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

YES H-Termination proof of ../tpdb/FP/full_haskell/Prelude_mapM__2.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((mapM_ :: (a -> [b]) -> [a] -> [()]) :: (a -> [b]) -> [a] -> [()])

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((mapM_ :: (b -> [a]) -> [b] -> [()]) :: (b -> [a]) -> [b] -> [()])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

mainModule Main

((mapM_ :: (a -> [b]) -> [a] -> [()]) :: (a -> [b]) -> [a] -> [()])

module Main where

import qualified Prelude

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

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(mapM_ :: (b -> [a]) -> [b] -> [()])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="mapM_\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mapM_ vy3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapM_ vy3 vy4\n",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence_ . map vy3\n",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence_ (map vy3 vy4)\n",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (map vy3 vy4)\n",fontsize=16,color="burlywood",shape="triangle"];37[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 8[label="",style="solid", color="burlywood", weight=3];
38[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr (>>) (return ()) (map vy3 (vy40 : vy41))\n",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldr (>>) (return ()) (map vy3 [])\n",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (vy3 vy40 : map vy3 vy41)\n",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) []\n",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12 -> 14[label="",style="dashed", color="red", weight=0];
12[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)\n",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
13[label="return ()\n",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
15 -> 7[label="",style="dashed", color="red", weight=0];
15[label="foldr (>>) (return ()) (map vy3 vy41)\n",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
14[label="(>>) vy3 vy40 vy5\n",fontsize=16,color="black",shape="triangle"];14 -> 18[label="",style="solid", color="black", weight=3];
16[label="() : []\n",fontsize=16,color="green",shape="box"];17[label="vy41\n",fontsize=16,color="green",shape="box"];18 -> 19[label="",style="dashed", color="red", weight=0];
18[label="vy3 vy40 >>= gtGt0 vy5\n",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
20[label="vy3 vy40\n",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
19[label="vy6 >>= gtGt0 vy5\n",fontsize=16,color="burlywood",shape="triangle"];42[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];19 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 22[label="",style="solid", color="burlywood", weight=3];
43[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 23[label="",style="solid", color="burlywood", weight=3];
24[label="vy40\n",fontsize=16,color="green",shape="box"];22[label="vy60 : vy61 >>= gtGt0 vy5\n",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="[] >>= gtGt0 vy5\n",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
25 -> 27[label="",style="dashed", color="red", weight=0];
25[label="gtGt0 vy5 vy60 ++ (vy61 >>= gtGt0 vy5)\n",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3];
26[label="[]\n",fontsize=16,color="green",shape="box"];28 -> 19[label="",style="dashed", color="red", weight=0];
28[label="vy61 >>= gtGt0 vy5\n",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
27[label="gtGt0 vy5 vy60 ++ vy7\n",fontsize=16,color="black",shape="triangle"];27 -> 30[label="",style="solid", color="black", weight=3];
29[label="vy61\n",fontsize=16,color="green",shape="box"];30[label="vy5 ++ vy7\n",fontsize=16,color="burlywood",shape="triangle"];46[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];30 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 31[label="",style="solid", color="burlywood", weight=3];
47[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 32[label="",style="solid", color="burlywood", weight=3];
31[label="(vy50 : vy51) ++ vy7\n",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="[] ++ vy7\n",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="vy50 : vy51 ++ vy7\n",fontsize=16,color="green",shape="box"];33 -> 35[label="",style="dashed", color="green", weight=3];
34[label="vy7\n",fontsize=16,color="green",shape="box"];35 -> 30[label="",style="dashed", color="red", weight=0];
35[label="vy51 ++ vy7\n",fontsize=16,color="magenta"];35 -> 36[label="",style="dashed", color="magenta", weight=3];
36[label="vy51\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_foldr(vy3, :(vy40, vy41), h, ba) → new_foldr(vy3, vy41, h, ba)

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_foldr(vy3, :(vy40, vy41), h, ba) → new_foldr(vy3, vy41, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_gtGtEs(:(vy60, vy61), vy5, h) → new_gtGtEs(vy61, vy5, h)

From the DPs we obtained the following set of size-change graphs:

new_gtGtEs(:(vy60, vy61), vy5, h) → new_gtGtEs(vy61, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

new_psPs(:(vy50, vy51), vy7) → new_psPs(vy51, vy7)

From the DPs we obtained the following set of size-change graphs:

new_psPs(:(vy50, vy51), vy7) → new_psPs(vy51, vy7)
The graph contains the following edges 1 > 1, 2 >= 2