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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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(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 ;

zip :: List a  ->  List b  ->  List (Tup2 a b);
zip = zipWith zip0;

zip0 a b = Tup2 a b;

zipWith :: (c  ->  a  ->  b)  ->  List c  ->  List a  ->  List b;
zipWith z (Cons a as) (Cons b bs) = Cons (z a b) (zipWith z as bs);
zipWith vv vw vx = Nil;

}

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

(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 b a = Tup2 b a ;

zip :: List b  ->  List a  ->  List (Tup2 b a);
zip = zipWith zip0;

zip0 a b = Tup2 a b;

zipWith :: (c  ->  a  ->  b)  ->  List c  ->  List a  ->  List b;
zipWith z (Cons a as) (Cons b bs) = Cons (z a b) (zipWith z as bs);
zipWith vv vw vx = Nil;

}

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

(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 ;

zip :: List a  ->  List b  ->  List (Tup2 a b);
zip = zipWith zip0;

zip0 a b = Tup2 a b;

zipWith :: (b  ->  c  ->  a)  ->  List b  ->  List c  ->  List a;
zipWith z (Cons a as) (Cons b bs) = Cons (z a b) (zipWith z as bs);
zipWith vv vw vx = Nil;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="zip",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="zip wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="zip wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="zipWith zip0 wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];18[label="wu3/Cons wu30 wu31",fontsize=10,color="white",style="solid",shape="box"];5 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 6[label="",style="solid", color="burlywood", weight=3];
19[label="wu3/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="zipWith zip0 (Cons wu30 wu31) wu4",fontsize=16,color="burlywood",shape="box"];20[label="wu4/Cons wu40 wu41",fontsize=10,color="white",style="solid",shape="box"];6 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 8[label="",style="solid", color="burlywood", weight=3];
21[label="wu4/Nil",fontsize=10,color="white",style="solid",shape="box"];6 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="zipWith zip0 Nil wu4",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="zipWith zip0 (Cons wu30 wu31) (Cons wu40 wu41)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="zipWith zip0 (Cons wu30 wu31) Nil",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="Nil",fontsize=16,color="green",shape="box"];11[label="Cons (zip0 wu30 wu40) (zipWith zip0 wu31 wu41)",fontsize=16,color="green",shape="box"];11 -> 13[label="",style="dashed", color="green", weight=3];
11 -> 14[label="",style="dashed", color="green", weight=3];
12[label="Nil",fontsize=16,color="green",shape="box"];13[label="zip0 wu30 wu40",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 5[label="",style="dashed", color="red", weight=0];
14[label="zipWith zip0 wu31 wu41",fontsize=16,color="magenta"];14 -> 16[label="",style="dashed", color="magenta", weight=3];
14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="Tup2 wu30 wu40",fontsize=16,color="green",shape="box"];16[label="wu31",fontsize=16,color="green",shape="box"];17[label="wu41",fontsize=16,color="green",shape="box"];}

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

   new_zipWith(Cons(wu30, wu31), Cons(wu40, wu41), h, ba) -> new_zipWith(wu31, wu41, 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_zipWith(Cons(wu30, wu31), Cons(wu40, wu41), h, ba) -> new_zipWith(wu31, wu41, h, ba)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4


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

(8)
YES