9.00/3.89	YES
10.93/4.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.93/4.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.93/4.38	
10.93/4.38	
10.93/4.38	H-Termination with start terms of the given HASKELL could be proven:
10.93/4.38	
10.93/4.38	(0) HASKELL
10.93/4.38	(1) BR [EQUIVALENT, 0 ms]
10.93/4.38	(2) HASKELL
10.93/4.38	(3) COR [EQUIVALENT, 0 ms]
10.93/4.38	(4) HASKELL
10.93/4.38	(5) Narrow [SOUND, 0 ms]
10.93/4.38	(6) AND
10.93/4.38	    (7) QDP
10.93/4.38	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.93/4.38	        (9) YES
10.93/4.38	    (10) QDP
10.93/4.38	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.93/4.38	        (12) YES
10.93/4.38	
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(0)
10.93/4.38	Obligation:
10.93/4.38	mainModule Main 
10.93/4.38	module Maybe where {
10.93/4.38	import qualified Main;
10.93/4.38	import qualified Monad;
10.93/4.38	import qualified Prelude;
10.93/4.38	}
10.93/4.38	module Main where {
10.93/4.38	import qualified Maybe;
10.93/4.38	import qualified Monad;
10.93/4.38	import qualified Prelude;
10.93/4.38	}
10.93/4.38	module Monad where {
10.93/4.38	import qualified Main;
10.93/4.38	import qualified Maybe;
10.93/4.38	import qualified Prelude;
10.93/4.38	class Monad a => MonadPlus a where {
10.93/4.38	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.93/4.38	mzero :: MonadPlus a => a b;
10.93/4.38	}
10.93/4.38	instance MonadPlus Maybe where { 
10.93/4.38	}
10.93/4.38	instance MonadPlus [] where { 
10.93/4.38	mplus = (++);
10.93/4.38	mzero = [];
10.93/4.38	}
10.93/4.38	msum :: MonadPlus a => [a b]  ->  a b;
10.93/4.38	msum = foldr mplus mzero;
10.93/4.38	
10.93/4.38	}
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(1) BR (EQUIVALENT)
10.93/4.38	Replaced joker patterns by fresh variables and removed binding patterns.
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(2)
10.93/4.38	Obligation:
10.93/4.38	mainModule Main 
10.93/4.38	module Maybe where {
10.93/4.38	import qualified Main;
10.93/4.38	import qualified Monad;
10.93/4.38	import qualified Prelude;
10.93/4.38	}
10.93/4.38	module Main where {
10.93/4.38	import qualified Maybe;
10.93/4.38	import qualified Monad;
10.93/4.38	import qualified Prelude;
10.93/4.38	}
10.93/4.38	module Monad where {
10.93/4.38	import qualified Main;
10.93/4.38	import qualified Maybe;
10.93/4.38	import qualified Prelude;
10.93/4.38	class Monad a => MonadPlus a where {
10.93/4.38	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.93/4.38	mzero :: MonadPlus a => a b;
10.93/4.38	}
10.93/4.38	instance MonadPlus Maybe where { 
10.93/4.38	}
10.93/4.38	instance MonadPlus [] where { 
10.93/4.38	mplus = (++);
10.93/4.38	mzero = [];
10.93/4.38	}
10.93/4.38	msum :: MonadPlus b => [b a]  ->  b a;
10.93/4.38	msum = foldr mplus mzero;
10.93/4.38	
10.93/4.38	}
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(3) COR (EQUIVALENT)
10.93/4.38	Cond Reductions:
10.93/4.38	The following Function with conditions
10.93/4.38	"undefined |Falseundefined;
10.93/4.38	"
10.93/4.38	is transformed to
10.93/4.38	"undefined  = undefined1;
10.93/4.38	"
10.93/4.38	"undefined0 True = undefined;
10.93/4.38	"
10.93/4.38	"undefined1  = undefined0 False;
10.93/4.38	"
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(4)
10.93/4.38	Obligation:
10.93/4.38	mainModule Main 
10.93/4.38	module Maybe where {
10.93/4.38	import qualified Main;
10.93/4.38	import qualified Monad;
10.93/4.38	import qualified Prelude;
10.93/4.38	}
10.93/4.38	module Main where {
10.93/4.38	import qualified Maybe;
10.93/4.38	import qualified Monad;
10.93/4.38	import qualified Prelude;
10.93/4.38	}
10.93/4.38	module Monad where {
10.93/4.38	import qualified Main;
10.93/4.38	import qualified Maybe;
10.93/4.38	import qualified Prelude;
10.93/4.38	class Monad a => MonadPlus a where {
10.93/4.38	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.93/4.38	mzero :: MonadPlus a => a b;
10.93/4.38	}
10.93/4.38	instance MonadPlus Maybe where { 
10.93/4.38	}
10.93/4.38	instance MonadPlus [] where { 
10.93/4.38	mplus = (++);
10.93/4.38	mzero = [];
10.93/4.38	}
10.93/4.38	msum :: MonadPlus b => [b a]  ->  b a;
10.93/4.38	msum = foldr mplus mzero;
10.93/4.38	
10.93/4.38	}
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(5) Narrow (SOUND)
10.93/4.38	Haskell To QDPs
10.93/4.38	
10.93/4.38	digraph dp_graph {
10.93/4.38	node [outthreshold=100, inthreshold=100];1[label="Monad.msum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.93/4.38	3[label="Monad.msum vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.93/4.38	4[label="foldr Monad.mplus Monad.mzero vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
10.93/4.38	20 -> 5[label="",style="solid", color="burlywood", weight=3];
10.93/4.38	21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
10.93/4.38	21 -> 6[label="",style="solid", color="burlywood", weight=3];
10.93/4.38	5[label="foldr Monad.mplus Monad.mzero (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.93/4.38	6[label="foldr Monad.mplus Monad.mzero []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.93/4.38	7 -> 9[label="",style="dashed", color="red", weight=0];
10.93/4.38	7[label="Monad.mplus vy30 (foldr Monad.mplus Monad.mzero vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
10.93/4.38	8[label="Monad.mzero",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
10.93/4.38	10 -> 4[label="",style="dashed", color="red", weight=0];
10.93/4.38	10[label="foldr Monad.mplus Monad.mzero vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
10.93/4.38	9[label="Monad.mplus vy30 vy4",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3];
10.93/4.38	11[label="[]",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="(++) vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9];
10.93/4.38	22 -> 14[label="",style="solid", color="burlywood", weight=3];
10.93/4.38	23[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 23[label="",style="solid", color="burlywood", weight=9];
10.93/4.38	23 -> 15[label="",style="solid", color="burlywood", weight=3];
10.93/4.38	14[label="(++) (vy300 : vy301) vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
10.93/4.38	15[label="(++) [] vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
10.93/4.38	16[label="vy300 : vy301 ++ vy4",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
10.93/4.38	17[label="vy4",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0];
10.93/4.38	18[label="vy301 ++ vy4",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
10.93/4.38	19[label="vy301",fontsize=16,color="green",shape="box"];}
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(6)
10.93/4.38	Complex Obligation (AND)
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(7)
10.93/4.38	Obligation:
10.93/4.38	Q DP problem:
10.93/4.38	The TRS P consists of the following rules:
10.93/4.38	
10.93/4.38	   new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
10.93/4.38	
10.93/4.38	R is empty.
10.93/4.38	Q is empty.
10.93/4.38	We have to consider all minimal (P,Q,R)-chains.
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(8) QDPSizeChangeProof (EQUIVALENT)
10.93/4.38	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. 
10.93/4.38	
10.93/4.38	From the DPs we obtained the following set of size-change graphs:
10.93/4.38	*new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
10.93/4.38	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.93/4.38	
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(9)
10.93/4.38	YES
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(10)
10.93/4.38	Obligation:
10.93/4.38	Q DP problem:
10.93/4.38	The TRS P consists of the following rules:
10.93/4.38	
10.93/4.38	   new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h)
10.93/4.38	
10.93/4.38	R is empty.
10.93/4.38	Q is empty.
10.93/4.38	We have to consider all minimal (P,Q,R)-chains.
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(11) QDPSizeChangeProof (EQUIVALENT)
10.93/4.38	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. 
10.93/4.38	
10.93/4.38	From the DPs we obtained the following set of size-change graphs:
10.93/4.38	*new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h)
10.93/4.38	The graph contains the following edges 1 > 1, 2 >= 2
10.93/4.38	
10.93/4.38	
10.93/4.38	----------------------------------------
10.93/4.38	
10.93/4.38	(12)
10.93/4.38	YES
11.05/4.47	EOF