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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

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

data Tup2 a b = Tup2 a b ;

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);

unzip :: List (Tup2 b a)  ->  Tup2 (List b) (List a);
unzip = foldr unzip0 (Tup2 Nil Nil);

unzip0 (Tup2 a b) vv = Tup2 (Cons a (unzip00 vv)) (Cons b (unzip01 vv));

unzip00 (Tup2 as bs) = as;

unzip01 (Tup2 as bs) = bs;

}

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

(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 Tup2 a b = Tup2 a b ;

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);

unzip :: List (Tup2 a b)  ->  Tup2 (List a) (List b);
unzip = foldr unzip0 (Tup2 Nil Nil);

unzip0 (Tup2 a b) vv = Tup2 (Cons a (unzip00 vv)) (Cons b (unzip01 vv));

unzip00 (Tup2 as bs) = as;

unzip01 (Tup2 as bs) = bs;

}

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

(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 Tup2 b a = Tup2 b a ;

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);

unzip :: List (Tup2 a b)  ->  Tup2 (List a) (List b);
unzip = foldr unzip0 (Tup2 Nil Nil);

unzip0 (Tup2 a b) vv = Tup2 (Cons a (unzip00 vv)) (Cons b (unzip01 vv));

unzip00 (Tup2 as bs) = as;

unzip01 (Tup2 as bs) = bs;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="unzip",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="unzip vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr unzip0 (Tup2 Nil Nil) vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 5[label="",style="solid", color="burlywood", weight=3];
21[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr unzip0 (Tup2 Nil Nil) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr unzip0 (Tup2 Nil 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="unzip0 vy30 (foldr unzip0 (Tup2 Nil Nil) vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="Tup2 Nil Nil",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr unzip0 (Tup2 Nil Nil) vy31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="unzip0 vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy30/Tup2 vy300 vy301",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="vy31",fontsize=16,color="green",shape="box"];12[label="unzip0 (Tup2 vy300 vy301) vy4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="Tup2 (Cons vy300 (unzip00 vy4)) (Cons vy301 (unzip01 vy4))",fontsize=16,color="green",shape="box"];13 -> 14[label="",style="dashed", color="green", weight=3];
13 -> 15[label="",style="dashed", color="green", weight=3];
14[label="unzip00 vy4",fontsize=16,color="burlywood",shape="box"];23[label="vy4/Tup2 vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];14 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 16[label="",style="solid", color="burlywood", weight=3];
15[label="unzip01 vy4",fontsize=16,color="burlywood",shape="box"];24[label="vy4/Tup2 vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];15 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="unzip00 (Tup2 vy40 vy41)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="unzip01 (Tup2 vy40 vy41)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="vy40",fontsize=16,color="green",shape="box"];19[label="vy41",fontsize=16,color="green",shape="box"];}

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

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

   new_foldr(Cons(vy30, vy31), h, ba) -> new_foldr(vy31, h, ba)

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

(7) 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, ba) -> new_foldr(vy31, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(8)
YES