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

MAYBE 1.667 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 :: Monad b => (a -> c -> b a) -> a -> [c] -> b a) :: Monad b => (a -> c -> b a) -> a -> [c] -> b a)

module Monad where

import qualified Maybe
import qualified Prelude

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

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

module Maybe where

import qualified Monad
import qualified Prelude

module Monad where

import qualified Maybe
import qualified Prelude

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

module Monad where

import qualified Maybe
import qualified Prelude

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

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

module Maybe where

import qualified Monad
import qualified Prelude

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

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldM\n  Monad a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldM vz3\n  Monad a",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldM vz3 vz4\n  Monad a",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldM vz3 vz4 vz5\n  Monad a",fontsize=16,color="burlywood",shape="triangle"];73[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 6[label="",style="solid", color="burlywood", weight=3];
74[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldM vz3 vz4 (vz50 : vz51)\n  Monad a",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldM vz3 vz4 []\n  Monad a",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="vz3 vz4 vz50 >>= foldM0 vz3 vz51\n  Monad a",fontsize=16,color="blue",shape="box"];75[label=">>= :: (IO a) -> (a -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];8 -> 75[label="",style="solid", color="blue", weight=9];
75 -> 10[label="",style="solid", color="blue", weight=3];
76[label=">>= :: (Maybe a) -> (a -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];8 -> 76[label="",style="solid", color="blue", weight=9];
76 -> 11[label="",style="solid", color="blue", weight=3];
77[label=">>= :: ([] a) -> (a -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];8 -> 77[label="",style="solid", color="blue", weight=9];
77 -> 12[label="",style="solid", color="blue", weight=3];
9[label="return vz4\n  Monad a",fontsize=16,color="blue",shape="box"];78[label="return :: a -> IO a",fontsize=10,color="white",style="solid",shape="box"];9 -> 78[label="",style="solid", color="blue", weight=9];
78 -> 13[label="",style="solid", color="blue", weight=3];
79[label="return :: a -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];9 -> 79[label="",style="solid", color="blue", weight=9];
79 -> 14[label="",style="solid", color="blue", weight=3];
80[label="return :: a -> [] a",fontsize=10,color="white",style="solid",shape="box"];9 -> 80[label="",style="solid", color="blue", weight=9];
80 -> 15[label="",style="solid", color="blue", weight=3];
10[label="vz3 vz4 vz50 >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];10 -> 16[label="",style="solid", color="black", weight=3];
11 -> 17[label="",style="dashed", color="red", weight=0];
11[label="vz3 vz4 vz50 >>= foldM0 vz3 vz51",fontsize=16,color="magenta"];11 -> 18[label="",style="dashed", color="magenta", weight=3];
12 -> 19[label="",style="dashed", color="red", weight=0];
12[label="vz3 vz4 vz50 >>= foldM0 vz3 vz51",fontsize=16,color="magenta"];12 -> 20[label="",style="dashed", color="magenta", weight=3];
13[label="return vz4",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="return vz4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="return vz4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="primbindIO (vz3 vz4 vz50) (foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
18[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];18 -> 25[label="",style="dashed", color="green", weight=3];
18 -> 26[label="",style="dashed", color="green", weight=3];
17[label="vz6 >>= foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];83[label="vz6/Nothing",fontsize=10,color="white",style="solid",shape="box"];17 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 27[label="",style="solid", color="burlywood", weight=3];
84[label="vz6/Just vz60",fontsize=10,color="white",style="solid",shape="box"];17 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 28[label="",style="solid", color="burlywood", weight=3];
20[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];20 -> 33[label="",style="dashed", color="green", weight=3];
20 -> 34[label="",style="dashed", color="green", weight=3];
19[label="vz7 >>= foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];85[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];19 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 31[label="",style="solid", color="burlywood", weight=3];
86[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 32[label="",style="solid", color="burlywood", weight=3];
21[label="primretIO vz4",fontsize=16,color="black",shape="box"];21 -> 35[label="",style="solid", color="black", weight=3];
22[label="Just vz4",fontsize=16,color="green",shape="box"];23[label="vz4 : []",fontsize=16,color="green",shape="box"];24[label="terminator (vz3 vz4 vz50) (foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];24 -> 36[label="",style="solid", color="black", weight=3];
25[label="vz4",fontsize=16,color="green",shape="box"];26[label="vz50",fontsize=16,color="green",shape="box"];27[label="Nothing >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];27 -> 37[label="",style="solid", color="black", weight=3];
28[label="Just vz60 >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];28 -> 38[label="",style="solid", color="black", weight=3];
33[label="vz4",fontsize=16,color="green",shape="box"];34[label="vz50",fontsize=16,color="green",shape="box"];31[label="vz70 : vz71 >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];31 -> 39[label="",style="solid", color="black", weight=3];
32[label="[] >>= foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
35[label="terminator vz4",fontsize=16,color="black",shape="box"];35 -> 41[label="",style="solid", color="black", weight=3];
36[label="ter0m (vz3 vz4 vz50) (foldM0 vz3 vz51)",fontsize=16,color="green",shape="box"];36 -> 42[label="",style="dashed", color="green", weight=3];
36 -> 43[label="",style="dashed", color="green", weight=3];
37[label="Nothing",fontsize=16,color="green",shape="box"];38[label="foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];38 -> 44[label="",style="solid", color="black", weight=3];
39 -> 56[label="",style="dashed", color="red", weight=0];
39[label="foldM0 vz3 vz51 vz70 ++ (vz71 >>= foldM0 vz3 vz51)",fontsize=16,color="magenta"];39 -> 57[label="",style="dashed", color="magenta", weight=3];
39 -> 58[label="",style="dashed", color="magenta", weight=3];
40[label="[]",fontsize=16,color="green",shape="box"];41[label="ter1m vz4",fontsize=16,color="green",shape="box"];41 -> 47[label="",style="dashed", color="green", weight=3];
42[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];42 -> 48[label="",style="dashed", color="green", weight=3];
42 -> 49[label="",style="dashed", color="green", weight=3];
43[label="foldM0 vz3 vz51",fontsize=16,color="grey",shape="box"];43 -> 50[label="",style="dashed", color="grey", weight=3];
44 -> 5[label="",style="dashed", color="red", weight=0];
44[label="foldM vz3 vz60 vz51",fontsize=16,color="magenta"];44 -> 51[label="",style="dashed", color="magenta", weight=3];
44 -> 52[label="",style="dashed", color="magenta", weight=3];
57[label="foldM0 vz3 vz51 vz70",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3];
58 -> 19[label="",style="dashed", color="red", weight=0];
58[label="vz71 >>= foldM0 vz3 vz51",fontsize=16,color="magenta"];58 -> 62[label="",style="dashed", color="magenta", weight=3];
56[label="vz10 ++ vz8",fontsize=16,color="burlywood",shape="triangle"];90[label="vz10/vz100 : vz101",fontsize=10,color="white",style="solid",shape="box"];56 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 63[label="",style="solid", color="burlywood", weight=3];
91[label="vz10/[]",fontsize=10,color="white",style="solid",shape="box"];56 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 64[label="",style="solid", color="burlywood", weight=3];
47[label="vz4",fontsize=16,color="green",shape="box"];48[label="vz4",fontsize=16,color="green",shape="box"];49[label="vz50",fontsize=16,color="green",shape="box"];50[label="foldM0 vz3 vz51 vz9",fontsize=16,color="black",shape="triangle"];50 -> 55[label="",style="solid", color="black", weight=3];
51[label="vz51",fontsize=16,color="green",shape="box"];52[label="vz60",fontsize=16,color="green",shape="box"];61 -> 5[label="",style="dashed", color="red", weight=0];
61[label="foldM vz3 vz70 vz51",fontsize=16,color="magenta"];61 -> 67[label="",style="dashed", color="magenta", weight=3];
61 -> 68[label="",style="dashed", color="magenta", weight=3];
62[label="vz71",fontsize=16,color="green",shape="box"];63[label="(vz100 : vz101) ++ vz8",fontsize=16,color="black",shape="box"];63 -> 69[label="",style="solid", color="black", weight=3];
64[label="[] ++ vz8",fontsize=16,color="black",shape="box"];64 -> 70[label="",style="solid", color="black", weight=3];
55 -> 5[label="",style="dashed", color="red", weight=0];
55[label="foldM vz3 vz9 vz51",fontsize=16,color="magenta"];55 -> 65[label="",style="dashed", color="magenta", weight=3];
55 -> 66[label="",style="dashed", color="magenta", weight=3];
67[label="vz51",fontsize=16,color="green",shape="box"];68[label="vz70",fontsize=16,color="green",shape="box"];69[label="vz100 : vz101 ++ vz8",fontsize=16,color="green",shape="box"];69 -> 71[label="",style="dashed", color="green", weight=3];
70[label="vz8",fontsize=16,color="green",shape="box"];65[label="vz51",fontsize=16,color="green",shape="box"];66[label="vz9",fontsize=16,color="green",shape="box"];71 -> 56[label="",style="dashed", color="red", weight=0];
71[label="vz101 ++ vz8",fontsize=16,color="magenta"];71 -> 72[label="",style="dashed", color="magenta", weight=3];
72[label="vz101",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(:(vz100, vz101), vz8, h) → new_psPs(vz101, vz8, 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(:(vz100, vz101), vz8, h) → new_psPs(vz101, vz8, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

              ↳ Narrow

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

new_foldM0(vz3, vz51, vz9, h, ba) → new_foldM(vz3, vz51, ty_IO, h, ba)
new_gtGtEs(vz3, vz51, h, ba) → new_foldM(vz3, vz51, ty_Maybe, h, ba)
new_gtGtEs0(vz3, vz51, h, ba) → new_foldM(vz3, vz51, ty_[], h, ba)
new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) → new_foldM(vz3, vz51, ty_IO, h, ba)
new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) → new_gtGtEs0(vz3, vz51, h, ba)
new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) → new_gtGtEs(vz3, vz51, h, ba)
new_gtGtEs0(vz3, vz51, h, ba) → new_gtGtEs0(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 3 SCCs with 1 less node.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPSizeChangeProof

                        ↳ QDP

                        ↳ QDP

              ↳ Narrow

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

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

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

new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) → new_foldM(vz3, vz51, ty_IO, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesReductionPairsProof

                        ↳ QDP

              ↳ Narrow

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

new_gtGtEs0(vz3, vz51, h, ba) → new_foldM(vz3, vz51, ty_[], h, ba)
new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) → new_gtGtEs0(vz3, vz51, h, ba)
new_gtGtEs0(vz3, vz51, h, ba) → new_gtGtEs0(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_gtGtEs0(vz3, vz51, h, ba) → new_foldM(vz3, vz51, ty_[], h, ba)
new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) → new_gtGtEs0(vz3, vz51, h, ba)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(:(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(new_foldM(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + 2·x₃ + x₄ + x₅
POL(new_gtGtEs0(x₁, x₂, x₃, x₄)) = 1 + x₁ + 2·x₂ + x₃ + x₄
POL(ty_[]) = 0

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ UsableRulesReductionPairsProof

                            ↳ QDP

                              ↳ NonTerminationProof

                        ↳ QDP

              ↳ Narrow

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

new_gtGtEs0(vz3, vz51, h, ba) → new_gtGtEs0(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_gtGtEs0(vz3, vz51, h, ba) → new_gtGtEs0(vz3, vz51, h, ba)

The TRS R consists of the following rules:none

s = new_gtGtEs0(vz3, vz51, h, ba) evaluates to t =new_gtGtEs0(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_gtGtEs0(vz3, vz51, h, ba) to new_gtGtEs0(vz3, vz51, h, ba).

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPSizeChangeProof

              ↳ Narrow

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

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

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

new_gtGtEs(vz3, vz51, h, ba) → new_foldM(vz3, vz51, ty_Maybe, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5
new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) → new_gtGtEs(vz3, vz51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 4 >= 3, 5 >= 4

Haskell To QDPs

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

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

new_foldM0(vz3, vz51, vz9, h, ba, []) → new_foldM(vz3, vz9, vz51, ty_IO, h, ba, [])
new_gtGtEs(Just(vz60), vz3, vz51, h, ba, []) → new_foldM(vz3, vz60, vz51, ty_Maybe, h, ba, [])

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.