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

YES 1.31 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 :: [Float] -> [Float] -> Bool) :: [Float] -> [Float] -> 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 :: [Float] -> [Float] -> Bool) :: [Float] -> [Float] -> 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 :: [Float] -> [Float] -> 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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553[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];350 -> 553[label="",style="solid", color="burlywood", weight=9];
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351[label="primEqInt (Neg (Succ ww1500)) (Pos ww1800)",fontsize=16,color="black",shape="box"];351 -> 366[label="",style="solid", color="black", weight=3];
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555[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];352 -> 555[label="",style="solid", color="burlywood", weight=9];
555 -> 368[label="",style="solid", color="burlywood", weight=3];
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557[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];353 -> 557[label="",style="solid", color="burlywood", weight=9];
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559[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];354 -> 559[label="",style="solid", color="burlywood", weight=9];
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355 -> 297[label="",style="dashed", color="red", weight=0];
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355 -> 374[label="",style="dashed", color="magenta", weight=3];
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358 -> 357[label="",style="dashed", color="red", weight=0];
358[label="ww151 * ww1801",fontsize=16,color="magenta"];358 -> 377[label="",style="dashed", color="magenta", weight=3];
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360[label="primEqInt (Pos (Succ ww1500)) (Pos Zero)",fontsize=16,color="black",shape="box"];360 -> 380[label="",style="solid", color="black", weight=3];
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363[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];363 -> 382[label="",style="solid", color="black", weight=3];
364[label="primEqInt (Pos Zero) (Neg (Succ ww18000))",fontsize=16,color="black",shape="box"];364 -> 383[label="",style="solid", color="black", weight=3];
365[label="primEqInt (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];365 -> 384[label="",style="solid", color="black", weight=3];
366[label="False",fontsize=16,color="green",shape="box"];367[label="primEqInt (Neg (Succ ww1500)) (Neg (Succ ww18000))",fontsize=16,color="black",shape="box"];367 -> 385[label="",style="solid", color="black", weight=3];
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370[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];370 -> 388[label="",style="solid", color="black", weight=3];
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372[label="primEqInt (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];372 -> 390[label="",style="solid", color="black", weight=3];
373[label="ww241\n  Eq a",fontsize=16,color="green",shape="box"];374[label="ww251\n  Eq a",fontsize=16,color="green",shape="box"];375[label="ww240 == ww250\n  Eq a",fontsize=16,color="blue",shape="box"];562[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 562[label="",style="solid", color="blue", weight=9];
562 -> 391[label="",style="solid", color="blue", weight=3];
563[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 563[label="",style="solid", color="blue", weight=9];
563 -> 392[label="",style="solid", color="blue", weight=3];
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564 -> 393[label="",style="solid", color="blue", weight=3];
565[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 565[label="",style="solid", color="blue", weight=9];
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567[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 567[label="",style="solid", color="blue", weight=9];
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568[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 568[label="",style="solid", color="blue", weight=9];
568 -> 397[label="",style="solid", color="blue", weight=3];
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571[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 571[label="",style="solid", color="blue", weight=9];
571 -> 400[label="",style="solid", color="blue", weight=3];
572[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 572[label="",style="solid", color="blue", weight=9];
572 -> 401[label="",style="solid", color="blue", weight=3];
573[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 573[label="",style="solid", color="blue", weight=9];
573 -> 402[label="",style="solid", color="blue", weight=3];
574[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 574[label="",style="solid", color="blue", weight=9];
574 -> 403[label="",style="solid", color="blue", weight=3];
575[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];375 -> 575[label="",style="solid", color="blue", weight=9];
575 -> 404[label="",style="solid", color="blue", weight=3];
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576 -> 405[label="",style="solid", color="burlywood", weight=3];
577[label="ww150/Neg ww1500",fontsize=10,color="white",style="solid",shape="box"];376 -> 577[label="",style="solid", color="burlywood", weight=9];
577 -> 406[label="",style="solid", color="burlywood", weight=3];
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578 -> 407[label="",style="solid", color="burlywood", weight=3];
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392 -> 302[label="",style="dashed", color="red", weight=0];
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393 -> 303[label="",style="dashed", color="red", weight=0];
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395 -> 305[label="",style="dashed", color="red", weight=0];
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398 -> 426[label="",style="dashed", color="magenta", weight=3];
399 -> 309[label="",style="dashed", color="red", weight=0];
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399 -> 428[label="",style="dashed", color="magenta", weight=3];
400 -> 310[label="",style="dashed", color="red", weight=0];
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402 -> 312[label="",style="dashed", color="red", weight=0];
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403 -> 313[label="",style="dashed", color="red", weight=0];
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600 -> 444[label="",style="solid", color="burlywood", weight=3];
408[label="primEqNat Zero ww18000",fontsize=16,color="burlywood",shape="box"];601[label="ww18000/Succ ww180000",fontsize=10,color="white",style="solid",shape="box"];408 -> 601[label="",style="solid", color="burlywood", weight=9];
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602 -> 446[label="",style="solid", color="burlywood", weight=3];
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459[label="ww15000",fontsize=16,color="green",shape="box"];460[label="ww180000",fontsize=16,color="green",shape="box"];461[label="primMulNat (Succ ww15000) ww18000",fontsize=16,color="burlywood",shape="box"];609[label="ww18000/Succ ww180000",fontsize=10,color="white",style="solid",shape="box"];461 -> 609[label="",style="solid", color="burlywood", weight=9];
609 -> 467[label="",style="solid", color="burlywood", weight=3];
610[label="ww18000/Zero",fontsize=10,color="white",style="solid",shape="box"];461 -> 610[label="",style="solid", color="burlywood", weight=9];
610 -> 468[label="",style="solid", color="burlywood", weight=3];
462[label="primMulNat Zero ww18000",fontsize=16,color="burlywood",shape="box"];611[label="ww18000/Succ ww180000",fontsize=10,color="white",style="solid",shape="box"];462 -> 611[label="",style="solid", color="burlywood", weight=9];
611 -> 469[label="",style="solid", color="burlywood", weight=3];
612[label="ww18000/Zero",fontsize=10,color="white",style="solid",shape="box"];462 -> 612[label="",style="solid", color="burlywood", weight=9];
612 -> 470[label="",style="solid", color="burlywood", weight=3];
463[label="ww18000",fontsize=16,color="green",shape="box"];464[label="ww1500",fontsize=16,color="green",shape="box"];465[label="ww1500",fontsize=16,color="green",shape="box"];466[label="ww18000",fontsize=16,color="green",shape="box"];467[label="primMulNat (Succ ww15000) (Succ ww180000)",fontsize=16,color="black",shape="box"];467 -> 471[label="",style="solid", color="black", weight=3];
468[label="primMulNat (Succ ww15000) Zero",fontsize=16,color="black",shape="box"];468 -> 472[label="",style="solid", color="black", weight=3];
469[label="primMulNat Zero (Succ ww180000)",fontsize=16,color="black",shape="box"];469 -> 473[label="",style="solid", color="black", weight=3];
470[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];470 -> 474[label="",style="solid", color="black", weight=3];
471 -> 475[label="",style="dashed", color="red", weight=0];
471[label="primPlusNat (primMulNat ww15000 (Succ ww180000)) (Succ ww180000)",fontsize=16,color="magenta"];471 -> 476[label="",style="dashed", color="magenta", weight=3];
472[label="Zero",fontsize=16,color="green",shape="box"];473[label="Zero",fontsize=16,color="green",shape="box"];474[label="Zero",fontsize=16,color="green",shape="box"];476 -> 455[label="",style="dashed", color="red", weight=0];
476[label="primMulNat ww15000 (Succ ww180000)",fontsize=16,color="magenta"];476 -> 477[label="",style="dashed", color="magenta", weight=3];
476 -> 478[label="",style="dashed", color="magenta", weight=3];
475[label="primPlusNat ww26 (Succ ww180000)",fontsize=16,color="burlywood",shape="triangle"];615[label="ww26/Succ ww260",fontsize=10,color="white",style="solid",shape="box"];475 -> 615[label="",style="solid", color="burlywood", weight=9];
615 -> 479[label="",style="solid", color="burlywood", weight=3];
616[label="ww26/Zero",fontsize=10,color="white",style="solid",shape="box"];475 -> 616[label="",style="solid", color="burlywood", weight=9];
616 -> 480[label="",style="solid", color="burlywood", weight=3];
477[label="ww15000",fontsize=16,color="green",shape="box"];478[label="Succ ww180000",fontsize=16,color="green",shape="box"];479[label="primPlusNat (Succ ww260) (Succ ww180000)",fontsize=16,color="black",shape="box"];479 -> 481[label="",style="solid", color="black", weight=3];
480[label="primPlusNat Zero (Succ ww180000)",fontsize=16,color="black",shape="box"];480 -> 482[label="",style="solid", color="black", weight=3];
481[label="Succ (Succ (primPlusNat ww260 ww180000))",fontsize=16,color="green",shape="box"];481 -> 483[label="",style="dashed", color="green", weight=3];
482[label="Succ ww180000",fontsize=16,color="green",shape="box"];483[label="primPlusNat ww260 ww180000",fontsize=16,color="burlywood",shape="triangle"];617[label="ww260/Succ ww2600",fontsize=10,color="white",style="solid",shape="box"];483 -> 617[label="",style="solid", color="burlywood", weight=9];
617 -> 484[label="",style="solid", color="burlywood", weight=3];
618[label="ww260/Zero",fontsize=10,color="white",style="solid",shape="box"];483 -> 618[label="",style="solid", color="burlywood", weight=9];
618 -> 485[label="",style="solid", color="burlywood", weight=3];
484[label="primPlusNat (Succ ww2600) ww180000",fontsize=16,color="burlywood",shape="box"];619[label="ww180000/Succ ww1800000",fontsize=10,color="white",style="solid",shape="box"];484 -> 619[label="",style="solid", color="burlywood", weight=9];
619 -> 486[label="",style="solid", color="burlywood", weight=3];
620[label="ww180000/Zero",fontsize=10,color="white",style="solid",shape="box"];484 -> 620[label="",style="solid", color="burlywood", weight=9];
620 -> 487[label="",style="solid", color="burlywood", weight=3];
485[label="primPlusNat Zero ww180000",fontsize=16,color="burlywood",shape="box"];621[label="ww180000/Succ ww1800000",fontsize=10,color="white",style="solid",shape="box"];485 -> 621[label="",style="solid", color="burlywood", weight=9];
621 -> 488[label="",style="solid", color="burlywood", weight=3];
622[label="ww180000/Zero",fontsize=10,color="white",style="solid",shape="box"];485 -> 622[label="",style="solid", color="burlywood", weight=9];
622 -> 489[label="",style="solid", color="burlywood", weight=3];
486[label="primPlusNat (Succ ww2600) (Succ ww1800000)",fontsize=16,color="black",shape="box"];486 -> 490[label="",style="solid", color="black", weight=3];
487[label="primPlusNat (Succ ww2600) Zero",fontsize=16,color="black",shape="box"];487 -> 491[label="",style="solid", color="black", weight=3];
488[label="primPlusNat Zero (Succ ww1800000)",fontsize=16,color="black",shape="box"];488 -> 492[label="",style="solid", color="black", weight=3];
489[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];489 -> 493[label="",style="solid", color="black", weight=3];
490[label="Succ (Succ (primPlusNat ww2600 ww1800000))",fontsize=16,color="green",shape="box"];490 -> 494[label="",style="dashed", color="green", weight=3];
491[label="Succ ww2600",fontsize=16,color="green",shape="box"];492[label="Succ ww1800000",fontsize=16,color="green",shape="box"];493[label="Zero",fontsize=16,color="green",shape="box"];494 -> 483[label="",style="dashed", color="red", weight=0];
494[label="primPlusNat ww2600 ww1800000",fontsize=16,color="magenta"];494 -> 495[label="",style="dashed", color="magenta", weight=3];
494 -> 496[label="",style="dashed", color="magenta", weight=3];
495[label="ww1800000",fontsize=16,color="green",shape="box"];496[label="ww2600",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_primPlusNat(Succ(ww2600), Succ(ww1800000)) → new_primPlusNat(ww2600, ww1800000)

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_primPlusNat(Succ(ww2600), Succ(ww1800000)) → new_primPlusNat(ww2600, ww1800000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_primMulNat(Succ(ww15000), Succ(ww180000)) → new_primMulNat(ww15000, Succ(ww180000))

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

new_primMulNat(Succ(ww15000), Succ(ww180000)) → new_primMulNat(ww15000, Succ(ww180000))
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

new_primEqNat(Succ(ww15000), Succ(ww180000)) → new_primEqNat(ww15000, ww180000)

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

new_primEqNat(Succ(ww15000), Succ(ww180000)) → new_primEqNat(ww15000, ww180000)
The graph contains the following edges 1 > 1, 2 > 2

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

new_asAs(True, :(ww240, ww241), :(ww250, ww251), ba) → new_asAs(new_esEs(ww240, ww250, ba), ww241, ww251, ba)

The TRS R consists of the following rules:

new_sr(Neg(ww1500), Neg(ww18000)) → Pos(new_primMulNat0(ww1500, ww18000))
new_esEs(ww240, ww250, app(ty_Ratio, bh)) → new_esEs10(ww240, ww250, bh)
new_sr(Pos(ww1500), Pos(ww18000)) → Pos(new_primMulNat0(ww1500, ww18000))
new_primEqNat0(Zero, Zero) → True
new_primPlusNat1(Succ(ww2600), Succ(ww1800000)) → Succ(Succ(new_primPlusNat1(ww2600, ww1800000)))
new_esEs4(ww15, ww180) → error([])
new_primMulNat0(Zero, Succ(ww180000)) → Zero
new_primMulNat0(Succ(ww15000), Zero) → Zero
new_esEs(ww240, ww250, app(ty_[], bg)) → new_esEs9(ww240, ww250, bg)
new_esEs0(Pos(Zero), Neg(Zero)) → True
new_esEs0(Neg(Zero), Pos(Zero)) → True
new_esEs7(Float(ww150, ww151), Float(ww1800, ww1801)) → new_esEs0(new_sr(ww150, ww1800), new_sr(ww151, ww1801))
new_esEs3(ww15, ww180, ce, cf, cg) → error([])
new_primPlusNat1(Zero, Zero) → Zero
new_primEqNat0(Zero, Succ(ww180000)) → False
new_primEqNat0(Succ(ww15000), Zero) → False
new_esEs6(ww15, ww180, de, df) → error([])
new_primMulNat0(Succ(ww15000), Succ(ww180000)) → new_primPlusNat0(new_primMulNat0(ww15000, Succ(ww180000)), ww180000)
new_esEs1(ww15, ww180) → error([])
new_esEs9(ww15, ww180, db) → error([])
new_esEs0(Pos(Succ(ww1500)), Neg(ww1800)) → False
new_esEs0(Neg(Succ(ww1500)), Pos(ww1800)) → False
new_esEs11(ww15, ww180) → error([])
new_esEs0(Neg(Zero), Neg(Zero)) → True
new_esEs(ww240, ww250, app(app(ty_Either, cb), cc)) → new_esEs13(ww240, ww250, cb, cc)
new_esEs0(Pos(Zero), Neg(Succ(ww18000))) → False
new_esEs0(Neg(Zero), Pos(Succ(ww18000))) → False
new_esEs10(ww15, ww180, da) → error([])
new_esEs(ww240, ww250, app(app(app(ty_@3, bb), bc), bd)) → new_esEs3(ww240, ww250, bb, bc, bd)
new_primMulNat0(Zero, Zero) → Zero
new_esEs0(Pos(Zero), Pos(Succ(ww18000))) → False
new_esEs0(Pos(Succ(ww1500)), Pos(Zero)) → False
new_esEs(ww240, ww250, app(ty_Maybe, ca)) → new_esEs12(ww240, ww250, ca)
new_esEs5(ww15, ww180) → error([])
new_primPlusNat0(Succ(ww260), ww180000) → Succ(Succ(new_primPlusNat1(ww260, ww180000)))
new_esEs0(Neg(Succ(ww1500)), Neg(Zero)) → False
new_esEs0(Neg(Zero), Neg(Succ(ww18000))) → False
new_esEs0(Neg(Succ(ww1500)), Neg(Succ(ww18000))) → new_primEqNat0(ww1500, ww18000)
new_esEs2(ww15, ww180) → error([])
new_primEqNat0(Succ(ww15000), Succ(ww180000)) → new_primEqNat0(ww15000, ww180000)
new_primPlusNat0(Zero, ww180000) → Succ(ww180000)
new_esEs(ww240, ww250, ty_Int) → new_esEs0(ww240, ww250)
new_esEs(ww240, ww250, ty_Bool) → new_esEs5(ww240, ww250)
new_esEs(ww240, ww250, ty_Float) → new_esEs7(ww240, ww250)
new_esEs8(ww15, ww180) → error([])
new_esEs(ww240, ww250, ty_@0) → new_esEs4(ww240, ww250)
new_esEs0(Pos(Zero), Pos(Zero)) → True
new_esEs0(Pos(Succ(ww1500)), Pos(Succ(ww18000))) → new_primEqNat0(ww1500, ww18000)
new_esEs(ww240, ww250, ty_Ordering) → new_esEs8(ww240, ww250)
new_primPlusNat1(Succ(ww2600), Zero) → Succ(ww2600)
new_primPlusNat1(Zero, Succ(ww1800000)) → Succ(ww1800000)
new_esEs13(ww15, ww180, dc, dd) → error([])
new_sr(Neg(ww1500), Pos(ww18000)) → Neg(new_primMulNat0(ww1500, ww18000))
new_sr(Pos(ww1500), Neg(ww18000)) → Neg(new_primMulNat0(ww1500, ww18000))
new_esEs(ww240, ww250, ty_Double) → new_esEs1(ww240, ww250)
new_esEs12(ww15, ww180, cd) → error([])
new_esEs(ww240, ww250, ty_Integer) → new_esEs2(ww240, ww250)
new_esEs(ww240, ww250, ty_Char) → new_esEs11(ww240, ww250)
new_esEs(ww240, ww250, app(app(ty_@2, be), bf)) → new_esEs6(ww240, ww250, be, bf)

The set Q consists of the following terms:

new_esEs11(x0, x1)
new_primPlusNat0(Succ(x0), x1)
new_esEs(x0, x1, app(ty_[], x2))
new_esEs(x0, x1, app(app(ty_Either, x2), x3))
new_esEs4(x0, x1)
new_primMulNat0(Zero, Zero)
new_primPlusNat1(Succ(x0), Zero)
new_esEs5(x0, x1)
new_esEs13(x0, x1, x2, x3)
new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
new_primPlusNat1(Zero, Succ(x0))
new_esEs2(x0, x1)
new_esEs(x0, x1, ty_Bool)
new_esEs(x0, x1, ty_Ordering)
new_esEs0(Neg(Zero), Pos(Succ(x0)))
new_esEs0(Pos(Zero), Neg(Succ(x0)))
new_esEs0(Neg(Succ(x0)), Neg(Succ(x1)))
new_esEs(x0, x1, ty_Double)
new_esEs7(Float(x0, x1), Float(x2, x3))
new_primPlusNat1(Zero, Zero)
new_esEs(x0, x1, ty_Int)
new_sr(Pos(x0), Pos(x1))
new_esEs(x0, x1, app(ty_Maybe, x2))
new_esEs0(Neg(Zero), Pos(Zero))
new_esEs0(Pos(Zero), Neg(Zero))
new_esEs0(Pos(Succ(x0)), Pos(Succ(x1)))
new_esEs0(Pos(Succ(x0)), Pos(Zero))
new_esEs(x0, x1, app(app(ty_@2, x2), x3))
new_esEs8(x0, x1)
new_esEs10(x0, x1, x2)
new_esEs0(Pos(Succ(x0)), Neg(x1))
new_esEs0(Neg(Succ(x0)), Pos(x1))
new_primMulNat0(Succ(x0), Zero)
new_esEs6(x0, x1, x2, x3)
new_primPlusNat0(Zero, x0)
new_esEs0(Neg(Zero), Neg(Zero))
new_esEs(x0, x1, app(ty_Ratio, x2))
new_esEs(x0, x1, ty_Float)
new_esEs(x0, x1, ty_Integer)
new_esEs1(x0, x1)
new_primMulNat0(Zero, Succ(x0))
new_esEs0(Pos(Zero), Pos(Zero))
new_primPlusNat1(Succ(x0), Succ(x1))
new_primMulNat0(Succ(x0), Succ(x1))
new_primEqNat0(Zero, Succ(x0))
new_esEs9(x0, x1, x2)
new_primEqNat0(Zero, Zero)
new_esEs(x0, x1, ty_Char)
new_esEs3(x0, x1, x2, x3, x4)
new_esEs0(Neg(Succ(x0)), Neg(Zero))
new_primEqNat0(Succ(x0), Zero)
new_primEqNat0(Succ(x0), Succ(x1))
new_esEs0(Neg(Zero), Neg(Succ(x0)))
new_sr(Neg(x0), Pos(x1))
new_sr(Pos(x0), Neg(x1))
new_esEs(x0, x1, ty_@0)
new_sr(Neg(x0), Neg(x1))
new_esEs0(Pos(Zero), Pos(Succ(x0)))
new_esEs12(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_asAs(True, :(ww240, ww241), :(ww250, ww251), ba) → new_asAs(new_esEs(ww240, ww250, ba), ww241, ww251, ba)
The graph contains the following edges 2 > 2, 3 > 3, 4 >= 4

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPSizeChangeProof

              ↳ QDP

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

new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) → new_isPrefixOf(ww15, ww14, new_flip(ww18, ww1710, ba), ww1711, ba)

The TRS R consists of the following rules:

new_flip(ww14, ww15, ba) → :(ww15, ww14)

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(ww15, ww14, ww18, :(ww1710, ww1711), ba) → new_isPrefixOf(ww15, ww14, new_flip(ww18, ww1710, ba), ww1711, 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

              ↳ QDP

                ↳ QDPSizeChangeProof

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

new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) → new_isPrefixOf0(new_flip(ww14, ww15, ba), ww160, ww161, ww17, ba)

The TRS R consists of the following rules:

new_flip(ww14, ww15, ba) → :(ww15, ww14)

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(ww14, ww15, :(ww160, ww161), ww17, ba) → new_isPrefixOf0(new_flip(ww14, ww15, ba), ww160, ww161, ww17, ba)
The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4, 5 >= 5