8.62/3.84	YES
10.62/4.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.62/4.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.62/4.35	
10.62/4.35	
10.62/4.35	H-Termination with start terms of the given HASKELL could be proven:
10.62/4.35	
10.62/4.35	(0) HASKELL
10.62/4.35	(1) BR [EQUIVALENT, 0 ms]
10.62/4.35	(2) HASKELL
10.62/4.35	(3) COR [EQUIVALENT, 0 ms]
10.62/4.35	(4) HASKELL
10.62/4.35	(5) Narrow [SOUND, 0 ms]
10.62/4.35	(6) QDP
10.62/4.35	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.62/4.35	(8) YES
10.62/4.35	
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(0)
10.62/4.35	Obligation:
10.62/4.35	mainModule Main 
10.62/4.35	module Maybe where {
10.62/4.35	import qualified Main;
10.62/4.35	import qualified Monad;
10.62/4.35	import qualified Prelude;
10.62/4.35	}
10.62/4.35	module Main where {
10.62/4.35	import qualified Maybe;
10.62/4.35	import qualified Monad;
10.62/4.35	import qualified Prelude;
10.62/4.35	}
10.62/4.35	module Monad where {
10.62/4.35	import qualified Main;
10.62/4.35	import qualified Maybe;
10.62/4.35	import qualified Prelude;
10.62/4.35	class Monad a => MonadPlus a where {
10.62/4.35	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.62/4.35	}
10.62/4.35	instance MonadPlus Maybe where { 
10.62/4.35	mplus Nothing ys = ys;
10.62/4.35	mplus xs _ys = xs;
10.62/4.35	}
10.62/4.35	instance MonadPlus [] where { 
10.62/4.35	mplus = (++);
10.62/4.35	}
10.62/4.35	}
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(1) BR (EQUIVALENT)
10.62/4.35	Replaced joker patterns by fresh variables and removed binding patterns.
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(2)
10.62/4.35	Obligation:
10.62/4.35	mainModule Main 
10.62/4.35	module Maybe where {
10.62/4.35	import qualified Main;
10.62/4.35	import qualified Monad;
10.62/4.35	import qualified Prelude;
10.62/4.35	}
10.62/4.35	module Main where {
10.62/4.35	import qualified Maybe;
10.62/4.35	import qualified Monad;
10.62/4.35	import qualified Prelude;
10.62/4.35	}
10.62/4.35	module Monad where {
10.62/4.35	import qualified Main;
10.62/4.35	import qualified Maybe;
10.62/4.35	import qualified Prelude;
10.62/4.35	class Monad a => MonadPlus a where {
10.62/4.35	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.62/4.35	}
10.62/4.35	instance MonadPlus Maybe where { 
10.62/4.35	mplus Nothing ys = ys;
10.62/4.35	mplus xs _ys = xs;
10.62/4.35	}
10.62/4.35	instance MonadPlus [] where { 
10.62/4.35	mplus = (++);
10.62/4.35	}
10.62/4.35	}
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(3) COR (EQUIVALENT)
10.62/4.35	Cond Reductions:
10.62/4.35	The following Function with conditions
10.62/4.35	"undefined |Falseundefined;
10.62/4.35	"
10.62/4.35	is transformed to
10.62/4.35	"undefined  = undefined1;
10.62/4.35	"
10.62/4.35	"undefined0 True = undefined;
10.62/4.35	"
10.62/4.35	"undefined1  = undefined0 False;
10.62/4.35	"
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(4)
10.62/4.35	Obligation:
10.62/4.35	mainModule Main 
10.62/4.35	module Maybe where {
10.62/4.35	import qualified Main;
10.62/4.35	import qualified Monad;
10.62/4.35	import qualified Prelude;
10.62/4.35	}
10.62/4.35	module Main where {
10.62/4.35	import qualified Maybe;
10.62/4.35	import qualified Monad;
10.62/4.35	import qualified Prelude;
10.62/4.35	}
10.62/4.35	module Monad where {
10.62/4.35	import qualified Main;
10.62/4.35	import qualified Maybe;
10.62/4.35	import qualified Prelude;
10.62/4.35	class Monad a => MonadPlus a where {
10.62/4.35	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.62/4.35	}
10.62/4.35	instance MonadPlus Maybe where { 
10.62/4.35	mplus Nothing ys = ys;
10.62/4.35	mplus xs _ys = xs;
10.62/4.35	}
10.62/4.35	instance MonadPlus [] where { 
10.62/4.35	mplus = (++);
10.62/4.35	}
10.62/4.35	}
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(5) Narrow (SOUND)
10.62/4.35	Haskell To QDPs
10.62/4.35	
10.62/4.35	digraph dp_graph {
10.62/4.35	node [outthreshold=100, inthreshold=100];1[label="Monad.mplus",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.62/4.35	3[label="Monad.mplus vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.62/4.35	4[label="Monad.mplus vy3 vy4",fontsize=16,color="blue",shape="box"];18[label="Monad.mplus :: (Maybe a) -> (Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="blue", weight=9];
10.62/4.35	18 -> 5[label="",style="solid", color="blue", weight=3];
10.62/4.35	19[label="Monad.mplus :: ([] a) -> ([] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="blue", weight=9];
10.62/4.35	19 -> 6[label="",style="solid", color="blue", weight=3];
10.62/4.35	5[label="Monad.mplus vy3 vy4",fontsize=16,color="burlywood",shape="box"];20[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];5 -> 20[label="",style="solid", color="burlywood", weight=9];
10.62/4.35	20 -> 7[label="",style="solid", color="burlywood", weight=3];
10.62/4.35	21[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];5 -> 21[label="",style="solid", color="burlywood", weight=9];
10.62/4.35	21 -> 8[label="",style="solid", color="burlywood", weight=3];
10.62/4.35	6[label="Monad.mplus vy3 vy4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
10.62/4.35	7[label="Monad.mplus Nothing vy4",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
10.62/4.35	8[label="Monad.mplus (Just vy30) vy4",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10.62/4.35	9[label="(++) vy3 vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9];
10.62/4.35	22 -> 12[label="",style="solid", color="burlywood", weight=3];
10.62/4.35	23[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
10.62/4.35	23 -> 13[label="",style="solid", color="burlywood", weight=3];
10.62/4.35	10[label="vy4",fontsize=16,color="green",shape="box"];11[label="Just vy30",fontsize=16,color="green",shape="box"];12[label="(++) (vy30 : vy31) vy4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.62/4.35	13[label="(++) [] vy4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
10.62/4.35	14[label="vy30 : vy31 ++ vy4",fontsize=16,color="green",shape="box"];14 -> 16[label="",style="dashed", color="green", weight=3];
10.62/4.35	15[label="vy4",fontsize=16,color="green",shape="box"];16 -> 9[label="",style="dashed", color="red", weight=0];
10.62/4.35	16[label="vy31 ++ vy4",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
10.62/4.35	17[label="vy31",fontsize=16,color="green",shape="box"];}
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(6)
10.62/4.35	Obligation:
10.62/4.35	Q DP problem:
10.62/4.35	The TRS P consists of the following rules:
10.62/4.35	
10.62/4.35	   new_psPs(:(vy30, vy31), vy4, h) -> new_psPs(vy31, vy4, h)
10.62/4.35	
10.62/4.35	R is empty.
10.62/4.35	Q is empty.
10.62/4.35	We have to consider all minimal (P,Q,R)-chains.
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(7) QDPSizeChangeProof (EQUIVALENT)
10.62/4.35	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. 
10.62/4.35	
10.62/4.35	From the DPs we obtained the following set of size-change graphs:
10.62/4.35	*new_psPs(:(vy30, vy31), vy4, h) -> new_psPs(vy31, vy4, h)
10.62/4.35	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.62/4.35	
10.62/4.35	
10.62/4.35	----------------------------------------
10.62/4.35	
10.62/4.35	(8)
10.62/4.35	YES
10.82/4.40	EOF