8.98/3.91	YES
10.78/4.40	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.78/4.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.78/4.40	
10.78/4.40	
10.78/4.40	H-Termination with start terms of the given HASKELL could be proven:
10.78/4.40	
10.78/4.40	(0) HASKELL
10.78/4.40	(1) BR [EQUIVALENT, 0 ms]
10.78/4.40	(2) HASKELL
10.78/4.40	(3) COR [EQUIVALENT, 0 ms]
10.78/4.40	(4) HASKELL
10.78/4.40	(5) Narrow [SOUND, 0 ms]
10.78/4.40	(6) QDP
10.78/4.40	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.78/4.40	(8) YES
10.78/4.40	
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(0)
10.78/4.40	Obligation:
10.78/4.40	mainModule Main 
10.78/4.40	module Maybe where {
10.78/4.40	import qualified Main;
10.78/4.40	import qualified Monad;
10.78/4.40	import qualified Prelude;
10.78/4.40	}
10.78/4.40	module Main where {
10.78/4.40	import qualified Maybe;
10.78/4.40	import qualified Monad;
10.78/4.40	import qualified Prelude;
10.78/4.40	}
10.78/4.40	module Monad where {
10.78/4.40	import qualified Main;
10.78/4.40	import qualified Maybe;
10.78/4.40	import qualified Prelude;
10.78/4.40	class Monad a => MonadPlus a where {
10.78/4.40	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.78/4.40	}
10.78/4.40	instance MonadPlus Maybe where { 
10.78/4.40	}
10.78/4.40	instance MonadPlus [] where { 
10.78/4.40	mplus = (++);
10.78/4.40	}
10.78/4.40	}
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(1) BR (EQUIVALENT)
10.78/4.40	Replaced joker patterns by fresh variables and removed binding patterns.
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(2)
10.78/4.40	Obligation:
10.78/4.40	mainModule Main 
10.78/4.40	module Maybe where {
10.78/4.40	import qualified Main;
10.78/4.40	import qualified Monad;
10.78/4.40	import qualified Prelude;
10.78/4.40	}
10.78/4.40	module Main where {
10.78/4.40	import qualified Maybe;
10.78/4.40	import qualified Monad;
10.78/4.40	import qualified Prelude;
10.78/4.40	}
10.78/4.40	module Monad where {
10.78/4.40	import qualified Main;
10.78/4.40	import qualified Maybe;
10.78/4.40	import qualified Prelude;
10.78/4.40	class Monad a => MonadPlus a where {
10.78/4.40	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.78/4.40	}
10.78/4.40	instance MonadPlus Maybe where { 
10.78/4.40	}
10.78/4.40	instance MonadPlus [] where { 
10.78/4.40	mplus = (++);
10.78/4.40	}
10.78/4.40	}
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(3) COR (EQUIVALENT)
10.78/4.40	Cond Reductions:
10.78/4.40	The following Function with conditions
10.78/4.40	"undefined |Falseundefined;
10.78/4.40	"
10.78/4.40	is transformed to
10.78/4.40	"undefined  = undefined1;
10.78/4.40	"
10.78/4.40	"undefined0 True = undefined;
10.78/4.40	"
10.78/4.40	"undefined1  = undefined0 False;
10.78/4.40	"
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(4)
10.78/4.40	Obligation:
10.78/4.40	mainModule Main 
10.78/4.40	module Maybe where {
10.78/4.40	import qualified Main;
10.78/4.40	import qualified Monad;
10.78/4.40	import qualified Prelude;
10.78/4.40	}
10.78/4.40	module Main where {
10.78/4.40	import qualified Maybe;
10.78/4.40	import qualified Monad;
10.78/4.40	import qualified Prelude;
10.78/4.40	}
10.78/4.40	module Monad where {
10.78/4.40	import qualified Main;
10.78/4.40	import qualified Maybe;
10.78/4.40	import qualified Prelude;
10.78/4.40	class Monad a => MonadPlus a where {
10.78/4.40	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.78/4.40	}
10.78/4.40	instance MonadPlus Maybe where { 
10.78/4.40	}
10.78/4.40	instance MonadPlus [] where { 
10.78/4.40	mplus = (++);
10.78/4.40	}
10.78/4.40	}
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(5) Narrow (SOUND)
10.78/4.40	Haskell To QDPs
10.78/4.40	
10.78/4.40	digraph dp_graph {
10.78/4.40	node [outthreshold=100, inthreshold=100];1[label="Monad.mplus",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.78/4.40	3[label="Monad.mplus vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.78/4.40	4[label="Monad.mplus vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.78/4.40	5[label="(++) vy3 vy4",fontsize=16,color="burlywood",shape="triangle"];12[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];5 -> 12[label="",style="solid", color="burlywood", weight=9];
10.78/4.40	12 -> 6[label="",style="solid", color="burlywood", weight=3];
10.78/4.40	13[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 13[label="",style="solid", color="burlywood", weight=9];
10.78/4.40	13 -> 7[label="",style="solid", color="burlywood", weight=3];
10.78/4.40	6[label="(++) (vy30 : vy31) vy4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.78/4.40	7[label="(++) [] vy4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.78/4.40	8[label="vy30 : vy31 ++ vy4",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
10.78/4.40	9[label="vy4",fontsize=16,color="green",shape="box"];10 -> 5[label="",style="dashed", color="red", weight=0];
10.78/4.40	10[label="vy31 ++ vy4",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
10.78/4.40	11[label="vy31",fontsize=16,color="green",shape="box"];}
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(6)
10.78/4.40	Obligation:
10.78/4.40	Q DP problem:
10.78/4.40	The TRS P consists of the following rules:
10.78/4.40	
10.78/4.40	   new_psPs(:(vy30, vy31), vy4, h) -> new_psPs(vy31, vy4, h)
10.78/4.40	
10.78/4.40	R is empty.
10.78/4.40	Q is empty.
10.78/4.40	We have to consider all minimal (P,Q,R)-chains.
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(7) QDPSizeChangeProof (EQUIVALENT)
10.78/4.40	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. 
10.78/4.40	
10.78/4.40	From the DPs we obtained the following set of size-change graphs:
10.78/4.40	*new_psPs(:(vy30, vy31), vy4, h) -> new_psPs(vy31, vy4, h)
10.78/4.40	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.78/4.40	
10.78/4.40	
10.78/4.40	----------------------------------------
10.78/4.40	
10.78/4.40	(8)
10.78/4.40	YES
11.14/4.51	EOF