/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 MyBool = MyTrue  | MyFalse ;

data Tup2 a b = Tup2 a b ;

otherwise :: MyBool;
otherwise = MyTrue;

span :: (a  ->  MyBool)  ->  List a  ->  Tup2 (List a) (List a);
span p Nil = span3 p Nil;
span p (Cons vv vw) = span2 p (Cons vv vw);

span2 p (Cons vv vw) = span2Span1 p vw p vv vw (p vv);

span2Span0 wx wy p vv vw MyTrue = Tup2 Nil (Cons vv vw);

span2Span1 wx wy p vv vw MyTrue = Tup2 (Cons vv (span2Ys wx wy)) (span2Zs wx wy);
span2Span1 wx wy p vv vw MyFalse = span2Span0 wx wy p vv vw otherwise;

span2Vu43 wx wy = span wx wy;

span2Ys wx wy = span2Ys0 wx wy (span2Vu43 wx wy);

span2Ys0 wx wy (Tup2 ys vx) = ys;

span2Zs wx wy = span2Zs0 wx wy (span2Vu43 wx wy);

span2Zs0 wx wy (Tup2 vy zs) = zs;

span3 p Nil = Tup2 Nil Nil;
span3 wv ww = span2 wv ww;

}

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

(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 MyBool = MyTrue  | MyFalse ;

data Tup2 b a = Tup2 b a ;

otherwise :: MyBool;
otherwise = MyTrue;

span :: (a  ->  MyBool)  ->  List a  ->  Tup2 (List a) (List a);
span p Nil = span3 p Nil;
span p (Cons vv vw) = span2 p (Cons vv vw);

span2 p (Cons vv vw) = span2Span1 p vw p vv vw (p vv);

span2Span0 wx wy p vv vw MyTrue = Tup2 Nil (Cons vv vw);

span2Span1 wx wy p vv vw MyTrue = Tup2 (Cons vv (span2Ys wx wy)) (span2Zs wx wy);
span2Span1 wx wy p vv vw MyFalse = span2Span0 wx wy p vv vw otherwise;

span2Vu43 wx wy = span wx wy;

span2Ys wx wy = span2Ys0 wx wy (span2Vu43 wx wy);

span2Ys0 wx wy (Tup2 ys vx) = ys;

span2Zs wx wy = span2Zs0 wx wy (span2Vu43 wx wy);

span2Zs0 wx wy (Tup2 vy zs) = zs;

span3 p Nil = Tup2 Nil Nil;
span3 wv ww = span2 wv ww;

}

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

(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 MyBool = MyTrue  | MyFalse ;

data Tup2 a b = Tup2 a b ;

otherwise :: MyBool;
otherwise = MyTrue;

span :: (a  ->  MyBool)  ->  List a  ->  Tup2 (List a) (List a);
span p Nil = span3 p Nil;
span p (Cons vv vw) = span2 p (Cons vv vw);

span2 p (Cons vv vw) = span2Span1 p vw p vv vw (p vv);

span2Span0 wx wy p vv vw MyTrue = Tup2 Nil (Cons vv vw);

span2Span1 wx wy p vv vw MyTrue = Tup2 (Cons vv (span2Ys wx wy)) (span2Zs wx wy);
span2Span1 wx wy p vv vw MyFalse = span2Span0 wx wy p vv vw otherwise;

span2Vu43 wx wy = span wx wy;

span2Ys wx wy = span2Ys0 wx wy (span2Vu43 wx wy);

span2Ys0 wx wy (Tup2 ys vx) = ys;

span2Zs wx wy = span2Zs0 wx wy (span2Vu43 wx wy);

span2Zs0 wx wy (Tup2 vy zs) = zs;

span3 p Nil = Tup2 Nil Nil;
span3 wv ww = span2 wv ww;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="span",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="span wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="span wz3 wz4",fontsize=16,color="burlywood",shape="triangle"];39[label="wz4/Cons wz40 wz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 5[label="",style="solid", color="burlywood", weight=3];
40[label="wz4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="span wz3 (Cons wz40 wz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="span wz3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="span2 wz3 (Cons wz40 wz41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="span3 wz3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="span2Span1 wz3 wz41 wz3 wz40 wz41 (wz3 wz40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="Tup2 Nil Nil",fontsize=16,color="green",shape="box"];12[label="wz3 wz40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
11[label="span2Span1 wz3 wz41 wz3 wz40 wz41 wz5",fontsize=16,color="burlywood",shape="triangle"];41[label="wz5/MyTrue",fontsize=10,color="white",style="solid",shape="box"];11 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 14[label="",style="solid", color="burlywood", weight=3];
42[label="wz5/MyFalse",fontsize=10,color="white",style="solid",shape="box"];11 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 15[label="",style="solid", color="burlywood", weight=3];
16[label="wz40",fontsize=16,color="green",shape="box"];14[label="span2Span1 wz3 wz41 wz3 wz40 wz41 MyTrue",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="span2Span1 wz3 wz41 wz3 wz40 wz41 MyFalse",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
17[label="Tup2 (Cons wz40 (span2Ys wz3 wz41)) (span2Zs wz3 wz41)",fontsize=16,color="green",shape="box"];17 -> 19[label="",style="dashed", color="green", weight=3];
17 -> 20[label="",style="dashed", color="green", weight=3];
18[label="span2Span0 wz3 wz41 wz3 wz40 wz41 otherwise",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
19[label="span2Ys wz3 wz41",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
20[label="span2Zs wz3 wz41",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
21[label="span2Span0 wz3 wz41 wz3 wz40 wz41 MyTrue",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22 -> 27[label="",style="dashed", color="red", weight=0];
22[label="span2Ys0 wz3 wz41 (span2Vu43 wz3 wz41)",fontsize=16,color="magenta"];22 -> 28[label="",style="dashed", color="magenta", weight=3];
23 -> 32[label="",style="dashed", color="red", weight=0];
23[label="span2Zs0 wz3 wz41 (span2Vu43 wz3 wz41)",fontsize=16,color="magenta"];23 -> 33[label="",style="dashed", color="magenta", weight=3];
24[label="Tup2 Nil (Cons wz40 wz41)",fontsize=16,color="green",shape="box"];28[label="span2Vu43 wz3 wz41",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
27[label="span2Ys0 wz3 wz41 wz6",fontsize=16,color="burlywood",shape="triangle"];43[label="wz6/Tup2 wz60 wz61",fontsize=10,color="white",style="solid",shape="box"];27 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 31[label="",style="solid", color="burlywood", weight=3];
33 -> 28[label="",style="dashed", color="red", weight=0];
33[label="span2Vu43 wz3 wz41",fontsize=16,color="magenta"];32[label="span2Zs0 wz3 wz41 wz7",fontsize=16,color="burlywood",shape="triangle"];44[label="wz7/Tup2 wz70 wz71",fontsize=10,color="white",style="solid",shape="box"];32 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 35[label="",style="solid", color="burlywood", weight=3];
30 -> 4[label="",style="dashed", color="red", weight=0];
30[label="span wz3 wz41",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3];
31[label="span2Ys0 wz3 wz41 (Tup2 wz60 wz61)",fontsize=16,color="black",shape="box"];31 -> 37[label="",style="solid", color="black", weight=3];
35[label="span2Zs0 wz3 wz41 (Tup2 wz70 wz71)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
36[label="wz41",fontsize=16,color="green",shape="box"];37[label="wz60",fontsize=16,color="green",shape="box"];38[label="wz71",fontsize=16,color="green",shape="box"];}

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

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

   new_span2Vu43(wz3, wz41, h) -> new_span(wz3, wz41, h)
   new_span(wz3, Cons(wz40, wz41), h) -> new_span2Span1(wz3, wz41, wz40, h)
   new_span2Span1(wz3, wz41, wz40, h) -> new_span(wz3, wz41, h)
   new_span2Span1(wz3, wz41, wz40, h) -> new_span2Vu43(wz3, wz41, h)

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_span(wz3, Cons(wz40, wz41), h) -> new_span2Span1(wz3, wz41, wz40, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4


*new_span2Span1(wz3, wz41, wz40, h) -> new_span2Vu43(wz3, wz41, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3


*new_span2Span1(wz3, wz41, wz40, h) -> new_span(wz3, wz41, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3


*new_span2Vu43(wz3, wz41, h) -> new_span(wz3, wz41, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3


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

(8)
YES