H-Termination proof of ../tpdb/FP/full_haskell/Prelude_BANGBANG

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_BANGBANG_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

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

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

The following Function with conditions

!! (x : vv) 0 = x

!! (vw : xs) n

| n > 0

= xs !! (n - 1)

!! (vx : vy) vz = error []

!! [] wu = error []

is transformed to

!! (x : vv) yv = emEm5 (x : vv) yv

!! (vw : xs) n = emEm3 (vw : xs) n

!! (vx : vy) vz = emEm1 (vx : vy) vz

!! [] wu = emEm0 [] wu

emEm0 [] wu = error []

emEm1 (vx : vy) vz = error []

emEm1 xv xw = emEm0 xv xw

emEm2 vw xs n True = xs !! (n - 1)

emEm2 vw xs n False = emEm1 (vw : xs) n

emEm3 (vw : xs) n = emEm2 vw xs n (n > 0)

emEm3 xy xz = emEm1 xy xz

emEm4 True (x : vv) yv = x

emEm4 yw yx yy = emEm3 yx yy

emEm5 (x : vv) yv = emEm4 (yv == 0) (x : vv) yv

emEm5 yz zu = emEm3 yz zu

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

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

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(!!)\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(!!) zv3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(!!) zv3 zv4\n",fontsize=16,color="burlywood",shape="triangle"];50[label="zv3/zv30 : zv31",fontsize=10,color="white",style="solid",shape="box"];4 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 5[label="",style="solid", color="burlywood", weight=3];
51[label="zv3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="(!!) (zv30 : zv31) zv4\n",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="(!!) [] zv4\n",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="emEm5 (zv30 : zv31) zv4\n",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="emEm0 [] zv4\n",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="emEm4 (zv4 == Pos Zero) (zv30 : zv31) zv4\n",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="error []\n",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="emEm4 (primEqInt zv4 (Pos Zero)) (zv30 : zv31) zv4\n",fontsize=16,color="burlywood",shape="box"];52[label="zv4/Pos zv40",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 13[label="",style="solid", color="burlywood", weight=3];
53[label="zv4/Neg zv40",fontsize=10,color="white",style="solid",shape="box"];11 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="error []\n",fontsize=16,color="red",shape="box"];13[label="emEm4 (primEqInt (Pos zv40) (Pos Zero)) (zv30 : zv31) (Pos zv40)\n",fontsize=16,color="burlywood",shape="box"];54[label="zv40/Succ zv400",fontsize=10,color="white",style="solid",shape="box"];13 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 15[label="",style="solid", color="burlywood", weight=3];
55[label="zv40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="emEm4 (primEqInt (Neg zv40) (Pos Zero)) (zv30 : zv31) (Neg zv40)\n",fontsize=16,color="burlywood",shape="box"];56[label="zv40/Succ zv400",fontsize=10,color="white",style="solid",shape="box"];14 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 17[label="",style="solid", color="burlywood", weight=3];
57[label="zv40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="emEm4 (primEqInt (Pos (Succ zv400)) (Pos Zero)) (zv30 : zv31) (Pos (Succ zv400))\n",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="emEm4 (primEqInt (Pos Zero) (Pos Zero)) (zv30 : zv31) (Pos Zero)\n",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="emEm4 (primEqInt (Neg (Succ zv400)) (Pos Zero)) (zv30 : zv31) (Neg (Succ zv400))\n",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="emEm4 (primEqInt (Neg Zero) (Pos Zero)) (zv30 : zv31) (Neg Zero)\n",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="emEm4 False (zv30 : zv31) (Pos (Succ zv400))\n",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="emEm4 True (zv30 : zv31) (Pos Zero)\n",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="emEm4 False (zv30 : zv31) (Neg (Succ zv400))\n",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="emEm4 True (zv30 : zv31) (Neg Zero)\n",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="emEm3 (zv30 : zv31) (Pos (Succ zv400))\n",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="zv30\n",fontsize=16,color="green",shape="box"];25[label="emEm3 (zv30 : zv31) (Neg (Succ zv400))\n",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
26[label="zv30\n",fontsize=16,color="green",shape="box"];27[label="emEm2 zv30 zv31 (Pos (Succ zv400)) (Pos (Succ zv400) > Pos Zero)\n",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="emEm2 zv30 zv31 (Neg (Succ zv400)) (Neg (Succ zv400) > Pos Zero)\n",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="emEm2 zv30 zv31 (Pos (Succ zv400)) (compare (Pos (Succ zv400)) (Pos Zero) == GT)\n",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="emEm2 zv30 zv31 (Neg (Succ zv400)) (compare (Neg (Succ zv400)) (Pos Zero) == GT)\n",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
31[label="emEm2 zv30 zv31 (Pos (Succ zv400)) (primCmpInt (Pos (Succ zv400)) (Pos Zero) == GT)\n",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="emEm2 zv30 zv31 (Neg (Succ zv400)) (primCmpInt (Neg (Succ zv400)) (Pos Zero) == GT)\n",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="emEm2 zv30 zv31 (Pos (Succ zv400)) (primCmpNat (Succ zv400) Zero == GT)\n",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="emEm2 zv30 zv31 (Neg (Succ zv400)) (LT == GT)\n",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="emEm2 zv30 zv31 (Pos (Succ zv400)) (GT == GT)\n",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="emEm2 zv30 zv31 (Neg (Succ zv400)) False\n",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="emEm2 zv30 zv31 (Pos (Succ zv400)) True\n",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="emEm1 (zv30 : zv31) (Neg (Succ zv400))\n",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39 -> 4[label="",style="dashed", color="red", weight=0];
39[label="zv31 !! (Pos (Succ zv400) - Pos (Succ Zero))\n",fontsize=16,color="magenta"];39 -> 41[label="",style="dashed", color="magenta", weight=3];
39 -> 42[label="",style="dashed", color="magenta", weight=3];
40 -> 10[label="",style="dashed", color="red", weight=0];
40[label="error []\n",fontsize=16,color="magenta"];41[label="Pos (Succ zv400) - Pos (Succ Zero)\n",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="zv31\n",fontsize=16,color="green",shape="box"];43[label="primMinusInt (Pos (Succ zv400)) (Pos (Succ Zero))\n",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="primMinusNat (Succ zv400) (Succ Zero)\n",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
45[label="primMinusNat zv400 Zero\n",fontsize=16,color="burlywood",shape="box"];60[label="zv400/Succ zv4000",fontsize=10,color="white",style="solid",shape="box"];45 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 46[label="",style="solid", color="burlywood", weight=3];
61[label="zv400/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 47[label="",style="solid", color="burlywood", weight=3];
46[label="primMinusNat (Succ zv4000) Zero\n",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="primMinusNat Zero Zero\n",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="Pos (Succ zv4000)\n",fontsize=16,color="green",shape="box"];49[label="Pos Zero\n",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_emEm(:(zv30, zv31), Pos(Succ(zv400)), h) → new_emEm(zv31, new_primMinusNat(zv400), h)

The TRS R consists of the following rules:

new_primMinusNat(Succ(zv4000)) → Pos(Succ(zv4000))
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_emEm(:(zv30, zv31), Pos(Succ(zv400)), h) → new_emEm(zv31, new_primMinusNat(zv400), h)
The graph contains the following edges 1 > 1, 3 >= 3

!!	(x : vv) yv	= emEm5 (x : vv) yv
!!	(vw : xs) n	= emEm3 (vw : xs) n
!!	(vx : vy) vz	= emEm1 (vx : vy) vz
!!	[] wu	= emEm0 [] wu

emEm2	vw xs n True	= xs !! (n - 1)
emEm2	vw xs n False	= emEm1 (vw : xs) n

emEm3	(vw : xs) n	= emEm2 vw xs n (n > 0)
emEm3	xy xz	= emEm1 xy xz

emEm5	(x : vv) yv	= emEm4 (yv == 0) (x : vv) yv
emEm5	yz zu	= emEm3 yz zu