/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES


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

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

concat :: List (List a)  ->  List a;
concat = foldr psPs Nil;

foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
foldr f z Nil = z;
foldr f z (Cons x xs) = f x (foldr f z xs);

psPs :: List a  ->  List a  ->  List a;
psPs Nil ys = ys;
psPs (Cons x xs) ys = Cons x (psPs xs ys);

}

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

(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 ;

concat :: List (List a)  ->  List a;
concat = foldr psPs Nil;

foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
foldr f z Nil = z;
foldr f z (Cons x xs) = f x (foldr f z xs);

psPs :: List a  ->  List a  ->  List a;
psPs Nil ys = ys;
psPs (Cons x xs) ys = Cons x (psPs xs ys);

}

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

(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 ;

concat :: List (List a)  ->  List a;
concat = foldr psPs Nil;

foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
foldr f z Nil = z;
foldr f z (Cons x xs) = f x (foldr f z xs);

psPs :: List a  ->  List a  ->  List a;
psPs Nil ys = ys;
psPs (Cons x xs) ys = Cons x (psPs xs ys);

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="concat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="concat vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr psPs Nil vx3",fontsize=16,color="burlywood",shape="triangle"];18[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 5[label="",style="solid", color="burlywood", weight=3];
19[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr psPs Nil (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr psPs Nil Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="psPs vx30 (foldr psPs Nil vx31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="Nil",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr psPs Nil vx31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="psPs vx30 vx4",fontsize=16,color="burlywood",shape="triangle"];20[label="vx30/Cons vx300 vx301",fontsize=10,color="white",style="solid",shape="box"];9 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 12[label="",style="solid", color="burlywood", weight=3];
21[label="vx30/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="vx31",fontsize=16,color="green",shape="box"];12[label="psPs (Cons vx300 vx301) vx4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="psPs Nil vx4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="Cons vx300 (psPs vx301 vx4)",fontsize=16,color="green",shape="box"];14 -> 16[label="",style="dashed", color="green", weight=3];
15[label="vx4",fontsize=16,color="green",shape="box"];16 -> 9[label="",style="dashed", color="red", weight=0];
16[label="psPs vx301 vx4",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
17[label="vx301",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(Cons(vx300, vx301), vx4, h) -> new_psPs(vx301, vx4, h)

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

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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(Cons(vx300, vx301), vx4, h) -> new_psPs(vx301, vx4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(9)
YES

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

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

   new_foldr(Cons(vx30, vx31), h) -> new_foldr(vx31, h)

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

(11) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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_foldr(Cons(vx30, vx31), h) -> new_foldr(vx31, h)
The graph contains the following edges 1 > 1, 2 >= 2


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

(12)
YES