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

YES 1.235 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 :: [Int] -> [Int] -> Bool) :: [Int] -> [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 :: [Int] -> [Int] -> Bool) :: [Int] -> [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 :: [Int] -> [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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485[label="False",fontsize=16,color="green",shape="box"];486[label="False",fontsize=16,color="green",shape="box"];487[label="True",fontsize=16,color="green",shape="box"];488[label="ww15000",fontsize=16,color="green",shape="box"];489[label="ww180000",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ AND

              ↳ 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)

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(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

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

The set Q consists of the following terms:

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

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

                ↳ 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

                ↳ 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