9.12/3.84	YES
11.02/4.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.02/4.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.02/4.37	
11.02/4.37	
11.02/4.37	H-Termination with start terms of the given HASKELL could be proven:
11.02/4.37	
11.02/4.37	(0) HASKELL
11.02/4.37	(1) LR [EQUIVALENT, 0 ms]
11.02/4.37	(2) HASKELL
11.02/4.37	(3) BR [EQUIVALENT, 0 ms]
11.02/4.37	(4) HASKELL
11.02/4.37	(5) COR [EQUIVALENT, 0 ms]
11.02/4.37	(6) HASKELL
11.02/4.37	(7) Narrow [SOUND, 0 ms]
11.02/4.37	(8) AND
11.02/4.37	    (9) QDP
11.02/4.37	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.02/4.37	        (11) YES
11.02/4.37	    (12) QDP
11.02/4.37	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.02/4.37	        (14) YES
11.02/4.37	    (15) QDP
11.02/4.37	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.02/4.37	        (17) YES
11.02/4.37	
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(0)
11.02/4.37	Obligation:
11.02/4.37	mainModule Main 
11.02/4.37	module Maybe where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Main where {
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Monad where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Prelude;
11.02/4.37	ap :: Monad a => a (b  ->  c)  ->  a b  ->  a c;
11.02/4.37	ap = liftM2 id;
11.02/4.37	
11.02/4.37	liftM2 :: Monad c => (d  ->  a  ->  b)  ->  c d  ->  c a  ->  c b;
11.02/4.37	liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));
11.02/4.37	
11.02/4.37	}
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(1) LR (EQUIVALENT)
11.02/4.37	Lambda Reductions:
11.02/4.37	The following Lambda expression
11.02/4.37	"\x2->return (f x1 x2)"
11.02/4.37	is transformed to
11.02/4.37	"liftM20 f x1 x2 = return (f x1 x2);
11.02/4.37	"
11.02/4.37	The following Lambda expression
11.02/4.37	"\x1->m2 >>= liftM20 f x1"
11.02/4.37	is transformed to
11.02/4.37	"liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.02/4.37	"
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(2)
11.02/4.37	Obligation:
11.02/4.37	mainModule Main 
11.02/4.37	module Maybe where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Main where {
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Monad where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Prelude;
11.02/4.37	ap :: Monad a => a (b  ->  c)  ->  a b  ->  a c;
11.02/4.37	ap = liftM2 id;
11.02/4.37	
11.02/4.37	liftM2 :: Monad b => (d  ->  a  ->  c)  ->  b d  ->  b a  ->  b c;
11.02/4.37	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.02/4.37	
11.02/4.37	liftM20 f x1 x2 = return (f x1 x2);
11.02/4.37	
11.02/4.37	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.02/4.37	
11.02/4.37	}
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(3) BR (EQUIVALENT)
11.02/4.37	Replaced joker patterns by fresh variables and removed binding patterns.
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(4)
11.02/4.37	Obligation:
11.02/4.37	mainModule Main 
11.02/4.37	module Maybe where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Main where {
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Monad where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Prelude;
11.02/4.37	ap :: Monad c => c (b  ->  a)  ->  c b  ->  c a;
11.02/4.37	ap = liftM2 id;
11.02/4.37	
11.02/4.37	liftM2 :: Monad c => (d  ->  b  ->  a)  ->  c d  ->  c b  ->  c a;
11.02/4.37	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.02/4.37	
11.02/4.37	liftM20 f x1 x2 = return (f x1 x2);
11.02/4.37	
11.02/4.37	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.02/4.37	
11.02/4.37	}
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(5) COR (EQUIVALENT)
11.02/4.37	Cond Reductions:
11.02/4.37	The following Function with conditions
11.02/4.37	"undefined |Falseundefined;
11.02/4.37	"
11.02/4.37	is transformed to
11.02/4.37	"undefined  = undefined1;
11.02/4.37	"
11.02/4.37	"undefined0 True = undefined;
11.02/4.37	"
11.02/4.37	"undefined1  = undefined0 False;
11.02/4.37	"
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(6)
11.02/4.37	Obligation:
11.02/4.37	mainModule Main 
11.02/4.37	module Maybe where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Main where {
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Monad;
11.02/4.37	import qualified Prelude;
11.02/4.37	}
11.02/4.37	module Monad where {
11.02/4.37	import qualified Main;
11.02/4.37	import qualified Maybe;
11.02/4.37	import qualified Prelude;
11.02/4.37	ap :: Monad c => c (b  ->  a)  ->  c b  ->  c a;
11.02/4.37	ap = liftM2 id;
11.02/4.37	
11.02/4.37	liftM2 :: Monad b => (c  ->  a  ->  d)  ->  b c  ->  b a  ->  b d;
11.02/4.37	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.02/4.37	
11.02/4.37	liftM20 f x1 x2 = return (f x1 x2);
11.02/4.37	
11.02/4.37	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.02/4.37	
11.02/4.37	}
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(7) Narrow (SOUND)
11.02/4.37	Haskell To QDPs
11.02/4.37	
11.02/4.37	digraph dp_graph {
11.02/4.37	node [outthreshold=100, inthreshold=100];1[label="Monad.ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.02/4.37	3[label="Monad.ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.02/4.37	4[label="Monad.ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.02/4.37	5[label="Monad.liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.02/4.37	6[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="triangle"];49[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	49 -> 7[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	50[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	50 -> 8[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	7[label="vy30 : vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.02/4.37	8[label="[] >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.02/4.37	9 -> 11[label="",style="dashed", color="red", weight=0];
11.02/4.37	9[label="Monad.liftM21 vy4 id vy30 ++ (vy31 >>= Monad.liftM21 vy4 id)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	10[label="[]",fontsize=16,color="green",shape="box"];12 -> 6[label="",style="dashed", color="red", weight=0];
11.02/4.37	12[label="vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="magenta"];12 -> 13[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	11[label="Monad.liftM21 vy4 id vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
11.02/4.37	13[label="vy31",fontsize=16,color="green",shape="box"];14[label="(vy4 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="burlywood",shape="box"];51[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];14 -> 51[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	51 -> 15[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	52[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];14 -> 52[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	52 -> 16[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	15[label="(vy40 : vy41 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
11.02/4.37	16[label="([] >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
11.02/4.37	17[label="(Monad.liftM20 id vy30 vy40 ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
11.02/4.37	18[label="[] ++ vy5",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
11.02/4.37	19[label="(return (id vy30 vy40) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
11.02/4.37	20[label="vy5",fontsize=16,color="green",shape="box"];21[label="((id vy30 vy40 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
11.02/4.37	22 -> 23[label="",style="dashed", color="red", weight=0];
11.02/4.37	22[label="(id vy30 vy40 : [] ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	24 -> 18[label="",style="dashed", color="red", weight=0];
11.02/4.37	24[label="[] ++ (vy41 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	23[label="(id vy30 vy40 : vy6) ++ vy5",fontsize=16,color="black",shape="triangle"];23 -> 26[label="",style="solid", color="black", weight=3];
11.02/4.37	25[label="vy41 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="triangle"];53[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];25 -> 53[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	53 -> 27[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	54[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 54[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	54 -> 28[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	26[label="id vy30 vy40 : vy6 ++ vy5",fontsize=16,color="green",shape="box"];26 -> 29[label="",style="dashed", color="green", weight=3];
11.02/4.37	26 -> 30[label="",style="dashed", color="green", weight=3];
11.02/4.37	27[label="vy410 : vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
11.02/4.37	28[label="[] >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
11.02/4.37	29[label="id vy30 vy40",fontsize=16,color="black",shape="triangle"];29 -> 33[label="",style="solid", color="black", weight=3];
11.02/4.37	30[label="vy6 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];55[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	55 -> 34[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	56[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 56[label="",style="solid", color="burlywood", weight=9];
11.02/4.37	56 -> 35[label="",style="solid", color="burlywood", weight=3];
11.02/4.37	31 -> 30[label="",style="dashed", color="red", weight=0];
11.02/4.37	31[label="Monad.liftM20 id vy30 vy410 ++ (vy411 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	31 -> 37[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	32[label="[]",fontsize=16,color="green",shape="box"];33[label="vy30 vy40",fontsize=16,color="green",shape="box"];33 -> 38[label="",style="dashed", color="green", weight=3];
11.02/4.37	34[label="(vy60 : vy61) ++ vy5",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
11.02/4.37	35[label="[] ++ vy5",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
11.02/4.37	36[label="Monad.liftM20 id vy30 vy410",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
11.02/4.37	37 -> 25[label="",style="dashed", color="red", weight=0];
11.02/4.37	37[label="vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	38[label="vy40",fontsize=16,color="green",shape="box"];39[label="vy60 : vy61 ++ vy5",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
11.02/4.37	40[label="vy5",fontsize=16,color="green",shape="box"];41 -> 44[label="",style="dashed", color="red", weight=0];
11.02/4.37	41[label="return (id vy30 vy410)",fontsize=16,color="magenta"];41 -> 45[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	42[label="vy411",fontsize=16,color="green",shape="box"];43 -> 30[label="",style="dashed", color="red", weight=0];
11.02/4.37	43[label="vy61 ++ vy5",fontsize=16,color="magenta"];43 -> 46[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	45 -> 29[label="",style="dashed", color="red", weight=0];
11.02/4.37	45[label="id vy30 vy410",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
11.02/4.37	44[label="return vy7",fontsize=16,color="black",shape="triangle"];44 -> 48[label="",style="solid", color="black", weight=3];
11.02/4.37	46[label="vy61",fontsize=16,color="green",shape="box"];47[label="vy410",fontsize=16,color="green",shape="box"];48[label="vy7 : []",fontsize=16,color="green",shape="box"];}
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(8)
11.02/4.37	Complex Obligation (AND)
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(9)
11.02/4.37	Obligation:
11.02/4.37	Q DP problem:
11.02/4.37	The TRS P consists of the following rules:
11.02/4.37	
11.02/4.37	   new_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)
11.02/4.37	
11.02/4.37	R is empty.
11.02/4.37	Q is empty.
11.02/4.37	We have to consider all minimal (P,Q,R)-chains.
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(10) QDPSizeChangeProof (EQUIVALENT)
11.02/4.37	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. 
11.02/4.37	
11.02/4.37	From the DPs we obtained the following set of size-change graphs:
11.02/4.37	*new_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)
11.02/4.37	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
11.02/4.37	
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(11)
11.02/4.37	YES
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(12)
11.02/4.37	Obligation:
11.02/4.37	Q DP problem:
11.02/4.37	The TRS P consists of the following rules:
11.02/4.37	
11.02/4.37	   new_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)
11.02/4.37	
11.02/4.37	R is empty.
11.02/4.37	Q is empty.
11.02/4.37	We have to consider all minimal (P,Q,R)-chains.
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(13) QDPSizeChangeProof (EQUIVALENT)
11.02/4.37	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. 
11.02/4.37	
11.02/4.37	From the DPs we obtained the following set of size-change graphs:
11.02/4.37	*new_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)
11.02/4.37	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
11.02/4.37	
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(14)
11.02/4.37	YES
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(15)
11.02/4.37	Obligation:
11.02/4.37	Q DP problem:
11.02/4.37	The TRS P consists of the following rules:
11.02/4.37	
11.02/4.37	   new_psPs(:(vy60, vy61), vy5, h) -> new_psPs(vy61, vy5, h)
11.02/4.37	
11.02/4.37	R is empty.
11.02/4.37	Q is empty.
11.02/4.37	We have to consider all minimal (P,Q,R)-chains.
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(16) QDPSizeChangeProof (EQUIVALENT)
11.02/4.37	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. 
11.02/4.37	
11.02/4.37	From the DPs we obtained the following set of size-change graphs:
11.02/4.37	*new_psPs(:(vy60, vy61), vy5, h) -> new_psPs(vy61, vy5, h)
11.02/4.37	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.02/4.37	
11.02/4.37	
11.02/4.37	----------------------------------------
11.02/4.37	
11.02/4.37	(17)
11.02/4.37	YES
11.24/4.41	EOF