8.96/3.91	YES
11.22/4.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.22/4.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.22/4.46	
11.22/4.46	
11.22/4.46	H-Termination with start terms of the given HASKELL could be proven:
11.22/4.46	
11.22/4.46	(0) HASKELL
11.22/4.46	(1) LR [EQUIVALENT, 0 ms]
11.22/4.46	(2) HASKELL
11.22/4.46	(3) BR [EQUIVALENT, 0 ms]
11.22/4.46	(4) HASKELL
11.22/4.46	(5) COR [EQUIVALENT, 0 ms]
11.22/4.46	(6) HASKELL
11.22/4.46	(7) Narrow [SOUND, 0 ms]
11.22/4.46	(8) QDP
11.22/4.46	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.22/4.46	(10) YES
11.22/4.46	
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(0)
11.22/4.46	Obligation:
11.22/4.46	mainModule Main 
11.22/4.46	module Maybe where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Main where {
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Monad where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Prelude;
11.22/4.46	foldM :: Monad b => (a  ->  c  ->  b a)  ->  a  ->  [c]  ->  b a;
11.22/4.46	foldM _ a [] = return a;
11.22/4.46	foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs);
11.22/4.46	
11.22/4.46	foldM_ :: Monad c => (a  ->  b  ->  c a)  ->  a  ->  [b]  ->  c ();
11.22/4.46	foldM_ f a xs = foldM f a xs >> return ();
11.22/4.46	
11.22/4.46	}
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(1) LR (EQUIVALENT)
11.22/4.46	Lambda Reductions:
11.22/4.46	The following Lambda expression
11.22/4.46	"\_->q"
11.22/4.46	is transformed to
11.22/4.46	"gtGt0 q _ = q;
11.22/4.46	"
11.22/4.46	The following Lambda expression
11.22/4.46	"\fax->foldM f fax xs"
11.22/4.46	is transformed to
11.22/4.46	"foldM0 f xs fax = foldM f fax xs;
11.22/4.46	"
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(2)
11.22/4.46	Obligation:
11.22/4.46	mainModule Main 
11.22/4.46	module Maybe where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Main where {
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Monad where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Prelude;
11.22/4.46	foldM :: Monad b => (c  ->  a  ->  b c)  ->  c  ->  [a]  ->  b c;
11.22/4.46	foldM _ a [] = return a;
11.22/4.46	foldM f a (x : xs) = f a x >>= foldM0 f xs;
11.22/4.46	
11.22/4.46	foldM0 f xs fax = foldM f fax xs;
11.22/4.46	
11.22/4.46	foldM_ :: Monad a => (c  ->  b  ->  a c)  ->  c  ->  [b]  ->  a ();
11.22/4.46	foldM_ f a xs = foldM f a xs >> return ();
11.22/4.46	
11.22/4.46	}
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(3) BR (EQUIVALENT)
11.22/4.46	Replaced joker patterns by fresh variables and removed binding patterns.
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(4)
11.22/4.46	Obligation:
11.22/4.46	mainModule Main 
11.22/4.46	module Maybe where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Main where {
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Monad where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Prelude;
11.22/4.46	foldM :: Monad b => (c  ->  a  ->  b c)  ->  c  ->  [a]  ->  b c;
11.22/4.46	foldM vz a [] = return a;
11.22/4.46	foldM f a (x : xs) = f a x >>= foldM0 f xs;
11.22/4.46	
11.22/4.46	foldM0 f xs fax = foldM f fax xs;
11.22/4.46	
11.22/4.46	foldM_ :: Monad c => (b  ->  a  ->  c b)  ->  b  ->  [a]  ->  c ();
11.22/4.46	foldM_ f a xs = foldM f a xs >> return ();
11.22/4.46	
11.22/4.46	}
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(5) COR (EQUIVALENT)
11.22/4.46	Cond Reductions:
11.22/4.46	The following Function with conditions
11.22/4.46	"undefined |Falseundefined;
11.22/4.46	"
11.22/4.46	is transformed to
11.22/4.46	"undefined  = undefined1;
11.22/4.46	"
11.22/4.46	"undefined0 True = undefined;
11.22/4.46	"
11.22/4.46	"undefined1  = undefined0 False;
11.22/4.46	"
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(6)
11.22/4.46	Obligation:
11.22/4.46	mainModule Main 
11.22/4.46	module Maybe where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Main where {
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Monad;
11.22/4.46	import qualified Prelude;
11.22/4.46	}
11.22/4.46	module Monad where {
11.22/4.46	import qualified Main;
11.22/4.46	import qualified Maybe;
11.22/4.46	import qualified Prelude;
11.22/4.46	foldM :: Monad b => (c  ->  a  ->  b c)  ->  c  ->  [a]  ->  b c;
11.22/4.46	foldM vz a [] = return a;
11.22/4.46	foldM f a (x : xs) = f a x >>= foldM0 f xs;
11.22/4.46	
11.22/4.46	foldM0 f xs fax = foldM f fax xs;
11.22/4.46	
11.22/4.46	foldM_ :: Monad a => (c  ->  b  ->  a c)  ->  c  ->  [b]  ->  a ();
11.22/4.46	foldM_ f a xs = foldM f a xs >> return ();
11.22/4.46	
11.22/4.46	}
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(7) Narrow (SOUND)
11.22/4.46	Haskell To QDPs
11.22/4.46	
11.22/4.46	digraph dp_graph {
11.22/4.46	node [outthreshold=100, inthreshold=100];1[label="Monad.foldM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.22/4.46	3[label="Monad.foldM_ wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.22/4.46	4[label="Monad.foldM_ wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.22/4.46	5[label="Monad.foldM_ wu3 wu4 wu5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
11.22/4.46	6[label="Monad.foldM wu3 wu4 wu5 >> return ()",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.22/4.46	7[label="Monad.foldM wu3 wu4 wu5 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
11.22/4.46	8[label="primbindIO (Monad.foldM wu3 wu4 wu5) (gtGt0 (return ()))",fontsize=16,color="burlywood",shape="triangle"];39[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];8 -> 39[label="",style="solid", color="burlywood", weight=9];
11.22/4.46	39 -> 9[label="",style="solid", color="burlywood", weight=3];
11.22/4.46	40[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 40[label="",style="solid", color="burlywood", weight=9];
11.22/4.46	40 -> 10[label="",style="solid", color="burlywood", weight=3];
11.22/4.46	9[label="primbindIO (Monad.foldM wu3 wu4 (wu50 : wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
11.22/4.46	10[label="primbindIO (Monad.foldM wu3 wu4 []) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11.22/4.46	11[label="primbindIO (wu3 wu4 wu50 >>= Monad.foldM0 wu3 wu51) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
11.22/4.46	12[label="primbindIO (return wu4) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.22/4.46	13 -> 15[label="",style="dashed", color="red", weight=0];
11.22/4.46	13[label="primbindIO (primbindIO (wu3 wu4 wu50) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
11.22/4.46	14[label="primbindIO (primretIO wu4) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
11.22/4.46	16[label="wu3 wu4 wu50",fontsize=16,color="green",shape="box"];16 -> 24[label="",style="dashed", color="green", weight=3];
11.22/4.46	16 -> 25[label="",style="dashed", color="green", weight=3];
11.22/4.46	15[label="primbindIO (primbindIO wu6 (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="burlywood",shape="triangle"];41[label="wu6/IO wu60",fontsize=10,color="white",style="solid",shape="box"];15 -> 41[label="",style="solid", color="burlywood", weight=9];
11.22/4.46	41 -> 20[label="",style="solid", color="burlywood", weight=3];
11.22/4.46	42[label="wu6/AProVE_IO wu60",fontsize=10,color="white",style="solid",shape="box"];15 -> 42[label="",style="solid", color="burlywood", weight=9];
11.22/4.46	42 -> 21[label="",style="solid", color="burlywood", weight=3];
11.22/4.46	43[label="wu6/AProVE_Exception wu60",fontsize=10,color="white",style="solid",shape="box"];15 -> 43[label="",style="solid", color="burlywood", weight=9];
11.22/4.46	43 -> 22[label="",style="solid", color="burlywood", weight=3];
11.22/4.46	44[label="wu6/AProVE_Error wu60",fontsize=10,color="white",style="solid",shape="box"];15 -> 44[label="",style="solid", color="burlywood", weight=9];
11.22/4.46	44 -> 23[label="",style="solid", color="burlywood", weight=3];
11.22/4.46	17[label="primbindIO (AProVE_IO wu4) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
11.22/4.46	24[label="wu4",fontsize=16,color="green",shape="box"];25[label="wu50",fontsize=16,color="green",shape="box"];20[label="primbindIO (primbindIO (IO wu60) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
11.22/4.46	21[label="primbindIO (primbindIO (AProVE_IO wu60) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];21 -> 28[label="",style="solid", color="black", weight=3];
11.22/4.46	22[label="primbindIO (primbindIO (AProVE_Exception wu60) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];22 -> 29[label="",style="solid", color="black", weight=3];
11.22/4.46	23[label="primbindIO (primbindIO (AProVE_Error wu60) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];23 -> 30[label="",style="solid", color="black", weight=3];
11.22/4.46	26[label="gtGt0 (return ()) wu4",fontsize=16,color="black",shape="box"];26 -> 31[label="",style="solid", color="black", weight=3];
11.22/4.46	27[label="error []",fontsize=16,color="red",shape="box"];28[label="primbindIO (Monad.foldM0 wu3 wu51 wu60) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
11.22/4.46	29[label="primbindIO (AProVE_Exception wu60) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
11.22/4.46	30[label="primbindIO (AProVE_Error wu60) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
11.22/4.46	31[label="return ()",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
11.22/4.46	32 -> 8[label="",style="dashed", color="red", weight=0];
11.22/4.46	32[label="primbindIO (Monad.foldM wu3 wu60 wu51) (gtGt0 (return ()))",fontsize=16,color="magenta"];32 -> 36[label="",style="dashed", color="magenta", weight=3];
11.22/4.46	32 -> 37[label="",style="dashed", color="magenta", weight=3];
11.22/4.46	33[label="AProVE_Exception wu60",fontsize=16,color="green",shape="box"];34[label="AProVE_Error wu60",fontsize=16,color="green",shape="box"];35[label="primretIO ()",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
11.22/4.46	36[label="wu51",fontsize=16,color="green",shape="box"];37[label="wu60",fontsize=16,color="green",shape="box"];38[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];}
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(8)
11.22/4.46	Obligation:
11.22/4.46	Q DP problem:
11.22/4.46	The TRS P consists of the following rules:
11.22/4.46	
11.22/4.46	   new_primbindIO(wu3, wu51, h, ba) -> new_primbindIO0(wu3, wu51, h, ba)
11.22/4.46	   new_primbindIO0(wu3, :(wu50, wu51), h, ba) -> new_primbindIO(wu3, wu51, h, ba)
11.22/4.46	
11.22/4.46	R is empty.
11.22/4.46	Q is empty.
11.22/4.46	We have to consider all minimal (P,Q,R)-chains.
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(9) QDPSizeChangeProof (EQUIVALENT)
11.22/4.46	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. 
11.22/4.46	
11.22/4.46	From the DPs we obtained the following set of size-change graphs:
11.22/4.46	*new_primbindIO0(wu3, :(wu50, wu51), h, ba) -> new_primbindIO(wu3, wu51, h, ba)
11.22/4.46	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
11.22/4.46	
11.22/4.46	
11.22/4.46	*new_primbindIO(wu3, wu51, h, ba) -> new_primbindIO0(wu3, wu51, h, ba)
11.22/4.46	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
11.22/4.46	
11.22/4.46	
11.22/4.46	----------------------------------------
11.22/4.46	
11.22/4.46	(10)
11.22/4.46	YES
11.22/4.50	EOF