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

YES 1.114 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

((genericTake :: Int -> [a] -> [a]) :: Int -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

genericTake :: Integral a => a -> [b] -> [b]

genericTake

0 _

[]

genericTake

_ []

[]

genericTake

n (x : xs)

n > 0

x : genericTake (n - 1) xs

genericTake

_ _

error []

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

((genericTake :: Int -> [a] -> [a]) :: Int -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

genericTake :: Integral a => a -> [b] -> [b]

genericTake

0 vw

[]

genericTake

vx []

[]

genericTake

n (x : xs)

n > 0

x : genericTake (n - 1) xs

genericTake

vy vz

error []

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

genericTake 0 vw = []

genericTake vx [] = []

genericTake n (x : xs)

| n > 0

= x : genericTake (n - 1) xs

genericTake vy vz = error []

is transformed to

genericTake zu vw = genericTake5 zu vw

genericTake vx [] = genericTake3 vx []

genericTake n (x : xs) = genericTake2 n (x : xs)

genericTake vy vz = genericTake0 vy vz

genericTake0 vy vz = error []

genericTake1 n x xs True = x : genericTake (n - 1) xs

genericTake1 n x xs False = genericTake0 n (x : xs)

genericTake2 n (x : xs) = genericTake1 n x xs (n > 0)

genericTake2 yv yw = genericTake0 yv yw

genericTake3 vx [] = []

genericTake3 yy yz = genericTake2 yy yz

genericTake4 True zu vw = []

genericTake4 zv zw zx = genericTake3 zw zx

genericTake5 zu vw = genericTake4 (zu == 0) zu vw

genericTake5 zy zz = genericTake3 zy zz

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule List

((genericTake :: Int -> [a] -> [a]) :: Int -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

genericTake :: Integral b => b -> [a] -> [a]

genericTake	zu vw	=	genericTake5 zu vw
genericTake	vx []	=	genericTake3 vx []
genericTake	n (x : xs)	=	genericTake2 n (x : xs)
genericTake	vy vz	=	genericTake0 vy vz

genericTake0

vy vz

error []

genericTake1	n x xs True	=	x : genericTake (n - 1) xs
genericTake1	n x xs False	=	genericTake0 n (x : xs)

genericTake2	n (x : xs)	=	genericTake1 n x xs (n > 0)
genericTake2	yv yw	=	genericTake0 yv yw

genericTake3	vx []	=	[]
genericTake3	yy yz	=	genericTake2 yy yz

genericTake4	True zu vw	=	[]
genericTake4	zv zw zx	=	genericTake3 zw zx

genericTake5	zu vw	=	genericTake4 (zu == 0) zu vw
genericTake5	zy zz	=	genericTake3 zy zz

module Maybe where

import qualified List
import qualified Prelude

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule List

(genericTake :: Int -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

genericTake :: Integral a => a -> [b] -> [b]

genericTake	zu vw	=	genericTake5 zu vw
genericTake	vx []	=	genericTake3 vx []
genericTake	n (x : xs)	=	genericTake2 n (x : xs)
genericTake	vy vz	=	genericTake0 vy vz

genericTake0

vy vz

error []

genericTake1	n x xs True	=	x : genericTake (n - fromInt (Pos (Succ Zero))) xs
genericTake1	n x xs False	=	genericTake0 n (x : xs)

genericTake2	n (x : xs)	=	genericTake1 n x xs (n > fromInt (Pos Zero))
genericTake2	yv yw	=	genericTake0 yv yw

genericTake3	vx []	=	[]
genericTake3	yy yz	=	genericTake2 yy yz

genericTake4	True zu vw	=	[]
genericTake4	zv zw zx	=	genericTake3 zw zx

genericTake5	zu vw	=	genericTake4 (zu == fromInt (Pos Zero)) zu vw
genericTake5	zy zz	=	genericTake3 zy zz

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="genericTake",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="genericTake vuu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="genericTake vuu3 vuu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="genericTake5 vuu3 vuu4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="genericTake4 (vuu3 == fromInt (Pos Zero)) vuu3 vuu4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="genericTake4 (primEqInt vuu3 (fromInt (Pos Zero))) vuu3 vuu4",fontsize=16,color="burlywood",shape="box"];62[label="vuu3/Pos vuu30",fontsize=10,color="white",style="solid",shape="box"];7 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 8[label="",style="solid", color="burlywood", weight=3];
63[label="vuu3/Neg vuu30",fontsize=10,color="white",style="solid",shape="box"];7 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="genericTake4 (primEqInt (Pos vuu30) (fromInt (Pos Zero))) (Pos vuu30) vuu4",fontsize=16,color="burlywood",shape="box"];64[label="vuu30/Succ vuu300",fontsize=10,color="white",style="solid",shape="box"];8 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 10[label="",style="solid", color="burlywood", weight=3];
65[label="vuu30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="genericTake4 (primEqInt (Neg vuu30) (fromInt (Pos Zero))) (Neg vuu30) vuu4",fontsize=16,color="burlywood",shape="box"];66[label="vuu30/Succ vuu300",fontsize=10,color="white",style="solid",shape="box"];9 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 12[label="",style="solid", color="burlywood", weight=3];
67[label="vuu30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="genericTake4 (primEqInt (Pos (Succ vuu300)) (fromInt (Pos Zero))) (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11[label="genericTake4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
12[label="genericTake4 (primEqInt (Neg (Succ vuu300)) (fromInt (Pos Zero))) (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
13[label="genericTake4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
14[label="genericTake4 (primEqInt (Pos (Succ vuu300)) (Pos Zero)) (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="genericTake4 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="genericTake4 (primEqInt (Neg (Succ vuu300)) (Pos Zero)) (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="genericTake4 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="genericTake4 False (Pos (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="genericTake4 True (Pos Zero) vuu4",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="genericTake4 False (Neg (Succ vuu300)) vuu4",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="genericTake4 True (Neg Zero) vuu4",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="genericTake3 (Pos (Succ vuu300)) vuu4",fontsize=16,color="burlywood",shape="box"];68[label="vuu4/vuu40 : vuu41",fontsize=10,color="white",style="solid",shape="box"];22 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 26[label="",style="solid", color="burlywood", weight=3];
69[label="vuu4/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 27[label="",style="solid", color="burlywood", weight=3];
23[label="[]",fontsize=16,color="green",shape="box"];24[label="genericTake3 (Neg (Succ vuu300)) vuu4",fontsize=16,color="burlywood",shape="box"];70[label="vuu4/vuu40 : vuu41",fontsize=10,color="white",style="solid",shape="box"];24 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 28[label="",style="solid", color="burlywood", weight=3];
71[label="vuu4/[]",fontsize=10,color="white",style="solid",shape="box"];24 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 29[label="",style="solid", color="burlywood", weight=3];
25[label="[]",fontsize=16,color="green",shape="box"];26[label="genericTake3 (Pos (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="genericTake3 (Pos (Succ vuu300)) []",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="genericTake3 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="genericTake3 (Neg (Succ vuu300)) []",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="genericTake2 (Pos (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="[]",fontsize=16,color="green",shape="box"];32[label="genericTake2 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
33[label="[]",fontsize=16,color="green",shape="box"];34[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (Pos (Succ vuu300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (Neg (Succ vuu300) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (compare (Pos (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (compare (Neg (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpInt (Pos (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (primCmpInt (Neg (Succ vuu300)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpInt (Pos (Succ vuu300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
41[label="genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (primCmpInt (Neg (Succ vuu300)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (primCmpNat (Succ vuu300) Zero == GT)",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 (LT == GT)",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 (GT == GT)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="genericTake1 (Neg (Succ vuu300)) vuu40 vuu41 False",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="genericTake1 (Pos (Succ vuu300)) vuu40 vuu41 True",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="genericTake0 (Neg (Succ vuu300)) (vuu40 : vuu41)",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="vuu40 : genericTake (Pos (Succ vuu300) - fromInt (Pos (Succ Zero))) vuu41",fontsize=16,color="green",shape="box"];48 -> 50[label="",style="dashed", color="green", weight=3];
49[label="error []",fontsize=16,color="black",shape="box"];49 -> 51[label="",style="solid", color="black", weight=3];
50 -> 4[label="",style="dashed", color="red", weight=0];
50[label="genericTake (Pos (Succ vuu300) - fromInt (Pos (Succ Zero))) vuu41",fontsize=16,color="magenta"];50 -> 52[label="",style="dashed", color="magenta", weight=3];
50 -> 53[label="",style="dashed", color="magenta", weight=3];
51[label="error []",fontsize=16,color="red",shape="box"];52[label="vuu41",fontsize=16,color="green",shape="box"];53[label="Pos (Succ vuu300) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];53 -> 54[label="",style="solid", color="black", weight=3];
54[label="primMinusInt (Pos (Succ vuu300)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];54 -> 55[label="",style="solid", color="black", weight=3];
55[label="primMinusInt (Pos (Succ vuu300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];55 -> 56[label="",style="solid", color="black", weight=3];
56[label="primMinusNat (Succ vuu300) (Succ Zero)",fontsize=16,color="black",shape="box"];56 -> 57[label="",style="solid", color="black", weight=3];
57[label="primMinusNat vuu300 Zero",fontsize=16,color="burlywood",shape="box"];73[label="vuu300/Succ vuu3000",fontsize=10,color="white",style="solid",shape="box"];57 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 58[label="",style="solid", color="burlywood", weight=3];
74[label="vuu300/Zero",fontsize=10,color="white",style="solid",shape="box"];57 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 59[label="",style="solid", color="burlywood", weight=3];
58[label="primMinusNat (Succ vuu3000) Zero",fontsize=16,color="black",shape="box"];58 -> 60[label="",style="solid", color="black", weight=3];
59[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
60[label="Pos (Succ vuu3000)",fontsize=16,color="green",shape="box"];61[label="Pos Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_genericTake(Pos(Succ(vuu300)), :(vuu40, vuu41), ba) → new_genericTake(new_primMinusNat(vuu300), vuu41, ba)

The TRS R consists of the following rules:

new_primMinusNat(Succ(vuu3000)) → Pos(Succ(vuu3000))
new_primMinusNat(Zero) → Pos(Zero)

The set Q consists of the following terms:

new_primMinusNat(Succ(x0))
new_primMinusNat(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:

new_genericTake(Pos(Succ(vuu300)), :(vuu40, vuu41), ba) → new_genericTake(new_primMinusNat(vuu300), vuu41, ba)
The graph contains the following edges 2 > 2, 3 >= 3

genericTake	zu vw	= genericTake5 zu vw
genericTake	vx []	= genericTake3 vx []
genericTake	n (x : xs)	= genericTake2 n (x : xs)
genericTake	vy vz	= genericTake0 vy vz

genericTake1	n x xs True	= x : genericTake (n - 1) xs
genericTake1	n x xs False	= genericTake0 n (x : xs)

genericTake2	n (x : xs)	= genericTake1 n x xs (n > 0)
genericTake2	yv yw	= genericTake0 yv yw

genericTake4	True zu vw	= []
genericTake4	zv zw zx	= genericTake3 zw zx

genericTake5	zu vw	= genericTake4 (zu == 0) zu vw
genericTake5	zy zz	= genericTake3 zy zz