/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) CR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [EQUIVALENT, 0 ms]
(8) YES


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

(0)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
elemFM key fm = case lookupFM fm key of { 
Nothing-> False; 
Just elt-> True; 
 } ;

lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
 | key_to_find > key = lookupFM fm_r key_to_find
 | otherwise = Just elt;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(1) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case lookupFM fm key of {
Nothing  -> False;
Just elt  -> True}
"
is transformed to
"elemFM0 Nothing = False;
elemFM0 (Just elt) = True;
"

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

(2)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
elemFM key fm = elemFM0 (lookupFM fm key);

elemFM0 Nothing = False;
elemFM0 (Just elt) = True;

lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
 | key_to_find > key = lookupFM fm_r key_to_find
 | otherwise = Just elt;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

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

(4)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
elemFM key fm = elemFM0 (lookupFM fm key);

elemFM0 Nothing = False;
elemFM0 (Just elt) = True;

lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
 | key_to_find > key = lookupFM fm_r key_to_find
 | otherwise = Just elt;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find|key_to_find < keylookupFM fm_l key_to_find|key_to_find > keylookupFM fm_r key_to_find|otherwiseJust elt;
"
is transformed to
"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
"
"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
"
"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
lookupFM2 key elt vy fm_l fm_r key_to_find False = lookupFM1 key elt vy fm_l fm_r key_to_find (key_to_find > key);
"
"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
"
"lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find = lookupFM2 key elt vy fm_l fm_r key_to_find (key_to_find < key);
"
"lookupFM4 EmptyFM key = Nothing;
lookupFM4 wv ww = lookupFM3 wv ww;
"

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

(6)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
elemFM key fm = elemFM0 (lookupFM fm key);

elemFM0 Nothing = False;
elemFM0 (Just elt) = True;

lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
lookupFM EmptyFM key = lookupFM4 EmptyFM key;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;

lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;

lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;

lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
lookupFM2 key elt vy fm_l fm_r key_to_find False = lookupFM1 key elt vy fm_l fm_r key_to_find (key_to_find > key);

lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find = lookupFM2 key elt vy fm_l fm_r key_to_find (key_to_find < key);

lookupFM4 EmptyFM key = Nothing;
lookupFM4 wv ww = lookupFM3 wv ww;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(7) Narrow (EQUIVALENT)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.elemFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.elemFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.elemFM wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="FiniteMap.elemFM0 (FiniteMap.lookupFM wx4 wx3)",fontsize=16,color="burlywood",shape="box"];26[label="wx4/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 6[label="",style="solid", color="burlywood", weight=3];
27[label="wx4/FiniteMap.Branch wx40 wx41 wx42 wx43 wx44",fontsize=10,color="white",style="solid",shape="box"];5 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="FiniteMap.elemFM0 (FiniteMap.lookupFM FiniteMap.EmptyFM wx3)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="FiniteMap.elemFM0 (FiniteMap.lookupFM (FiniteMap.Branch wx40 wx41 wx42 wx43 wx44) wx3)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.elemFM0 (FiniteMap.lookupFM4 FiniteMap.EmptyFM wx3)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.elemFM0 (FiniteMap.lookupFM3 (FiniteMap.Branch wx40 wx41 wx42 wx43 wx44) wx3)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.elemFM0 Nothing",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (wx3 < wx40))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="False",fontsize=16,color="green",shape="box"];13[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (compare wx3 wx40 == LT))",fontsize=16,color="burlywood",shape="box"];28[label="wx3/()",fontsize=10,color="white",style="solid",shape="box"];13 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 14[label="",style="solid", color="burlywood", weight=3];
14[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 () (compare () wx40 == LT))",fontsize=16,color="burlywood",shape="box"];29[label="wx40/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 15[label="",style="solid", color="burlywood", weight=3];
15[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 () wx41 wx42 wx43 wx44 () (compare () () == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 () wx41 wx42 wx43 wx44 () (EQ == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 () wx41 wx42 wx43 wx44 () False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 () wx41 wx42 wx43 wx44 () (() > ()))",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 () wx41 wx42 wx43 wx44 () (compare () () == GT))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 () wx41 wx42 wx43 wx44 () (EQ == GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 () wx41 wx42 wx43 wx44 () False)",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="FiniteMap.elemFM0 (FiniteMap.lookupFM0 () wx41 wx42 wx43 wx44 () otherwise)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="FiniteMap.elemFM0 (FiniteMap.lookupFM0 () wx41 wx42 wx43 wx44 () True)",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="FiniteMap.elemFM0 (Just wx41)",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3];
25[label="True",fontsize=16,color="green",shape="box"];}

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

(8)
YES