9.18/3.91	YES
11.39/4.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.39/4.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.39/4.50	
11.39/4.50	
11.39/4.50	H-Termination with start terms of the given HASKELL could be proven:
11.39/4.50	
11.39/4.50	(0) HASKELL
11.39/4.50	(1) LR [EQUIVALENT, 0 ms]
11.39/4.50	(2) HASKELL
11.39/4.50	(3) BR [EQUIVALENT, 0 ms]
11.39/4.50	(4) HASKELL
11.39/4.50	(5) COR [EQUIVALENT, 0 ms]
11.39/4.50	(6) HASKELL
11.39/4.50	(7) Narrow [EQUIVALENT, 22 ms]
11.39/4.50	(8) YES
11.39/4.50	
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(0)
11.39/4.50	Obligation:
11.39/4.50	mainModule Main 
11.39/4.50	module Maybe where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Main where {
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Monad where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Prelude;
11.39/4.50	ap :: Monad a => a (b  ->  c)  ->  a b  ->  a c;
11.39/4.50	ap = liftM2 id;
11.39/4.50	
11.39/4.50	liftM2 :: Monad c => (d  ->  a  ->  b)  ->  c d  ->  c a  ->  c b;
11.39/4.50	liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));
11.39/4.50	
11.39/4.50	}
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(1) LR (EQUIVALENT)
11.39/4.50	Lambda Reductions:
11.39/4.50	The following Lambda expression
11.39/4.50	"\x2->return (f x1 x2)"
11.39/4.50	is transformed to
11.39/4.50	"liftM20 f x1 x2 = return (f x1 x2);
11.39/4.50	"
11.39/4.50	The following Lambda expression
11.39/4.50	"\x1->m2 >>= liftM20 f x1"
11.39/4.50	is transformed to
11.39/4.50	"liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.39/4.50	"
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(2)
11.39/4.50	Obligation:
11.39/4.50	mainModule Main 
11.39/4.50	module Maybe where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Main where {
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Monad where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Prelude;
11.39/4.50	ap :: Monad a => a (c  ->  b)  ->  a c  ->  a b;
11.39/4.50	ap = liftM2 id;
11.39/4.50	
11.39/4.50	liftM2 :: Monad b => (c  ->  a  ->  d)  ->  b c  ->  b a  ->  b d;
11.39/4.50	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.39/4.50	
11.39/4.50	liftM20 f x1 x2 = return (f x1 x2);
11.39/4.50	
11.39/4.50	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.39/4.50	
11.39/4.50	}
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(3) BR (EQUIVALENT)
11.39/4.50	Replaced joker patterns by fresh variables and removed binding patterns.
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(4)
11.39/4.50	Obligation:
11.39/4.50	mainModule Main 
11.39/4.50	module Maybe where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Main where {
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Monad where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Prelude;
11.39/4.50	ap :: Monad c => c (b  ->  a)  ->  c b  ->  c a;
11.39/4.50	ap = liftM2 id;
11.39/4.50	
11.39/4.50	liftM2 :: Monad a => (d  ->  c  ->  b)  ->  a d  ->  a c  ->  a b;
11.39/4.50	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.39/4.50	
11.39/4.50	liftM20 f x1 x2 = return (f x1 x2);
11.39/4.50	
11.39/4.50	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.39/4.50	
11.39/4.50	}
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(5) COR (EQUIVALENT)
11.39/4.50	Cond Reductions:
11.39/4.50	The following Function with conditions
11.39/4.50	"undefined |Falseundefined;
11.39/4.50	"
11.39/4.50	is transformed to
11.39/4.50	"undefined  = undefined1;
11.39/4.50	"
11.39/4.50	"undefined0 True = undefined;
11.39/4.50	"
11.39/4.50	"undefined1  = undefined0 False;
11.39/4.50	"
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(6)
11.39/4.50	Obligation:
11.39/4.50	mainModule Main 
11.39/4.50	module Maybe where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Main where {
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Monad;
11.39/4.50	import qualified Prelude;
11.39/4.50	}
11.39/4.50	module Monad where {
11.39/4.50	import qualified Main;
11.39/4.50	import qualified Maybe;
11.39/4.50	import qualified Prelude;
11.39/4.50	ap :: Monad c => c (a  ->  b)  ->  c a  ->  c b;
11.39/4.50	ap = liftM2 id;
11.39/4.50	
11.39/4.50	liftM2 :: Monad a => (d  ->  c  ->  b)  ->  a d  ->  a c  ->  a b;
11.39/4.50	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.39/4.50	
11.39/4.50	liftM20 f x1 x2 = return (f x1 x2);
11.39/4.50	
11.39/4.50	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.39/4.50	
11.39/4.50	}
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(7) Narrow (EQUIVALENT)
11.39/4.50	Haskell To QDPs
11.39/4.50	
11.39/4.50	digraph dp_graph {
11.39/4.50	node [outthreshold=100, inthreshold=100];1[label="Monad.ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.39/4.50	3[label="Monad.ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.39/4.50	4[label="Monad.ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.39/4.50	5[label="Monad.liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.39/4.50	6[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="box"];21[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 21[label="",style="solid", color="burlywood", weight=9];
11.39/4.50	21 -> 7[label="",style="solid", color="burlywood", weight=3];
11.39/4.50	22[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];6 -> 22[label="",style="solid", color="burlywood", weight=9];
11.39/4.50	22 -> 8[label="",style="solid", color="burlywood", weight=3];
11.39/4.50	7[label="Nothing >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.39/4.50	8[label="Just vy30 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.39/4.50	9[label="Nothing",fontsize=16,color="green",shape="box"];10[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11.39/4.50	11[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="box"];23[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];11 -> 23[label="",style="solid", color="burlywood", weight=9];
11.39/4.50	23 -> 12[label="",style="solid", color="burlywood", weight=3];
11.39/4.50	24[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 24[label="",style="solid", color="burlywood", weight=9];
11.39/4.50	24 -> 13[label="",style="solid", color="burlywood", weight=3];
11.39/4.50	12[label="Nothing >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.39/4.50	13[label="Just vy40 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
11.39/4.50	14[label="Nothing",fontsize=16,color="green",shape="box"];15[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
11.39/4.50	16[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
11.39/4.50	17[label="Just (id vy30 vy40)",fontsize=16,color="green",shape="box"];17 -> 18[label="",style="dashed", color="green", weight=3];
11.39/4.50	18[label="id vy30 vy40",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
11.39/4.50	19[label="vy30 vy40",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="green", weight=3];
11.39/4.50	20[label="vy40",fontsize=16,color="green",shape="box"];}
11.39/4.50	
11.39/4.50	----------------------------------------
11.39/4.50	
11.39/4.50	(8)
11.39/4.50	YES
11.39/4.54	EOF