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

YES 0.6980000000000001 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

((mapM :: (b -> IO a) -> [b] -> IO [a]) :: (b -> IO a) -> [b] -> IO [a])

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\xs→return (x : xs)

is transformed to

sequence0 x xs = return (x : xs)

The following Lambda expression

\x→sequence cs >>= sequence0 x

is transformed to

sequence1 cs x = sequence cs >>= sequence0 x

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((mapM :: (a -> IO b) -> [a] -> IO [b]) :: (a -> IO b) -> [a] -> IO [b])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((mapM :: (a -> IO b) -> [a] -> IO [b]) :: (a -> IO b) -> [a] -> IO [b])

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

              ↳ Narrow

mainModule Main

(mapM :: (a -> IO b) -> [a] -> IO [b])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];39[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 7[label="",style="solid", color="burlywood", weight=3];
40[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="sequence (map vx3 [])",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="sequence (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="sequence []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="vx3 vx40 >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="return []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="primbindIO (vx3 vx40) (sequence1 (map vx3 vx41))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="primretIO []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="terminator (vx3 vx40) (sequence1 (map vx3 vx41))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="terminator []",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="ter0m (vx3 vx40) (sequence1 (map vx3 vx41))",fontsize=16,color="green",shape="box"];17 -> 19[label="",style="dashed", color="green", weight=3];
17 -> 20[label="",style="dashed", color="green", weight=3];
18[label="ter1m []",fontsize=16,color="green",shape="box"];18 -> 21[label="",style="dashed", color="green", weight=3];
19[label="vx3 vx40",fontsize=16,color="green",shape="box"];19 -> 22[label="",style="dashed", color="green", weight=3];
20[label="sequence1 (map vx3 vx41)",fontsize=16,color="grey",shape="box"];20 -> 23[label="",style="dashed", color="grey", weight=3];
21[label="[]",fontsize=16,color="green",shape="box"];22[label="vx40",fontsize=16,color="green",shape="box"];23[label="sequence1 (map vx3 vx41) vx5",fontsize=16,color="black",shape="triangle"];23 -> 24[label="",style="solid", color="black", weight=3];
24 -> 25[label="",style="dashed", color="red", weight=0];
24[label="sequence (map vx3 vx41) >>= sequence0 vx5",fontsize=16,color="magenta"];24 -> 26[label="",style="dashed", color="magenta", weight=3];
26 -> 6[label="",style="dashed", color="red", weight=0];
26[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
25[label="vx6 >>= sequence0 vx5",fontsize=16,color="black",shape="triangle"];25 -> 28[label="",style="solid", color="black", weight=3];
27[label="vx41",fontsize=16,color="green",shape="box"];28[label="primbindIO vx6 (sequence0 vx5)",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3];
29[label="terminator vx6 (sequence0 vx5)",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="ter2m vx6 (sequence0 vx5)",fontsize=16,color="green",shape="box"];30 -> 31[label="",style="dashed", color="green", weight=3];
30 -> 32[label="",style="dashed", color="green", weight=3];
31[label="vx6",fontsize=16,color="green",shape="box"];32[label="sequence0 vx5",fontsize=16,color="grey",shape="box"];32 -> 33[label="",style="dashed", color="grey", weight=3];
33[label="sequence0 vx5 vx7",fontsize=16,color="black",shape="triangle"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="return (vx5 : vx7)",fontsize=16,color="black",shape="box"];34 -> 35[label="",style="solid", color="black", weight=3];
35[label="primretIO (vx5 : vx7)",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="terminator (vx5 : vx7)",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="ter3m (vx5 : vx7)",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3];
38[label="vx5 : vx7",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

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

new_sequence(vx3, :(vx40, vx41), ba, bb) → new_sequence(vx3, vx41, ba, bb)
new_sequence1(vx3, vx41, vx5, ba, bb) → new_sequence(vx3, vx41, ba, bb)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_sequence(vx3, :(vx40, vx41), ba, bb) → new_sequence(vx3, vx41, ba, bb)

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_sequence(vx3, :(vx40, vx41), ba, bb) → new_sequence(vx3, vx41, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4