8.69/3.83	YES
10.80/4.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.80/4.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.80/4.40	
10.80/4.40	
10.80/4.40	H-Termination with start terms of the given HASKELL could be proven:
10.80/4.40	
10.80/4.40	(0) HASKELL
10.80/4.40	(1) LR [EQUIVALENT, 0 ms]
10.80/4.40	(2) HASKELL
10.80/4.40	(3) BR [EQUIVALENT, 0 ms]
10.80/4.40	(4) HASKELL
10.80/4.40	(5) COR [EQUIVALENT, 0 ms]
10.80/4.40	(6) HASKELL
10.80/4.40	(7) Narrow [SOUND, 0 ms]
10.80/4.40	(8) QDP
10.80/4.40	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.80/4.40	(10) YES
10.80/4.40	
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(0)
10.80/4.40	Obligation:
10.80/4.40	mainModule Main 
10.80/4.40	module Maybe where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Main where {
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Monad where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Prelude;
10.80/4.40	zipWithM_ :: Monad c => (a  ->  b  ->  c d)  ->  [a]  ->  [b]  ->  c ();
10.80/4.40	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.80/4.40	
10.80/4.40	}
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(1) LR (EQUIVALENT)
10.80/4.40	Lambda Reductions:
10.80/4.40	The following Lambda expression
10.80/4.40	"\_->q"
10.80/4.40	is transformed to
10.80/4.40	"gtGt0 q _ = q;
10.80/4.40	"
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(2)
10.80/4.40	Obligation:
10.80/4.40	mainModule Main 
10.80/4.40	module Maybe where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Main where {
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Monad where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Prelude;
10.80/4.40	zipWithM_ :: Monad c => (b  ->  a  ->  c d)  ->  [b]  ->  [a]  ->  c ();
10.80/4.40	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.80/4.40	
10.80/4.40	}
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(3) BR (EQUIVALENT)
10.80/4.40	Replaced joker patterns by fresh variables and removed binding patterns.
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(4)
10.80/4.40	Obligation:
10.80/4.40	mainModule Main 
10.80/4.40	module Maybe where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Main where {
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Monad where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Prelude;
10.80/4.40	zipWithM_ :: Monad a => (c  ->  b  ->  a d)  ->  [c]  ->  [b]  ->  a ();
10.80/4.40	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.80/4.40	
10.80/4.40	}
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(5) COR (EQUIVALENT)
10.80/4.40	Cond Reductions:
10.80/4.40	The following Function with conditions
10.80/4.40	"undefined |Falseundefined;
10.80/4.40	"
10.80/4.40	is transformed to
10.80/4.40	"undefined  = undefined1;
10.80/4.40	"
10.80/4.40	"undefined0 True = undefined;
10.80/4.40	"
10.80/4.40	"undefined1  = undefined0 False;
10.80/4.40	"
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(6)
10.80/4.40	Obligation:
10.80/4.40	mainModule Main 
10.80/4.40	module Maybe where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Main where {
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Monad;
10.80/4.40	import qualified Prelude;
10.80/4.40	}
10.80/4.40	module Monad where {
10.80/4.40	import qualified Main;
10.80/4.40	import qualified Maybe;
10.80/4.40	import qualified Prelude;
10.80/4.40	zipWithM_ :: Monad d => (b  ->  a  ->  d c)  ->  [b]  ->  [a]  ->  d ();
10.80/4.40	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
10.80/4.40	
10.80/4.40	}
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(7) Narrow (SOUND)
10.80/4.40	Haskell To QDPs
10.80/4.40	
10.80/4.40	digraph dp_graph {
10.80/4.40	node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.80/4.40	3[label="Monad.zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.80/4.40	4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
10.80/4.40	5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
10.80/4.40	6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.80/4.40	7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];34[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 34[label="",style="solid", color="burlywood", weight=9];
10.80/4.40	34 -> 8[label="",style="solid", color="burlywood", weight=3];
10.80/4.40	35[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
10.80/4.40	35 -> 9[label="",style="solid", color="burlywood", weight=3];
10.80/4.40	8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];36[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 36[label="",style="solid", color="burlywood", weight=9];
10.80/4.40	36 -> 10[label="",style="solid", color="burlywood", weight=3];
10.80/4.40	37[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 37[label="",style="solid", color="burlywood", weight=9];
10.80/4.40	37 -> 11[label="",style="solid", color="burlywood", weight=3];
10.80/4.40	9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10.80/4.40	10[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) (ww50 : ww51))",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
10.80/4.40	11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
10.80/4.40	12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
10.80/4.40	13[label="foldr (>>) (return ()) (ww3 ww40 ww50 : zipWith ww3 ww41 ww51)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
10.80/4.40	14 -> 12[label="",style="dashed", color="red", weight=0];
10.80/4.40	14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
10.80/4.40	16 -> 18[label="",style="dashed", color="red", weight=0];
10.80/4.40	16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
10.80/4.40	17[label="Just ()",fontsize=16,color="green",shape="box"];19 -> 7[label="",style="dashed", color="red", weight=0];
10.80/4.40	19[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
10.80/4.40	19 -> 21[label="",style="dashed", color="magenta", weight=3];
10.80/4.40	18[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];18 -> 22[label="",style="solid", color="black", weight=3];
10.80/4.40	20[label="ww51",fontsize=16,color="green",shape="box"];21[label="ww41",fontsize=16,color="green",shape="box"];22 -> 23[label="",style="dashed", color="red", weight=0];
10.80/4.40	22[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
10.80/4.40	24[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];24 -> 29[label="",style="dashed", color="green", weight=3];
10.80/4.40	24 -> 30[label="",style="dashed", color="green", weight=3];
10.80/4.40	23[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];38[label="ww8/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 38[label="",style="solid", color="burlywood", weight=9];
10.80/4.40	38 -> 27[label="",style="solid", color="burlywood", weight=3];
10.80/4.40	39[label="ww8/Just ww80",fontsize=10,color="white",style="solid",shape="box"];23 -> 39[label="",style="solid", color="burlywood", weight=9];
10.80/4.40	39 -> 28[label="",style="solid", color="burlywood", weight=3];
10.80/4.40	29[label="ww40",fontsize=16,color="green",shape="box"];30[label="ww50",fontsize=16,color="green",shape="box"];27[label="Nothing >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.80/4.40	28[label="Just ww80 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.80/4.40	31[label="Nothing",fontsize=16,color="green",shape="box"];32[label="gtGt0 ww6 ww80",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
10.80/4.40	33[label="ww6",fontsize=16,color="green",shape="box"];}
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(8)
10.80/4.40	Obligation:
10.80/4.40	Q DP problem:
10.80/4.40	The TRS P consists of the following rules:
10.80/4.40	
10.80/4.40	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
10.80/4.40	
10.80/4.40	R is empty.
10.80/4.40	Q is empty.
10.80/4.40	We have to consider all minimal (P,Q,R)-chains.
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(9) QDPSizeChangeProof (EQUIVALENT)
10.80/4.40	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. 
10.80/4.40	
10.80/4.40	From the DPs we obtained the following set of size-change graphs:
10.80/4.40	*new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
10.80/4.40	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6
10.80/4.40	
10.80/4.40	
10.80/4.40	----------------------------------------
10.80/4.40	
10.80/4.40	(10)
10.80/4.40	YES
11.01/4.48	EOF