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

MAYBE 1.584 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Monad.hs
H-Termination of the given Haskell-Program with start terms could not be shown:

↳ HASKELL

  ↳ LR

mainModule Monad

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

module Monad where

import qualified Maybe
import qualified Prelude

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

foldM	_ a []	=	return a
foldM	f a (x : xs)	=	f a x >>= (\fax ->foldM f fax xs)

module Maybe where

import qualified Monad
import qualified Prelude

Lambda Reductions:
The following Lambda expression

\fax→foldM f fax xs

is transformed to

foldM0 f xs fax = foldM f fax xs

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Monad

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

foldM	_ a []	=	return a
foldM	f a (x : xs)	=	f a x >>= foldM0 f xs

foldM0

f xs fax

foldM f fax xs

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Monad

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

module Monad where

import qualified Maybe
import qualified Prelude

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

foldM	vw a []	=	return a
foldM	f a (x : xs)	=	f a x >>= foldM0 f xs

foldM0

f xs fax

foldM f fax xs

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

              ↳ Narrow

mainModule Monad

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

foldM	vw a []	=	return a
foldM	f a (x : xs)	=	f a x >>= foldM0 f xs

foldM0

f xs fax

foldM f fax xs

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];40[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 6[label="",style="solid", color="burlywood", weight=3];
41[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 10[label="",style="dashed", color="red", weight=0];
8[label="vz3 vz4 vz50 >>= foldM0 vz3 vz51",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="return vz4",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
11[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];11 -> 17[label="",style="dashed", color="green", weight=3];
11 -> 18[label="",style="dashed", color="green", weight=3];
10[label="vz6 >>= foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];43[label="vz6/vz60 : vz61",fontsize=10,color="white",style="solid",shape="box"];10 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 15[label="",style="solid", color="burlywood", weight=3];
44[label="vz6/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 16[label="",style="solid", color="burlywood", weight=3];
12[label="vz4 : []",fontsize=16,color="green",shape="box"];17[label="vz4",fontsize=16,color="green",shape="box"];18[label="vz50",fontsize=16,color="green",shape="box"];15[label="vz60 : vz61 >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="[] >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
19 -> 25[label="",style="dashed", color="red", weight=0];
19[label="foldM0 vz3 vz51 vz60 ++ (vz61 >>= foldM0 vz3 vz51)",fontsize=16,color="magenta"];19 -> 26[label="",style="dashed", color="magenta", weight=3];
19 -> 27[label="",style="dashed", color="magenta", weight=3];
20[label="[]",fontsize=16,color="green",shape="box"];26[label="foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27 -> 10[label="",style="dashed", color="red", weight=0];
27[label="vz61 >>= foldM0 vz3 vz51",fontsize=16,color="magenta"];27 -> 31[label="",style="dashed", color="magenta", weight=3];
25[label="vz8 ++ vz7",fontsize=16,color="burlywood",shape="triangle"];47[label="vz8/vz80 : vz81",fontsize=10,color="white",style="solid",shape="box"];25 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 32[label="",style="solid", color="burlywood", weight=3];
48[label="vz8/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 33[label="",style="solid", color="burlywood", weight=3];
30 -> 5[label="",style="dashed", color="red", weight=0];
30[label="foldM vz3 vz60 vz51",fontsize=16,color="magenta"];30 -> 34[label="",style="dashed", color="magenta", weight=3];
30 -> 35[label="",style="dashed", color="magenta", weight=3];
31[label="vz61",fontsize=16,color="green",shape="box"];32[label="(vz80 : vz81) ++ vz7",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="[] ++ vz7",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="vz60",fontsize=16,color="green",shape="box"];35[label="vz51",fontsize=16,color="green",shape="box"];36[label="vz80 : vz81 ++ vz7",fontsize=16,color="green",shape="box"];36 -> 38[label="",style="dashed", color="green", weight=3];
37[label="vz7",fontsize=16,color="green",shape="box"];38 -> 25[label="",style="dashed", color="red", weight=0];
38[label="vz81 ++ vz7",fontsize=16,color="magenta"];38 -> 39[label="",style="dashed", color="magenta", weight=3];
39[label="vz81",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

              ↳ Narrow

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

new_psPs(:(vz80, vz81), vz7, h) → new_psPs(vz81, vz7, h)

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(:(vz80, vz81), vz7, h) → new_psPs(vz81, vz7, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesReductionPairsProof

              ↳ Narrow

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

new_gtGtEs(vz3, vz51, h, ba) → new_gtGtEs(vz3, vz51, h, ba)
new_foldM(vz3, :(vz50, vz51), h, ba) → new_gtGtEs(vz3, vz51, h, ba)
new_gtGtEs(vz3, vz51, h, ba) → new_foldM(vz3, vz51, h, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

new_foldM(vz3, :(vz50, vz51), h, ba) → new_gtGtEs(vz3, vz51, h, ba)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(:(x₁, x₂)) = x₁ + 2·x₂
POL(new_foldM(x₁, x₂, x₃, x₄)) = 2 + x₁ + x₂ + x₃ + x₄
POL(new_gtGtEs(x₁, x₂, x₃, x₄)) = 2 + x₁ + 2·x₂ + x₃ + x₄

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesReductionPairsProof

                      ↳ QDP

                        ↳ DependencyGraphProof

              ↳ Narrow

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

new_gtGtEs(vz3, vz51, h, ba) → new_gtGtEs(vz3, vz51, h, ba)
new_gtGtEs(vz3, vz51, h, ba) → new_foldM(vz3, vz51, h, ba)

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

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesReductionPairsProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ NonTerminationProof

              ↳ Narrow

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

new_gtGtEs(vz3, vz51, h, ba) → new_gtGtEs(vz3, vz51, h, ba)

R is empty.
Q is empty.
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_gtGtEs(vz3, vz51, h, ba) → new_gtGtEs(vz3, vz51, h, ba)

The TRS R consists of the following rules:none

s = new_gtGtEs(vz3, vz51, h, ba) evaluates to t =new_gtGtEs(vz3, vz51, h, ba)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_gtGtEs(vz3, vz51, h, ba) to new_gtGtEs(vz3, vz51, h, ba).

Haskell To QDPs