10.20/4.25	YES
12.48/4.82	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
12.48/4.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.48/4.82	
12.48/4.82	
12.48/4.82	H-Termination with start terms of the given HASKELL could be proven:
12.48/4.82	
12.48/4.82	(0) HASKELL
12.48/4.82	(1) CR [EQUIVALENT, 0 ms]
12.48/4.82	(2) HASKELL
12.48/4.82	(3) BR [EQUIVALENT, 0 ms]
12.48/4.82	(4) HASKELL
12.48/4.82	(5) COR [EQUIVALENT, 0 ms]
12.48/4.82	(6) HASKELL
12.48/4.82	(7) Narrow [EQUIVALENT, 0 ms]
12.48/4.82	(8) YES
12.48/4.82	
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(0)
12.48/4.82	Obligation:
12.48/4.82	mainModule Main 
12.48/4.82	module Maybe where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	module List where {
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	insert :: Ord a => a  ->  [a]  ->  [a];
12.48/4.82	insert e ls = insertBy compare e ls;
12.48/4.82	
12.48/4.82	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.48/4.82	insertBy _ x [] = x : [];
12.48/4.82	insertBy cmp x ys@(y : ys') = case cmp x y of { 
12.48/4.82	GT-> y : insertBy cmp x ys'; 
12.48/4.82	_-> x : ys; 
12.48/4.82	 } ;
12.48/4.82	
12.48/4.82	}
12.48/4.82	module Main where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(1) CR (EQUIVALENT)
12.48/4.82	Case Reductions:
12.48/4.82	The following Case expression
12.48/4.82	"case cmp x y of {
12.48/4.82	GT  -> y : insertBy cmp x ys';
12.48/4.82	_  -> x : ys}
12.48/4.82	"
12.48/4.82	is transformed to
12.48/4.82	"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.48/4.82	insertBy0 y cmp x ys' ys _ = x : ys;
12.48/4.82	"
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(2)
12.48/4.82	Obligation:
12.48/4.82	mainModule Main 
12.48/4.82	module Maybe where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	module List where {
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	insert :: Ord a => a  ->  [a]  ->  [a];
12.48/4.82	insert e ls = insertBy compare e ls;
12.48/4.82	
12.48/4.82	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.48/4.82	insertBy _ x [] = x : [];
12.48/4.82	insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);
12.48/4.82	
12.48/4.82	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.48/4.82	insertBy0 y cmp x ys' ys _ = x : ys;
12.48/4.82	
12.48/4.82	}
12.48/4.82	module Main where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(3) BR (EQUIVALENT)
12.48/4.82	Replaced joker patterns by fresh variables and removed binding patterns.
12.48/4.82	
12.48/4.82	Binding Reductions:
12.48/4.82	The bind variable of the following binding Pattern
12.48/4.82	"ys@(wu : wv)"
12.48/4.82	is replaced by the following term
12.48/4.82	"wu : wv"
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(4)
12.48/4.82	Obligation:
12.48/4.82	mainModule Main 
12.48/4.82	module Maybe where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	module List where {
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	insert :: Ord a => a  ->  [a]  ->  [a];
12.48/4.82	insert e ls = insertBy compare e ls;
12.48/4.82	
12.48/4.82	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.48/4.82	insertBy vz x [] = x : [];
12.48/4.82	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
12.48/4.82	
12.48/4.82	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.48/4.82	insertBy0 y cmp x ys' ys vy = x : ys;
12.48/4.82	
12.48/4.82	}
12.48/4.82	module Main where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(5) COR (EQUIVALENT)
12.48/4.82	Cond Reductions:
12.48/4.82	The following Function with conditions
12.48/4.82	"undefined |Falseundefined;
12.48/4.82	"
12.48/4.82	is transformed to
12.48/4.82	"undefined  = undefined1;
12.48/4.82	"
12.48/4.82	"undefined0 True = undefined;
12.48/4.82	"
12.48/4.82	"undefined1  = undefined0 False;
12.48/4.82	"
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(6)
12.48/4.82	Obligation:
12.48/4.82	mainModule Main 
12.48/4.82	module Maybe where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	module List where {
12.48/4.82	import qualified Main;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	insert :: Ord a => a  ->  [a]  ->  [a];
12.48/4.82	insert e ls = insertBy compare e ls;
12.48/4.82	
12.48/4.82	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
12.48/4.82	insertBy vz x [] = x : [];
12.48/4.82	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
12.48/4.82	
12.48/4.82	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
12.48/4.82	insertBy0 y cmp x ys' ys vy = x : ys;
12.48/4.82	
12.48/4.82	}
12.48/4.82	module Main where {
12.48/4.82	import qualified List;
12.48/4.82	import qualified Maybe;
12.48/4.82	import qualified Prelude;
12.48/4.82	}
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(7) Narrow (EQUIVALENT)
12.48/4.82	Haskell To QDPs
12.48/4.82	
12.48/4.82	digraph dp_graph {
12.48/4.82	node [outthreshold=100, inthreshold=100];1[label="List.insert",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.48/4.82	3[label="List.insert ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.48/4.82	4[label="List.insert ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.48/4.82	5[label="List.insertBy compare ww3 ww4",fontsize=16,color="burlywood",shape="box"];14[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 14[label="",style="solid", color="burlywood", weight=9];
12.48/4.82	14 -> 6[label="",style="solid", color="burlywood", weight=3];
12.48/4.82	15[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 15[label="",style="solid", color="burlywood", weight=9];
12.48/4.82	15 -> 7[label="",style="solid", color="burlywood", weight=3];
12.48/4.82	6[label="List.insertBy compare ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.48/4.82	7[label="List.insertBy compare ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.48/4.82	8[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare ww3 ww40)",fontsize=16,color="burlywood",shape="box"];16[label="ww3/()",fontsize=10,color="white",style="solid",shape="box"];8 -> 16[label="",style="solid", color="burlywood", weight=9];
12.48/4.82	16 -> 10[label="",style="solid", color="burlywood", weight=3];
12.48/4.82	9[label="ww3 : []",fontsize=16,color="green",shape="box"];10[label="List.insertBy0 ww40 compare () ww41 (ww40 : ww41) (compare () ww40)",fontsize=16,color="burlywood",shape="box"];17[label="ww40/()",fontsize=10,color="white",style="solid",shape="box"];10 -> 17[label="",style="solid", color="burlywood", weight=9];
12.48/4.82	17 -> 11[label="",style="solid", color="burlywood", weight=3];
12.48/4.82	11[label="List.insertBy0 () compare () ww41 (() : ww41) (compare () ())",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.48/4.82	12[label="List.insertBy0 () compare () ww41 (() : ww41) EQ",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.48/4.82	13[label="() : () : ww41",fontsize=16,color="green",shape="box"];}
12.48/4.82	
12.48/4.82	----------------------------------------
12.48/4.82	
12.48/4.82	(8)
12.48/4.82	YES
12.63/7.39	EOF