10.57/4.44	YES
12.46/4.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.46/4.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.46/4.98	
12.46/4.98	
12.46/4.98	H-Termination with start terms of the given HASKELL could be proven:
12.46/4.98	
12.46/4.98	(0) HASKELL
12.46/4.98	(1) CR [EQUIVALENT, 0 ms]
12.46/4.98	(2) HASKELL
12.46/4.98	(3) BR [EQUIVALENT, 0 ms]
12.46/4.98	(4) HASKELL
12.46/4.98	(5) COR [EQUIVALENT, 13 ms]
12.46/4.98	(6) HASKELL
12.46/4.98	(7) Narrow [SOUND, 0 ms]
12.46/4.98	(8) QDP
12.46/4.98	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.46/4.98	(10) YES
12.46/4.98	
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(0)
12.46/4.98	Obligation:
12.46/4.98	mainModule Main 
12.46/4.98	module Maybe where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	module List where {
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	insert :: Ord a => a  ->  [a]  ->  [a];
12.46/4.98	insert e ls = insertBy compare e ls;
12.46/4.98	
12.46/4.98	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.46/4.98	insertBy _ x [] = x : [];
12.46/4.98	insertBy cmp x ys@(y : ys') = case cmp x y of { 
12.46/4.98	GT-> y : insertBy cmp x ys'; 
12.46/4.98	_-> x : ys; 
12.46/4.98	 } ;
12.46/4.98	
12.46/4.98	}
12.46/4.98	module Main where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(1) CR (EQUIVALENT)
12.46/4.98	Case Reductions:
12.46/4.98	The following Case expression
12.46/4.98	"case cmp x y of {
12.46/4.98	GT  -> y : insertBy cmp x ys';
12.46/4.98	_  -> x : ys}
12.46/4.98	"
12.46/4.98	is transformed to
12.46/4.98	"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.46/4.98	insertBy0 y cmp x ys' ys _ = x : ys;
12.46/4.98	"
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(2)
12.46/4.98	Obligation:
12.46/4.98	mainModule Main 
12.46/4.98	module Maybe where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	module List where {
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	insert :: Ord a => a  ->  [a]  ->  [a];
12.46/4.98	insert e ls = insertBy compare e ls;
12.46/4.98	
12.46/4.98	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.46/4.98	insertBy _ x [] = x : [];
12.46/4.98	insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);
12.46/4.98	
12.46/4.98	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.46/4.98	insertBy0 y cmp x ys' ys _ = x : ys;
12.46/4.98	
12.46/4.98	}
12.46/4.98	module Main where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(3) BR (EQUIVALENT)
12.46/4.98	Replaced joker patterns by fresh variables and removed binding patterns.
12.46/4.98	
12.46/4.98	Binding Reductions:
12.46/4.98	The bind variable of the following binding Pattern
12.46/4.98	"ys@(wu : wv)"
12.46/4.98	is replaced by the following term
12.46/4.98	"wu : wv"
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(4)
12.46/4.98	Obligation:
12.46/4.98	mainModule Main 
12.46/4.98	module Maybe where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	module List where {
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	insert :: Ord a => a  ->  [a]  ->  [a];
12.46/4.98	insert e ls = insertBy compare e ls;
12.46/4.98	
12.46/4.98	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.46/4.98	insertBy vz x [] = x : [];
12.46/4.98	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
12.46/4.98	
12.46/4.98	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.46/4.98	insertBy0 y cmp x ys' ys vy = x : ys;
12.46/4.98	
12.46/4.98	}
12.46/4.98	module Main where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(5) COR (EQUIVALENT)
12.46/4.98	Cond Reductions:
12.46/4.98	The following Function with conditions
12.46/4.98	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
12.46/4.98	"
12.46/4.98	is transformed to
12.46/4.98	"compare x y = compare3 x y;
12.46/4.98	"
12.46/4.98	"compare2 x y True = EQ;
12.46/4.98	compare2 x y False = compare1 x y (x <= y);
12.46/4.98	"
12.46/4.98	"compare0 x y True = GT;
12.46/4.98	"
12.46/4.98	"compare1 x y True = LT;
12.46/4.98	compare1 x y False = compare0 x y otherwise;
12.46/4.98	"
12.46/4.98	"compare3 x y = compare2 x y (x == y);
12.46/4.98	"
12.46/4.98	The following Function with conditions
12.46/4.98	"undefined |Falseundefined;
12.46/4.98	"
12.46/4.98	is transformed to
12.46/4.98	"undefined  = undefined1;
12.46/4.98	"
12.46/4.98	"undefined0 True = undefined;
12.46/4.98	"
12.46/4.98	"undefined1  = undefined0 False;
12.46/4.98	"
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(6)
12.46/4.98	Obligation:
12.46/4.98	mainModule Main 
12.46/4.98	module Maybe where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	module List where {
12.46/4.98	import qualified Main;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	insert :: Ord a => a  ->  [a]  ->  [a];
12.46/4.98	insert e ls = insertBy compare e ls;
12.46/4.98	
12.46/4.98	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.46/4.98	insertBy vz x [] = x : [];
12.46/4.98	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
12.46/4.98	
12.46/4.98	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.46/4.98	insertBy0 y cmp x ys' ys vy = x : ys;
12.46/4.98	
12.46/4.98	}
12.46/4.98	module Main where {
12.46/4.98	import qualified List;
12.46/4.98	import qualified Maybe;
12.46/4.98	import qualified Prelude;
12.46/4.98	}
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(7) Narrow (SOUND)
12.46/4.98	Haskell To QDPs
12.46/4.98	
12.46/4.98	digraph dp_graph {
12.46/4.98	node [outthreshold=100, inthreshold=100];1[label="List.insert",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.46/4.98	3[label="List.insert ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.46/4.98	4[label="List.insert ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.46/4.98	5[label="List.insertBy compare ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];39[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 39[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	39 -> 6[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	40[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 40[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	40 -> 7[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	6[label="List.insertBy compare ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.46/4.98	7[label="List.insertBy compare ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.46/4.98	8[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare ww3 ww40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.46/4.98	9[label="ww3 : []",fontsize=16,color="green",shape="box"];10[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare3 ww3 ww40)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
12.46/4.98	11[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare2 ww3 ww40 (ww3 == ww40))",fontsize=16,color="burlywood",shape="box"];41[label="ww3/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 41[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	41 -> 12[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	42[label="ww3/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 42[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	42 -> 13[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	12[label="List.insertBy0 ww40 compare False ww41 (ww40 : ww41) (compare2 False ww40 (False == ww40))",fontsize=16,color="burlywood",shape="box"];43[label="ww40/False",fontsize=10,color="white",style="solid",shape="box"];12 -> 43[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	43 -> 14[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	44[label="ww40/True",fontsize=10,color="white",style="solid",shape="box"];12 -> 44[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	44 -> 15[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	13[label="List.insertBy0 ww40 compare True ww41 (ww40 : ww41) (compare2 True ww40 (True == ww40))",fontsize=16,color="burlywood",shape="box"];45[label="ww40/False",fontsize=10,color="white",style="solid",shape="box"];13 -> 45[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	45 -> 16[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	46[label="ww40/True",fontsize=10,color="white",style="solid",shape="box"];13 -> 46[label="",style="solid", color="burlywood", weight=9];
12.46/4.98	46 -> 17[label="",style="solid", color="burlywood", weight=3];
12.46/4.98	14[label="List.insertBy0 False compare False ww41 (False : ww41) (compare2 False False (False == False))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
12.46/4.98	15[label="List.insertBy0 True compare False ww41 (True : ww41) (compare2 False True (False == True))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
12.46/4.98	16[label="List.insertBy0 False compare True ww41 (False : ww41) (compare2 True False (True == False))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
12.46/4.98	17[label="List.insertBy0 True compare True ww41 (True : ww41) (compare2 True True (True == True))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
12.46/4.98	18[label="List.insertBy0 False compare False ww41 (False : ww41) (compare2 False False True)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
12.46/4.98	19[label="List.insertBy0 True compare False ww41 (True : ww41) (compare2 False True False)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.46/4.98	20[label="List.insertBy0 False compare True ww41 (False : ww41) (compare2 True False False)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.46/4.98	21[label="List.insertBy0 True compare True ww41 (True : ww41) (compare2 True True True)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.46/4.98	22[label="List.insertBy0 False compare False ww41 (False : ww41) EQ",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.46/4.98	23[label="List.insertBy0 True compare False ww41 (True : ww41) (compare1 False True (False <= True))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.46/4.98	24[label="List.insertBy0 False compare True ww41 (False : ww41) (compare1 True False (True <= False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.46/4.98	25[label="List.insertBy0 True compare True ww41 (True : ww41) EQ",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.46/4.98	26[label="False : False : ww41",fontsize=16,color="green",shape="box"];27[label="List.insertBy0 True compare False ww41 (True : ww41) (compare1 False True True)",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3];
12.46/4.98	28[label="List.insertBy0 False compare True ww41 (False : ww41) (compare1 True False False)",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3];
12.46/4.98	29[label="True : True : ww41",fontsize=16,color="green",shape="box"];30[label="List.insertBy0 True compare False ww41 (True : ww41) LT",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
12.46/4.98	31[label="List.insertBy0 False compare True ww41 (False : ww41) (compare0 True False otherwise)",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
12.46/4.98	32[label="False : True : ww41",fontsize=16,color="green",shape="box"];33[label="List.insertBy0 False compare True ww41 (False : ww41) (compare0 True False True)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
12.46/4.98	34[label="List.insertBy0 False compare True ww41 (False : ww41) GT",fontsize=16,color="black",shape="box"];34 -> 35[label="",style="solid", color="black", weight=3];
12.46/4.98	35[label="False : List.insertBy compare True ww41",fontsize=16,color="green",shape="box"];35 -> 36[label="",style="dashed", color="green", weight=3];
12.46/4.98	36 -> 5[label="",style="dashed", color="red", weight=0];
12.46/4.98	36[label="List.insertBy compare True ww41",fontsize=16,color="magenta"];36 -> 37[label="",style="dashed", color="magenta", weight=3];
12.46/4.98	36 -> 38[label="",style="dashed", color="magenta", weight=3];
12.46/4.98	37[label="ww41",fontsize=16,color="green",shape="box"];38[label="True",fontsize=16,color="green",shape="box"];}
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(8)
12.46/4.98	Obligation:
12.46/4.98	Q DP problem:
12.46/4.98	The TRS P consists of the following rules:
12.46/4.98	
12.46/4.98	   new_insertBy(True, :(False, ww41)) -> new_insertBy(True, ww41)
12.46/4.98	
12.46/4.98	R is empty.
12.46/4.98	Q is empty.
12.46/4.98	We have to consider all minimal (P,Q,R)-chains.
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(9) QDPSizeChangeProof (EQUIVALENT)
12.46/4.98	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. 
12.46/4.98	
12.46/4.98	From the DPs we obtained the following set of size-change graphs:
12.46/4.98	*new_insertBy(True, :(False, ww41)) -> new_insertBy(True, ww41)
12.46/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.46/4.98	
12.46/4.98	
12.46/4.98	----------------------------------------
12.46/4.98	
12.46/4.98	(10)
12.46/4.98	YES
12.67/5.07	EOF