9.40/3.93	YES
11.24/4.41	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.24/4.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.24/4.41	
11.24/4.41	
11.24/4.41	H-Termination with start terms of the given HASKELL could be proven:
11.24/4.41	
11.24/4.41	(0) HASKELL
11.24/4.41	(1) BR [EQUIVALENT, 0 ms]
11.24/4.41	(2) HASKELL
11.24/4.41	(3) COR [EQUIVALENT, 0 ms]
11.24/4.41	(4) HASKELL
11.24/4.41	(5) Narrow [SOUND, 0 ms]
11.24/4.41	(6) QDP
11.24/4.41	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.24/4.41	(8) YES
11.24/4.41	
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(0)
11.24/4.41	Obligation:
11.24/4.41	mainModule Main 
11.24/4.41	module Maybe where {
11.24/4.41	import qualified Main;
11.24/4.41	import qualified Monad;
11.24/4.41	import qualified Prelude;
11.24/4.41	}
11.24/4.41	module Main where {
11.24/4.41	import qualified Maybe;
11.24/4.41	import qualified Monad;
11.24/4.41	import qualified Prelude;
11.24/4.41	}
11.24/4.41	module Monad where {
11.24/4.41	import qualified Main;
11.24/4.41	import qualified Maybe;
11.24/4.41	import qualified Prelude;
11.24/4.41	class Monad a => MonadPlus a where {
11.24/4.41	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
11.24/4.41	mzero :: MonadPlus a => a b;
11.24/4.41	}
11.24/4.41	instance MonadPlus Maybe where { 
11.24/4.41	mplus Nothing ys = ys;
11.24/4.41	mplus xs _ys = xs;
11.24/4.41	mzero = Nothing;
11.24/4.41	}
11.24/4.41	instance MonadPlus [] where { 
11.24/4.41	}
11.24/4.41	msum :: MonadPlus a => [a b]  ->  a b;
11.24/4.41	msum = foldr mplus mzero;
11.24/4.41	
11.24/4.41	}
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(1) BR (EQUIVALENT)
11.24/4.41	Replaced joker patterns by fresh variables and removed binding patterns.
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(2)
11.24/4.41	Obligation:
11.24/4.41	mainModule Main 
11.24/4.41	module Maybe where {
11.24/4.41	import qualified Main;
11.24/4.41	import qualified Monad;
11.24/4.41	import qualified Prelude;
11.24/4.41	}
11.24/4.41	module Main where {
11.24/4.41	import qualified Maybe;
11.24/4.41	import qualified Monad;
11.24/4.41	import qualified Prelude;
11.24/4.41	}
11.24/4.41	module Monad where {
11.24/4.41	import qualified Main;
11.24/4.41	import qualified Maybe;
11.24/4.41	import qualified Prelude;
11.24/4.41	class Monad a => MonadPlus a where {
11.24/4.41	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
11.24/4.41	mzero :: MonadPlus a => a b;
11.24/4.41	}
11.24/4.41	instance MonadPlus Maybe where { 
11.24/4.41	mplus Nothing ys = ys;
11.24/4.41	mplus xs _ys = xs;
11.24/4.41	mzero = Nothing;
11.24/4.41	}
11.24/4.41	instance MonadPlus [] where { 
11.24/4.41	}
11.24/4.41	msum :: MonadPlus b => [b a]  ->  b a;
11.24/4.41	msum = foldr mplus mzero;
11.24/4.41	
11.24/4.41	}
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(3) COR (EQUIVALENT)
11.24/4.41	Cond Reductions:
11.24/4.41	The following Function with conditions
11.24/4.41	"undefined |Falseundefined;
11.24/4.41	"
11.24/4.41	is transformed to
11.24/4.41	"undefined  = undefined1;
11.24/4.41	"
11.24/4.41	"undefined0 True = undefined;
11.24/4.41	"
11.24/4.41	"undefined1  = undefined0 False;
11.24/4.41	"
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(4)
11.24/4.41	Obligation:
11.24/4.41	mainModule Main 
11.24/4.41	module Maybe where {
11.24/4.41	import qualified Main;
11.24/4.41	import qualified Monad;
11.24/4.41	import qualified Prelude;
11.24/4.41	}
11.24/4.41	module Main where {
11.24/4.41	import qualified Maybe;
11.24/4.41	import qualified Monad;
11.24/4.41	import qualified Prelude;
11.24/4.41	}
11.24/4.41	module Monad where {
11.24/4.41	import qualified Main;
11.24/4.41	import qualified Maybe;
11.24/4.41	import qualified Prelude;
11.24/4.41	class Monad a => MonadPlus a where {
11.24/4.41	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
11.24/4.41	mzero :: MonadPlus a => a b;
11.24/4.41	}
11.24/4.41	instance MonadPlus Maybe where { 
11.24/4.41	mplus Nothing ys = ys;
11.24/4.41	mplus xs _ys = xs;
11.24/4.41	mzero = Nothing;
11.24/4.41	}
11.24/4.41	instance MonadPlus [] where { 
11.24/4.41	}
11.24/4.41	msum :: MonadPlus b => [b a]  ->  b a;
11.24/4.41	msum = foldr mplus mzero;
11.24/4.41	
11.24/4.41	}
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(5) Narrow (SOUND)
11.24/4.41	Haskell To QDPs
11.24/4.41	
11.24/4.41	digraph dp_graph {
11.24/4.41	node [outthreshold=100, inthreshold=100];1[label="Monad.msum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.24/4.41	3[label="Monad.msum vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
11.24/4.41	4[label="foldr Monad.mplus Monad.mzero vy3",fontsize=16,color="burlywood",shape="triangle"];17[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
11.24/4.41	17 -> 5[label="",style="solid", color="burlywood", weight=3];
11.24/4.41	18[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
11.24/4.41	18 -> 6[label="",style="solid", color="burlywood", weight=3];
11.24/4.41	5[label="foldr Monad.mplus Monad.mzero (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.24/4.41	6[label="foldr Monad.mplus Monad.mzero []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.24/4.41	7 -> 9[label="",style="dashed", color="red", weight=0];
11.24/4.41	7[label="Monad.mplus vy30 (foldr Monad.mplus Monad.mzero vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
11.24/4.41	8[label="Monad.mzero",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
11.24/4.41	10 -> 4[label="",style="dashed", color="red", weight=0];
11.24/4.41	10[label="foldr Monad.mplus Monad.mzero vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
11.24/4.41	9[label="Monad.mplus vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];19[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 19[label="",style="solid", color="burlywood", weight=9];
11.24/4.41	19 -> 13[label="",style="solid", color="burlywood", weight=3];
11.24/4.41	20[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];9 -> 20[label="",style="solid", color="burlywood", weight=9];
11.24/4.41	20 -> 14[label="",style="solid", color="burlywood", weight=3];
11.24/4.41	11[label="Nothing",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="Monad.mplus Nothing vy4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
11.24/4.41	14[label="Monad.mplus (Just vy300) vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
11.24/4.41	15[label="vy4",fontsize=16,color="green",shape="box"];16[label="Just vy300",fontsize=16,color="green",shape="box"];}
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(6)
11.24/4.41	Obligation:
11.24/4.41	Q DP problem:
11.24/4.41	The TRS P consists of the following rules:
11.24/4.41	
11.24/4.41	   new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h)
11.24/4.41	
11.24/4.41	R is empty.
11.24/4.41	Q is empty.
11.24/4.41	We have to consider all minimal (P,Q,R)-chains.
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(7) QDPSizeChangeProof (EQUIVALENT)
11.24/4.41	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. 
11.24/4.41	
11.24/4.41	From the DPs we obtained the following set of size-change graphs:
11.24/4.41	*new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h)
11.24/4.41	The graph contains the following edges 1 > 1, 2 >= 2
11.24/4.41	
11.24/4.41	
11.24/4.41	----------------------------------------
11.24/4.41	
11.24/4.41	(8)
11.24/4.41	YES
11.24/4.45	EOF