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

MAYBE 1.974 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:

↳ HASKELL

  ↳ BR

mainModule Main

((enumFrom :: Int -> [Int]) :: Int -> [Int])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((enumFrom :: Int -> [Int]) :: Int -> [Int])

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

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((enumFrom :: Int -> [Int]) :: Int -> [Int])

module Main where

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

              ↳ Narrow

mainModule Main

(enumFrom :: Int -> [Int])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="enumFrom",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="enumFrom vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="numericEnumFrom vx3",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="vx3 : (numericEnumFrom $! vx3 + fromInt (Pos (Succ Zero)))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
6[label="(numericEnumFrom $! vx3 + fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7 -> 8[label="",style="dashed", color="red", weight=0];
7[label="(vx3 + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (vx3 + fromInt (Pos (Succ Zero))))",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
9 -> 4[label="",style="dashed", color="red", weight=0];
9[label="numericEnumFrom (vx3 + fromInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="(vx3 + fromInt (Pos (Succ Zero)) `seq` vx4)",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
10[label="vx3 + fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="enforceWHNF (WHNF (vx3 + fromInt (Pos (Succ Zero)))) vx4",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="primPlusInt vx3 (fromInt (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];43[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];12 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 15[label="",style="solid", color="burlywood", weight=3];
44[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];12 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 16[label="",style="solid", color="burlywood", weight=3];
14 -> 10[label="",style="dashed", color="red", weight=0];
14[label="vx3 + fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];13[label="enforceWHNF (WHNF vx5) vx4",fontsize=16,color="black",shape="triangle"];13 -> 17[label="",style="solid", color="black", weight=3];
15[label="primPlusInt (Pos vx30) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
16[label="primPlusInt (Neg vx30) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="vx4",fontsize=16,color="green",shape="box"];18[label="primPlusInt (Pos vx30) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="primPlusInt (Neg vx30) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="Pos (primPlusNat vx30 (Succ Zero))",fontsize=16,color="green",shape="box"];20 -> 22[label="",style="dashed", color="green", weight=3];
21[label="primMinusNat (Succ Zero) vx30",fontsize=16,color="burlywood",shape="box"];46[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];21 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 23[label="",style="solid", color="burlywood", weight=3];
47[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 24[label="",style="solid", color="burlywood", weight=3];
22[label="primPlusNat vx30 (Succ Zero)",fontsize=16,color="burlywood",shape="box"];48[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 25[label="",style="solid", color="burlywood", weight=3];
49[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 26[label="",style="solid", color="burlywood", weight=3];
23[label="primMinusNat (Succ Zero) (Succ vx300)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="primPlusNat (Succ vx300) (Succ Zero)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="primMinusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];50[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];27 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 31[label="",style="solid", color="burlywood", weight=3];
51[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 32[label="",style="solid", color="burlywood", weight=3];
28[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];29[label="Succ (Succ (primPlusNat vx300 Zero))",fontsize=16,color="green",shape="box"];29 -> 33[label="",style="dashed", color="green", weight=3];
30[label="Succ Zero",fontsize=16,color="green",shape="box"];31[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
32[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
33[label="primPlusNat vx300 Zero",fontsize=16,color="burlywood",shape="box"];52[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];33 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 36[label="",style="solid", color="burlywood", weight=3];
53[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 37[label="",style="solid", color="burlywood", weight=3];
34[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];35[label="Pos Zero",fontsize=16,color="green",shape="box"];36[label="primPlusNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="Succ vx3000",fontsize=16,color="green",shape="box"];39[label="Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ RuleRemovalProof

              ↳ Narrow

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

new_numericEnumFrom(vx3) → new_numericEnumFrom(new_ps(vx3))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_ps(Neg(Succ(Zero))) → Pos(Zero)
new_ps(Neg(Zero)) → Pos(Succ(Zero))
new_ps(Neg(Succ(Succ(vx3000)))) → Neg(Succ(vx3000))
new_primPlusNat0(Succ(vx3000)) → Succ(vx3000)
new_primPlusNat(Succ(vx300)) → Succ(Succ(new_primPlusNat0(vx300)))
new_ps(Pos(vx30)) → Pos(new_primPlusNat(vx30))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_ps(Neg(Succ(Zero)))
new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_ps(Neg(Succ(Succ(x0))))
new_primPlusNat0(Zero)
new_ps(Pos(x0))
new_ps(Neg(Zero))
new_primPlusNat0(Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

new_ps(Neg(Succ(Zero))) → Pos(Zero)
new_ps(Neg(Zero)) → Pos(Succ(Zero))

Used ordering: POLO with Polynomial interpretation [25]:

POL(Neg(x₁)) = 1 + x₁
POL(Pos(x₁)) = 2·x₁
POL(Succ(x₁)) = x₁
POL(Zero) = 0
POL(new_numericEnumFrom(x₁)) = 2·x₁
POL(new_primPlusNat(x₁)) = x₁
POL(new_primPlusNat0(x₁)) = x₁
POL(new_ps(x₁)) = x₁

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ RuleRemovalProof

                    ↳ QDP

                      ↳ NonTerminationProof

              ↳ Narrow

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

new_numericEnumFrom(vx3) → new_numericEnumFrom(new_ps(vx3))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_ps(Neg(Succ(Succ(vx3000)))) → Neg(Succ(vx3000))
new_primPlusNat0(Succ(vx3000)) → Succ(vx3000)
new_primPlusNat(Succ(vx300)) → Succ(Succ(new_primPlusNat0(vx300)))
new_ps(Pos(vx30)) → Pos(new_primPlusNat(vx30))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_ps(Neg(Succ(Zero)))
new_primPlusNat(Succ(x0))
new_primPlusNat(Zero)
new_ps(Neg(Succ(Succ(x0))))
new_primPlusNat0(Zero)
new_ps(Pos(x0))
new_ps(Neg(Zero))
new_primPlusNat0(Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_numericEnumFrom(vx3) → new_numericEnumFrom(new_ps(vx3))

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_ps(Neg(Succ(Succ(vx3000)))) → Neg(Succ(vx3000))
new_primPlusNat0(Succ(vx3000)) → Succ(vx3000)
new_primPlusNat(Succ(vx300)) → Succ(Succ(new_primPlusNat0(vx300)))
new_ps(Pos(vx30)) → Pos(new_primPlusNat(vx30))
new_primPlusNat0(Zero) → Zero

s = new_numericEnumFrom(vx3) evaluates to t =new_numericEnumFrom(new_ps(vx3))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [vx3 / new_ps(vx3)]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom(vx3) to new_numericEnumFrom(new_ps(vx3)).

Haskell To QDPs