8.79/3.85	YES
11.33/4.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.33/4.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.33/4.46	
11.33/4.46	
11.33/4.46	H-Termination with start terms of the given HASKELL could be proven:
11.33/4.46	
11.33/4.46	(0) HASKELL
11.33/4.46	(1) LR [EQUIVALENT, 0 ms]
11.33/4.46	(2) HASKELL
11.33/4.46	(3) BR [EQUIVALENT, 0 ms]
11.33/4.46	(4) HASKELL
11.33/4.46	(5) COR [EQUIVALENT, 0 ms]
11.33/4.46	(6) HASKELL
11.33/4.46	(7) Narrow [EQUIVALENT, 19 ms]
11.33/4.46	(8) YES
11.33/4.46	
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(0)
11.33/4.46	Obligation:
11.33/4.46	mainModule Main 
11.33/4.46	module Maybe where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Main where {
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Monad where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Prelude;
11.33/4.46	ap :: Monad a => a (b  ->  c)  ->  a b  ->  a c;
11.33/4.46	ap = liftM2 id;
11.33/4.46	
11.33/4.46	liftM2 :: Monad d => (a  ->  c  ->  b)  ->  d a  ->  d c  ->  d b;
11.33/4.46	liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));
11.33/4.46	
11.33/4.46	}
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(1) LR (EQUIVALENT)
11.33/4.46	Lambda Reductions:
11.33/4.46	The following Lambda expression
11.33/4.46	"\x2->return (f x1 x2)"
11.33/4.46	is transformed to
11.33/4.46	"liftM20 f x1 x2 = return (f x1 x2);
11.33/4.46	"
11.33/4.46	The following Lambda expression
11.33/4.46	"\x1->m2 >>= liftM20 f x1"
11.33/4.46	is transformed to
11.33/4.46	"liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.33/4.46	"
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(2)
11.33/4.46	Obligation:
11.33/4.46	mainModule Main 
11.33/4.46	module Maybe where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Main where {
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Monad where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Prelude;
11.33/4.46	ap :: Monad b => b (c  ->  a)  ->  b c  ->  b a;
11.33/4.46	ap = liftM2 id;
11.33/4.46	
11.33/4.46	liftM2 :: Monad a => (c  ->  d  ->  b)  ->  a c  ->  a d  ->  a b;
11.33/4.46	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.33/4.46	
11.33/4.46	liftM20 f x1 x2 = return (f x1 x2);
11.33/4.46	
11.33/4.46	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.33/4.46	
11.33/4.46	}
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(3) BR (EQUIVALENT)
11.33/4.46	Replaced joker patterns by fresh variables and removed binding patterns.
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(4)
11.33/4.46	Obligation:
11.33/4.46	mainModule Main 
11.33/4.46	module Maybe where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Main where {
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Monad where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Prelude;
11.33/4.46	ap :: Monad a => a (c  ->  b)  ->  a c  ->  a b;
11.33/4.46	ap = liftM2 id;
11.33/4.46	
11.33/4.46	liftM2 :: Monad a => (b  ->  d  ->  c)  ->  a b  ->  a d  ->  a c;
11.33/4.46	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.33/4.46	
11.33/4.46	liftM20 f x1 x2 = return (f x1 x2);
11.33/4.46	
11.33/4.46	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.33/4.46	
11.33/4.46	}
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(5) COR (EQUIVALENT)
11.33/4.46	Cond Reductions:
11.33/4.46	The following Function with conditions
11.33/4.46	"undefined |Falseundefined;
11.33/4.46	"
11.33/4.46	is transformed to
11.33/4.46	"undefined  = undefined1;
11.33/4.46	"
11.33/4.46	"undefined0 True = undefined;
11.33/4.46	"
11.33/4.46	"undefined1  = undefined0 False;
11.33/4.46	"
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(6)
11.33/4.46	Obligation:
11.33/4.46	mainModule Main 
11.33/4.46	module Maybe where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Main where {
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Monad;
11.33/4.46	import qualified Prelude;
11.33/4.46	}
11.33/4.46	module Monad where {
11.33/4.46	import qualified Main;
11.33/4.46	import qualified Maybe;
11.33/4.46	import qualified Prelude;
11.33/4.46	ap :: Monad b => b (a  ->  c)  ->  b a  ->  b c;
11.33/4.46	ap = liftM2 id;
11.33/4.46	
11.33/4.46	liftM2 :: Monad a => (b  ->  d  ->  c)  ->  a b  ->  a d  ->  a c;
11.33/4.46	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.33/4.46	
11.33/4.46	liftM20 f x1 x2 = return (f x1 x2);
11.33/4.46	
11.33/4.46	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.33/4.46	
11.33/4.46	}
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(7) Narrow (EQUIVALENT)
11.33/4.46	Haskell To QDPs
11.33/4.46	
11.33/4.46	digraph dp_graph {
11.33/4.46	node [outthreshold=100, inthreshold=100];1[label="Monad.ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.33/4.46	3[label="Monad.ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.33/4.46	4[label="Monad.ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.33/4.46	5[label="Monad.liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.33/4.46	6[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.33/4.46	7[label="primbindIO vy3 (Monad.liftM21 vy4 id)",fontsize=16,color="burlywood",shape="box"];32[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];7 -> 32[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	32 -> 8[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	33[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];7 -> 33[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	33 -> 9[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	34[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];7 -> 34[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	34 -> 10[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	35[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	35 -> 11[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	8[label="primbindIO (IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];8 -> 12[label="",style="solid", color="black", weight=3];
11.33/4.46	9[label="primbindIO (AProVE_IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];9 -> 13[label="",style="solid", color="black", weight=3];
11.33/4.46	10[label="primbindIO (AProVE_Exception vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11.33/4.46	11[label="primbindIO (AProVE_Error vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
11.33/4.46	12[label="error []",fontsize=16,color="red",shape="box"];13[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
11.33/4.46	14[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];15[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];16[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
11.33/4.46	17[label="primbindIO vy4 (Monad.liftM20 id vy30)",fontsize=16,color="burlywood",shape="box"];36[label="vy4/IO vy40",fontsize=10,color="white",style="solid",shape="box"];17 -> 36[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	36 -> 18[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	37[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];17 -> 37[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	37 -> 19[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	38[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];17 -> 38[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	38 -> 20[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	39[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];17 -> 39[label="",style="solid", color="burlywood", weight=9];
11.33/4.46	39 -> 21[label="",style="solid", color="burlywood", weight=3];
11.33/4.46	18[label="primbindIO (IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
11.33/4.46	19[label="primbindIO (AProVE_IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
11.33/4.46	20[label="primbindIO (AProVE_Exception vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
11.33/4.46	21[label="primbindIO (AProVE_Error vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
11.33/4.46	22[label="error []",fontsize=16,color="red",shape="box"];23[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
11.33/4.46	24[label="AProVE_Exception vy40",fontsize=16,color="green",shape="box"];25[label="AProVE_Error vy40",fontsize=16,color="green",shape="box"];26[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
11.33/4.46	27[label="primretIO (id vy30 vy40)",fontsize=16,color="black",shape="box"];27 -> 28[label="",style="solid", color="black", weight=3];
11.33/4.46	28[label="AProVE_IO (id vy30 vy40)",fontsize=16,color="green",shape="box"];28 -> 29[label="",style="dashed", color="green", weight=3];
11.33/4.46	29[label="id vy30 vy40",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
11.33/4.46	30[label="vy30 vy40",fontsize=16,color="green",shape="box"];30 -> 31[label="",style="dashed", color="green", weight=3];
11.33/4.46	31[label="vy40",fontsize=16,color="green",shape="box"];}
11.33/4.46	
11.33/4.46	----------------------------------------
11.33/4.46	
11.33/4.46	(8)
11.33/4.46	YES
11.39/4.51	EOF