/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) MRRProof [EQUIVALENT, 40 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) TRUE


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

scanr1 :: (a  ->  a  ->  a)  ->  List a  ->  List a;
scanr1 f Nil = Nil;
scanr1 f (Cons x Nil) = Cons x Nil;
scanr1 f (Cons x xs) = Cons (f x (scanr1Q f xs)) (scanr1Qs f xs);

scanr1Q vw vx = scanr1Q1 vw vx (scanr1Vu41 vw vx);

scanr1Q1 vw vx (Cons q vv) = q;

scanr1Qs vw vx = scanr1Qs0 vw vx (scanr1Vu41 vw vx);

scanr1Qs0 vw vx qs = qs;

scanr1Vu41 vw vx = scanr1 vw vx;

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

scanr1 :: (a  ->  a  ->  a)  ->  List a  ->  List a;
scanr1 f Nil = Nil;
scanr1 f (Cons x Nil) = Cons x Nil;
scanr1 f (Cons x xs) = Cons (f x (scanr1Q f xs)) (scanr1Qs f xs);

scanr1Q vw vx = scanr1Q1 vw vx (scanr1Vu41 vw vx);

scanr1Q1 vw vx (Cons q vv) = q;

scanr1Qs vw vx = scanr1Qs0 vw vx (scanr1Vu41 vw vx);

scanr1Qs0 vw vx qs = qs;

scanr1Vu41 vw vx = scanr1 vw vx;

}

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

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

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

scanr1 :: (a  ->  a  ->  a)  ->  List a  ->  List a;
scanr1 f Nil = Nil;
scanr1 f (Cons x Nil) = Cons x Nil;
scanr1 f (Cons x xs) = Cons (f x (scanr1Q f xs)) (scanr1Qs f xs);

scanr1Q vw vx = scanr1Q1 vw vx (scanr1Vu41 vw vx);

scanr1Q1 vw vx (Cons q vv) = q;

scanr1Qs vw vx = scanr1Qs0 vw vx (scanr1Vu41 vw vx);

scanr1Qs0 vw vx qs = qs;

scanr1Vu41 vw vx = scanr1 vw vx;

}

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

(5) 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/Cons 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/Nil",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 (Cons wu40 wu41)",fontsize=16,color="burlywood",shape="box"];31[label="wu41/Cons 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/Nil",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 Nil",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="scanr1 wu3 (Cons wu40 (Cons wu410 wu411))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="scanr1 wu3 (Cons wu40 Nil)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="Nil",fontsize=16,color="green",shape="box"];10[label="Cons (wu3 wu40 (scanr1Q wu3 (Cons wu410 wu411))) (scanr1Qs wu3 (Cons 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="Cons wu40 Nil",fontsize=16,color="green",shape="box"];12[label="wu3 wu40 (scanr1Q wu3 (Cons 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 (Cons 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 (Cons wu410 wu411)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="scanr1Qs0 wu3 (Cons wu410 wu411) (scanr1Vu41 wu3 (Cons 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 (Cons wu410 wu411) (scanr1Vu41 wu3 (Cons wu410 wu411))",fontsize=16,color="magenta"];17 -> 22[label="",style="dashed", color="magenta", weight=3];
18[label="scanr1Vu41 wu3 (Cons 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 (Cons wu410 wu411)",fontsize=16,color="magenta"];21[label="scanr1Q1 wu3 (Cons wu410 wu411) wu5",fontsize=16,color="burlywood",shape="triangle"];33[label="wu5/Cons 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/Nil",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 (Cons wu410 wu411)",fontsize=16,color="magenta"];20 -> 26[label="",style="dashed", color="magenta", weight=3];
24[label="scanr1Q1 wu3 (Cons wu410 wu411) (Cons wu50 wu51)",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="scanr1Q1 wu3 (Cons wu410 wu411) Nil",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
26[label="Cons 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"];}

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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


Used ordering: Polynomial interpretation [POLO]:

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


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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

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

(10)
TRUE