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

YES 0.784 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Monad.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Monad

((zipWithM_ :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a ()) :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a ())

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM_ :: Monad d => (c -> a -> d b) -> [c] -> [a] -> d ()

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

((zipWithM_ :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a ()) :: Monad a => (b -> c -> a d) -> [b] -> [c] -> a ())

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM_ :: Monad b => (a -> d -> b c) -> [a] -> [d] -> b ()

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

((zipWithM_ :: Monad c => (d -> b -> c a) -> [d] -> [b] -> c ()) :: Monad c => (d -> b -> c a) -> [d] -> [b] -> c ())

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM_ :: Monad a => (c -> d -> a b) -> [c] -> [d] -> a ()

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

module Maybe where

import qualified Monad
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 Monad

(zipWithM_ :: Monad d => (b -> c -> d a) -> [b] -> [c] -> d ())

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

zipWithM_ :: Monad d => (a -> c -> d b) -> [a] -> [c] -> d ()

zipWithM_

f xs ys

sequence_ (zipWith f xs ys)

Haskell To QDPs

digraph dp_graph {
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86 -> 8[label="",style="solid", color="burlywood", weight=3];
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42 -> 54[label="",style="dashed", color="green", weight=3];
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105 -> 55[label="",style="solid", color="burlywood", weight=3];
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106 -> 56[label="",style="solid", color="burlywood", weight=3];
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51 -> 63[label="",style="dashed", color="green", weight=3];
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107 -> 60[label="",style="solid", color="burlywood", weight=3];
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108 -> 61[label="",style="solid", color="burlywood", weight=3];
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57[label="terminator (ww3 ww40 ww50) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];57 -> 66[label="",style="solid", color="black", weight=3];
62[label="ww40",fontsize=16,color="green",shape="box"];63[label="ww50",fontsize=16,color="green",shape="box"];60[label="Nothing >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];60 -> 67[label="",style="solid", color="black", weight=3];
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64 -> 69[label="",style="dashed", color="red", weight=0];
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66 -> 72[label="",style="dashed", color="green", weight=3];
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70 -> 41[label="",style="dashed", color="red", weight=0];
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69[label="gtGt0 ww6 ww70 ++ ww9",fontsize=16,color="black",shape="triangle"];69 -> 75[label="",style="solid", color="black", weight=3];
71[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];71 -> 76[label="",style="dashed", color="green", weight=3];
71 -> 77[label="",style="dashed", color="green", weight=3];
72[label="gtGt0 ww6",fontsize=16,color="grey",shape="box"];72 -> 78[label="",style="dashed", color="grey", weight=3];
73[label="ww6",fontsize=16,color="green",shape="box"];74[label="ww71",fontsize=16,color="green",shape="box"];75[label="ww6 ++ ww9",fontsize=16,color="burlywood",shape="triangle"];111[label="ww6/ww60 : ww61",fontsize=10,color="white",style="solid",shape="box"];75 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 79[label="",style="solid", color="burlywood", weight=3];
112[label="ww6/[]",fontsize=10,color="white",style="solid",shape="box"];75 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 80[label="",style="solid", color="burlywood", weight=3];
76[label="ww40",fontsize=16,color="green",shape="box"];77[label="ww50",fontsize=16,color="green",shape="box"];78[label="gtGt0 ww6 ww10",fontsize=16,color="black",shape="triangle"];78 -> 81[label="",style="solid", color="black", weight=3];
79[label="(ww60 : ww61) ++ ww9",fontsize=16,color="black",shape="box"];79 -> 82[label="",style="solid", color="black", weight=3];
80[label="[] ++ ww9",fontsize=16,color="black",shape="box"];80 -> 83[label="",style="solid", color="black", weight=3];
81[label="ww6",fontsize=16,color="green",shape="box"];82[label="ww60 : ww61 ++ ww9",fontsize=16,color="green",shape="box"];82 -> 84[label="",style="dashed", color="green", weight=3];
83[label="ww9",fontsize=16,color="green",shape="box"];84 -> 75[label="",style="dashed", color="red", weight=0];
84[label="ww61 ++ ww9",fontsize=16,color="magenta"];84 -> 85[label="",style="dashed", color="magenta", weight=3];
85[label="ww61",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

new_psPs(:(ww60, ww61), ww9) → new_psPs(ww61, ww9)

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_psPs(:(ww60, ww61), ww9) → new_psPs(ww61, ww9)
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

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

new_gtGtEs(:(ww70, ww71), ww6, h) → new_gtGtEs(ww71, ww6, h)

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

new_gtGtEs(:(ww70, ww71), ww6, h) → new_gtGtEs(ww71, ww6, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_IO, h, 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 3 SCCs.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPSizeChangeProof

                        ↳ QDP

                        ↳ QDP

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_IO, h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_IO, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPSizeChangeProof

                        ↳ QDP

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_Maybe, h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_Maybe, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPSizeChangeProof

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)

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

new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), ty_[], h, ba, bb) → new_foldr(ww3, ww41, ww51, ty_[], h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7