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


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl f z Nil = z;
foldl f z (Cons x xs) = foldl f (f z x) xs;

}

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

(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 ;

foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl f z Nil = z;
foldl f z (Cons x xs) = foldl f (f z x) xs;

}

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

(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 ;

foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl f z Nil = z;
foldl f z (Cons x xs) = foldl f (f z x) xs;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="foldl",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="foldl vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="foldl vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="foldl vx3 vx4 vx5",fontsize=16,color="burlywood",shape="triangle"];14[label="vx5/Cons vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];5 -> 14[label="",style="solid", color="burlywood", weight=9];
14 -> 6[label="",style="solid", color="burlywood", weight=3];
15[label="vx5/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldl vx3 vx4 (Cons vx50 vx51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl vx3 vx4 Nil",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 5[label="",style="dashed", color="red", weight=0];
8[label="foldl vx3 (vx3 vx4 vx50) vx51",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="vx4",fontsize=16,color="green",shape="box"];10[label="vx3 vx4 vx50",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 13[label="",style="dashed", color="green", weight=3];
11[label="vx51",fontsize=16,color="green",shape="box"];12[label="vx4",fontsize=16,color="green",shape="box"];13[label="vx50",fontsize=16,color="green",shape="box"];}

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

   new_foldl(vx3, Cons(vx50, vx51), h, ba) -> new_foldl(vx3, vx51, h, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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_foldl(vx3, Cons(vx50, vx51), h, ba) -> new_foldl(vx3, vx51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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(8)
YES