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

YES 0.707 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

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

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\n_→n + 1

is transformed to

length0 n _ = n + 1

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

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

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

mainModule Main

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

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

(length :: [a] -> Int)

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="length",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="length vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl' length0 (Pos Zero) vy3",fontsize=16,color="burlywood",shape="box"];54[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 5[label="",style="solid", color="burlywood", weight=3];
55[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl' length0 (Pos Zero) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl' length0 (Pos Zero) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(foldl' length0 $! length0 (Pos Zero) vy30)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="Pos Zero",fontsize=16,color="green",shape="box"];9[label="(length0 (Pos Zero) vy30 `seq` foldl' length0 (length0 (Pos Zero) vy30))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10 -> 23[label="",style="dashed", color="red", weight=0];
10[label="enforceWHNF (WHNF (length0 (Pos Zero) vy30)) (foldl' length0 (length0 (Pos Zero) vy30)) vy31",fontsize=16,color="magenta"];10 -> 24[label="",style="dashed", color="magenta", weight=3];
10 -> 25[label="",style="dashed", color="magenta", weight=3];
10 -> 26[label="",style="dashed", color="magenta", weight=3];
10 -> 27[label="",style="dashed", color="magenta", weight=3];
24[label="Zero",fontsize=16,color="green",shape="box"];25[label="vy30",fontsize=16,color="green",shape="box"];26[label="Zero",fontsize=16,color="green",shape="box"];27[label="vy31",fontsize=16,color="green",shape="box"];23[label="enforceWHNF (WHNF (length0 (Pos vy5) vy310)) (foldl' length0 (length0 (Pos vy4) vy310)) vy311",fontsize=16,color="black",shape="triangle"];23 -> 30[label="",style="solid", color="black", weight=3];
30[label="enforceWHNF (WHNF (Pos vy5 + Pos (Succ Zero))) (foldl' length0 (Pos vy5 + Pos (Succ Zero))) vy311",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
31[label="enforceWHNF (WHNF (primPlusInt (Pos vy5) (Pos (Succ Zero)))) (foldl' length0 (primPlusInt (Pos vy5) (Pos (Succ Zero)))) vy311",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32[label="enforceWHNF (WHNF (Pos (primPlusNat vy5 (Succ Zero)))) (foldl' length0 (Pos (primPlusNat vy5 (Succ Zero)))) vy311",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
33[label="foldl' length0 (Pos (primPlusNat vy5 (Succ Zero))) vy311",fontsize=16,color="burlywood",shape="box"];57[label="vy311/vy3110 : vy3111",fontsize=10,color="white",style="solid",shape="box"];33 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 34[label="",style="solid", color="burlywood", weight=3];
58[label="vy311/[]",fontsize=10,color="white",style="solid",shape="box"];33 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 35[label="",style="solid", color="burlywood", weight=3];
34[label="foldl' length0 (Pos (primPlusNat vy5 (Succ Zero))) (vy3110 : vy3111)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="foldl' length0 (Pos (primPlusNat vy5 (Succ Zero))) []",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="(foldl' length0 $! length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="Pos (primPlusNat vy5 (Succ Zero))",fontsize=16,color="green",shape="box"];37 -> 39[label="",style="dashed", color="green", weight=3];
38[label="(length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110 `seq` foldl' length0 (length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="primPlusNat vy5 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];59[label="vy5/Succ vy50",fontsize=10,color="white",style="solid",shape="box"];39 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 41[label="",style="solid", color="burlywood", weight=3];
60[label="vy5/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 42[label="",style="solid", color="burlywood", weight=3];
40 -> 23[label="",style="dashed", color="red", weight=0];
40[label="enforceWHNF (WHNF (length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110)) (foldl' length0 (length0 (Pos (primPlusNat vy5 (Succ Zero))) vy3110)) vy3111",fontsize=16,color="magenta"];40 -> 43[label="",style="dashed", color="magenta", weight=3];
40 -> 44[label="",style="dashed", color="magenta", weight=3];
40 -> 45[label="",style="dashed", color="magenta", weight=3];
40 -> 46[label="",style="dashed", color="magenta", weight=3];
41[label="primPlusNat (Succ vy50) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
42[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
43 -> 39[label="",style="dashed", color="red", weight=0];
43[label="primPlusNat vy5 (Succ Zero)",fontsize=16,color="magenta"];44[label="vy3110",fontsize=16,color="green",shape="box"];45 -> 39[label="",style="dashed", color="red", weight=0];
45[label="primPlusNat vy5 (Succ Zero)",fontsize=16,color="magenta"];46[label="vy3111",fontsize=16,color="green",shape="box"];47[label="Succ (Succ (primPlusNat vy50 Zero))",fontsize=16,color="green",shape="box"];47 -> 49[label="",style="dashed", color="green", weight=3];
48[label="Succ Zero",fontsize=16,color="green",shape="box"];49[label="primPlusNat vy50 Zero",fontsize=16,color="burlywood",shape="box"];64[label="vy50/Succ vy500",fontsize=10,color="white",style="solid",shape="box"];49 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 50[label="",style="solid", color="burlywood", weight=3];
65[label="vy50/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 51[label="",style="solid", color="burlywood", weight=3];
50[label="primPlusNat (Succ vy500) Zero",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3];
51[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
52[label="Succ vy500",fontsize=16,color="green",shape="box"];53[label="Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ NumRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_enforceWHNF(vy5, vy310, vy4, :(vy3110, vy3111), ba) → new_enforceWHNF(new_primPlusNat(vy5), vy3110, new_primPlusNat(vy5), vy3111, ba)

The TRS R consists of the following rules:

new_primPlusNat(Zero) → Succ(Zero)
new_primPlusNat0(Succ(vy500)) → Succ(vy500)
new_primPlusNat(Succ(vy50)) → Succ(Succ(new_primPlusNat0(vy50)))
new_primPlusNat0(Zero) → Zero

The set Q consists of the following terms:

new_primPlusNat0(Succ(x0))
new_primPlusNat(Zero)
new_primPlusNat0(Zero)
new_primPlusNat(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_enforceWHNF(vy5, vy310, vy4, :(vy3110, vy3111), ba) → new_enforceWHNF(new_primPlusNat(vy5), vy3110, new_primPlusNat(vy5), vy3111, ba)
The graph contains the following edges 4 > 2, 4 > 4, 5 >= 5