9.43/4.19	YES
11.63/4.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.63/4.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.63/4.77	
11.63/4.77	
11.63/4.77	H-Termination with start terms of the given HASKELL could be proven:
11.63/4.77	
11.63/4.77	(0) HASKELL
11.63/4.77	(1) LR [EQUIVALENT, 0 ms]
11.63/4.77	(2) HASKELL
11.63/4.77	(3) BR [EQUIVALENT, 0 ms]
11.63/4.77	(4) HASKELL
11.63/4.77	(5) COR [EQUIVALENT, 0 ms]
11.63/4.77	(6) HASKELL
11.63/4.77	(7) Narrow [SOUND, 0 ms]
11.63/4.77	(8) AND
11.63/4.77	    (9) QDP
11.63/4.77	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.63/4.77	        (11) YES
11.63/4.77	    (12) QDP
11.63/4.77	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.63/4.77	        (14) YES
11.63/4.77	    (15) QDP
11.63/4.77	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.63/4.77	        (17) YES
11.63/4.77	
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(0)
11.63/4.77	Obligation:
11.63/4.77	mainModule Main 
11.63/4.77	module Maybe where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Main where {
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Monad where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Prelude;
11.63/4.77	liftM2 :: Monad b => (d  ->  a  ->  c)  ->  b d  ->  b a  ->  b c;
11.63/4.77	liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));
11.63/4.77	
11.63/4.77	}
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(1) LR (EQUIVALENT)
11.63/4.77	Lambda Reductions:
11.63/4.77	The following Lambda expression
11.63/4.77	"\x2->return (f x1 x2)"
11.63/4.77	is transformed to
11.63/4.77	"liftM20 f x1 x2 = return (f x1 x2);
11.63/4.77	"
11.63/4.77	The following Lambda expression
11.63/4.77	"\x1->m2 >>= liftM20 f x1"
11.63/4.77	is transformed to
11.63/4.77	"liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.63/4.77	"
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(2)
11.63/4.77	Obligation:
11.63/4.77	mainModule Main 
11.63/4.77	module Maybe where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Main where {
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Monad where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Prelude;
11.63/4.77	liftM2 :: Monad b => (d  ->  a  ->  c)  ->  b d  ->  b a  ->  b c;
11.63/4.77	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.63/4.77	
11.63/4.77	liftM20 f x1 x2 = return (f x1 x2);
11.63/4.77	
11.63/4.77	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.63/4.77	
11.63/4.77	}
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(3) BR (EQUIVALENT)
11.63/4.77	Replaced joker patterns by fresh variables and removed binding patterns.
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(4)
11.63/4.77	Obligation:
11.63/4.77	mainModule Main 
11.63/4.77	module Maybe where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Main where {
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Monad where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Prelude;
11.63/4.77	liftM2 :: Monad b => (a  ->  d  ->  c)  ->  b a  ->  b d  ->  b c;
11.63/4.77	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.63/4.77	
11.63/4.77	liftM20 f x1 x2 = return (f x1 x2);
11.63/4.77	
11.63/4.77	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.63/4.77	
11.63/4.77	}
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(5) COR (EQUIVALENT)
11.63/4.77	Cond Reductions:
11.63/4.77	The following Function with conditions
11.63/4.77	"undefined |Falseundefined;
11.63/4.77	"
11.63/4.77	is transformed to
11.63/4.77	"undefined  = undefined1;
11.63/4.77	"
11.63/4.77	"undefined0 True = undefined;
11.63/4.77	"
11.63/4.77	"undefined1  = undefined0 False;
11.63/4.77	"
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(6)
11.63/4.77	Obligation:
11.63/4.77	mainModule Main 
11.63/4.77	module Maybe where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Main where {
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Monad;
11.63/4.77	import qualified Prelude;
11.63/4.77	}
11.63/4.77	module Monad where {
11.63/4.77	import qualified Main;
11.63/4.77	import qualified Maybe;
11.63/4.77	import qualified Prelude;
11.63/4.77	liftM2 :: Monad a => (d  ->  c  ->  b)  ->  a d  ->  a c  ->  a b;
11.63/4.77	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.63/4.77	
11.63/4.77	liftM20 f x1 x2 = return (f x1 x2);
11.63/4.77	
11.63/4.77	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.63/4.77	
11.63/4.77	}
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(7) Narrow (SOUND)
11.63/4.77	Haskell To QDPs
11.63/4.77	
11.63/4.77	digraph dp_graph {
11.63/4.77	node [outthreshold=100, inthreshold=100];1[label="Monad.liftM2",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.63/4.77	3[label="Monad.liftM2 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.63/4.77	4[label="Monad.liftM2 vy3 vy4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.63/4.77	5[label="Monad.liftM2 vy3 vy4 vy5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
11.63/4.77	6[label="vy4 >>= Monad.liftM21 vy5 vy3",fontsize=16,color="burlywood",shape="triangle"];49[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	49 -> 7[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	50[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	50 -> 8[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	7[label="vy40 : vy41 >>= Monad.liftM21 vy5 vy3",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.63/4.77	8[label="[] >>= Monad.liftM21 vy5 vy3",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.63/4.77	9 -> 11[label="",style="dashed", color="red", weight=0];
11.63/4.77	9[label="Monad.liftM21 vy5 vy3 vy40 ++ (vy41 >>= Monad.liftM21 vy5 vy3)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	10[label="[]",fontsize=16,color="green",shape="box"];12 -> 6[label="",style="dashed", color="red", weight=0];
11.63/4.77	12[label="vy41 >>= Monad.liftM21 vy5 vy3",fontsize=16,color="magenta"];12 -> 13[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	11[label="Monad.liftM21 vy5 vy3 vy40 ++ vy6",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
11.63/4.77	13[label="vy41",fontsize=16,color="green",shape="box"];14[label="(vy5 >>= Monad.liftM20 vy3 vy40) ++ vy6",fontsize=16,color="burlywood",shape="box"];51[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];14 -> 51[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	51 -> 15[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	52[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];14 -> 52[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	52 -> 16[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	15[label="(vy50 : vy51 >>= Monad.liftM20 vy3 vy40) ++ vy6",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
11.63/4.77	16[label="([] >>= Monad.liftM20 vy3 vy40) ++ vy6",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
11.63/4.77	17[label="(Monad.liftM20 vy3 vy40 vy50 ++ (vy51 >>= Monad.liftM20 vy3 vy40)) ++ vy6",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
11.63/4.77	18[label="[] ++ vy6",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
11.63/4.77	19[label="(return (vy3 vy40 vy50) ++ (vy51 >>= Monad.liftM20 vy3 vy40)) ++ vy6",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
11.63/4.77	20[label="vy6",fontsize=16,color="green",shape="box"];21[label="((vy3 vy40 vy50 : []) ++ (vy51 >>= Monad.liftM20 vy3 vy40)) ++ vy6",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
11.63/4.77	22 -> 23[label="",style="dashed", color="red", weight=0];
11.63/4.77	22[label="(vy3 vy40 vy50 : [] ++ (vy51 >>= Monad.liftM20 vy3 vy40)) ++ vy6",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	24 -> 18[label="",style="dashed", color="red", weight=0];
11.63/4.77	24[label="[] ++ (vy51 >>= Monad.liftM20 vy3 vy40)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	23[label="(vy3 vy40 vy50 : vy7) ++ vy6",fontsize=16,color="black",shape="triangle"];23 -> 26[label="",style="solid", color="black", weight=3];
11.63/4.77	25[label="vy51 >>= Monad.liftM20 vy3 vy40",fontsize=16,color="burlywood",shape="triangle"];53[label="vy51/vy510 : vy511",fontsize=10,color="white",style="solid",shape="box"];25 -> 53[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	53 -> 27[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	54[label="vy51/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 54[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	54 -> 28[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	26[label="vy3 vy40 vy50 : vy7 ++ vy6",fontsize=16,color="green",shape="box"];26 -> 29[label="",style="dashed", color="green", weight=3];
11.63/4.77	26 -> 30[label="",style="dashed", color="green", weight=3];
11.63/4.77	27[label="vy510 : vy511 >>= Monad.liftM20 vy3 vy40",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
11.63/4.77	28[label="[] >>= Monad.liftM20 vy3 vy40",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
11.63/4.77	29[label="vy3 vy40 vy50",fontsize=16,color="green",shape="box"];29 -> 33[label="",style="dashed", color="green", weight=3];
11.63/4.77	29 -> 34[label="",style="dashed", color="green", weight=3];
11.63/4.77	30[label="vy7 ++ vy6",fontsize=16,color="burlywood",shape="triangle"];55[label="vy7/vy70 : vy71",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	55 -> 35[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	56[label="vy7/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 56[label="",style="solid", color="burlywood", weight=9];
11.63/4.77	56 -> 36[label="",style="solid", color="burlywood", weight=3];
11.63/4.77	31 -> 30[label="",style="dashed", color="red", weight=0];
11.63/4.77	31[label="Monad.liftM20 vy3 vy40 vy510 ++ (vy511 >>= Monad.liftM20 vy3 vy40)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	31 -> 38[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	32[label="[]",fontsize=16,color="green",shape="box"];33[label="vy40",fontsize=16,color="green",shape="box"];34[label="vy50",fontsize=16,color="green",shape="box"];35[label="(vy70 : vy71) ++ vy6",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
11.63/4.77	36[label="[] ++ vy6",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
11.63/4.77	37 -> 25[label="",style="dashed", color="red", weight=0];
11.63/4.77	37[label="vy511 >>= Monad.liftM20 vy3 vy40",fontsize=16,color="magenta"];37 -> 41[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	38[label="Monad.liftM20 vy3 vy40 vy510",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
11.63/4.77	39[label="vy70 : vy71 ++ vy6",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
11.63/4.77	40[label="vy6",fontsize=16,color="green",shape="box"];41[label="vy511",fontsize=16,color="green",shape="box"];42[label="return (vy3 vy40 vy510)",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
11.63/4.77	43 -> 30[label="",style="dashed", color="red", weight=0];
11.63/4.77	43[label="vy71 ++ vy6",fontsize=16,color="magenta"];43 -> 45[label="",style="dashed", color="magenta", weight=3];
11.63/4.77	44[label="vy3 vy40 vy510 : []",fontsize=16,color="green",shape="box"];44 -> 46[label="",style="dashed", color="green", weight=3];
11.63/4.77	45[label="vy71",fontsize=16,color="green",shape="box"];46[label="vy3 vy40 vy510",fontsize=16,color="green",shape="box"];46 -> 47[label="",style="dashed", color="green", weight=3];
11.63/4.77	46 -> 48[label="",style="dashed", color="green", weight=3];
11.63/4.77	47[label="vy40",fontsize=16,color="green",shape="box"];48[label="vy510",fontsize=16,color="green",shape="box"];}
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(8)
11.63/4.77	Complex Obligation (AND)
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(9)
11.63/4.77	Obligation:
11.63/4.77	Q DP problem:
11.63/4.77	The TRS P consists of the following rules:
11.63/4.77	
11.63/4.77	   new_gtGtEs0(:(vy40, vy41), vy5, vy3, h, ba, bb) -> new_gtGtEs0(vy41, vy5, vy3, h, ba, bb)
11.63/4.77	
11.63/4.77	R is empty.
11.63/4.77	Q is empty.
11.63/4.77	We have to consider all minimal (P,Q,R)-chains.
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(10) QDPSizeChangeProof (EQUIVALENT)
11.63/4.77	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. 
11.63/4.77	
11.63/4.77	From the DPs we obtained the following set of size-change graphs:
11.63/4.77	*new_gtGtEs0(:(vy40, vy41), vy5, vy3, h, ba, bb) -> new_gtGtEs0(vy41, vy5, vy3, h, ba, bb)
11.63/4.77	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6
11.63/4.77	
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(11)
11.63/4.77	YES
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(12)
11.63/4.77	Obligation:
11.63/4.77	Q DP problem:
11.63/4.77	The TRS P consists of the following rules:
11.63/4.77	
11.63/4.77	   new_gtGtEs(:(vy510, vy511), vy3, vy40, h, ba, bb) -> new_gtGtEs(vy511, vy3, vy40, h, ba, bb)
11.63/4.77	
11.63/4.77	R is empty.
11.63/4.77	Q is empty.
11.63/4.77	We have to consider all minimal (P,Q,R)-chains.
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(13) QDPSizeChangeProof (EQUIVALENT)
11.63/4.77	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. 
11.63/4.77	
11.63/4.77	From the DPs we obtained the following set of size-change graphs:
11.63/4.77	*new_gtGtEs(:(vy510, vy511), vy3, vy40, h, ba, bb) -> new_gtGtEs(vy511, vy3, vy40, h, ba, bb)
11.63/4.77	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6
11.63/4.77	
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(14)
11.63/4.77	YES
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(15)
11.63/4.77	Obligation:
11.63/4.77	Q DP problem:
11.63/4.77	The TRS P consists of the following rules:
11.63/4.77	
11.63/4.77	   new_psPs(:(vy70, vy71), vy6, h) -> new_psPs(vy71, vy6, h)
11.63/4.77	
11.63/4.77	R is empty.
11.63/4.77	Q is empty.
11.63/4.77	We have to consider all minimal (P,Q,R)-chains.
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(16) QDPSizeChangeProof (EQUIVALENT)
11.63/4.77	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. 
11.63/4.77	
11.63/4.77	From the DPs we obtained the following set of size-change graphs:
11.63/4.77	*new_psPs(:(vy70, vy71), vy6, h) -> new_psPs(vy71, vy6, h)
11.63/4.77	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.63/4.77	
11.63/4.77	
11.63/4.77	----------------------------------------
11.63/4.77	
11.63/4.77	(17)
11.63/4.77	YES
11.83/6.40	EOF