9.68/4.43	YES
11.42/4.90	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
11.42/4.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.42/4.90	
11.42/4.90	
11.42/4.90	H-Termination with start terms of the given HASKELL could be proven:
11.42/4.90	
11.42/4.90	(0) HASKELL
11.42/4.90	(1) BR [EQUIVALENT, 0 ms]
11.42/4.90	(2) HASKELL
11.42/4.90	(3) COR [EQUIVALENT, 0 ms]
11.42/4.90	(4) HASKELL
11.42/4.90	(5) Narrow [EQUIVALENT, 0 ms]
11.42/4.90	(6) YES
11.42/4.90	
11.42/4.90	
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(0)
11.42/4.90	Obligation:
11.42/4.90	mainModule Main 
11.42/4.90	module FiniteMap where {
11.42/4.90	import qualified Main;
11.42/4.90	import qualified Maybe;
11.42/4.90	import qualified Prelude;
11.42/4.90	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.42/4.90	
11.42/4.90	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
11.42/4.90	}
11.42/4.90	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
11.42/4.90	lookupFM EmptyFM key = Nothing;
11.42/4.90	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
11.42/4.90	 | key_to_find > key = lookupFM fm_r key_to_find
11.42/4.90	 | otherwise = Just elt;
11.42/4.90	
11.42/4.90	}
11.42/4.90	module Maybe where {
11.42/4.90	import qualified FiniteMap;
11.42/4.90	import qualified Main;
11.42/4.90	import qualified Prelude;
11.42/4.90	}
11.42/4.90	module Main where {
11.42/4.90	import qualified FiniteMap;
11.42/4.90	import qualified Maybe;
11.42/4.90	import qualified Prelude;
11.42/4.90	}
11.42/4.90	
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(1) BR (EQUIVALENT)
11.42/4.90	Replaced joker patterns by fresh variables and removed binding patterns.
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(2)
11.42/4.90	Obligation:
11.42/4.90	mainModule Main 
11.42/4.90	module FiniteMap where {
11.42/4.90	import qualified Main;
11.42/4.90	import qualified Maybe;
11.42/4.90	import qualified Prelude;
11.42/4.90	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.42/4.90	
11.42/4.90	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
11.42/4.90	}
11.42/4.90	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
11.42/4.90	lookupFM EmptyFM key = Nothing;
11.42/4.90	lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
11.42/4.90	 | key_to_find > key = lookupFM fm_r key_to_find
11.42/4.90	 | otherwise = Just elt;
11.42/4.90	
11.42/4.90	}
11.42/4.90	module Maybe where {
11.42/4.90	import qualified FiniteMap;
11.42/4.90	import qualified Main;
11.42/4.90	import qualified Prelude;
11.42/4.90	}
11.42/4.90	module Main where {
11.42/4.90	import qualified FiniteMap;
11.42/4.90	import qualified Maybe;
11.42/4.90	import qualified Prelude;
11.42/4.90	}
11.42/4.90	
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(3) COR (EQUIVALENT)
11.42/4.90	Cond Reductions:
11.42/4.90	The following Function with conditions
11.42/4.90	"undefined |Falseundefined;
11.42/4.90	"
11.42/4.90	is transformed to
11.42/4.90	"undefined  = undefined1;
11.42/4.90	"
11.42/4.90	"undefined0 True = undefined;
11.42/4.90	"
11.42/4.90	"undefined1  = undefined0 False;
11.42/4.90	"
11.42/4.90	The following Function with conditions
11.42/4.90	"lookupFM EmptyFM key = Nothing;
11.42/4.90	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;
11.42/4.90	"
11.42/4.90	is transformed to
11.42/4.90	"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
11.42/4.90	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
11.42/4.90	"
11.42/4.90	"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
11.42/4.90	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
11.42/4.90	"
11.42/4.90	"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
11.42/4.90	"
11.42/4.90	"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
11.42/4.90	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);
11.42/4.90	"
11.42/4.90	"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);
11.42/4.90	"
11.42/4.90	"lookupFM4 EmptyFM key = Nothing;
11.42/4.90	lookupFM4 wv ww = lookupFM3 wv ww;
11.42/4.90	"
11.42/4.90	
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(4)
11.42/4.90	Obligation:
11.42/4.90	mainModule Main 
11.42/4.90	module FiniteMap where {
11.42/4.90	import qualified Main;
11.42/4.90	import qualified Maybe;
11.42/4.90	import qualified Prelude;
11.42/4.90	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.42/4.90	
11.42/4.90	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
11.42/4.90	}
11.42/4.90	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
11.42/4.90	lookupFM EmptyFM key = lookupFM4 EmptyFM key;
11.42/4.90	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
11.42/4.90	
11.42/4.90	lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
11.42/4.90	
11.42/4.90	lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
11.42/4.90	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
11.42/4.90	
11.42/4.90	lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
11.42/4.90	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);
11.42/4.90	
11.42/4.90	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);
11.42/4.90	
11.42/4.90	lookupFM4 EmptyFM key = Nothing;
11.42/4.90	lookupFM4 wv ww = lookupFM3 wv ww;
11.42/4.90	
11.42/4.90	}
11.42/4.90	module Maybe where {
11.42/4.90	import qualified FiniteMap;
11.42/4.90	import qualified Main;
11.42/4.90	import qualified Prelude;
11.42/4.90	}
11.42/4.90	module Main where {
11.42/4.90	import qualified FiniteMap;
11.42/4.90	import qualified Maybe;
11.42/4.90	import qualified Prelude;
11.42/4.90	}
11.42/4.90	
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(5) Narrow (EQUIVALENT)
11.42/4.90	Haskell To QDPs
11.42/4.90	
11.42/4.90	digraph dp_graph {
11.42/4.90	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.lookupFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.42/4.90	3[label="FiniteMap.lookupFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.42/4.90	4[label="FiniteMap.lookupFM wx3 wx4",fontsize=16,color="burlywood",shape="triangle"];23[label="wx3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
11.42/4.90	23 -> 5[label="",style="solid", color="burlywood", weight=3];
11.42/4.90	24[label="wx3/FiniteMap.Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
11.42/4.90	24 -> 6[label="",style="solid", color="burlywood", weight=3];
11.42/4.90	5[label="FiniteMap.lookupFM FiniteMap.EmptyFM wx4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.42/4.90	6[label="FiniteMap.lookupFM (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.42/4.90	7[label="FiniteMap.lookupFM4 FiniteMap.EmptyFM wx4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.42/4.90	8[label="FiniteMap.lookupFM3 (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx4",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.42/4.90	9[label="Nothing",fontsize=16,color="green",shape="box"];10[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (wx4 < wx30)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11.42/4.90	11[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare wx4 wx30 == LT)",fontsize=16,color="burlywood",shape="box"];25[label="wx4/()",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
11.42/4.90	25 -> 12[label="",style="solid", color="burlywood", weight=3];
11.42/4.90	12[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 () (compare () wx30 == LT)",fontsize=16,color="burlywood",shape="box"];26[label="wx30/()",fontsize=10,color="white",style="solid",shape="box"];12 -> 26[label="",style="solid", color="burlywood", weight=9];
11.42/4.90	26 -> 13[label="",style="solid", color="burlywood", weight=3];
11.42/4.90	13[label="FiniteMap.lookupFM2 () wx31 wx32 wx33 wx34 () (compare () () == LT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
11.42/4.90	14[label="FiniteMap.lookupFM2 () wx31 wx32 wx33 wx34 () (EQ == LT)",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
11.42/4.90	15[label="FiniteMap.lookupFM2 () wx31 wx32 wx33 wx34 () False",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
11.42/4.90	16[label="FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () (() > ())",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
11.42/4.90	17[label="FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () (compare () () == GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
11.42/4.90	18[label="FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () (EQ == GT)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
11.42/4.90	19[label="FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () False",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
11.42/4.90	20[label="FiniteMap.lookupFM0 () wx31 wx32 wx33 wx34 () otherwise",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
11.42/4.90	21[label="FiniteMap.lookupFM0 () wx31 wx32 wx33 wx34 () True",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
11.42/4.90	22[label="Just wx31",fontsize=16,color="green",shape="box"];}
11.42/4.90	
11.42/4.90	----------------------------------------
11.42/4.90	
11.42/4.90	(6)
11.42/4.90	YES
11.49/6.58	EOF