H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 1.308 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

  ↳ BR

mainModule List

((isSuffixOf :: [Ratio Int] -> [Ratio Int] -> Bool) :: [Ratio Int] -> [Ratio Int] -> Bool)

module List where

import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] _	=	True
isPrefixOf	_ []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

isSuffixOf :: Eq a => [a] -> [a] -> Bool

isSuffixOf

x y

reverse x `isPrefixOf` reverse y

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

((isSuffixOf :: [Ratio Int] -> [Ratio Int] -> Bool) :: [Ratio Int] -> [Ratio Int] -> Bool)

module List where

import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] vw	=	True
isPrefixOf	vx []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

isSuffixOf :: Eq a => [a] -> [a] -> Bool

isSuffixOf

x y

reverse x `isPrefixOf` reverse y

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule List

(isSuffixOf :: [Ratio Int] -> [Ratio Int] -> Bool)

module List where

import qualified Maybe
import qualified Prelude

isPrefixOf :: Eq a => [a] -> [a] -> Bool

isPrefixOf	[] vw	=	True
isPrefixOf	vx []	=	False
isPrefixOf	(x : xs) (y : ys)	=	x == y && isPrefixOf xs ys

isSuffixOf :: Eq a => [a] -> [a] -> Bool

isSuffixOf

x y

reverse x `isPrefixOf` reverse y

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

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560[label="ww1900/Zero",fontsize=10,color="white",style="solid",shape="box"];357 -> 560[label="",style="solid", color="burlywood", weight=9];
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561 -> 374[label="",style="solid", color="burlywood", weight=3];
562[label="ww1900/Zero",fontsize=10,color="white",style="solid",shape="box"];359 -> 562[label="",style="solid", color="burlywood", weight=9];
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564[label="ww1900/Zero",fontsize=10,color="white",style="solid",shape="box"];360 -> 564[label="",style="solid", color="burlywood", weight=9];
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361[label="primEqInt (Neg (Succ ww1600)) (Pos ww1900)",fontsize=16,color="black",shape="box"];361 -> 378[label="",style="solid", color="black", weight=3];
362[label="primEqInt (Neg (Succ ww1600)) (Neg ww1900)",fontsize=16,color="burlywood",shape="box"];565[label="ww1900/Succ ww19000",fontsize=10,color="white",style="solid",shape="box"];362 -> 565[label="",style="solid", color="burlywood", weight=9];
565 -> 379[label="",style="solid", color="burlywood", weight=3];
566[label="ww1900/Zero",fontsize=10,color="white",style="solid",shape="box"];362 -> 566[label="",style="solid", color="burlywood", weight=9];
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567 -> 381[label="",style="solid", color="burlywood", weight=3];
568[label="ww1900/Zero",fontsize=10,color="white",style="solid",shape="box"];363 -> 568[label="",style="solid", color="burlywood", weight=9];
568 -> 382[label="",style="solid", color="burlywood", weight=3];
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570[label="ww1900/Zero",fontsize=10,color="white",style="solid",shape="box"];364 -> 570[label="",style="solid", color="burlywood", weight=9];
570 -> 384[label="",style="solid", color="burlywood", weight=3];
365 -> 304[label="",style="dashed", color="red", weight=0];
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365 -> 386[label="",style="dashed", color="magenta", weight=3];
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368 -> 315[label="",style="dashed", color="red", weight=0];
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368 -> 391[label="",style="dashed", color="magenta", weight=3];
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575 -> 407[label="",style="solid", color="blue", weight=3];
576[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 576[label="",style="solid", color="blue", weight=9];
576 -> 408[label="",style="solid", color="blue", weight=3];
577[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 577[label="",style="solid", color="blue", weight=9];
577 -> 409[label="",style="solid", color="blue", weight=3];
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578 -> 410[label="",style="solid", color="blue", weight=3];
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580[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 580[label="",style="solid", color="blue", weight=9];
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582[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 582[label="",style="solid", color="blue", weight=9];
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584[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 584[label="",style="solid", color="blue", weight=9];
584 -> 416[label="",style="solid", color="blue", weight=3];
585[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 585[label="",style="solid", color="blue", weight=9];
585 -> 417[label="",style="solid", color="blue", weight=3];
586[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];387 -> 586[label="",style="solid", color="blue", weight=9];
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590[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];393 -> 590[label="",style="solid", color="blue", weight=9];
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591[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];393 -> 591[label="",style="solid", color="blue", weight=9];
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432 -> 491[label="",style="dashed", color="magenta", weight=3];
433 -> 321[label="",style="dashed", color="red", weight=0];
433[label="ww32 == ww33\n  Eq a",fontsize=16,color="magenta"];433 -> 492[label="",style="dashed", color="magenta", weight=3];
433 -> 493[label="",style="dashed", color="magenta", weight=3];
434[label="primEqNat (Succ ww16000) ww19000",fontsize=16,color="burlywood",shape="box"];633[label="ww19000/Succ ww190000",fontsize=10,color="white",style="solid",shape="box"];434 -> 633[label="",style="solid", color="burlywood", weight=9];
633 -> 494[label="",style="solid", color="burlywood", weight=3];
634[label="ww19000/Zero",fontsize=10,color="white",style="solid",shape="box"];434 -> 634[label="",style="solid", color="burlywood", weight=9];
634 -> 495[label="",style="solid", color="burlywood", weight=3];
435[label="primEqNat Zero ww19000",fontsize=16,color="burlywood",shape="box"];635[label="ww19000/Succ ww190000",fontsize=10,color="white",style="solid",shape="box"];435 -> 635[label="",style="solid", color="burlywood", weight=9];
635 -> 496[label="",style="solid", color="burlywood", weight=3];
636[label="ww19000/Zero",fontsize=10,color="white",style="solid",shape="box"];435 -> 636[label="",style="solid", color="burlywood", weight=9];
636 -> 497[label="",style="solid", color="burlywood", weight=3];
436[label="ww1600",fontsize=16,color="green",shape="box"];437[label="ww19000",fontsize=16,color="green",shape="box"];438[label="ww250\n  Integral a",fontsize=16,color="green",shape="box"];439[label="ww260\n  Integral a",fontsize=16,color="green",shape="box"];440[label="ww250\n  Eq a",fontsize=16,color="green",shape="box"];441[label="ww260\n  Eq a",fontsize=16,color="green",shape="box"];442[label="ww250",fontsize=16,color="green",shape="box"];443[label="ww260",fontsize=16,color="green",shape="box"];444[label="ww250",fontsize=16,color="green",shape="box"];445[label="ww260",fontsize=16,color="green",shape="box"];446[label="ww250\n  Eq a  Eq b  Eq c",fontsize=16,color="green",shape="box"];447[label="ww260\n  Eq a  Eq b  Eq c",fontsize=16,color="green",shape="box"];448[label="ww250",fontsize=16,color="green",shape="box"];449[label="ww260",fontsize=16,color="green",shape="box"];450[label="ww250",fontsize=16,color="green",shape="box"];451[label="ww260",fontsize=16,color="green",shape="box"];452[label="ww250",fontsize=16,color="green",shape="box"];453[label="ww260",fontsize=16,color="green",shape="box"];454[label="ww250",fontsize=16,color="green",shape="box"];455[label="ww260",fontsize=16,color="green",shape="box"];456[label="ww250\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];457[label="ww260\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];458[label="ww250",fontsize=16,color="green",shape="box"];459[label="ww260",fontsize=16,color="green",shape="box"];460[label="ww250",fontsize=16,color="green",shape="box"];461[label="ww260",fontsize=16,color="green",shape="box"];462[label="ww250\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];463[label="ww260\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];464[label="ww250\n  Eq a",fontsize=16,color="green",shape="box"];465[label="ww260\n  Eq a",fontsize=16,color="green",shape="box"];466[label="ww32\n  Integral a",fontsize=16,color="green",shape="box"];467[label="ww33\n  Integral a",fontsize=16,color="green",shape="box"];468[label="ww32\n  Eq a",fontsize=16,color="green",shape="box"];469[label="ww33\n  Eq a",fontsize=16,color="green",shape="box"];470[label="ww32",fontsize=16,color="green",shape="box"];471[label="ww33",fontsize=16,color="green",shape="box"];472[label="ww32",fontsize=16,color="green",shape="box"];473[label="ww33",fontsize=16,color="green",shape="box"];474[label="ww32\n  Eq a  Eq b  Eq c",fontsize=16,color="green",shape="box"];475[label="ww33\n  Eq a  Eq b  Eq c",fontsize=16,color="green",shape="box"];476[label="ww32",fontsize=16,color="green",shape="box"];477[label="ww33",fontsize=16,color="green",shape="box"];478[label="ww32",fontsize=16,color="green",shape="box"];479[label="ww33",fontsize=16,color="green",shape="box"];480[label="ww32",fontsize=16,color="green",shape="box"];481[label="ww33",fontsize=16,color="green",shape="box"];482[label="ww32",fontsize=16,color="green",shape="box"];483[label="ww33",fontsize=16,color="green",shape="box"];484[label="ww32\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];485[label="ww33\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];486[label="ww32",fontsize=16,color="green",shape="box"];487[label="ww33",fontsize=16,color="green",shape="box"];488[label="ww32",fontsize=16,color="green",shape="box"];489[label="ww33",fontsize=16,color="green",shape="box"];490[label="ww32\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];491[label="ww33\n  Eq a  Eq b",fontsize=16,color="green",shape="box"];492[label="ww32\n  Eq a",fontsize=16,color="green",shape="box"];493[label="ww33\n  Eq a",fontsize=16,color="green",shape="box"];494[label="primEqNat (Succ ww16000) (Succ ww190000)",fontsize=16,color="black",shape="box"];494 -> 498[label="",style="solid", color="black", weight=3];
495[label="primEqNat (Succ ww16000) Zero",fontsize=16,color="black",shape="box"];495 -> 499[label="",style="solid", color="black", weight=3];
496[label="primEqNat Zero (Succ ww190000)",fontsize=16,color="black",shape="box"];496 -> 500[label="",style="solid", color="black", weight=3];
497[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];497 -> 501[label="",style="solid", color="black", weight=3];
498 -> 394[label="",style="dashed", color="red", weight=0];
498[label="primEqNat ww16000 ww190000",fontsize=16,color="magenta"];498 -> 502[label="",style="dashed", color="magenta", weight=3];
498 -> 503[label="",style="dashed", color="magenta", weight=3];
499[label="False",fontsize=16,color="green",shape="box"];500[label="False",fontsize=16,color="green",shape="box"];501[label="True",fontsize=16,color="green",shape="box"];502[label="ww16000",fontsize=16,color="green",shape="box"];503[label="ww190000",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_primEqNat(Succ(ww16000), Succ(ww190000)) → new_primEqNat(ww16000, ww190000)

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:

new_primEqNat(Succ(ww16000), Succ(ww190000)) → new_primEqNat(ww16000, ww190000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_asAs(True, ww32, ww33, app(ty_Ratio, ba)) → new_esEs(ww32, ww33, ba)
new_esEs(:%(ww160, ww161), :%(ww1900, ww1901), bb) → new_asAs(new_esEs0(ww160, ww1900, bb), ww161, ww1901, bb)

The TRS R consists of the following rules:

new_primEqNat0(Zero, Succ(ww190000)) → False
new_primEqNat0(Succ(ww16000), Zero) → False
new_esEs2(Neg(Succ(ww1600)), Neg(Zero)) → False
new_esEs2(Neg(Zero), Neg(Succ(ww19000))) → False
new_esEs2(Pos(Zero), Pos(Zero)) → True
new_esEs2(Neg(Succ(ww1600)), Neg(Succ(ww19000))) → new_primEqNat0(ww1600, ww19000)
new_esEs2(Neg(Zero), Neg(Zero)) → True
new_esEs0(ww160, ww1900, ty_Integer) → new_esEs1(ww160, ww1900)
new_esEs1(ww16, ww190) → error([])
new_esEs2(Pos(Succ(ww1600)), Pos(Succ(ww19000))) → new_primEqNat0(ww1600, ww19000)
new_esEs0(ww160, ww1900, ty_Int) → new_esEs2(ww160, ww1900)
new_primEqNat0(Zero, Zero) → True
new_primEqNat0(Succ(ww16000), Succ(ww190000)) → new_primEqNat0(ww16000, ww190000)
new_esEs2(Neg(Succ(ww1600)), Pos(ww1900)) → False
new_esEs2(Pos(Zero), Neg(Succ(ww19000))) → False
new_esEs2(Pos(Succ(ww1600)), Neg(ww1900)) → False
new_esEs2(Neg(Zero), Pos(Succ(ww19000))) → False
new_esEs2(Pos(Zero), Neg(Zero)) → True
new_esEs2(Neg(Zero), Pos(Zero)) → True
new_esEs2(Pos(Zero), Pos(Succ(ww19000))) → False
new_esEs2(Pos(Succ(ww1600)), Pos(Zero)) → False

The set Q consists of the following terms:

new_esEs2(Neg(Zero), Neg(Succ(x0)))
new_esEs1(x0, x1)
new_primEqNat0(Succ(x0), Zero)
new_esEs2(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs2(Pos(Zero), Neg(Zero))
new_esEs2(Neg(Zero), Pos(Zero))
new_primEqNat0(Zero, Zero)
new_esEs0(x0, x1, ty_Int)
new_esEs0(x0, x1, ty_Integer)
new_primEqNat0(Zero, Succ(x0))
new_esEs2(Neg(Succ(x0)), Neg(Zero))
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs2(Pos(Succ(x0)), Pos(Zero))
new_esEs2(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs2(Pos(Zero), Pos(Zero))
new_esEs2(Neg(Zero), Neg(Zero))
new_esEs2(Neg(Succ(x0)), Pos(x1))
new_esEs2(Pos(Succ(x0)), Neg(x1))
new_esEs2(Pos(Zero), Neg(Succ(x0)))
new_esEs2(Neg(Zero), Pos(Succ(x0)))
new_esEs2(Pos(Zero), Pos(Succ(x0)))

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:

new_asAs(True, ww32, ww33, app(ty_Ratio, ba)) → new_esEs(ww32, ww33, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3
new_esEs(:%(ww160, ww161), :%(ww1900, ww1901), bb) → new_asAs(new_esEs0(ww160, ww1900, bb), ww161, ww1901, bb)
The graph contains the following edges 1 > 2, 2 > 3, 3 >= 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

new_asAs0(True, :(ww250, ww251), :(ww260, ww261), ba) → new_asAs0(new_esEs3(ww250, ww260, ba), ww251, ww261, ba)

The TRS R consists of the following rules:

new_asAs1(True, ww32, ww33, app(app(app(ty_@3, bf), bg), bh)) → new_esEs8(ww32, ww33, bf, bg, bh)
new_asAs1(True, ww32, ww33, ty_Float) → new_esEs4(ww32, ww33)
new_esEs14(ww16, ww190, eg, eh) → error([])
new_esEs3(ww250, ww260, ty_Float) → new_esEs4(ww250, ww260)
new_esEs6(:%(ww160, ww161), :%(ww1900, ww1901), fa) → new_asAs1(new_esEs0(ww160, ww1900, fa), ww161, ww1901, fa)
new_esEs0(ww160, ww1900, ty_Integer) → new_esEs1(ww160, ww1900)
new_esEs7(ww16, ww190) → error([])
new_esEs3(ww250, ww260, app(ty_Ratio, db)) → new_esEs6(ww250, ww260, db)
new_asAs1(False, ww32, ww33, bc) → False
new_esEs5(ww16, ww190, bb) → error([])
new_esEs0(ww160, ww1900, ty_Int) → new_esEs2(ww160, ww1900)
new_primEqNat0(Zero, Zero) → True
new_esEs4(ww16, ww190) → error([])
new_esEs2(Pos(Zero), Neg(Zero)) → True
new_esEs2(Neg(Zero), Pos(Zero)) → True
new_esEs3(ww250, ww260, ty_Bool) → new_esEs9(ww250, ww260)
new_asAs1(True, ww32, ww33, app(ty_Ratio, bd)) → new_esEs6(ww32, ww33, bd)
new_esEs8(ww16, ww190, ed, ee, ef) → error([])
new_esEs3(ww250, ww260, ty_@0) → new_esEs12(ww250, ww260)
new_esEs13(ww16, ww190) → error([])
new_asAs1(True, ww32, ww33, ty_Int) → new_esEs2(ww32, ww33)
new_asAs1(True, ww32, ww33, app(app(ty_@2, ca), cb)) → new_esEs11(ww32, ww33, ca, cb)
new_esEs12(ww16, ww190) → error([])
new_asAs1(True, ww32, ww33, ty_Ordering) → new_esEs13(ww32, ww33)
new_primEqNat0(Zero, Succ(ww190000)) → False
new_primEqNat0(Succ(ww16000), Zero) → False
new_asAs1(True, ww32, ww33, ty_@0) → new_esEs12(ww32, ww33)
new_asAs1(True, ww32, ww33, ty_Bool) → new_esEs9(ww32, ww33)
new_esEs2(Neg(Zero), Neg(Zero)) → True
new_esEs3(ww250, ww260, ty_Double) → new_esEs7(ww250, ww260)
new_esEs1(ww16, ww190) → error([])
new_asAs1(True, ww32, ww33, ty_Double) → new_esEs7(ww32, ww33)
new_esEs10(ww16, ww190) → error([])
new_esEs3(ww250, ww260, app(ty_Maybe, ec)) → new_esEs15(ww250, ww260, ec)
new_esEs3(ww250, ww260, app(app(ty_Either, ea), eb)) → new_esEs14(ww250, ww260, ea, eb)
new_esEs3(ww250, ww260, ty_Integer) → new_esEs1(ww250, ww260)
new_asAs1(True, ww32, ww33, app(ty_Maybe, ce)) → new_esEs15(ww32, ww33, ce)
new_esEs3(ww250, ww260, app(app(ty_@2, dg), dh)) → new_esEs11(ww250, ww260, dg, dh)
new_esEs3(ww250, ww260, ty_Int) → new_esEs2(ww250, ww260)
new_esEs2(Neg(Succ(ww1600)), Neg(Zero)) → False
new_esEs2(Neg(Zero), Neg(Succ(ww19000))) → False
new_esEs11(ww16, ww190, cf, cg) → error([])
new_esEs2(Neg(Succ(ww1600)), Neg(Succ(ww19000))) → new_primEqNat0(ww1600, ww19000)
new_primEqNat0(Succ(ww16000), Succ(ww190000)) → new_primEqNat0(ww16000, ww190000)
new_esEs3(ww250, ww260, app(app(app(ty_@3, dd), de), df)) → new_esEs8(ww250, ww260, dd, de, df)
new_esEs2(Pos(Zero), Neg(Succ(ww19000))) → False
new_esEs2(Neg(Zero), Pos(Succ(ww19000))) → False
new_esEs3(ww250, ww260, ty_Char) → new_esEs10(ww250, ww260)
new_asAs1(True, ww32, ww33, app(app(ty_Either, cc), cd)) → new_esEs14(ww32, ww33, cc, cd)
new_esEs2(Pos(Zero), Pos(Succ(ww19000))) → False
new_esEs2(Pos(Succ(ww1600)), Pos(Zero)) → False
new_asAs1(True, ww32, ww33, ty_Char) → new_esEs10(ww32, ww33)
new_esEs2(Pos(Zero), Pos(Zero)) → True
new_asAs1(True, ww32, ww33, app(ty_[], be)) → new_esEs5(ww32, ww33, be)
new_esEs3(ww250, ww260, ty_Ordering) → new_esEs13(ww250, ww260)
new_esEs9(ww16, ww190) → error([])
new_esEs2(Pos(Succ(ww1600)), Pos(Succ(ww19000))) → new_primEqNat0(ww1600, ww19000)
new_esEs2(Pos(Succ(ww1600)), Neg(ww1900)) → False
new_esEs2(Neg(Succ(ww1600)), Pos(ww1900)) → False
new_asAs1(True, ww32, ww33, ty_Integer) → new_esEs1(ww32, ww33)
new_esEs15(ww16, ww190, da) → error([])
new_esEs3(ww250, ww260, app(ty_[], dc)) → new_esEs5(ww250, ww260, dc)

The set Q consists of the following terms:

new_esEs15(x0, x1, x2)
new_primEqNat0(Succ(x0), Zero)
new_esEs0(x0, x1, ty_Integer)
new_esEs3(x0, x1, ty_Bool)
new_asAs1(True, x0, x1, app(ty_Ratio, x2))
new_esEs14(x0, x1, x2, x3)
new_esEs2(Neg(Succ(x0)), Neg(Zero))
new_esEs3(x0, x1, ty_Integer)
new_esEs3(x0, x1, ty_Float)
new_esEs9(x0, x1)
new_asAs1(True, x0, x1, ty_Double)
new_asAs1(True, x0, x1, ty_Int)
new_esEs8(x0, x1, x2, x3, x4)
new_esEs2(Neg(Zero), Neg(Succ(x0)))
new_esEs12(x0, x1)
new_esEs3(x0, x1, app(ty_[], x2))
new_esEs2(Pos(Zero), Neg(Zero))
new_esEs2(Neg(Zero), Pos(Zero))
new_esEs5(x0, x1, x2)
new_esEs4(x0, x1)
new_asAs1(True, x0, x1, ty_@0)
new_asAs1(True, x0, x1, app(ty_[], x2))
new_asAs1(True, x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs2(Neg(Zero), Pos(Succ(x0)))
new_esEs2(Pos(Zero), Neg(Succ(x0)))
new_asAs1(True, x0, x1, ty_Char)
new_asAs1(True, x0, x1, ty_Integer)
new_esEs6(:%(x0, x1), :%(x2, x3), x4)
new_asAs1(True, x0, x1, app(app(ty_@2, x2), x3))
new_esEs3(x0, x1, app(ty_Ratio, x2))
new_esEs11(x0, x1, x2, x3)
new_esEs3(x0, x1, app(app(ty_@2, x2), x3))
new_asAs1(True, x0, x1, ty_Float)
new_esEs2(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs3(x0, x1, ty_Ordering)
new_esEs0(x0, x1, ty_Int)
new_primEqNat0(Zero, Succ(x0))
new_asAs1(True, x0, x1, app(app(ty_Either, x2), x3))
new_esEs3(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_esEs2(Pos(Succ(x0)), Pos(Zero))
new_esEs2(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs2(Pos(Zero), Pos(Zero))
new_asAs1(False, x0, x1, x2)
new_asAs1(True, x0, x1, app(ty_Maybe, x2))
new_esEs7(x0, x1)
new_esEs3(x0, x1, ty_Double)
new_esEs3(x0, x1, app(ty_Maybe, x2))
new_esEs1(x0, x1)
new_asAs1(True, x0, x1, ty_Ordering)
new_esEs3(x0, x1, ty_@0)
new_esEs3(x0, x1, app(app(ty_Either, x2), x3))
new_primEqNat0(Zero, Zero)
new_esEs13(x0, x1)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs2(Neg(Zero), Neg(Zero))
new_esEs10(x0, x1)
new_esEs2(Pos(Succ(x0)), Neg(x1))
new_esEs2(Neg(Succ(x0)), Pos(x1))
new_esEs3(x0, x1, ty_Char)
new_asAs1(True, x0, x1, ty_Bool)
new_esEs2(Pos(Zero), Pos(Succ(x0)))
new_esEs3(x0, x1, ty_Int)

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:

new_asAs0(True, :(ww250, ww251), :(ww260, ww261), ba) → new_asAs0(new_esEs3(ww250, ww260, ba), ww251, ww261, ba)
The graph contains the following edges 2 > 2, 3 > 3, 4 >= 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_isPrefixOf(ww16, ww15, ww19, :(ww1810, ww1811), ba) → new_isPrefixOf(ww16, ww15, new_flip(ww19, ww1810, ba), ww1811, ba)

The TRS R consists of the following rules:

new_flip(ww15, ww16, ba) → :(ww16, ww15)

The set Q consists of the following terms:

new_flip(x0, x1, x2)

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:

new_isPrefixOf(ww16, ww15, ww19, :(ww1810, ww1811), ba) → new_isPrefixOf(ww16, ww15, new_flip(ww19, ww1810, ba), ww1811, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 > 4, 5 >= 5

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_isPrefixOf0(ww15, ww16, :(ww170, ww171), ww18, ba) → new_isPrefixOf0(new_flip(ww15, ww16, ba), ww170, ww171, ww18, ba)

The TRS R consists of the following rules:

new_flip(ww15, ww16, ba) → :(ww16, ww15)

The set Q consists of the following terms:

new_flip(x0, x1, x2)

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:

new_isPrefixOf0(ww15, ww16, :(ww170, ww171), ww18, ba) → new_isPrefixOf0(new_flip(ww15, ww16, ba), ww170, ww171, ww18, ba)
The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4, 5 >= 5