/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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;

concatMap :: (a  ->  List b)  ->  List a  ->  List b;
concatMap f = pt concat (map f);

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);

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

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

pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
pt f g x = f (g x);

}

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

(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;

concatMap :: (b  ->  List a)  ->  List b  ->  List a;
concatMap f = pt concat (map f);

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

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

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

pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
pt f g x = f (g x);

}

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

(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;

concatMap :: (a  ->  List b)  ->  List a  ->  List b;
concatMap f = pt concat (map f);

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

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

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

pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
pt f g x = f (g x);

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="concatMap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="concatMap vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="concatMap vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt concat (map vx3) vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concat (map vx3 vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr psPs Nil (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];30[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 8[label="",style="solid", color="burlywood", weight=3];
31[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr psPs Nil (map vx3 (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldr psPs Nil (map vx3 Nil)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldr psPs Nil (Cons (vx3 vx40) (map vx3 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr psPs Nil Nil",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12 -> 17[label="",style="dashed", color="red", weight=0];
12[label="psPs (vx3 vx40) (foldr psPs Nil (map vx3 vx41))",fontsize=16,color="magenta"];12 -> 18[label="",style="dashed", color="magenta", weight=3];
12 -> 19[label="",style="dashed", color="magenta", weight=3];
13[label="Nil",fontsize=16,color="green",shape="box"];18[label="vx3 vx40",fontsize=16,color="green",shape="box"];18 -> 21[label="",style="dashed", color="green", weight=3];
19 -> 7[label="",style="dashed", color="red", weight=0];
19[label="foldr psPs Nil (map vx3 vx41)",fontsize=16,color="magenta"];19 -> 22[label="",style="dashed", color="magenta", weight=3];
17[label="psPs vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];32[label="vx6/Cons vx60 vx61",fontsize=10,color="white",style="solid",shape="box"];17 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 23[label="",style="solid", color="burlywood", weight=3];
33[label="vx6/Nil",fontsize=10,color="white",style="solid",shape="box"];17 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 24[label="",style="solid", color="burlywood", weight=3];
21[label="vx40",fontsize=16,color="green",shape="box"];22[label="vx41",fontsize=16,color="green",shape="box"];23[label="psPs (Cons vx60 vx61) vx5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="psPs Nil vx5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
26[label="Cons vx60 (psPs vx61 vx5)",fontsize=16,color="green",shape="box"];26 -> 28[label="",style="dashed", color="green", weight=3];
27[label="vx5",fontsize=16,color="green",shape="box"];28 -> 17[label="",style="dashed", color="red", weight=0];
28[label="psPs vx61 vx5",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
29[label="vx61",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(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, 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(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, 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(vx3, Cons(vx40, vx41), h, ba) -> new_foldr(vx3, vx41, h, ba)

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(vx3, Cons(vx40, vx41), h, ba) -> new_foldr(vx3, vx41, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(12)
YES