9.12/3.92	YES
11.32/4.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.32/4.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.32/4.50	
11.32/4.50	
11.32/4.50	H-Termination with start terms of the given HASKELL could be proven:
11.32/4.50	
11.32/4.50	(0) HASKELL
11.32/4.50	(1) LR [EQUIVALENT, 0 ms]
11.32/4.50	(2) HASKELL
11.32/4.50	(3) BR [EQUIVALENT, 0 ms]
11.32/4.50	(4) HASKELL
11.32/4.50	(5) COR [EQUIVALENT, 0 ms]
11.32/4.50	(6) HASKELL
11.32/4.50	(7) Narrow [SOUND, 0 ms]
11.32/4.50	(8) QDP
11.32/4.50	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.32/4.50	(10) YES
11.32/4.50	
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(0)
11.32/4.50	Obligation:
11.32/4.50	mainModule Main 
11.32/4.50	module Maybe where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Main where {
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Monad where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Prelude;
11.32/4.50	zipWithM_ :: Monad b => (d  ->  a  ->  b c)  ->  [d]  ->  [a]  ->  b ();
11.32/4.50	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.50	
11.32/4.50	}
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(1) LR (EQUIVALENT)
11.32/4.50	Lambda Reductions:
11.32/4.50	The following Lambda expression
11.32/4.50	"\_->q"
11.32/4.50	is transformed to
11.32/4.50	"gtGt0 q _ = q;
11.32/4.50	"
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(2)
11.32/4.50	Obligation:
11.32/4.50	mainModule Main 
11.32/4.50	module Maybe where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Main where {
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Monad where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Prelude;
11.32/4.50	zipWithM_ :: Monad c => (b  ->  a  ->  c d)  ->  [b]  ->  [a]  ->  c ();
11.32/4.50	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.50	
11.32/4.50	}
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(3) BR (EQUIVALENT)
11.32/4.50	Replaced joker patterns by fresh variables and removed binding patterns.
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(4)
11.32/4.50	Obligation:
11.32/4.50	mainModule Main 
11.32/4.50	module Maybe where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Main where {
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Monad where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Prelude;
11.32/4.50	zipWithM_ :: Monad a => (b  ->  d  ->  a c)  ->  [b]  ->  [d]  ->  a ();
11.32/4.50	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.50	
11.32/4.50	}
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(5) COR (EQUIVALENT)
11.32/4.50	Cond Reductions:
11.32/4.50	The following Function with conditions
11.32/4.50	"undefined |Falseundefined;
11.32/4.50	"
11.32/4.50	is transformed to
11.32/4.50	"undefined  = undefined1;
11.32/4.50	"
11.32/4.50	"undefined0 True = undefined;
11.32/4.50	"
11.32/4.50	"undefined1  = undefined0 False;
11.32/4.50	"
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(6)
11.32/4.50	Obligation:
11.32/4.50	mainModule Main 
11.32/4.50	module Maybe where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Main where {
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Monad;
11.32/4.50	import qualified Prelude;
11.32/4.50	}
11.32/4.50	module Monad where {
11.32/4.50	import qualified Main;
11.32/4.50	import qualified Maybe;
11.32/4.50	import qualified Prelude;
11.32/4.50	zipWithM_ :: Monad c => (a  ->  b  ->  c d)  ->  [a]  ->  [b]  ->  c ();
11.32/4.50	zipWithM_ f xs ys = sequence_ (zipWith f xs ys);
11.32/4.50	
11.32/4.50	}
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(7) Narrow (SOUND)
11.32/4.50	Haskell To QDPs
11.32/4.50	
11.32/4.50	digraph dp_graph {
11.32/4.50	node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.32/4.50	3[label="Monad.zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.32/4.50	4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.32/4.50	5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
11.32/4.50	6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.32/4.50	7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];40[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 40[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	40 -> 8[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	41[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 41[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	41 -> 9[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];42[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 42[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	42 -> 10[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	43[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 43[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	43 -> 11[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
11.32/4.50	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];
11.32/4.50	11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
11.32/4.50	12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
11.32/4.50	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];
11.32/4.50	14 -> 12[label="",style="dashed", color="red", weight=0];
11.32/4.50	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];
11.32/4.50	16 -> 18[label="",style="dashed", color="red", weight=0];
11.32/4.50	16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
11.32/4.50	17[label="primretIO ()",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
11.32/4.50	19 -> 7[label="",style="dashed", color="red", weight=0];
11.32/4.50	19[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
11.32/4.50	19 -> 22[label="",style="dashed", color="magenta", weight=3];
11.32/4.50	18[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];18 -> 23[label="",style="solid", color="black", weight=3];
11.32/4.50	20[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];21[label="ww41",fontsize=16,color="green",shape="box"];22[label="ww51",fontsize=16,color="green",shape="box"];23[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
11.32/4.50	24 -> 25[label="",style="dashed", color="red", weight=0];
11.32/4.50	24[label="primbindIO (ww3 ww40 ww50) (gtGt0 ww6)",fontsize=16,color="magenta"];24 -> 26[label="",style="dashed", color="magenta", weight=3];
11.32/4.50	26[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];26 -> 33[label="",style="dashed", color="green", weight=3];
11.32/4.50	26 -> 34[label="",style="dashed", color="green", weight=3];
11.32/4.50	25[label="primbindIO ww8 (gtGt0 ww6)",fontsize=16,color="burlywood",shape="triangle"];44[label="ww8/IO ww80",fontsize=10,color="white",style="solid",shape="box"];25 -> 44[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	44 -> 29[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	45[label="ww8/AProVE_IO ww80",fontsize=10,color="white",style="solid",shape="box"];25 -> 45[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	45 -> 30[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	46[label="ww8/AProVE_Exception ww80",fontsize=10,color="white",style="solid",shape="box"];25 -> 46[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	46 -> 31[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	47[label="ww8/AProVE_Error ww80",fontsize=10,color="white",style="solid",shape="box"];25 -> 47[label="",style="solid", color="burlywood", weight=9];
11.32/4.50	47 -> 32[label="",style="solid", color="burlywood", weight=3];
11.32/4.50	33[label="ww40",fontsize=16,color="green",shape="box"];34[label="ww50",fontsize=16,color="green",shape="box"];29[label="primbindIO (IO ww80) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3];
11.32/4.50	30[label="primbindIO (AProVE_IO ww80) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];30 -> 36[label="",style="solid", color="black", weight=3];
11.32/4.50	31[label="primbindIO (AProVE_Exception ww80) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];31 -> 37[label="",style="solid", color="black", weight=3];
11.32/4.50	32[label="primbindIO (AProVE_Error ww80) (gtGt0 ww6)",fontsize=16,color="black",shape="box"];32 -> 38[label="",style="solid", color="black", weight=3];
11.32/4.50	35[label="error []",fontsize=16,color="red",shape="box"];36[label="gtGt0 ww6 ww80",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
11.32/4.50	37[label="AProVE_Exception ww80",fontsize=16,color="green",shape="box"];38[label="AProVE_Error ww80",fontsize=16,color="green",shape="box"];39[label="ww6",fontsize=16,color="green",shape="box"];}
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(8)
11.32/4.50	Obligation:
11.32/4.50	Q DP problem:
11.32/4.50	The TRS P consists of the following rules:
11.32/4.50	
11.32/4.50	   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
11.32/4.50	
11.32/4.50	R is empty.
11.32/4.50	Q is empty.
11.32/4.50	We have to consider all minimal (P,Q,R)-chains.
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(9) QDPSizeChangeProof (EQUIVALENT)
11.32/4.50	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. 
11.32/4.50	
11.32/4.50	From the DPs we obtained the following set of size-change graphs:
11.32/4.50	*new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
11.32/4.50	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6
11.32/4.50	
11.32/4.50	
11.32/4.50	----------------------------------------
11.32/4.50	
11.32/4.50	(10)
11.32/4.50	YES
11.32/4.55	EOF