEs18(yv310, yv4010)
new_esEs20(yv302, yv4002, ty_Bool) → new_esEs11(yv302, yv4002)
new_sr(Neg(yv3010), Neg(yv40010)) → Pos(new_primMulNat0(yv3010, yv40010))
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs21(yv301, yv4001, app(ty_[], fg)) → new_esEs4(yv301, yv4001, fg)
new_esEs17(Left(yv300), Left(yv4000), ty_Double, bca) → new_esEs10(yv300, yv4000)
new_esEs7(yv300, yv4000, ty_Integer) → new_esEs13(yv300, yv4000)
new_asAs(False, yv53) → False
new_sr(Pos(yv3010), Pos(yv40010)) → Pos(new_primMulNat0(yv3010, yv40010))
new_primEqNat0(Zero, Zero) → True
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Bool) → new_esEs11(yv300, yv4000)
new_primMulNat0(Zero, Succ(yv400100)) → Zero
new_primMulNat0(Succ(yv30100), Zero) → Zero
new_esEs19(Just(yv300), Just(yv4000), ty_Char) → new_esEs18(yv300, yv4000)
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_@0) → new_esEs9(yv300, yv4000)
new_esEs17(Left(yv300), Left(yv4000), ty_@0, bca) → new_esEs9(yv300, yv4000)
new_esEs6(yv301, yv4001, app(ty_Ratio, bh)) → new_esEs16(yv301, yv4001, bh)
new_esEs20(yv302, yv4002, app(ty_Maybe, ff)) → new_esEs19(yv302, yv4002, ff)
new_esEs6(yv301, yv4001, ty_Char) → new_esEs18(yv301, yv4001)
new_esEs25(yv300, yv4000, ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs11(True, True) → True
new_esEs16(:%(yv300, yv301), :%(yv4000, yv4001), bbg) → new_asAs(new_esEs25(yv300, yv4000, bbg), new_esEs24(yv301, yv4001, bbg))
new_primPlusNat0(Succ(yv540), yv400100) → Succ(Succ(new_primPlusNat1(yv540, yv400100)))
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Char) → new_esEs18(yv300, yv4000)
new_esEs22(yv300, yv4000, ty_Float) → new_esEs8(yv300, yv4000)
new_esEs20(yv302, yv4002, app(app(app(ty_@3, ee), ef), eg)) → new_esEs14(yv302, yv4002, ee, ef, eg)
new_esEs21(yv301, yv4001, app(ty_Ratio, gc)) → new_esEs16(yv301, yv4001, gc)
new_esEs23(yv310, yv4010, ty_Int) → new_esEs12(yv310, yv4010)
new_esEs13(Integer(yv300), Integer(yv4000)) → new_primEqInt(yv300, yv4000)
new_esEs20(yv302, yv4002, app(app(ty_Either, fa), fb)) → new_esEs17(yv302, yv4002, fa, fb)
new_esEs6(yv301, yv4001, ty_Ordering) → new_esEs15(yv301, yv4001)
new_primEqInt(Neg(Succ(yv3000)), Neg(Succ(yv40000))) → new_primEqNat0(yv3000, yv40000)
new_esEs4([], [], ba) → True
new_esEs22(yv300, yv4000, app(ty_Maybe, bab)) → new_esEs19(yv300, yv4000, bab)
new_esEs7(yv300, yv4000, app(app(app(ty_@3, cg), da), db)) → new_esEs14(yv300, yv4000, cg, da, db)
new_primPlusNat1(Succ(yv5400), Zero) → Succ(yv5400)
new_primPlusNat1(Zero, Succ(yv4001000)) → Succ(yv4001000)
new_esEs17(Right(yv300), Right(yv4000), bbh, app(app(ty_Either, bea), beb)) → new_esEs17(yv300, yv4000, bea, beb)
new_esEs23(yv310, yv4010, app(app(ty_Either, bbh), bca)) → new_esEs17(yv310, yv4010, bbh, bca)
new_esEs20(yv302, yv4002, ty_Int) → new_esEs12(yv302, yv4002)
new_esEs17(Left(yv300), Left(yv4000), ty_Bool, bca) → new_esEs11(yv300, yv4000)
new_esEs21(yv301, yv4001, ty_@0) → new_esEs9(yv301, yv4001)
new_esEs23(yv310, yv4010, ty_Double) → new_esEs10(yv310, yv4010)
new_primEqInt(Neg(Zero), Neg(Zero)) → True
new_esEs6(yv301, yv4001, ty_@0) → new_esEs9(yv301, yv4001)
new_esEs7(yv300, yv4000, ty_Double) → new_esEs10(yv300, yv4000)
new_esEs6(yv301, yv4001, app(app(ty_Either, ca), cb)) → new_esEs17(yv301, yv4001, ca, cb)
new_esEs14(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), ea, eb, ec) → new_asAs(new_esEs22(yv300, yv4000, ea), new_asAs(new_esEs21(yv301, yv4001, eb), new_esEs20(yv302, yv4002, ec)))
new_primEqInt(Neg(Zero), Neg(Succ(yv40000))) → False
new_primEqInt(Neg(Succ(yv3000)), Neg(Zero)) → False
new_esEs21(yv301, yv4001, ty_Ordering) → new_esEs15(yv301, yv4001)
new_esEs4([], :(yv4010, yv4011), ba) → False
new_esEs4(:(yv310, yv311), [], ba) → False
new_esEs21(yv301, yv4001, ty_Float) → new_esEs8(yv301, yv4001)
new_esEs23(yv310, yv4010, ty_Float) → new_esEs8(yv310, yv4010)
new_esEs19(Just(yv300), Just(yv4000), app(ty_Maybe, bbe)) → new_esEs19(yv300, yv4000, bbe)
new_esEs7(yv300, yv4000, app(app(ty_Either, dd), de)) → new_esEs17(yv300, yv4000, dd, de)
new_esEs22(yv300, yv4000, ty_Int) → new_esEs12(yv300, yv4000)
new_primPlusNat1(Zero, Zero) → Zero
new_esEs7(yv300, yv4000, ty_Char) → new_esEs18(yv300, yv4000)
new_asAs(True, yv53) → yv53
new_esEs6(yv301, yv4001, ty_Integer) → new_esEs13(yv301, yv4001)
new_esEs22(yv300, yv4000, ty_@0) → new_esEs9(yv300, yv4000)
new_primMulNat0(Succ(yv30100), Succ(yv400100)) → new_primPlusNat0(new_primMulNat0(yv30100, Succ(yv400100)), yv400100)
new_esEs11(False, False) → True
new_esEs22(yv300, yv4000, app(ty_[], ha)) → new_esEs4(yv300, yv4000, ha)
new_primEqInt(Pos(Succ(yv3000)), Pos(Succ(yv40000))) → new_primEqNat0(yv3000, yv40000)
new_esEs19(Just(yv300), Just(yv4000), app(ty_Ratio, bah)) → new_esEs16(yv300, yv4000, bah)
new_esEs17(Right(yv300), Right(yv4000), bbh, app(app(ty_@2, bec), bed)) → new_esEs5(yv300, yv4000, bec, bed)
new_esEs6(yv301, yv4001, ty_Double) → new_esEs10(yv301, yv4001)
new_esEs17(Left(yv300), Left(yv4000), ty_Ordering, bca) → new_esEs15(yv300, yv4000)
new_esEs22(yv300, yv4000, app(app(app(ty_@3, hb), hc), hd)) → new_esEs14(yv300, yv4000, hb, hc, hd)
new_primEqNat0(Succ(yv3000), Succ(yv40000)) → new_primEqNat0(yv3000, yv40000)
new_esEs17(Right(yv300), Right(yv4000), bbh, app(ty_Maybe, bee)) → new_esEs19(yv300, yv4000, bee)
new_esEs15(GT, GT) → True
new_esEs20(yv302, yv4002, ty_Ordering) → new_esEs15(yv302, yv4002)
new_esEs23(yv310, yv4010, app(app(app(ty_@3, ea), eb), ec)) → new_esEs14(yv310, yv4010, ea, eb, ec)
new_esEs19(Just(yv300), Just(yv4000), ty_Double) → new_esEs10(yv300, yv4000)
new_esEs9(@0, @0) → True
new_esEs22(yv300, yv4000, ty_Double) → new_esEs10(yv300, yv4000)
new_esEs20(yv302, yv4002, ty_Double) → new_esEs10(yv302, yv4002)
new_esEs7(yv300, yv4000, app(ty_Ratio, dc)) → new_esEs16(yv300, yv4000, dc)
new_esEs17(Right(yv300), Right(yv4000), bbh, ty_Ordering) → new_esEs15(yv300, yv4000)
new_primEqInt(Pos(Zero), Pos(Succ(yv40000))) → False
new_primEqInt(Pos(Succ(yv3000)), Pos(Zero)) → False
new_esEs8(Float(yv300, yv301), Float(yv4000, yv4001)) → new_esEs12(new_sr(yv300, yv4000), new_sr(yv301, yv4001))
new_esEs22(yv300, yv4000, ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs22(yv300, yv4000, app(app(ty_Either, hf), hg)) → new_esEs17(yv300, yv4000, hf, hg)
new_esEs17(Left(yv300), Left(yv4000), ty_Integer, bca) → new_esEs13(yv300, yv4000)
new_primEqInt(Neg(Zero), Pos(Zero)) → True
new_primEqInt(Pos(Zero), Neg(Zero)) → True
new_esEs19(Just(yv300), Just(yv4000), app(app(ty_@2, bbc), bbd)) → new_esEs5(yv300, yv4000, bbc, bbd)
new_esEs17(Right(yv300), Right(yv4000), bbh, app(ty_[], bdd)) → new_esEs4(yv300, yv4000, bdd)
new_esEs6(yv301, yv4001, app(app(app(ty_@3, be), bf), bg)) → new_esEs14(yv301, yv4001, be, bf, bg)
new_esEs17(Left(yv300), Left(yv4000), ty_Char, bca) → new_esEs18(yv300, yv4000)
new_esEs23(yv310, yv4010, app(ty_[], bbf)) → new_esEs4(yv310, yv4010, bbf)

The set Q consists of the following terms:

new_esEs7(x0, x1, app(ty_Maybe, x2))
new_esEs23(x0, x1, ty_Ordering)
new_esEs8(Float(x0, x1), Float(x2, x3))
new_esEs17(Left(x0), Left(x1), app(app(ty_Either, x2), x3), x4)
new_primEqInt(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs17(Right(x0), Right(x1), x2, app(ty_Ratio, x3))
new_esEs11(True, True)
new_esEs21(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Char)
new_primEqInt(Pos(Zero), Neg(Zero))
new_primEqInt(Neg(Zero), Pos(Zero))
new_esEs17(Left(x0), Left(x1), app(app(app(ty_@3, x2), x3), x4), x5)
new_esEs20(x0, x1, ty_Char)
new_esEs23(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), app(ty_Maybe, x2), x3)
new_esEs17(Right(x0), Right(x1), x2, ty_Ordering)
new_esEs22(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), app(app(ty_@2, x2), x3), x4)
new_esEs20(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), ty_Char)
new_esEs21(x0, x1, app(app(ty_Either, x2), x3))
new_esEs4(:(x0, x1), [], x2)
new_esEs17(Right(x0), Right(x1), x2, app(ty_Maybe, x3))
new_esEs20(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Char)
new_esEs19(Just(x0), Just(x1), ty_Float)
new_esEs22(x0, x1, app(ty_[], x2))
new_esEs17(Right(x0), Right(x1), x2, ty_Int)
new_primEqInt(Neg(Zero), Neg(Zero))
new_esEs19(Just(x0), Just(x1), ty_@0)
new_esEs6(x0, x1, ty_Int)
new_esEs13(Integer(x0), Integer(x1))
new_esEs5(@2(x0, x1), @2(x2, x3), x4, x5)
new_esEs19(Nothing, Nothing, x0)
new_sr(Neg(x0), Neg(x1))
new_primEqInt(Pos(Succ(x0)), Pos(Zero))
new_esEs22(x0, x1, app(app(ty_Either, x2), x3))
new_esEs18(Char(x0), Char(x1))
new_esEs22(x0, x1, ty_Integer)
new_esEs19(Just(x0), Just(x1), ty_Bool)
new_esEs20(x0, x1, ty_Float)
new_primMulNat0(Succ(x0), Succ(x1))
new_primEqInt(Pos(Succ(x0)), Neg(x1))
new_primEqInt(Neg(Succ(x0)), Pos(x1))
new_esEs23(x0, x1, app(ty_[], x2))
new_esEs11(True, False)
new_esEs11(False, True)
new_esEs19(Just(x0), Just(x1), ty_Ordering)
new_esEs21(x0, x1, app(ty_Maybe, x2))
new_esEs6(x0, x1, app(ty_Ratio, x2))
new_esEs19(Just(x0), Just(x1), app(app(app(ty_@3, x2), x3), x4))
new_esEs20(x0, x1, app(ty_Ratio, x2))
new_esEs17(Left(x0), Left(x1), ty_Integer, x2)
new_esEs19(Just(x0), Just(x1), ty_Int)
new_primPlusNat0(Zero, x0)
new_esEs21(x0, x1, ty_Integer)
new_esEs17(Left(x0), Left(x1), ty_Char, x2)
new_esEs20(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs19(Just(x0), Nothing, x1)
new_esEs25(x0, x1, ty_Int)
new_esEs17(Left(x0), Left(x1), ty_Float, x2)
new_esEs25(x0, x1, ty_Integer)
new_esEs4([], :(x0, x1), x2)
new_esEs9(@0, @0)
new_esEs7(x0, x1, ty_Bool)
new_esEs6(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Succ(x0)), Neg(Zero))
new_esEs14(@3(x0, x1, x2), @3(x3, x4, x5), x6, x7, x8)
new_primEqInt(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs20(x0, x1, app(ty_Maybe, x2))
new_esEs17(Right(x0), Right(x1), x2, app(ty_[], x3))
new_esEs6(x0, x1, ty_Integer)
new_esEs20(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Int)
new_esEs7(x0, x1, app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Double)
new_esEs6(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Integer)
new_asAs(False, x0)
new_esEs17(Right(x0), Right(x1), x2, ty_Integer)
new_esEs23(x0, x1, app(app(ty_@2, x2), x3))
new_esEs24(x0, x1, ty_Int)
new_esEs16(:%(x0, x1), :%(x2, x3), x4)
new_esEs23(x0, x1, ty_Double)
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_Either, x3), x4))
new_esEs23(x0, x1, ty_Float)
new_esEs6(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs23(x0, x1, app(app(ty_Either, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_@2, x3), x4))
new_primEqNat0(Zero, Zero)
new_esEs17(Right(x0), Right(x1), x2, ty_@0)
new_esEs6(x0, x1, app(app(ty_Either, x2), x3))
new_esEs19(Just(x0), Just(x1), app(ty_[], x2))
new_primEqInt(Neg(Zero), Neg(Succ(x0)))
new_esEs19(Nothing, Just(x0), x1)
new_esEs20(x0, x1, app(ty_[], x2))
new_esEs15(GT, GT)
new_esEs22(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), ty_Int, x2)
new_primEqNat0(Zero, Succ(x0))
new_esEs12(x0, x1)
new_esEs22(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs15(GT, LT)
new_esEs15(LT, GT)
new_primMulNat0(Zero, Zero)
new_primPlusNat1(Succ(x0), Succ(x1))
new_esEs22(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Int)
new_esEs23(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs6(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Bool)
new_esEs17(Right(x0), Right(x1), x2, ty_Bool)
new_esEs21(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_sr(Pos(x0), Pos(x1))
new_esEs21(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(ty_Maybe, x2))
new_esEs22(x0, x1, app(ty_Maybe, x2))
new_esEs17(Left(x0), Left(x1), ty_Ordering, x2)
new_esEs17(Left(x0), Left(x1), ty_Double, x2)
new_esEs7(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), ty_@0, x2)
new_asAs(True, x0)
new_esEs15(EQ, EQ)
new_primPlusNat0(Succ(x0), x1)
new_primEqInt(Pos(Zero), Pos(Succ(x0)))
new_esEs7(x0, x1, app(ty_[], x2))
new_esEs22(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(app(ty_Either, x2), x3))
new_esEs20(x0, x1, ty_Double)
new_esEs22(x0, x1, app(ty_Ratio, x2))
new_esEs23(x0, x1, ty_@0)
new_esEs17(Right(x0), Right(x1), x2, ty_Float)
new_esEs17(Right(x0), Right(x1), x2, app(app(app(ty_@3, x3), x4), x5))
new_esEs23(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), ty_Double)
new_esEs10(Double(x0, x1), Double(x2, x3))
new_esEs17(Left(x0), Left(x1), app(ty_Ratio, x2), x3)
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs20(x0, x1, app(app(ty_Either, x2), x3))
new_esEs7(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_primMulNat0(Zero, Succ(x0))
new_esEs6(x0, x1, ty_@0)
new_esEs23(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Zero), Pos(Succ(x0)))
new_primEqInt(Pos(Zero), Neg(Succ(x0)))
new_esEs17(Left(x0), Left(x1), app(ty_[], x2), x3)
new_esEs4([], [], x0)
new_primPlusNat1(Succ(x0), Zero)
new_esEs19(Just(x0), Just(x1), ty_Integer)
new_esEs11(False, False)
new_esEs4(:(x0, x1), :(x2, x3), x4)
new_esEs6(x0, x1, ty_Double)
new_primPlusNat1(Zero, Succ(x0))
new_esEs17(Left(x0), Left(x1), ty_Bool, x2)
new_esEs22(x0, x1, ty_Ordering)
new_esEs24(x0, x1, ty_Integer)
new_esEs22(x0, x1, ty_@0)
new_esEs23(x0, x1, ty_Integer)
new_esEs6(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Integer)
new_esEs23(x0, x1, ty_Bool)
new_esEs6(x0, x1, ty_Ordering)
new_esEs21(x0, x1, ty_Char)
new_primPlusNat1(Zero, Zero)
new_esEs20(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Ordering)
new_esEs15(EQ, LT)
new_esEs15(LT, EQ)
new_esEs22(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, ty_Char)
new_esEs19(Just(x0), Just(x1), app(app(ty_@2, x2), x3))
new_primEqInt(Pos(Zero), Pos(Zero))
new_esEs23(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_@0)
new_esEs17(Left(x0), Right(x1), x2, x3)
new_esEs17(Right(x0), Left(x1), x2, x3)
new_esEs15(LT, LT)
new_esEs21(x0, x1, ty_Int)
new_esEs22(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Float)
new_esEs7(x0, x1, app(app(ty_Either, x2), x3))
new_esEs15(GT, EQ)
new_esEs15(EQ, GT)
new_primMulNat0(Succ(x0), Zero)
new_esEs17(Right(x0), Right(x1), x2, ty_Double)
new_sr(Neg(x0), Pos(x1))
new_sr(Pos(x0), Neg(x1))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_listToMaybe(yv35, False, :(yv41110, yv41111), ba) → new_listToMaybe(new_primPlusNat0(yv35, Zero), new_esEs4([], yv41110, ba), yv41111, ba)

The TRS R consists of the following rules:

new_primPlusNat0(Zero, yv400100) → Succ(yv400100)
new_primPlusNat0(Succ(yv540), yv400100) → Succ(Succ(new_primPlusNat1(yv540, yv400100)))
new_esEs4([], [], ba) → True
new_esEs4([], :(yv4010, yv4011), ba) → False
new_primPlusNat1(Succ(yv5400), Succ(yv4001000)) → Succ(Succ(new_primPlusNat1(yv5400, yv4001000)))
new_primPlusNat1(Succ(yv5400), Zero) → Succ(yv5400)
new_primPlusNat1(Zero, Succ(yv4001000)) → Succ(yv4001000)
new_primPlusNat1(Zero, Zero) → Zero

The set Q consists of the following terms:

new_esEs7(x0, x1, app(ty_Maybe, x2))
new_esEs23(x0, x1, ty_Ordering)
new_esEs8(Float(x0, x1), Float(x2, x3))
new_esEs17(Left(x0), Left(x1), app(app(ty_Either, x2), x3), x4)
new_primEqInt(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs17(Right(x0), Right(x1), x2, app(ty_Ratio, x3))
new_esEs11(True, True)
new_esEs21(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Char)
new_primEqInt(Pos(Zero), Neg(Zero))
new_primEqInt(Neg(Zero), Pos(Zero))
new_esEs17(Left(x0), Left(x1), app(app(app(ty_@3, x2), x3), x4), x5)
new_esEs20(x0, x1, ty_Char)
new_esEs23(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), app(ty_Maybe, x2), x3)
new_esEs17(Right(x0), Right(x1), x2, ty_Ordering)
new_esEs22(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), app(app(ty_@2, x2), x3), x4)
new_esEs20(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), ty_Char)
new_esEs21(x0, x1, app(app(ty_Either, x2), x3))
new_esEs4(:(x0, x1), [], x2)
new_esEs17(Right(x0), Right(x1), x2, app(ty_Maybe, x3))
new_esEs20(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Char)
new_esEs19(Just(x0), Just(x1), ty_Float)
new_esEs22(x0, x1, app(ty_[], x2))
new_esEs17(Right(x0), Right(x1), x2, ty_Int)
new_primEqInt(Neg(Zero), Neg(Zero))
new_esEs19(Just(x0), Just(x1), ty_@0)
new_esEs6(x0, x1, ty_Int)
new_esEs13(Integer(x0), Integer(x1))
new_esEs5(@2(x0, x1), @2(x2, x3), x4, x5)
new_esEs19(Nothing, Nothing, x0)
new_sr(Neg(x0), Neg(x1))
new_primEqInt(Pos(Succ(x0)), Pos(Zero))
new_esEs22(x0, x1, app(app(ty_Either, x2), x3))
new_esEs18(Char(x0), Char(x1))
new_esEs22(x0, x1, ty_Integer)
new_esEs19(Just(x0), Just(x1), ty_Bool)
new_esEs20(x0, x1, ty_Float)
new_primMulNat0(Succ(x0), Succ(x1))
new_primEqInt(Pos(Succ(x0)), Neg(x1))
new_primEqInt(Neg(Succ(x0)), Pos(x1))
new_esEs23(x0, x1, app(ty_[], x2))
new_esEs11(True, False)
new_esEs11(False, True)
new_esEs19(Just(x0), Just(x1), ty_Ordering)
new_esEs21(x0, x1, app(ty_Maybe, x2))
new_esEs6(x0, x1, app(ty_Ratio, x2))
new_esEs19(Just(x0), Just(x1), app(app(app(ty_@3, x2), x3), x4))
new_esEs20(x0, x1, app(ty_Ratio, x2))
new_esEs17(Left(x0), Left(x1), ty_Integer, x2)
new_esEs19(Just(x0), Just(x1), ty_Int)
new_primPlusNat0(Zero, x0)
new_esEs21(x0, x1, ty_Integer)
new_esEs17(Left(x0), Left(x1), ty_Char, x2)
new_esEs20(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs19(Just(x0), Nothing, x1)
new_esEs25(x0, x1, ty_Int)
new_esEs17(Left(x0), Left(x1), ty_Float, x2)
new_esEs25(x0, x1, ty_Integer)
new_esEs4([], :(x0, x1), x2)
new_esEs9(@0, @0)
new_esEs7(x0, x1, ty_Bool)
new_esEs6(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Succ(x0)), Neg(Zero))
new_esEs14(@3(x0, x1, x2), @3(x3, x4, x5), x6, x7, x8)
new_primEqInt(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs20(x0, x1, app(ty_Maybe, x2))
new_esEs17(Right(x0), Right(x1), x2, app(ty_[], x3))
new_esEs6(x0, x1, ty_Integer)
new_esEs20(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Int)
new_esEs7(x0, x1, app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Double)
new_esEs6(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Integer)
new_asAs(False, x0)
new_esEs17(Right(x0), Right(x1), x2, ty_Integer)
new_esEs23(x0, x1, app(app(ty_@2, x2), x3))
new_esEs24(x0, x1, ty_Int)
new_esEs16(:%(x0, x1), :%(x2, x3), x4)
new_esEs23(x0, x1, ty_Double)
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_Either, x3), x4))
new_esEs23(x0, x1, ty_Float)
new_esEs6(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs23(x0, x1, app(app(ty_Either, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_@2, x3), x4))
new_primEqNat0(Zero, Zero)
new_esEs17(Right(x0), Right(x1), x2, ty_@0)
new_esEs6(x0, x1, app(app(ty_Either, x2), x3))
new_esEs19(Just(x0), Just(x1), app(ty_[], x2))
new_primEqInt(Neg(Zero), Neg(Succ(x0)))
new_esEs19(Nothing, Just(x0), x1)
new_esEs20(x0, x1, app(ty_[], x2))
new_esEs15(GT, GT)
new_esEs22(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), ty_Int, x2)
new_primEqNat0(Zero, Succ(x0))
new_esEs12(x0, x1)
new_esEs22(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs15(GT, LT)
new_esEs15(LT, GT)
new_primMulNat0(Zero, Zero)
new_primPlusNat1(Succ(x0), Succ(x1))
new_esEs22(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Int)
new_esEs23(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs6(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Bool)
new_esEs17(Right(x0), Right(x1), x2, ty_Bool)
new_esEs21(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_sr(Pos(x0), Pos(x1))
new_esEs21(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(ty_Maybe, x2))
new_esEs22(x0, x1, app(ty_Maybe, x2))
new_esEs17(Left(x0), Left(x1), ty_Ordering, x2)
new_esEs17(Left(x0), Left(x1), ty_Double, x2)
new_esEs7(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), ty_@0, x2)
new_asAs(True, x0)
new_esEs15(EQ, EQ)
new_primPlusNat0(Succ(x0), x1)
new_primEqInt(Pos(Zero), Pos(Succ(x0)))
new_esEs7(x0, x1, app(ty_[], x2))
new_esEs22(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(app(ty_Either, x2), x3))
new_esEs20(x0, x1, ty_Double)
new_esEs22(x0, x1, app(ty_Ratio, x2))
new_esEs23(x0, x1, ty_@0)
new_esEs17(Right(x0), Right(x1), x2, ty_Float)
new_esEs17(Right(x0), Right(x1), x2, app(app(app(ty_@3, x3), x4), x5))
new_esEs23(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), ty_Double)
new_esEs10(Double(x0, x1), Double(x2, x3))
new_esEs17(Left(x0), Left(x1), app(ty_Ratio, x2), x3)
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs20(x0, x1, app(app(ty_Either, x2), x3))
new_esEs7(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_primMulNat0(Zero, Succ(x0))
new_esEs6(x0, x1, ty_@0)
new_esEs23(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Zero), Pos(Succ(x0)))
new_primEqInt(Pos(Zero), Neg(Succ(x0)))
new_esEs17(Left(x0), Left(x1), app(ty_[], x2), x3)
new_esEs4([], [], x0)
new_primPlusNat1(Succ(x0), Zero)
new_esEs19(Just(x0), Just(x1), ty_Integer)
new_esEs11(False, False)
new_esEs4(:(x0, x1), :(x2, x3), x4)
new_esEs6(x0, x1, ty_Double)
new_primPlusNat1(Zero, Succ(x0))
new_esEs17(Left(x0), Left(x1), ty_Bool, x2)
new_esEs22(x0, x1, ty_Ordering)
new_esEs24(x0, x1, ty_Integer)
new_esEs22(x0, x1, ty_@0)
new_esEs23(x0, x1, ty_Integer)
new_esEs6(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Integer)
new_esEs23(x0, x1, ty_Bool)
new_esEs6(x0, x1, ty_Ordering)
new_esEs21(x0, x1, ty_Char)
new_primPlusNat1(Zero, Zero)
new_esEs20(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Ordering)
new_esEs15(EQ, LT)
new_esEs15(LT, EQ)
new_esEs22(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, ty_Char)
new_esEs19(Just(x0), Just(x1), app(app(ty_@2, x2), x3))
new_primEqInt(Pos(Zero), Pos(Zero))
new_esEs23(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_@0)
new_esEs17(Left(x0), Right(x1), x2, x3)
new_esEs17(Right(x0), Left(x1), x2, x3)
new_esEs15(LT, LT)
new_esEs21(x0, x1, ty_Int)
new_esEs22(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Float)
new_esEs7(x0, x1, app(app(ty_Either, x2), x3))
new_esEs15(GT, EQ)
new_esEs15(EQ, GT)
new_primMulNat0(Succ(x0), Zero)
new_esEs17(Right(x0), Right(x1), x2, ty_Double)
new_sr(Neg(x0), Pos(x1))
new_sr(Pos(x0), Neg(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_esEs7(x0, x1, app(ty_Maybe, x2))
new_esEs23(x0, x1, ty_Ordering)
new_esEs8(Float(x0, x1), Float(x2, x3))
new_esEs17(Left(x0), Left(x1), app(app(ty_Either, x2), x3), x4)
new_primEqInt(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs17(Right(x0), Right(x1), x2, app(ty_Ratio, x3))
new_esEs11(True, True)
new_esEs21(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Char)
new_primEqInt(Pos(Zero), Neg(Zero))
new_primEqInt(Neg(Zero), Pos(Zero))
new_esEs17(Left(x0), Left(x1), app(app(app(ty_@3, x2), x3), x4), x5)
new_esEs20(x0, x1, ty_Char)
new_esEs23(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), app(ty_Maybe, x2), x3)
new_esEs17(Right(x0), Right(x1), x2, ty_Ordering)
new_esEs22(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), app(app(ty_@2, x2), x3), x4)
new_esEs20(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), ty_Char)
new_esEs21(x0, x1, app(app(ty_Either, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, app(ty_Maybe, x3))
new_esEs20(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Char)
new_esEs19(Just(x0), Just(x1), ty_Float)
new_esEs22(x0, x1, app(ty_[], x2))
new_esEs17(Right(x0), Right(x1), x2, ty_Int)
new_primEqInt(Neg(Zero), Neg(Zero))
new_esEs19(Just(x0), Just(x1), ty_@0)
new_esEs6(x0, x1, ty_Int)
new_esEs13(Integer(x0), Integer(x1))
new_esEs5(@2(x0, x1), @2(x2, x3), x4, x5)
new_esEs19(Nothing, Nothing, x0)
new_sr(Neg(x0), Neg(x1))
new_primEqInt(Pos(Succ(x0)), Pos(Zero))
new_esEs22(x0, x1, app(app(ty_Either, x2), x3))
new_esEs18(Char(x0), Char(x1))
new_esEs22(x0, x1, ty_Integer)
new_esEs19(Just(x0), Just(x1), ty_Bool)
new_esEs20(x0, x1, ty_Float)
new_primMulNat0(Succ(x0), Succ(x1))
new_primEqInt(Pos(Succ(x0)), Neg(x1))
new_primEqInt(Neg(Succ(x0)), Pos(x1))
new_esEs23(x0, x1, app(ty_[], x2))
new_esEs11(True, False)
new_esEs11(False, True)
new_esEs19(Just(x0), Just(x1), ty_Ordering)
new_esEs21(x0, x1, app(ty_Maybe, x2))
new_esEs6(x0, x1, app(ty_Ratio, x2))
new_esEs19(Just(x0), Just(x1), app(app(app(ty_@3, x2), x3), x4))
new_esEs20(x0, x1, app(ty_Ratio, x2))
new_esEs17(Left(x0), Left(x1), ty_Integer, x2)
new_esEs19(Just(x0), Just(x1), ty_Int)
new_esEs21(x0, x1, ty_Integer)
new_esEs17(Left(x0), Left(x1), ty_Char, x2)
new_esEs20(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs19(Just(x0), Nothing, x1)
new_esEs25(x0, x1, ty_Int)
new_esEs17(Left(x0), Left(x1), ty_Float, x2)
new_esEs25(x0, x1, ty_Integer)
new_esEs9(@0, @0)
new_esEs7(x0, x1, ty_Bool)
new_esEs6(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Succ(x0)), Neg(Zero))
new_esEs14(@3(x0, x1, x2), @3(x3, x4, x5), x6, x7, x8)
new_primEqInt(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs20(x0, x1, app(ty_Maybe, x2))
new_esEs17(Right(x0), Right(x1), x2, app(ty_[], x3))
new_esEs6(x0, x1, ty_Integer)
new_esEs20(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Int)
new_esEs7(x0, x1, app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Double)
new_esEs6(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Integer)
new_asAs(False, x0)
new_esEs17(Right(x0), Right(x1), x2, ty_Integer)
new_esEs23(x0, x1, app(app(ty_@2, x2), x3))
new_esEs24(x0, x1, ty_Int)
new_esEs16(:%(x0, x1), :%(x2, x3), x4)
new_esEs23(x0, x1, ty_Double)
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_Either, x3), x4))
new_esEs23(x0, x1, ty_Float)
new_esEs6(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs23(x0, x1, app(app(ty_Either, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_@2, x3), x4))
new_primEqNat0(Zero, Zero)
new_esEs17(Right(x0), Right(x1), x2, ty_@0)
new_esEs6(x0, x1, app(app(ty_Either, x2), x3))
new_esEs19(Just(x0), Just(x1), app(ty_[], x2))
new_primEqInt(Neg(Zero), Neg(Succ(x0)))
new_esEs19(Nothing, Just(x0), x1)
new_esEs20(x0, x1, app(ty_[], x2))
new_esEs15(GT, GT)
new_esEs22(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), ty_Int, x2)
new_primEqNat0(Zero, Succ(x0))
new_esEs12(x0, x1)
new_esEs22(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs15(GT, LT)
new_esEs15(LT, GT)
new_primMulNat0(Zero, Zero)
new_esEs22(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Int)
new_esEs23(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs6(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Bool)
new_esEs17(Right(x0), Right(x1), x2, ty_Bool)
new_esEs21(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_sr(Pos(x0), Pos(x1))
new_esEs21(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(ty_Maybe, x2))
new_esEs22(x0, x1, app(ty_Maybe, x2))
new_esEs17(Left(x0), Left(x1), ty_Ordering, x2)
new_esEs17(Left(x0), Left(x1), ty_Double, x2)
new_esEs7(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), ty_@0, x2)
new_asAs(True, x0)
new_esEs15(EQ, EQ)
new_primEqInt(Pos(Zero), Pos(Succ(x0)))
new_esEs7(x0, x1, app(ty_[], x2))
new_esEs22(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(app(ty_Either, x2), x3))
new_esEs20(x0, x1, ty_Double)
new_esEs22(x0, x1, app(ty_Ratio, x2))
new_esEs23(x0, x1, ty_@0)
new_esEs17(Right(x0), Right(x1), x2, ty_Float)
new_esEs17(Right(x0), Right(x1), x2, app(app(app(ty_@3, x3), x4), x5))
new_esEs23(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), ty_Double)
new_esEs10(Double(x0, x1), Double(x2, x3))
new_esEs17(Left(x0), Left(x1), app(ty_Ratio, x2), x3)
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs20(x0, x1, app(app(ty_Either, x2), x3))
new_esEs7(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_primMulNat0(Zero, Succ(x0))
new_esEs6(x0, x1, ty_@0)
new_esEs23(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Zero), Pos(Succ(x0)))
new_primEqInt(Pos(Zero), Neg(Succ(x0)))
new_esEs17(Left(x0), Left(x1), app(ty_[], x2), x3)
new_esEs19(Just(x0), Just(x1), ty_Integer)
new_esEs11(False, False)
new_esEs6(x0, x1, ty_Double)
new_esEs17(Left(x0), Left(x1), ty_Bool, x2)
new_esEs22(x0, x1, ty_Ordering)
new_esEs24(x0, x1, ty_Integer)
new_esEs22(x0, x1, ty_@0)
new_esEs23(x0, x1, ty_Integer)
new_esEs6(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Integer)
new_esEs23(x0, x1, ty_Bool)
new_esEs6(x0, x1, ty_Ordering)
new_esEs21(x0, x1, ty_Char)
new_esEs20(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Ordering)
new_esEs15(EQ, LT)
new_esEs15(LT, EQ)
new_esEs22(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Right(x0), Right(x1), x2, ty_Char)
new_esEs19(Just(x0), Just(x1), app(app(ty_@2, x2), x3))
new_primEqInt(Pos(Zero), Pos(Zero))
new_esEs23(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_@0)
new_esEs17(Left(x0), Right(x1), x2, x3)
new_esEs17(Right(x0), Left(x1), x2, x3)
new_esEs15(LT, LT)
new_esEs21(x0, x1, ty_Int)
new_esEs22(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Float)
new_esEs7(x0, x1, app(app(ty_Either, x2), x3))
new_esEs15(GT, EQ)
new_esEs15(EQ, GT)
new_primMulNat0(Succ(x0), Zero)
new_esEs17(Right(x0), Right(x1), x2, ty_Double)
new_sr(Neg(x0), Pos(x1))
new_sr(Pos(x0), Neg(x1))



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPSizeChangeProof
                              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_listToMaybe(yv35, False, :(yv41110, yv41111), ba) → new_listToMaybe(new_primPlusNat0(yv35, Zero), new_esEs4([], yv41110, ba), yv41111, ba)

The TRS R consists of the following rules:

new_primPlusNat0(Zero, yv400100) → Succ(yv400100)
new_primPlusNat0(Succ(yv540), yv400100) → Succ(Succ(new_primPlusNat1(yv540, yv400100)))
new_esEs4([], [], ba) → True
new_esEs4([], :(yv4010, yv4011), ba) → False
new_primPlusNat1(Succ(yv5400), Succ(yv4001000)) → Succ(Succ(new_primPlusNat1(yv5400, yv4001000)))
new_primPlusNat1(Succ(yv5400), Zero) → Succ(yv5400)
new_primPlusNat1(Zero, Succ(yv4001000)) → Succ(yv4001000)
new_primPlusNat1(Zero, Zero) → Zero

The set Q consists of the following terms:

new_esEs4(:(x0, x1), [], x2)
new_primPlusNat0(Zero, x0)
new_esEs4([], :(x0, x1), x2)
new_primPlusNat1(Succ(x0), Succ(x1))
new_primPlusNat0(Succ(x0), x1)
new_esEs4([], [], x0)
new_primPlusNat1(Succ(x0), Zero)
new_esEs4(:(x0, x1), :(x2, x3), x4)
new_primPlusNat1(Zero, Succ(x0))
new_primPlusNat1(Zero, Zero)

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ HASKELL
  ↳ LR
    ↳ HASKELL
      ↳ CR
        ↳ HASKELL
          ↳ IFR
            ↳ HASKELL
              ↳ BR
                ↳ HASKELL
                  ↳ COR
                    ↳ HASKELL
                      ↳ NumRed
                        ↳ HASKELL
                          ↳ Narrow
                            ↳ AND
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                              ↳ QDP
                                ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_listToMaybe0(yv37, False, yv14, yv12, :(yv15110, yv15111), ba) → new_listToMaybe0(new_primPlusNat0(yv37, Zero), new_esEs4(:(yv14, yv12), yv15110, ba), yv14, yv12, yv15111, ba)

The TRS R consists of the following rules:

new_esEs23(yv310, yv4010, app(app(ty_@2, bb), bc)) → new_esEs5(yv310, yv4010, bb, bc)
new_esEs23(yv310, yv4010, ty_Integer) → new_esEs13(yv310, yv4010)
new_esEs6(yv301, yv4001, ty_Int) → new_esEs12(yv301, yv4001)
new_esEs18(Char(yv300), Char(yv4000)) → new_primEqNat0(yv300, yv4000)
new_primPlusNat1(Succ(yv5400), Succ(yv4001000)) → Succ(Succ(new_primPlusNat1(yv5400, yv4001000)))
new_primEqInt(Neg(Succ(yv3000)), Pos(yv4000)) → False
new_primEqInt(Pos(Succ(yv3000)), Neg(yv4000)) → False
new_esEs6(yv301, yv4001, ty_Float) → new_esEs8(yv301, yv4001)
new_esEs17(Right(yv300), Right(yv4000), bca, app(ty_Ratio, bea)) → new_esEs16(yv300, yv4000, bea)
new_esEs22(yv300, yv4000, ty_Ordering) → new_esEs15(yv300, yv4000)
new_esEs17(Left(yv300), Left(yv4000), app(ty_Ratio, bcg), bcb) → new_esEs16(yv300, yv4000, bcg)
new_esEs19(Just(yv300), Just(yv4000), app(ty_[], bad)) → new_esEs4(yv300, yv4000, bad)
new_esEs17(Left(yv300), Left(yv4000), app(ty_Maybe, bdd), bcb) → new_esEs19(yv300, yv4000, bdd)
new_esEs20(yv302, yv4002, ty_Integer) → new_esEs13(yv302, yv4002)
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Double) → new_esEs10(yv300, yv4000)
new_primEqInt(Neg(Zero), Pos(Succ(yv40000))) → False
new_primEqInt(Pos(Zero), Neg(Succ(yv40000))) → False
new_esEs20(yv302, yv4002, ty_Char) → new_esEs18(yv302, yv4002)
new_esEs15(EQ, EQ) → True
new_esEs23(yv310, yv4010, ty_Bool) → new_esEs11(yv310, yv4010)
new_esEs17(Left(yv300), Left(yv4000), ty_Int, bcb) → new_esEs12(yv300, yv4000)
new_esEs7(yv300, yv4000, ty_Bool) → new_esEs11(yv300, yv4000)
new_esEs21(yv301, yv4001, app(ty_Maybe, gh)) → new_esEs19(yv301, yv4001, gh)
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Int) → new_esEs12(yv300, yv4000)
new_primMulNat0(Zero, Zero) → Zero
new_esEs21(yv301, yv4001, ty_Int) → new_esEs12(yv301, yv4001)
new_esEs21(yv301, yv4001, ty_Char) → new_esEs18(yv301, yv4001)
new_esEs24(yv301, yv4001, ty_Integer) → new_esEs13(yv301, yv4001)
new_esEs17(Right(yv300), Left(yv4000), bca, bcb) → False
new_esEs17(Left(yv300), Right(yv4000), bca, bcb) → False
new_esEs21(yv301, yv4001, app(app(app(ty_@3, fh), ga), gb)) → new_esEs14(yv301, yv4001, fh, ga, gb)
new_esEs17(Left(yv300), Left(yv4000), ty_Float, bcb) → new_esEs8(yv300, yv4000)
new_esEs21(yv301, yv4001, app(app(ty_@2, gf), gg)) → new_esEs5(yv301, yv4001, gf, gg)
new_esEs17(Left(yv300), Left(yv4000), app(app(app(ty_@3, bcd), bce), bcf), bcb) → new_esEs14(yv300, yv4000, bcd, bce, bcf)
new_esEs21(yv301, yv4001, ty_Double) → new_esEs10(yv301, yv4001)
new_esEs19(Just(yv300), Just(yv4000), app(app(ty_Either, bba), bbb)) → new_esEs17(yv300, yv4000, bba, bbb)
new_esEs20(yv302, yv4002, app(ty_Ratio, eh)) → new_esEs16(yv302, yv4002, eh)
new_primPlusNat0(Zero, yv400100) → Succ(yv400100)
new_esEs19(Just(yv300), Just(yv4000), ty_@0) → new_esEs9(yv300, yv4000)
new_esEs5(@2(yv300, yv301), @2(yv4000, yv4001), bb, bc) → new_asAs(new_esEs7(yv300, yv4000, bb), new_esEs6(yv301, yv4001, bc))
new_esEs19(Just(yv300), Just(yv4000), ty_Bool) → new_esEs11(yv300, yv4000)
new_esEs21(yv301, yv4001, ty_Integer) → new_esEs13(yv301, yv4001)
new_esEs7(yv300, yv4000, app(app(ty_@2, df), dg)) → new_esEs5(yv300, yv4000, df, dg)
new_sr(Pos(yv3010), Neg(yv40010)) → Neg(new_primMulNat0(yv3010, yv40010))
new_sr(Neg(yv3010), Pos(yv40010)) → Neg(new_primMulNat0(yv3010, yv40010))
new_esEs23(yv310, yv4010, app(ty_Maybe, bac)) → new_esEs19(yv310, yv4010, bac)
new_esEs23(yv310, yv4010, ty_@0) → new_esEs9(yv310, yv4010)
new_esEs19(Just(yv300), Just(yv4000), ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs6(yv301, yv4001, app(ty_Maybe, ce)) → new_esEs19(yv301, yv4001, ce)
new_esEs4(:(yv310, yv311), :(yv4010, yv4011), bbf) → new_asAs(new_esEs23(yv310, yv4010, bbf), new_esEs4(yv311, yv4011, bbf))
new_esEs22(yv300, yv4000, ty_Bool) → new_esEs11(yv300, yv4000)
new_esEs20(yv302, yv4002, ty_@0) → new_esEs9(yv302, yv4002)
new_esEs19(Nothing, Just(yv4000), bac) → False
new_esEs19(Just(yv300), Nothing, bac) → False
new_esEs22(yv300, yv4000, app(app(ty_@2, hh), baa)) → new_esEs5(yv300, yv4000, hh, baa)
new_esEs7(yv300, yv4000, ty_Ordering) → new_esEs15(yv300, yv4000)
new_esEs22(yv300, yv4000, ty_Char) → new_esEs18(yv300, yv4000)
new_esEs20(yv302, yv4002, ty_Float) → new_esEs8(yv302, yv4002)
new_esEs6(yv301, yv4001, ty_Bool) → new_esEs11(yv301, yv4001)
new_esEs19(Just(yv300), Just(yv4000), app(app(app(ty_@3, bae), baf), bag)) → new_esEs14(yv300, yv4000, bae, baf, bag)
new_esEs23(yv310, yv4010, ty_Ordering) → new_esEs15(yv310, yv4010)
new_esEs19(Nothing, Nothing, bac) → True
new_esEs7(yv300, yv4000, app(ty_[], cf)) → new_esEs4(yv300, yv4000, cf)
new_esEs17(Left(yv300), Left(yv4000), app(ty_[], bcc), bcb) → new_esEs4(yv300, yv4000, bcc)
new_primEqNat0(Zero, Succ(yv40000)) → False
new_primEqNat0(Succ(yv3000), Zero) → False
new_esEs15(GT, LT) → False
new_esEs15(LT, GT) → False
new_esEs19(Just(yv300), Just(yv4000), ty_Float) → new_esEs8(yv300, yv4000)
new_primEqInt(Pos(Zero), Pos(Zero)) → True
new_esEs25(yv300, yv4000, ty_Int) → new_esEs12(yv300, yv4000)
new_esEs15(LT, LT) → True
new_esEs20(yv302, yv4002, app(ty_[], ed)) → new_esEs4(yv302, yv4002, ed)
new_esEs21(yv301, yv4001, app(app(ty_Either, gd), ge)) → new_esEs17(yv301, yv4001, gd, ge)
new_esEs7(yv300, yv4000, ty_Int) → new_esEs12(yv300, yv4000)
new_esEs6(yv301, yv4001, app(ty_[], bd)) → new_esEs4(yv301, yv4001, bd)
new_esEs17(Left(yv300), Left(yv4000), app(app(ty_Either, bch), bda), bcb) → new_esEs17(yv300, yv4000, bch, bda)
new_esEs7(yv300, yv4000, ty_@0) → new_esEs9(yv300, yv4000)
new_esEs19(Just(yv300), Just(yv4000), ty_Ordering) → new_esEs15(yv300, yv4000)
new_esEs11(True, False) → False
new_esEs11(False, True) → False
new_esEs12(yv30, yv400) → new_primEqInt(yv30, yv400)
new_esEs15(EQ, LT) → False
new_esEs15(LT, EQ) → False
new_esEs17(Right(yv300), Right(yv4000), bca, app(app(app(ty_@3, bdf), bdg), bdh)) → new_esEs14(yv300, yv4000, bdf, bdg, bdh)
new_esEs20(yv302, yv4002, app(app(ty_@2, fc), fd)) → new_esEs5(yv302, yv4002, fc, fd)
new_esEs6(yv301, yv4001, app(app(ty_@2, cc), cd)) → new_esEs5(yv301, yv4001, cc, cd)
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Float) → new_esEs8(yv300, yv4000)
new_esEs7(yv300, yv4000, app(ty_Maybe, dh)) → new_esEs19(yv300, yv4000, dh)
new_esEs19(Just(yv300), Just(yv4000), ty_Int) → new_esEs12(yv300, yv4000)
new_esEs23(yv310, yv4010, app(ty_Ratio, bbh)) → new_esEs16(yv310, yv4010, bbh)
new_esEs15(GT, EQ) → False
new_esEs15(EQ, GT) → False
new_esEs22(yv300, yv4000, app(ty_Ratio, he)) → new_esEs16(yv300, yv4000, he)
new_esEs21(yv301, yv4001, ty_Bool) → new_esEs11(yv301, yv4001)
new_esEs10(Double(yv300, yv301), Double(yv4000, yv4001)) → new_esEs12(new_sr(yv300, yv4000), new_sr(yv301, yv4001))
new_esEs7(yv300, yv4000, ty_Float) → new_esEs8(yv300, yv4000)
new_esEs17(Left(yv300), Left(yv4000), app(app(ty_@2, bdb), bdc), bcb) → new_esEs5(yv300, yv4000, bdb, bdc)
new_esEs24(yv301, yv4001, ty_Int) → new_esEs12(yv301, yv4001)
new_esEs23(yv310, yv4010, ty_Char) → new_esEs18(yv310, yv4010)
new_esEs20(yv302, yv4002, ty_Bool) → new_esEs11(yv302, yv4002)
new_sr(Neg(yv3010), Neg(yv40010)) → Pos(new_primMulNat0(yv3010, yv40010))
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs21(yv301, yv4001, app(ty_[], fg)) → new_esEs4(yv301, yv4001, fg)
new_esEs17(Left(yv300), Left(yv4000), ty_Double, bcb) → new_esEs10(yv300, yv4000)
new_esEs7(yv300, yv4000, ty_Integer) → new_esEs13(yv300, yv4000)
new_asAs(False, yv53) → False
new_sr(Pos(yv3010), Pos(yv40010)) → Pos(new_primMulNat0(yv3010, yv40010))
new_primEqNat0(Zero, Zero) → True
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Bool) → new_esEs11(yv300, yv4000)
new_primMulNat0(Zero, Succ(yv400100)) → Zero
new_primMulNat0(Succ(yv30100), Zero) → Zero
new_esEs19(Just(yv300), Just(yv4000), ty_Char) → new_esEs18(yv300, yv4000)
new_esEs17(Right(yv300), Right(yv4000), bca, ty_@0) → new_esEs9(yv300, yv4000)
new_esEs17(Left(yv300), Left(yv4000), ty_@0, bcb) → new_esEs9(yv300, yv4000)
new_esEs6(yv301, yv4001, app(ty_Ratio, bh)) → new_esEs16(yv301, yv4001, bh)
new_esEs20(yv302, yv4002, app(ty_Maybe, ff)) → new_esEs19(yv302, yv4002, ff)
new_esEs6(yv301, yv4001, ty_Char) → new_esEs18(yv301, yv4001)
new_esEs25(yv300, yv4000, ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs11(True, True) → True
new_esEs16(:%(yv300, yv301), :%(yv4000, yv4001), bbh) → new_asAs(new_esEs25(yv300, yv4000, bbh), new_esEs24(yv301, yv4001, bbh))
new_primPlusNat0(Succ(yv540), yv400100) → Succ(Succ(new_primPlusNat1(yv540, yv400100)))
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Char) → new_esEs18(yv300, yv4000)
new_esEs22(yv300, yv4000, ty_Float) → new_esEs8(yv300, yv4000)
new_esEs20(yv302, yv4002, app(app(app(ty_@3, ee), ef), eg)) → new_esEs14(yv302, yv4002, ee, ef, eg)
new_esEs21(yv301, yv4001, app(ty_Ratio, gc)) → new_esEs16(yv301, yv4001, gc)
new_esEs23(yv310, yv4010, ty_Int) → new_esEs12(yv310, yv4010)
new_esEs13(Integer(yv300), Integer(yv4000)) → new_primEqInt(yv300, yv4000)
new_esEs20(yv302, yv4002, app(app(ty_Either, fa), fb)) → new_esEs17(yv302, yv4002, fa, fb)
new_esEs6(yv301, yv4001, ty_Ordering) → new_esEs15(yv301, yv4001)
new_primEqInt(Neg(Succ(yv3000)), Neg(Succ(yv40000))) → new_primEqNat0(yv3000, yv40000)
new_esEs4([], [], bbf) → True
new_esEs22(yv300, yv4000, app(ty_Maybe, bab)) → new_esEs19(yv300, yv4000, bab)
new_esEs7(yv300, yv4000, app(app(app(ty_@3, cg), da), db)) → new_esEs14(yv300, yv4000, cg, da, db)
new_primPlusNat1(Succ(yv5400), Zero) → Succ(yv5400)
new_primPlusNat1(Zero, Succ(yv4001000)) → Succ(yv4001000)
new_esEs17(Right(yv300), Right(yv4000), bca, app(app(ty_Either, beb), bec)) → new_esEs17(yv300, yv4000, beb, bec)
new_esEs23(yv310, yv4010, app(app(ty_Either, bca), bcb)) → new_esEs17(yv310, yv4010, bca, bcb)
new_esEs20(yv302, yv4002, ty_Int) → new_esEs12(yv302, yv4002)
new_esEs17(Left(yv300), Left(yv4000), ty_Bool, bcb) → new_esEs11(yv300, yv4000)
new_esEs21(yv301, yv4001, ty_@0) → new_esEs9(yv301, yv4001)
new_esEs23(yv310, yv4010, ty_Double) → new_esEs10(yv310, yv4010)
new_primEqInt(Neg(Zero), Neg(Zero)) → True
new_esEs6(yv301, yv4001, ty_@0) → new_esEs9(yv301, yv4001)
new_esEs7(yv300, yv4000, ty_Double) → new_esEs10(yv300, yv4000)
new_esEs6(yv301, yv4001, app(app(ty_Either, ca), cb)) → new_esEs17(yv301, yv4001, ca, cb)
new_esEs14(@3(yv300, yv301, yv302), @3(yv4000, yv4001, yv4002), ea, eb, ec) → new_asAs(new_esEs22(yv300, yv4000, ea), new_asAs(new_esEs21(yv301, yv4001, eb), new_esEs20(yv302, yv4002, ec)))
new_primEqInt(Neg(Zero), Neg(Succ(yv40000))) → False
new_primEqInt(Neg(Succ(yv3000)), Neg(Zero)) → False
new_esEs21(yv301, yv4001, ty_Ordering) → new_esEs15(yv301, yv4001)
new_esEs4([], :(yv4010, yv4011), bbf) → False
new_esEs4(:(yv310, yv311), [], bbf) → False
new_esEs21(yv301, yv4001, ty_Float) → new_esEs8(yv301, yv4001)
new_esEs23(yv310, yv4010, ty_Float) → new_esEs8(yv310, yv4010)
new_esEs19(Just(yv300), Just(yv4000), app(ty_Maybe, bbe)) → new_esEs19(yv300, yv4000, bbe)
new_esEs7(yv300, yv4000, app(app(ty_Either, dd), de)) → new_esEs17(yv300, yv4000, dd, de)
new_esEs22(yv300, yv4000, ty_Int) → new_esEs12(yv300, yv4000)
new_primPlusNat1(Zero, Zero) → Zero
new_esEs7(yv300, yv4000, ty_Char) → new_esEs18(yv300, yv4000)
new_asAs(True, yv53) → yv53
new_esEs6(yv301, yv4001, ty_Integer) → new_esEs13(yv301, yv4001)
new_esEs22(yv300, yv4000, ty_@0) → new_esEs9(yv300, yv4000)
new_primMulNat0(Succ(yv30100), Succ(yv400100)) → new_primPlusNat0(new_primMulNat0(yv30100, Succ(yv400100)), yv400100)
new_esEs11(False, False) → True
new_esEs22(yv300, yv4000, app(ty_[], ha)) → new_esEs4(yv300, yv4000, ha)
new_primEqInt(Pos(Succ(yv3000)), Pos(Succ(yv40000))) → new_primEqNat0(yv3000, yv40000)
new_esEs19(Just(yv300), Just(yv4000), app(ty_Ratio, bah)) → new_esEs16(yv300, yv4000, bah)
new_esEs17(Right(yv300), Right(yv4000), bca, app(app(ty_@2, bed), bee)) → new_esEs5(yv300, yv4000, bed, bee)
new_esEs6(yv301, yv4001, ty_Double) → new_esEs10(yv301, yv4001)
new_esEs17(Left(yv300), Left(yv4000), ty_Ordering, bcb) → new_esEs15(yv300, yv4000)
new_esEs22(yv300, yv4000, app(app(app(ty_@3, hb), hc), hd)) → new_esEs14(yv300, yv4000, hb, hc, hd)
new_primEqNat0(Succ(yv3000), Succ(yv40000)) → new_primEqNat0(yv3000, yv40000)
new_esEs17(Right(yv300), Right(yv4000), bca, app(ty_Maybe, bef)) → new_esEs19(yv300, yv4000, bef)
new_esEs15(GT, GT) → True
new_esEs20(yv302, yv4002, ty_Ordering) → new_esEs15(yv302, yv4002)
new_esEs23(yv310, yv4010, app(app(app(ty_@3, ea), eb), ec)) → new_esEs14(yv310, yv4010, ea, eb, ec)
new_esEs19(Just(yv300), Just(yv4000), ty_Double) → new_esEs10(yv300, yv4000)
new_esEs9(@0, @0) → True
new_esEs22(yv300, yv4000, ty_Double) → new_esEs10(yv300, yv4000)
new_esEs20(yv302, yv4002, ty_Double) → new_esEs10(yv302, yv4002)
new_esEs7(yv300, yv4000, app(ty_Ratio, dc)) → new_esEs16(yv300, yv4000, dc)
new_esEs17(Right(yv300), Right(yv4000), bca, ty_Ordering) → new_esEs15(yv300, yv4000)
new_primEqInt(Pos(Zero), Pos(Succ(yv40000))) → False
new_primEqInt(Pos(Succ(yv3000)), Pos(Zero)) → False
new_esEs8(Float(yv300, yv301), Float(yv4000, yv4001)) → new_esEs12(new_sr(yv300, yv4000), new_sr(yv301, yv4001))
new_esEs22(yv300, yv4000, ty_Integer) → new_esEs13(yv300, yv4000)
new_esEs22(yv300, yv4000, app(app(ty_Either, hf), hg)) → new_esEs17(yv300, yv4000, hf, hg)
new_esEs17(Left(yv300), Left(yv4000), ty_Integer, bcb) → new_esEs13(yv300, yv4000)
new_primEqInt(Neg(Zero), Pos(Zero)) → True
new_primEqInt(Pos(Zero), Neg(Zero)) → True
new_esEs19(Just(yv300), Just(yv4000), app(app(ty_@2, bbc), bbd)) → new_esEs5(yv300, yv4000, bbc, bbd)
new_esEs17(Right(yv300), Right(yv4000), bca, app(ty_[], bde)) → new_esEs4(yv300, yv4000, bde)
new_esEs6(yv301, yv4001, app(app(app(ty_@3, be), bf), bg)) → new_esEs14(yv301, yv4001, be, bf, bg)
new_esEs17(Left(yv300), Left(yv4000), ty_Char, bcb) → new_esEs18(yv300, yv4000)
new_esEs23(yv310, yv4010, app(ty_[], bbg)) → new_esEs4(yv310, yv4010, bbg)

The set Q consists of the following terms:

new_esEs17(Left(x0), Left(x1), ty_Char, x2)
new_esEs7(x0, x1, app(ty_Maybe, x2))
new_esEs23(x0, x1, ty_Ordering)
new_esEs8(Float(x0, x1), Float(x2, x3))
new_primEqInt(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs11(True, True)
new_esEs21(x0, x1, ty_Float)
new_esEs17(Right(x0), Right(x1), x2, ty_Float)
new_esEs7(x0, x1, ty_Char)
new_primEqInt(Pos(Zero), Neg(Zero))
new_primEqInt(Neg(Zero), Pos(Zero))
new_esEs23(x0, x1, app(app(ty_Either, x2), x3))
new_esEs20(x0, x1, ty_Char)
new_esEs17(Right(x0), Right(x1), x2, ty_Char)
new_esEs21(x0, x1, app(ty_Ratio, x2))
new_esEs21(x0, x1, app(app(ty_@2, x2), x3))
new_esEs17(Left(x0), Left(x1), app(app(app(ty_@3, x2), x3), x4), x5)
new_esEs17(Right(x0), Right(x1), x2, ty_@0)
new_esEs22(x0, x1, ty_Double)
new_esEs20(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), ty_Char)
new_esEs17(Left(x0), Left(x1), ty_Int, x2)
new_esEs21(x0, x1, app(app(ty_Either, x2), x3))
new_esEs20(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Char)
new_esEs17(Right(x0), Right(x1), x2, app(app(app(ty_@3, x3), x4), x5))
new_esEs19(Just(x0), Just(x1), ty_Float)
new_esEs22(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Zero), Neg(Zero))
new_esEs19(Just(x0), Just(x1), ty_@0)
new_esEs6(x0, x1, ty_Int)
new_esEs13(Integer(x0), Integer(x1))
new_esEs5(@2(x0, x1), @2(x2, x3), x4, x5)
new_esEs19(Nothing, Nothing, x0)
new_sr(Neg(x0), Neg(x1))
new_primEqInt(Pos(Succ(x0)), Pos(Zero))
new_esEs22(x0, x1, app(app(ty_Either, x2), x3))
new_esEs18(Char(x0), Char(x1))
new_esEs22(x0, x1, ty_Integer)
new_esEs19(Just(x0), Just(x1), ty_Bool)
new_esEs23(x0, x1, app(ty_Ratio, x2))
new_esEs20(x0, x1, ty_Float)
new_primMulNat0(Succ(x0), Succ(x1))
new_esEs17(Left(x0), Left(x1), app(app(ty_Either, x2), x3), x4)
new_primEqInt(Pos(Succ(x0)), Neg(x1))
new_primEqInt(Neg(Succ(x0)), Pos(x1))
new_esEs11(True, False)
new_esEs11(False, True)
new_esEs19(Just(x0), Just(x1), ty_Ordering)
new_esEs21(x0, x1, app(ty_Maybe, x2))
new_esEs17(Right(x0), Right(x1), x2, ty_Integer)
new_esEs6(x0, x1, app(ty_Ratio, x2))
new_esEs19(Just(x0), Just(x1), app(app(app(ty_@3, x2), x3), x4))
new_esEs20(x0, x1, app(ty_Ratio, x2))
new_esEs19(Just(x0), Just(x1), ty_Int)
new_esEs17(Right(x0), Right(x1), x2, ty_Int)
new_primPlusNat0(Zero, x0)
new_esEs16(:%(x0, x1), :%(x2, x3), x4)
new_esEs21(x0, x1, ty_Integer)
new_esEs20(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs19(Just(x0), Nothing, x1)
new_esEs25(x0, x1, ty_Int)
new_esEs25(x0, x1, ty_Integer)
new_esEs9(@0, @0)
new_esEs7(x0, x1, ty_Bool)
new_esEs6(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Succ(x0)), Neg(Zero))
new_esEs14(@3(x0, x1, x2), @3(x3, x4, x5), x6, x7, x8)
new_primEqInt(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs20(x0, x1, app(ty_Maybe, x2))
new_esEs17(Left(x0), Left(x1), ty_Double, x2)
new_esEs6(x0, x1, ty_Integer)
new_esEs20(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Int)
new_esEs7(x0, x1, app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Double)
new_esEs6(x0, x1, ty_Float)
new_esEs17(Right(x0), Right(x1), x2, app(ty_Ratio, x3))
new_esEs7(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Integer)
new_asAs(False, x0)
new_esEs23(x0, x1, app(app(ty_@2, x2), x3))
new_esEs24(x0, x1, ty_Int)
new_esEs17(Left(x0), Left(x1), ty_@0, x2)
new_esEs23(x0, x1, ty_Double)
new_esEs4([], :(x0, x1), x2)
new_esEs23(x0, x1, ty_Float)
new_esEs6(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs17(Left(x0), Left(x1), ty_Bool, x2)
new_esEs23(x0, x1, app(ty_[], x2))
new_esEs17(Left(x0), Left(x1), ty_Ordering, x2)
new_primEqNat0(Zero, Zero)
new_esEs6(x0, x1, app(app(ty_Either, x2), x3))
new_esEs19(Just(x0), Just(x1), app(ty_[], x2))
new_primEqInt(Neg(Zero), Neg(Succ(x0)))
new_esEs19(Nothing, Just(x0), x1)
new_esEs20(x0, x1, app(ty_[], x2))
new_esEs15(GT, GT)
new_esEs22(x0, x1, ty_Char)
new_esEs17(Right(x0), Right(x1), x2, app(ty_Maybe, x3))
new_esEs21(x0, x1, ty_Double)
new_primEqNat0(Zero, Succ(x0))
new_esEs12(x0, x1)
new_esEs17(Right(x0), Right(x1), x2, app(ty_[], x3))
new_esEs4(:(x0, x1), :(x2, x3), x4)
new_esEs17(Right(x0), Right(x1), x2, ty_Double)
new_esEs22(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs15(GT, LT)
new_esEs15(LT, GT)
new_primMulNat0(Zero, Zero)
new_primPlusNat1(Succ(x0), Succ(x1))
new_esEs22(x0, x1, ty_Float)
new_esEs7(x0, x1, ty_Int)
new_esEs23(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs6(x0, x1, app(app(ty_@2, x2), x3))
new_esEs6(x0, x1, ty_Bool)
new_esEs17(Left(x0), Left(x1), app(ty_Maybe, x2), x3)
new_esEs21(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_sr(Pos(x0), Pos(x1))
new_esEs21(x0, x1, ty_Bool)
new_esEs17(Left(x0), Right(x1), x2, x3)
new_esEs17(Right(x0), Left(x1), x2, x3)
new_esEs19(Just(x0), Just(x1), app(ty_Maybe, x2))
new_esEs22(x0, x1, app(ty_Maybe, x2))
new_esEs7(x0, x1, app(app(ty_@2, x2), x3))
new_asAs(True, x0)
new_esEs15(EQ, EQ)
new_primPlusNat0(Succ(x0), x1)
new_primEqInt(Pos(Zero), Pos(Succ(x0)))
new_esEs7(x0, x1, app(ty_[], x2))
new_esEs22(x0, x1, ty_Bool)
new_esEs19(Just(x0), Just(x1), app(app(ty_Either, x2), x3))
new_esEs20(x0, x1, ty_Double)
new_esEs22(x0, x1, app(ty_Ratio, x2))
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_Either, x3), x4))
new_esEs23(x0, x1, ty_@0)
new_esEs23(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), ty_Double)
new_esEs10(Double(x0, x1), Double(x2, x3))
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs20(x0, x1, app(app(ty_Either, x2), x3))
new_esEs7(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs17(Left(x0), Left(x1), app(ty_Ratio, x2), x3)
new_primMulNat0(Zero, Succ(x0))
new_esEs6(x0, x1, ty_@0)
new_esEs23(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, app(ty_[], x2))
new_primEqInt(Neg(Zero), Pos(Succ(x0)))
new_primEqInt(Pos(Zero), Neg(Succ(x0)))
new_primPlusNat1(Succ(x0), Zero)
new_esEs19(Just(x0), Just(x1), ty_Integer)
new_esEs11(False, False)
new_esEs6(x0, x1, ty_Double)
new_primPlusNat1(Zero, Succ(x0))
new_esEs4([], [], x0)
new_esEs22(x0, x1, ty_Ordering)
new_esEs24(x0, x1, ty_Integer)
new_esEs22(x0, x1, ty_@0)
new_esEs23(x0, x1, ty_Integer)
new_esEs17(Left(x0), Left(x1), ty_Float, x2)
new_esEs6(x0, x1, app(ty_Maybe, x2))
new_esEs21(x0, x1, ty_Ordering)
new_esEs20(x0, x1, ty_Integer)
new_esEs23(x0, x1, ty_Bool)
new_esEs6(x0, x1, ty_Ordering)
new_esEs21(x0, x1, ty_Char)
new_esEs17(Left(x0), Left(x1), app(app(ty_@2, x2), x3), x4)
new_esEs17(Left(x0), Left(x1), app(ty_[], x2), x3)
new_primPlusNat1(Zero, Zero)
new_esEs20(x0, x1, ty_@0)
new_esEs7(x0, x1, ty_Ordering)
new_esEs17(Left(x0), Left(x1), ty_Integer, x2)
new_esEs15(EQ, LT)
new_esEs15(LT, EQ)
new_esEs22(x0, x1, app(app(ty_@2, x2), x3))
new_esEs19(Just(x0), Just(x1), app(app(ty_@2, x2), x3))
new_primEqInt(Pos(Zero), Pos(Zero))
new_esEs4(:(x0, x1), [], x2)
new_esEs23(x0, x1, ty_Char)
new_esEs21(x0, x1, ty_@0)
new_esEs15(LT, LT)
new_esEs21(x0, x1, ty_Int)
new_esEs22(x0, x1, ty_Int)
new_esEs19(Just(x0), Just(x1), app(ty_Ratio, x2))
new_esEs7(x0, x1, ty_Float)
new_esEs17(Right(x0), Right(x1), x2, ty_Ordering)
new_esEs17(Right(x0), Right(x1), x2, app(app(ty_@2, x3), x4))
new_esEs7(x0, x1, app(app(ty_Either, x2), x3))
new_esEs15(GT, EQ)
new_esEs15(EQ, GT)
new_esEs17(Right(x0), Right(x1), x2, ty_Bool)
new_primMulNat0(Succ(x0), Zero)
new_sr(Neg(x0), Pos(x1))
new_sr(Pos(x0), Neg(x1))

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: