/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) QDP
(11) MRRProof [EQUIVALENT, 29 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


----------------------------------------

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\(q : _)->q"
is transformed to
"q1 (q : _) = q;
"
The following Lambda expression
"\qs->qs"
is transformed to
"qs0 qs = qs;
"

----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"f x q : qs where {
q  = q1 vu41;
;
q1 (q : vv) = q;
;
qs  = qs0 vu41;
;
qs0 qs = qs;
;
vu41  = scanr1 f xs;
} 
"
are unpacked to the following functions on top level
"scanr1Q1 vy vz (q : vv) = q;
"
"scanr1Qs0 vy vz qs = qs;
"
"scanr1Vu41 vy vz = scanr1 vy vz;
"
"scanr1Q vy vz = scanr1Q1 vy vz (scanr1Vu41 vy vz);
"
"scanr1Qs vy vz = scanr1Qs0 vy vz (scanr1Vu41 vy vz);
"

----------------------------------------

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

----------------------------------------

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="scanr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="scanr1 wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="scanr1 wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];29[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 5[label="",style="solid", color="burlywood", weight=3];
30[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="scanr1 wu3 (wu40 : wu41)",fontsize=16,color="burlywood",shape="box"];31[label="wu41/wu410 : wu411",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 7[label="",style="solid", color="burlywood", weight=3];
32[label="wu41/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="scanr1 wu3 []",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="scanr1 wu3 (wu40 : wu410 : wu411)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="scanr1 wu3 (wu40 : [])",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];10[label="wu3 wu40 (scanr1Q wu3 (wu410 : wu411)) : scanr1Qs wu3 (wu410 : wu411)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 13[label="",style="dashed", color="green", weight=3];
11[label="wu40 : []",fontsize=16,color="green",shape="box"];12[label="wu3 wu40 (scanr1Q wu3 (wu410 : wu411))",fontsize=16,color="green",shape="box"];12 -> 14[label="",style="dashed", color="green", weight=3];
12 -> 15[label="",style="dashed", color="green", weight=3];
13[label="scanr1Qs wu3 (wu410 : wu411)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="wu40",fontsize=16,color="green",shape="box"];15[label="scanr1Q wu3 (wu410 : wu411)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="scanr1Qs0 wu3 (wu410 : wu411) (scanr1Vu41 wu3 (wu410 : wu411))",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17 -> 21[label="",style="dashed", color="red", weight=0];
17[label="scanr1Q1 wu3 (wu410 : wu411) (scanr1Vu41 wu3 (wu410 : wu411))",fontsize=16,color="magenta"];17 -> 22[label="",style="dashed", color="magenta", weight=3];
18[label="scanr1Vu41 wu3 (wu410 : wu411)",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
22 -> 18[label="",style="dashed", color="red", weight=0];
22[label="scanr1Vu41 wu3 (wu410 : wu411)",fontsize=16,color="magenta"];21[label="scanr1Q1 wu3 (wu410 : wu411) wu5",fontsize=16,color="burlywood",shape="triangle"];33[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];21 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 24[label="",style="solid", color="burlywood", weight=3];
34[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 25[label="",style="solid", color="burlywood", weight=3];
20 -> 4[label="",style="dashed", color="red", weight=0];
20[label="scanr1 wu3 (wu410 : wu411)",fontsize=16,color="magenta"];20 -> 26[label="",style="dashed", color="magenta", weight=3];
24[label="scanr1Q1 wu3 (wu410 : wu411) (wu50 : wu51)",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="scanr1Q1 wu3 (wu410 : wu411) []",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
26[label="wu410 : wu411",fontsize=16,color="green",shape="box"];27[label="wu50",fontsize=16,color="green",shape="box"];28[label="error []",fontsize=16,color="red",shape="box"];}

----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1Vu41(wu3, wu410, wu411, h)
   new_scanr1Vu41(wu3, wu410, wu411, h) -> new_scanr1(wu3, :(wu410, wu411), h)
   new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1(wu3, :(wu410, wu411), h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1Vu41(wu3, wu410, wu411, h)
   new_scanr1(wu3, :(wu40, :(wu410, wu411)), h) -> new_scanr1(wu3, :(wu410, wu411), h)


Used ordering: Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(new_scanr1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(new_scanr1Vu41(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + 2*x_3 + x_4


----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_scanr1Vu41(wu3, wu410, wu411, h) -> new_scanr1(wu3, :(wu410, wu411), h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(14)
TRUE