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

MAYBE 1.629 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could not be shown:

↳ HASKELL

  ↳ CR

mainModule List

((unfoldr :: (a -> Maybe (b,a)) -> a -> [b]) :: (a -> Maybe (b,a)) -> a -> [b])

module List where

import qualified Maybe
import qualified Prelude

unfoldr :: (a -> Maybe (b,a)) -> a -> [b]

unfoldr

f b

case	f b of
	Just (a,new_b)	->	a : unfoldr f new_b
	Nothing	->	[]

module Maybe where

import qualified List
import qualified Prelude

Case Reductions:
The following Case expression

case f b of
Just (a,new_b) → a : unfoldr f new_b

Nothing → []

is transformed to

unfoldr0 f (Just (a,new_b)) = a : unfoldr f new_b

unfoldr0 f Nothing = []

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule List

((unfoldr :: (b -> Maybe (a,b)) -> b -> [a]) :: (b -> Maybe (a,b)) -> b -> [a])

module List where

import qualified Maybe
import qualified Prelude

unfoldr :: (a -> Maybe (b,a)) -> a -> [b]

unfoldr

f b

unfoldr0 f (f b)

unfoldr0	f (Just (a,new_b))	=	a : unfoldr f new_b
unfoldr0	f Nothing	=	[]

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule List

((unfoldr :: (a -> Maybe (b,a)) -> a -> [b]) :: (a -> Maybe (b,a)) -> a -> [b])

module List where

import qualified Maybe
import qualified Prelude

unfoldr :: (b -> Maybe (a,b)) -> b -> [a]

unfoldr

f b

unfoldr0 f (f b)

unfoldr0	f (Just (a,new_b))	=	a : unfoldr f new_b
unfoldr0	f Nothing	=	[]

module Maybe where

import qualified List
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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

mainModule List

(unfoldr :: (b -> Maybe (a,b)) -> b -> [a])

module List where

import qualified Maybe
import qualified Prelude

unfoldr :: (b -> Maybe (a,b)) -> b -> [a]

unfoldr

f b

unfoldr0 f (f b)

unfoldr0	f (Just (a,new_b))	=	a : unfoldr f new_b
unfoldr0	f Nothing	=	[]

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="unfoldr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="unfoldr vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="unfoldr vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5 -> 6[label="",style="dashed", color="red", weight=0];
5[label="unfoldr0 vy3 (vy3 vy4)",fontsize=16,color="magenta"];5 -> 7[label="",style="dashed", color="magenta", weight=3];
7[label="vy3 vy4",fontsize=16,color="green",shape="box"];7 -> 11[label="",style="dashed", color="green", weight=3];
6[label="unfoldr0 vy3 vy5",fontsize=16,color="burlywood",shape="triangle"];18[label="vy5/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 9[label="",style="solid", color="burlywood", weight=3];
19[label="vy5/Just vy50",fontsize=10,color="white",style="solid",shape="box"];6 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 10[label="",style="solid", color="burlywood", weight=3];
11[label="vy4",fontsize=16,color="green",shape="box"];9[label="unfoldr0 vy3 Nothing",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="unfoldr0 vy3 (Just vy50)",fontsize=16,color="burlywood",shape="box"];20[label="vy50/(vy500,vy501)",fontsize=10,color="white",style="solid",shape="box"];10 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 13[label="",style="solid", color="burlywood", weight=3];
12[label="[]",fontsize=16,color="green",shape="box"];13[label="unfoldr0 vy3 (Just (vy500,vy501))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="vy500 : unfoldr vy3 vy501",fontsize=16,color="green",shape="box"];14 -> 15[label="",style="dashed", color="green", weight=3];
15 -> 4[label="",style="dashed", color="red", weight=0];
15[label="unfoldr vy3 vy501",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
16[label="vy501",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ NonTerminationProof

              ↳ Narrow

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

new_unfoldr(vy3, ba, bb) → new_unfoldr0(vy3, ba, bb)
new_unfoldr0(vy3, ba, bb) → new_unfoldr(vy3, ba, bb)

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 narrowing to the left:

The TRS P consists of the following rules:

new_unfoldr(vy3, ba, bb) → new_unfoldr0(vy3, ba, bb)
new_unfoldr0(vy3, ba, bb) → new_unfoldr(vy3, ba, bb)

The TRS R consists of the following rules:none

s = new_unfoldr0(vy3', ba', bb') evaluates to t =new_unfoldr0(vy3', ba', bb')

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

new_unfoldr0(vy3', ba', bb') → new_unfoldr(vy3', ba', bb')
with rule new_unfoldr0(vy3'', ba'', bb'') → new_unfoldr(vy3'', ba'', bb'') at position [] and matcher [vy3'' / vy3', bb'' / bb', ba'' / ba']

new_unfoldr(vy3', ba', bb') → new_unfoldr0(vy3', ba', bb')
with rule new_unfoldr(vy3, ba, bb) → new_unfoldr0(vy3, ba, bb)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

Haskell To QDPs

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

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

new_unfoldr0(vy3, Just(@2(vy500, vy501)), ba, bb, []) → new_unfoldr(vy3, vy501, ba, bb, [])

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 1 less node.

unfoldr0	f (Just (a,new_b))	= a : unfoldr f new_b
unfoldr0	f Nothing	= []