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

MAYBE 1.971 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 :: Float -> [Float]) :: Float -> [Float])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

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

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 :: Float -> [Float]) :: Float -> [Float])

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 :: Float -> [Float])

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="primPlusFloat vx3 (fromInt (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];62[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];12 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 15[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 -> 16[label="",style="solid", color="black", weight=3];
15[label="primPlusFloat (Float vx30 vx31) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vx4",fontsize=16,color="green",shape="box"];17[label="primPlusFloat (Float vx30 vx31) (primIntToFloat (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="primPlusFloat (Float vx30 vx31) (Float (Pos (Succ Zero)) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="Float (vx30 + Pos (Succ Zero)) (vx31 * Pos (Succ Zero))",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="green", weight=3];
19 -> 21[label="",style="dashed", color="green", weight=3];
20[label="vx30 + Pos (Succ Zero)",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="vx31 * Pos (Succ Zero)",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="primPlusInt vx30 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];64[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 24[label="",style="solid", color="burlywood", weight=3];
65[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 25[label="",style="solid", color="burlywood", weight=3];
23[label="primMulInt vx31 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];66[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];23 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 26[label="",style="solid", color="burlywood", weight=3];
67[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];23 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 27[label="",style="solid", color="burlywood", weight=3];
24[label="primPlusInt (Pos vx300) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="primPlusInt (Neg vx300) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="primMulInt (Pos vx310) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="primMulInt (Neg vx310) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="Pos (primPlusNat vx300 (Succ Zero))",fontsize=16,color="green",shape="box"];28 -> 32[label="",style="dashed", color="green", weight=3];
29[label="primMinusNat (Succ Zero) vx300",fontsize=16,color="burlywood",shape="box"];68[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];29 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 33[label="",style="solid", color="burlywood", weight=3];
69[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 34[label="",style="solid", color="burlywood", weight=3];
30[label="Pos (primMulNat vx310 (Succ Zero))",fontsize=16,color="green",shape="box"];30 -> 35[label="",style="dashed", color="green", weight=3];
31[label="Neg (primMulNat vx310 (Succ Zero))",fontsize=16,color="green",shape="box"];31 -> 36[label="",style="dashed", color="green", weight=3];
32[label="primPlusNat vx300 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];70[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];32 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 37[label="",style="solid", color="burlywood", weight=3];
71[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 38[label="",style="solid", color="burlywood", weight=3];
33[label="primMinusNat (Succ Zero) (Succ vx3000)",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3];
34[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
35[label="primMulNat vx310 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];72[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];35 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 41[label="",style="solid", color="burlywood", weight=3];
73[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 42[label="",style="solid", color="burlywood", weight=3];
36 -> 35[label="",style="dashed", color="red", weight=0];
36[label="primMulNat vx310 (Succ Zero)",fontsize=16,color="magenta"];36 -> 43[label="",style="dashed", color="magenta", weight=3];
37[label="primPlusNat (Succ vx3000) (Succ Zero)",fontsize=16,color="black",shape="box"];37 -> 44[label="",style="solid", color="black", weight=3];
38[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="primMinusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];75[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];39 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 46[label="",style="solid", color="burlywood", weight=3];
76[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 47[label="",style="solid", color="burlywood", weight=3];
40[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];41[label="primMulNat (Succ vx3100) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="primMulNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="vx310",fontsize=16,color="green",shape="box"];44[label="Succ (Succ (primPlusNat vx3000 Zero))",fontsize=16,color="green",shape="box"];44 -> 50[label="",style="dashed", color="green", weight=3];
45[label="Succ Zero",fontsize=16,color="green",shape="box"];46[label="primMinusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];47 -> 52[label="",style="solid", color="black", weight=3];
48 -> 32[label="",style="dashed", color="red", weight=0];
48[label="primPlusNat (primMulNat vx3100 (Succ Zero)) (Succ Zero)",fontsize=16,color="magenta"];48 -> 53[label="",style="dashed", color="magenta", weight=3];
49[label="Zero",fontsize=16,color="green",shape="box"];50[label="primPlusNat vx3000 Zero",fontsize=16,color="burlywood",shape="box"];78[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];50 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 54[label="",style="solid", color="burlywood", weight=3];
79[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 55[label="",style="solid", color="burlywood", weight=3];
51[label="Neg (Succ vx30000)",fontsize=16,color="green",shape="box"];52[label="Pos Zero",fontsize=16,color="green",shape="box"];53 -> 35[label="",style="dashed", color="red", weight=0];
53[label="primMulNat vx3100 (Succ Zero)",fontsize=16,color="magenta"];53 -> 56[label="",style="dashed", color="magenta", weight=3];
54[label="primPlusNat (Succ vx30000) Zero",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="vx3100",fontsize=16,color="green",shape="box"];57[label="Succ vx30000",fontsize=16,color="green",shape="box"];58[label="Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

              ↳ Narrow

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

new_primMulNat(Succ(vx3100)) → new_primMulNat(vx3100)

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_primMulNat(Succ(vx3100)) → new_primMulNat(vx3100)
The graph contains the following edges 1 > 1

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ 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_primPlusInt(Neg(Succ(Zero))) → Pos(Zero)
new_primPlusInt(Neg(Succ(Succ(vx30000)))) → Neg(Succ(vx30000))
new_primPlusNat(Succ(vx3000)) → Succ(Succ(new_primPlusNat0(vx3000)))
new_primPlusInt(Pos(vx300)) → Pos(new_primPlusNat(vx300))
new_primPlusNat0(Zero) → Zero
new_primPlusNat(Zero) → Succ(Zero)
new_ps(Float(vx30, vx31)) → Float(new_primPlusInt(vx30), new_primMulInt(vx31))
new_primMulNat0(Zero) → Zero
new_primMulInt(Pos(vx310)) → Pos(new_primMulNat0(vx310))
new_primPlusNat0(Succ(vx30000)) → Succ(vx30000)
new_primMulNat0(Succ(vx3100)) → new_primPlusNat(new_primMulNat0(vx3100))
new_primMulInt(Neg(vx310)) → Neg(new_primMulNat0(vx310))
new_primPlusInt(Neg(Zero)) → Pos(Succ(Zero))

The set Q consists of the following terms:

new_primPlusInt(Neg(Succ(Succ(x0))))
new_primPlusInt(Neg(Zero))
new_primMulInt(Neg(x0))
new_ps(Float(x0, x1))
new_primPlusNat(Succ(x0))
new_primPlusInt(Pos(x0))
new_primMulNat0(Succ(x0))
new_primMulInt(Pos(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primPlusInt(Neg(Succ(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_primPlusInt(Neg(Succ(Zero))) → Pos(Zero)
new_primPlusInt(Neg(Zero)) → Pos(Succ(Zero))

Used ordering: POLO with Polynomial interpretation [25]:

POL(Float(x₁, x₂)) = 2·x₁ + 2·x₂
POL(Neg(x₁)) = 2 + 2·x₁
POL(Pos(x₁)) = 2·x₁
POL(Succ(x₁)) = x₁
POL(Zero) = 0
POL(new_numericEnumFrom(x₁)) = x₁
POL(new_primMulInt(x₁)) = x₁
POL(new_primMulNat0(x₁)) = x₁
POL(new_primPlusInt(x₁)) = x₁
POL(new_primPlusNat(x₁)) = x₁
POL(new_primPlusNat0(x₁)) = x₁
POL(new_ps(x₁)) = x₁

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ 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_primPlusInt(Neg(Succ(Succ(vx30000)))) → Neg(Succ(vx30000))
new_primPlusNat(Succ(vx3000)) → Succ(Succ(new_primPlusNat0(vx3000)))
new_primPlusInt(Pos(vx300)) → Pos(new_primPlusNat(vx300))
new_primPlusNat0(Zero) → Zero
new_primPlusNat(Zero) → Succ(Zero)
new_ps(Float(vx30, vx31)) → Float(new_primPlusInt(vx30), new_primMulInt(vx31))
new_primMulNat0(Zero) → Zero
new_primMulInt(Pos(vx310)) → Pos(new_primMulNat0(vx310))
new_primPlusNat0(Succ(vx30000)) → Succ(vx30000)
new_primMulNat0(Succ(vx3100)) → new_primPlusNat(new_primMulNat0(vx3100))
new_primMulInt(Neg(vx310)) → Neg(new_primMulNat0(vx310))

The set Q consists of the following terms:

new_primPlusInt(Neg(Succ(Succ(x0))))
new_primPlusInt(Neg(Zero))
new_primMulInt(Neg(x0))
new_ps(Float(x0, x1))
new_primPlusNat(Succ(x0))
new_primPlusInt(Pos(x0))
new_primMulNat0(Succ(x0))
new_primMulInt(Pos(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primPlusInt(Neg(Succ(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_primPlusInt(Neg(Succ(Succ(vx30000)))) → Neg(Succ(vx30000))
new_primPlusNat(Succ(vx3000)) → Succ(Succ(new_primPlusNat0(vx3000)))
new_primPlusInt(Pos(vx300)) → Pos(new_primPlusNat(vx300))
new_primPlusNat0(Zero) → Zero
new_primPlusNat(Zero) → Succ(Zero)
new_ps(Float(vx30, vx31)) → Float(new_primPlusInt(vx30), new_primMulInt(vx31))
new_primMulNat0(Zero) → Zero
new_primMulInt(Pos(vx310)) → Pos(new_primMulNat0(vx310))
new_primPlusNat0(Succ(vx30000)) → Succ(vx30000)
new_primMulNat0(Succ(vx3100)) → new_primPlusNat(new_primMulNat0(vx3100))
new_primMulInt(Neg(vx310)) → Neg(new_primMulNat0(vx310))

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