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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

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

data Tup0 = Tup0 ;

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);

gtGt0 q vv = q;

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

gtGtNil :: List a  ->  List b  ->  List b;
gtGtNil p q = gtGtEsNil p (gtGt0 q);

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

returnNil :: a  ->  List a;
returnNil x = Cons x Nil;

sequence_Nil :: List (List a)  ->  List Tup0;
sequence_Nil = foldr gtGtNil (returnNil Tup0);

}

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

(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 ;

data Tup0 = Tup0 ;

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);

gtGt0 q vv = q;

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

gtGtNil :: List b  ->  List a  ->  List a;
gtGtNil p q = gtGtEsNil p (gtGt0 q);

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

returnNil :: a  ->  List a;
returnNil x = Cons x Nil;

sequence_Nil :: List (List a)  ->  List Tup0;
sequence_Nil = foldr gtGtNil (returnNil Tup0);

}

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

(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 ;

data Tup0 = Tup0 ;

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);

gtGt0 q vv = q;

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

gtGtNil :: List b  ->  List a  ->  List a;
gtGtNil p q = gtGtEsNil p (gtGt0 q);

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

returnNil :: a  ->  List a;
returnNil x = Cons x Nil;

sequence_Nil :: List (List a)  ->  List Tup0;
sequence_Nil = foldr gtGtNil (returnNil Tup0);

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence_Nil",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence_Nil vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr gtGtNil (returnNil Tup0) vy3",fontsize=16,color="burlywood",shape="triangle"];28[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 5[label="",style="solid", color="burlywood", weight=3];
29[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr gtGtNil (returnNil Tup0) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr gtGtNil (returnNil Tup0) 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="gtGtNil vy30 (foldr gtGtNil (returnNil Tup0) vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="returnNil Tup0",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr gtGtNil (returnNil Tup0) vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9[label="gtGtNil vy30 vy4",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3];
11[label="Cons Tup0 Nil",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="gtGtEsNil vy30 (gtGt0 vy4)",fontsize=16,color="burlywood",shape="triangle"];30[label="vy30/Cons vy300 vy301",fontsize=10,color="white",style="solid",shape="box"];13 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 14[label="",style="solid", color="burlywood", weight=3];
31[label="vy30/Nil",fontsize=10,color="white",style="solid",shape="box"];13 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="gtGtEsNil (Cons vy300 vy301) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="gtGtEsNil Nil (gtGt0 vy4)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="psPs (gtGt0 vy4 vy300) (gtGtEsNil vy301 (gtGt0 vy4))",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="Nil",fontsize=16,color="green",shape="box"];19 -> 13[label="",style="dashed", color="red", weight=0];
19[label="gtGtEsNil vy301 (gtGt0 vy4)",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
18[label="psPs (gtGt0 vy4 vy300) vy5",fontsize=16,color="black",shape="triangle"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="vy301",fontsize=16,color="green",shape="box"];21[label="psPs vy4 vy5",fontsize=16,color="burlywood",shape="triangle"];32[label="vy4/Cons vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];21 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 22[label="",style="solid", color="burlywood", weight=3];
33[label="vy4/Nil",fontsize=10,color="white",style="solid",shape="box"];21 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 23[label="",style="solid", color="burlywood", weight=3];
22[label="psPs (Cons vy40 vy41) vy5",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="psPs Nil vy5",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="Cons vy40 (psPs vy41 vy5)",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
25[label="vy5",fontsize=16,color="green",shape="box"];26 -> 21[label="",style="dashed", color="red", weight=0];
26[label="psPs vy41 vy5",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
27[label="vy41",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_foldr(Cons(vy30, vy31), h) -> new_foldr(vy31, 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_foldr(Cons(vy30, vy31), h) -> new_foldr(vy31, h)
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

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

   new_psPs(Cons(vy40, vy41), vy5) -> new_psPs(vy41, vy5)

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_psPs(Cons(vy40, vy41), vy5) -> new_psPs(vy41, vy5)
The graph contains the following edges 1 > 1, 2 >= 2


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

(12)
YES

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEsNil(Cons(vy300, vy301), vy4, h) -> new_gtGtEsNil(vy301, vy4, h)

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

(14) 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_gtGtEsNil(Cons(vy300, vy301), vy4, h) -> new_gtGtEsNil(vy301, vy4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(15)
YES