/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Main.Char = Char MyInt ;

data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

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

unlines :: List (List Main.Char)  ->  List Main.Char;
unlines Nil = Nil;
unlines (Cons l ls) = psPs l (Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))) (unlines ls));

}

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

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Main.Char = Char MyInt ;

data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

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

unlines :: List (List Main.Char)  ->  List Main.Char;
unlines Nil = Nil;
unlines (Cons l ls) = psPs l (Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))) (unlines ls));

}

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

(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 Main.Char = Char MyInt ;

data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

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

unlines :: List (List Main.Char)  ->  List Main.Char;
unlines Nil = Nil;
unlines (Cons l ls) = psPs l (Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))) (unlines ls));

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="unlines",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="unlines vx3",fontsize=16,color="burlywood",shape="triangle"];24[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 4[label="",style="solid", color="burlywood", weight=3];
25[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];3 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="unlines (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="unlines Nil",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6 -> 12[label="",style="dashed", color="red", weight=0];
6[label="psPs vx30 (Cons (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))) (unlines vx31))",fontsize=16,color="magenta"];6 -> 13[label="",style="dashed", color="magenta", weight=3];
6 -> 14[label="",style="dashed", color="magenta", weight=3];
6 -> 15[label="",style="dashed", color="magenta", weight=3];
7[label="Nil",fontsize=16,color="green",shape="box"];13[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];14[label="vx30",fontsize=16,color="green",shape="box"];15 -> 3[label="",style="dashed", color="red", weight=0];
15[label="unlines vx31",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
12[label="psPs vx5 (Cons (Char (Pos (Succ vx6))) vx8)",fontsize=16,color="burlywood",shape="triangle"];26[label="vx5/Cons vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];12 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 18[label="",style="solid", color="burlywood", weight=3];
27[label="vx5/Nil",fontsize=10,color="white",style="solid",shape="box"];12 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="vx31",fontsize=16,color="green",shape="box"];18[label="psPs (Cons vx50 vx51) (Cons (Char (Pos (Succ vx6))) vx8)",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="psPs Nil (Cons (Char (Pos (Succ vx6))) vx8)",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="Cons vx50 (psPs vx51 (Cons (Char (Pos (Succ vx6))) vx8))",fontsize=16,color="green",shape="box"];20 -> 22[label="",style="dashed", color="green", weight=3];
21[label="Cons (Char (Pos (Succ vx6))) vx8",fontsize=16,color="green",shape="box"];22 -> 12[label="",style="dashed", color="red", weight=0];
22[label="psPs vx51 (Cons (Char (Pos (Succ vx6))) vx8)",fontsize=16,color="magenta"];22 -> 23[label="",style="dashed", color="magenta", weight=3];
23[label="vx51",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(vx50, vx51), vx6, vx8) -> new_psPs(vx51, vx6, vx8)

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(vx50, vx51), vx6, vx8) -> new_psPs(vx51, vx6, vx8)
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_unlines(Cons(vx30, vx31)) -> new_unlines(vx31)

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_unlines(Cons(vx30, vx31)) -> new_unlines(vx31)
The graph contains the following edges 1 > 1


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

(12)
YES