9.15/3.88	YES
10.87/4.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.87/4.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.87/4.37	
10.87/4.37	
10.87/4.37	H-Termination with start terms of the given HASKELL could be proven:
10.87/4.37	
10.87/4.37	(0) HASKELL
10.87/4.37	(1) BR [EQUIVALENT, 0 ms]
10.87/4.37	(2) HASKELL
10.87/4.37	(3) COR [EQUIVALENT, 0 ms]
10.87/4.37	(4) HASKELL
10.87/4.37	(5) Narrow [SOUND, 0 ms]
10.87/4.37	(6) AND
10.87/4.37	    (7) QDP
10.87/4.37	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.87/4.37	        (9) YES
10.87/4.37	    (10) QDP
10.87/4.37	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.87/4.37	        (12) YES
10.87/4.37	
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(0)
10.87/4.37	Obligation:
10.87/4.37	mainModule Main 
10.87/4.37	module Maybe where {
10.87/4.37	import qualified Main;
10.87/4.37	import qualified Monad;
10.87/4.37	import qualified Prelude;
10.87/4.37	}
10.87/4.37	module Main where {
10.87/4.37	import qualified Maybe;
10.87/4.37	import qualified Monad;
10.87/4.37	import qualified Prelude;
10.87/4.37	}
10.87/4.37	module Monad where {
10.87/4.37	import qualified Main;
10.87/4.37	import qualified Maybe;
10.87/4.37	import qualified Prelude;
10.87/4.37	join :: Monad b => b (b a)  ->  b a;
10.87/4.37	join x = x >>= id;
10.87/4.37	
10.87/4.37	}
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(1) BR (EQUIVALENT)
10.87/4.37	Replaced joker patterns by fresh variables and removed binding patterns.
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(2)
10.87/4.37	Obligation:
10.87/4.37	mainModule Main 
10.87/4.37	module Maybe where {
10.87/4.37	import qualified Main;
10.87/4.37	import qualified Monad;
10.87/4.37	import qualified Prelude;
10.87/4.37	}
10.87/4.37	module Main where {
10.87/4.37	import qualified Maybe;
10.87/4.37	import qualified Monad;
10.87/4.37	import qualified Prelude;
10.87/4.37	}
10.87/4.37	module Monad where {
10.87/4.37	import qualified Main;
10.87/4.37	import qualified Maybe;
10.87/4.37	import qualified Prelude;
10.87/4.37	join :: Monad a => a (a b)  ->  a b;
10.87/4.37	join x = x >>= id;
10.87/4.37	
10.87/4.37	}
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(3) COR (EQUIVALENT)
10.87/4.37	Cond Reductions:
10.87/4.37	The following Function with conditions
10.87/4.37	"undefined |Falseundefined;
10.87/4.37	"
10.87/4.37	is transformed to
10.87/4.37	"undefined  = undefined1;
10.87/4.37	"
10.87/4.37	"undefined0 True = undefined;
10.87/4.37	"
10.87/4.37	"undefined1  = undefined0 False;
10.87/4.37	"
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(4)
10.87/4.37	Obligation:
10.87/4.37	mainModule Main 
10.87/4.37	module Maybe where {
10.87/4.37	import qualified Main;
10.87/4.37	import qualified Monad;
10.87/4.37	import qualified Prelude;
10.87/4.37	}
10.87/4.37	module Main where {
10.87/4.37	import qualified Maybe;
10.87/4.37	import qualified Monad;
10.87/4.37	import qualified Prelude;
10.87/4.37	}
10.87/4.37	module Monad where {
10.87/4.37	import qualified Main;
10.87/4.37	import qualified Maybe;
10.87/4.37	import qualified Prelude;
10.87/4.37	join :: Monad b => b (b a)  ->  b a;
10.87/4.37	join x = x >>= id;
10.87/4.37	
10.87/4.37	}
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(5) Narrow (SOUND)
10.87/4.37	Haskell To QDPs
10.87/4.37	
10.87/4.37	digraph dp_graph {
10.87/4.37	node [outthreshold=100, inthreshold=100];1[label="Monad.join",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.87/4.37	3[label="Monad.join vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.87/4.37	4[label="vy3 >>= id",fontsize=16,color="blue",shape="box"];37[label=">>= :: (IO (IO a)) -> ((IO a) -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];4 -> 37[label="",style="solid", color="blue", weight=9];
10.87/4.37	37 -> 5[label="",style="solid", color="blue", weight=3];
10.87/4.37	38[label=">>= :: (Maybe (Maybe a)) -> ((Maybe a) -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];4 -> 38[label="",style="solid", color="blue", weight=9];
10.87/4.37	38 -> 6[label="",style="solid", color="blue", weight=3];
10.87/4.37	39[label=">>= :: ([] ([] a)) -> (([] a) -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="blue", weight=9];
10.87/4.37	39 -> 7[label="",style="solid", color="blue", weight=3];
10.87/4.37	5[label="vy3 >>= id",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
10.87/4.37	6[label="vy3 >>= id",fontsize=16,color="burlywood",shape="box"];40[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 40[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	40 -> 9[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	41[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];6 -> 41[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	41 -> 10[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	7[label="vy3 >>= id",fontsize=16,color="burlywood",shape="triangle"];42[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];7 -> 42[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	42 -> 11[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	43[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 43[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	43 -> 12[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	8[label="primbindIO vy3 id",fontsize=16,color="burlywood",shape="box"];44[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 44[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	44 -> 13[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	45[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 45[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	45 -> 14[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	46[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 46[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	46 -> 15[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	47[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 47[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	47 -> 16[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	9[label="Nothing >>= id",fontsize=16,color="black",shape="box"];9 -> 17[label="",style="solid", color="black", weight=3];
10.87/4.37	10[label="Just vy30 >>= id",fontsize=16,color="black",shape="box"];10 -> 18[label="",style="solid", color="black", weight=3];
10.87/4.37	11[label="vy30 : vy31 >>= id",fontsize=16,color="black",shape="box"];11 -> 19[label="",style="solid", color="black", weight=3];
10.87/4.37	12[label="[] >>= id",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
10.87/4.37	13[label="primbindIO (IO vy30) id",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
10.87/4.37	14[label="primbindIO (AProVE_IO vy30) id",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.87/4.37	15[label="primbindIO (AProVE_Exception vy30) id",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.87/4.37	16[label="primbindIO (AProVE_Error vy30) id",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
10.87/4.37	17[label="Nothing",fontsize=16,color="green",shape="box"];18[label="id vy30",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
10.87/4.37	19 -> 26[label="",style="dashed", color="red", weight=0];
10.87/4.37	19[label="id vy30 ++ (vy31 >>= id)",fontsize=16,color="magenta"];19 -> 27[label="",style="dashed", color="magenta", weight=3];
10.87/4.37	20[label="[]",fontsize=16,color="green",shape="box"];21[label="error []",fontsize=16,color="red",shape="box"];22[label="id vy30",fontsize=16,color="black",shape="box"];22 -> 28[label="",style="solid", color="black", weight=3];
10.87/4.37	23[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];24[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];25[label="vy30",fontsize=16,color="green",shape="box"];27 -> 7[label="",style="dashed", color="red", weight=0];
10.87/4.37	27[label="vy31 >>= id",fontsize=16,color="magenta"];27 -> 29[label="",style="dashed", color="magenta", weight=3];
10.87/4.37	26[label="id vy30 ++ vy4",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
10.87/4.37	28[label="vy30",fontsize=16,color="green",shape="box"];29[label="vy31",fontsize=16,color="green",shape="box"];30[label="vy30 ++ vy4",fontsize=16,color="burlywood",shape="triangle"];48[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];30 -> 48[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	48 -> 31[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	49[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 49[label="",style="solid", color="burlywood", weight=9];
10.87/4.37	49 -> 32[label="",style="solid", color="burlywood", weight=3];
10.87/4.37	31[label="(vy300 : vy301) ++ vy4",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
10.87/4.37	32[label="[] ++ vy4",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
10.87/4.37	33[label="vy300 : vy301 ++ vy4",fontsize=16,color="green",shape="box"];33 -> 35[label="",style="dashed", color="green", weight=3];
10.87/4.37	34[label="vy4",fontsize=16,color="green",shape="box"];35 -> 30[label="",style="dashed", color="red", weight=0];
10.87/4.37	35[label="vy301 ++ vy4",fontsize=16,color="magenta"];35 -> 36[label="",style="dashed", color="magenta", weight=3];
10.87/4.37	36[label="vy301",fontsize=16,color="green",shape="box"];}
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(6)
10.87/4.37	Complex Obligation (AND)
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(7)
10.87/4.37	Obligation:
10.87/4.37	Q DP problem:
10.87/4.37	The TRS P consists of the following rules:
10.87/4.37	
10.87/4.37	   new_gtGtEs(:(vy30, vy31), h) -> new_gtGtEs(vy31, h)
10.87/4.37	
10.87/4.37	R is empty.
10.87/4.37	Q is empty.
10.87/4.37	We have to consider all minimal (P,Q,R)-chains.
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(8) QDPSizeChangeProof (EQUIVALENT)
10.87/4.37	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. 
10.87/4.37	
10.87/4.37	From the DPs we obtained the following set of size-change graphs:
10.87/4.37	*new_gtGtEs(:(vy30, vy31), h) -> new_gtGtEs(vy31, h)
10.87/4.37	The graph contains the following edges 1 > 1, 2 >= 2
10.87/4.37	
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(9)
10.87/4.37	YES
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(10)
10.87/4.37	Obligation:
10.87/4.37	Q DP problem:
10.87/4.37	The TRS P consists of the following rules:
10.87/4.37	
10.87/4.37	   new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
10.87/4.37	
10.87/4.37	R is empty.
10.87/4.37	Q is empty.
10.87/4.37	We have to consider all minimal (P,Q,R)-chains.
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(11) QDPSizeChangeProof (EQUIVALENT)
10.87/4.37	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. 
10.87/4.37	
10.87/4.37	From the DPs we obtained the following set of size-change graphs:
10.87/4.37	*new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
10.87/4.37	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.87/4.37	
10.87/4.37	
10.87/4.37	----------------------------------------
10.87/4.37	
10.87/4.37	(12)
10.87/4.37	YES
11.13/4.50	EOF