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

YES 1.113 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

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

module List where

import qualified Maybe
import qualified Prelude

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

genericIndex

(x : _) 0

genericIndex

(_ : xs) n

n > 0

genericIndex xs (n - 1)

otherwise

error []

genericIndex

_ _

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

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

module List where

import qualified Maybe
import qualified Prelude

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

genericIndex

(x : vw) 0

genericIndex

(vx : xs) n

n > 0

genericIndex xs (n - 1)

otherwise

error []

genericIndex

vy vz

error []

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

genericIndex (x : vw) 0 = x

genericIndex (vx : xs) n

| n > 0

= genericIndex xs (n - 1)

| otherwise

= error []

genericIndex vy vz = error []

is transformed to

genericIndex (x : vw) yy = genericIndex5 (x : vw) yy

genericIndex (vx : xs) n = genericIndex3 (vx : xs) n

genericIndex vy vz = genericIndex0 vy vz

genericIndex0 vy vz = error []

genericIndex2 vx xs n True = genericIndex xs (n - 1)

genericIndex2 vx xs n False = genericIndex1 vx xs n otherwise

genericIndex1 vx xs n True = error []

genericIndex1 vx xs n False = genericIndex0 (vx : xs) n

genericIndex3 (vx : xs) n = genericIndex2 vx xs n (n > 0)

genericIndex3 yv yw = genericIndex0 yv yw

genericIndex4 True (x : vw) yy = x

genericIndex4 yz zu zv = genericIndex3 zu zv

genericIndex5 (x : vw) yy = genericIndex4 (yy == 0) (x : vw) yy

genericIndex5 zw zx = genericIndex3 zw zx

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

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

module List where

import qualified Maybe
import qualified Prelude

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

genericIndex	(x : vw) yy	=	genericIndex5 (x : vw) yy
genericIndex	(vx : xs) n	=	genericIndex3 (vx : xs) n
genericIndex	vy vz	=	genericIndex0 vy vz

genericIndex0

vy vz

error []

genericIndex1	vx xs n True	=	error []
genericIndex1	vx xs n False	=	genericIndex0 (vx : xs) n

genericIndex2	vx xs n True	=	genericIndex xs (n - 1)
genericIndex2	vx xs n False	=	genericIndex1 vx xs n otherwise

genericIndex3	(vx : xs) n	=	genericIndex2 vx xs n (n > 0)
genericIndex3	yv yw	=	genericIndex0 yv yw

genericIndex4	True (x : vw) yy	=	x
genericIndex4	yz zu zv	=	genericIndex3 zu zv

genericIndex5	(x : vw) yy	=	genericIndex4 (yy == 0) (x : vw) yy
genericIndex5	zw zx	=	genericIndex3 zw zx

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

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

module List where

import qualified Maybe
import qualified Prelude

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

genericIndex	(x : vw) yy	=	genericIndex5 (x : vw) yy
genericIndex	(vx : xs) n	=	genericIndex3 (vx : xs) n
genericIndex	vy vz	=	genericIndex0 vy vz

genericIndex0

vy vz

error []

genericIndex1	vx xs n True	=	error []
genericIndex1	vx xs n False	=	genericIndex0 (vx : xs) n

genericIndex2	vx xs n True	=	genericIndex xs (n - fromInt (Pos (Succ Zero)))
genericIndex2	vx xs n False	=	genericIndex1 vx xs n otherwise

genericIndex3	(vx : xs) n	=	genericIndex2 vx xs n (n > fromInt (Pos Zero))
genericIndex3	yv yw	=	genericIndex0 yv yw

genericIndex4	True (x : vw) yy	=	x
genericIndex4	yz zu zv	=	genericIndex3 zu zv

genericIndex5	(x : vw) yy	=	genericIndex4 (yy == fromInt (Pos Zero)) (x : vw) yy
genericIndex5	zw zx	=	genericIndex3 zw zx

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="genericIndex",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="genericIndex zy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="genericIndex zy3 zy4",fontsize=16,color="burlywood",shape="triangle"];58[label="zy3/zy30 : zy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 5[label="",style="solid", color="burlywood", weight=3];
59[label="zy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="genericIndex (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="genericIndex [] zy4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="genericIndex5 (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="genericIndex0 [] zy4",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="genericIndex4 (zy4 == fromInt (Pos Zero)) (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="error []",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="genericIndex4 (primEqInt zy4 (fromInt (Pos Zero))) (zy30 : zy31) zy4",fontsize=16,color="burlywood",shape="box"];60[label="zy4/Pos zy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 13[label="",style="solid", color="burlywood", weight=3];
61[label="zy4/Neg zy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="error []",fontsize=16,color="red",shape="box"];13[label="genericIndex4 (primEqInt (Pos zy40) (fromInt (Pos Zero))) (zy30 : zy31) (Pos zy40)",fontsize=16,color="burlywood",shape="box"];62[label="zy40/Succ zy400",fontsize=10,color="white",style="solid",shape="box"];13 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 15[label="",style="solid", color="burlywood", weight=3];
63[label="zy40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="genericIndex4 (primEqInt (Neg zy40) (fromInt (Pos Zero))) (zy30 : zy31) (Neg zy40)",fontsize=16,color="burlywood",shape="box"];64[label="zy40/Succ zy400",fontsize=10,color="white",style="solid",shape="box"];14 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 17[label="",style="solid", color="burlywood", weight=3];
65[label="zy40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="genericIndex4 (primEqInt (Pos (Succ zy400)) (fromInt (Pos Zero))) (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="genericIndex4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="genericIndex4 (primEqInt (Neg (Succ zy400)) (fromInt (Pos Zero))) (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="genericIndex4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="genericIndex4 (primEqInt (Pos (Succ zy400)) (Pos Zero)) (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="genericIndex4 (primEqInt (Pos Zero) (Pos Zero)) (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="genericIndex4 (primEqInt (Neg (Succ zy400)) (Pos Zero)) (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="genericIndex4 (primEqInt (Neg Zero) (Pos Zero)) (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="genericIndex4 False (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="genericIndex4 True (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="genericIndex4 False (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="genericIndex4 True (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="genericIndex3 (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="zy30",fontsize=16,color="green",shape="box"];29[label="genericIndex3 (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="zy30",fontsize=16,color="green",shape="box"];31[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) (Pos (Succ zy400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="genericIndex2 zy30 zy31 (Neg (Succ zy400)) (Neg (Succ zy400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) (compare (Pos (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="genericIndex2 zy30 zy31 (Neg (Succ zy400)) (compare (Neg (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpInt (Pos (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="genericIndex2 zy30 zy31 (Neg (Succ zy400)) (primCmpInt (Neg (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpInt (Pos (Succ zy400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="genericIndex2 zy30 zy31 (Neg (Succ zy400)) (primCmpInt (Neg (Succ zy400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpNat (Succ zy400) Zero == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="genericIndex2 zy30 zy31 (Neg (Succ zy400)) (LT == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
41[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) (GT == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="genericIndex2 zy30 zy31 (Neg (Succ zy400)) False",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="genericIndex2 zy30 zy31 (Pos (Succ zy400)) True",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="genericIndex1 zy30 zy31 (Neg (Succ zy400)) otherwise",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45 -> 4[label="",style="dashed", color="red", weight=0];
45[label="genericIndex zy31 (Pos (Succ zy400) - fromInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
45 -> 48[label="",style="dashed", color="magenta", weight=3];
46[label="genericIndex1 zy30 zy31 (Neg (Succ zy400)) True",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3];
47[label="zy31",fontsize=16,color="green",shape="box"];48[label="Pos (Succ zy400) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
49 -> 10[label="",style="dashed", color="red", weight=0];
49[label="error []",fontsize=16,color="magenta"];50[label="primMinusInt (Pos (Succ zy400)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];50 -> 51[label="",style="solid", color="black", weight=3];
51[label="primMinusInt (Pos (Succ zy400)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];51 -> 52[label="",style="solid", color="black", weight=3];
52[label="primMinusNat (Succ zy400) (Succ Zero)",fontsize=16,color="black",shape="box"];52 -> 53[label="",style="solid", color="black", weight=3];
53[label="primMinusNat zy400 Zero",fontsize=16,color="burlywood",shape="box"];68[label="zy400/Succ zy4000",fontsize=10,color="white",style="solid",shape="box"];53 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 54[label="",style="solid", color="burlywood", weight=3];
69[label="zy400/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 55[label="",style="solid", color="burlywood", weight=3];
54[label="primMinusNat (Succ zy4000) Zero",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
55[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
56[label="Pos (Succ zy4000)",fontsize=16,color="green",shape="box"];57[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_genericIndex(:(zy30, zy31), Pos(Succ(zy400)), ba) → new_genericIndex(zy31, new_primMinusNat(zy400), ba)

The TRS R consists of the following rules:

new_primMinusNat(Succ(zy4000)) → Pos(Succ(zy4000))
new_primMinusNat(Zero) → Pos(Zero)

The set Q consists of the following terms:

new_primMinusNat(Zero)
new_primMinusNat(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_genericIndex(:(zy30, zy31), Pos(Succ(zy400)), ba) → new_genericIndex(zy31, new_primMinusNat(zy400), ba)
The graph contains the following edges 1 > 1, 3 >= 3

genericIndex	(x : vw) yy	= genericIndex5 (x : vw) yy
genericIndex	(vx : xs) n	= genericIndex3 (vx : xs) n
genericIndex	vy vz	= genericIndex0 vy vz

genericIndex2	vx xs n True	= genericIndex xs (n - 1)
genericIndex2	vx xs n False	= genericIndex1 vx xs n otherwise

genericIndex1	vx xs n True	= error []
genericIndex1	vx xs n False	= genericIndex0 (vx : xs) n

genericIndex3	(vx : xs) n	= genericIndex2 vx xs n (n > 0)
genericIndex3	yv yw	= genericIndex0 yv yw

genericIndex5	(x : vw) yy	= genericIndex4 (yy == 0) (x : vw) yy
genericIndex5	zw zx	= genericIndex3 zw zx