9.09/3.91	YES
10.84/4.41	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.84/4.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.84/4.41	
10.84/4.41	
10.84/4.41	H-Termination with start terms of the given HASKELL could be proven:
10.84/4.41	
10.84/4.41	(0) HASKELL
10.84/4.41	(1) BR [EQUIVALENT, 0 ms]
10.84/4.41	(2) HASKELL
10.84/4.41	(3) COR [EQUIVALENT, 0 ms]
10.84/4.41	(4) HASKELL
10.84/4.41	(5) Narrow [SOUND, 0 ms]
10.84/4.41	(6) AND
10.84/4.41	    (7) QDP
10.84/4.41	        (8) DependencyGraphProof [EQUIVALENT, 0 ms]
10.84/4.41	        (9) AND
10.84/4.41	            (10) QDP
10.84/4.41	                (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.84/4.41	                (12) YES
10.84/4.41	            (13) QDP
10.84/4.41	                (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.84/4.41	                (15) YES
10.84/4.41	    (16) QDP
10.84/4.41	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.84/4.41	        (18) YES
10.84/4.41	
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(0)
10.84/4.41	Obligation:
10.84/4.41	mainModule Main 
10.84/4.41	module Maybe where {
10.84/4.41	import qualified Main;
10.84/4.41	import qualified Monad;
10.84/4.41	import qualified Prelude;
10.84/4.41	}
10.84/4.41	module Main where {
10.84/4.41	import qualified Maybe;
10.84/4.41	import qualified Monad;
10.84/4.41	import qualified Prelude;
10.84/4.41	}
10.84/4.41	module Monad where {
10.84/4.41	import qualified Main;
10.84/4.41	import qualified Maybe;
10.84/4.41	import qualified Prelude;
10.84/4.41	class Monad a => MonadPlus a where {
10.84/4.41	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.84/4.41	mzero :: MonadPlus a => a b;
10.84/4.41	}
10.84/4.41	instance MonadPlus Maybe where { 
10.84/4.41	mplus Nothing ys = ys;
10.84/4.41	mplus xs _ys = xs;
10.84/4.41	mzero = Nothing;
10.84/4.41	}
10.84/4.41	instance MonadPlus [] where { 
10.84/4.41	mplus = (++);
10.84/4.41	mzero = [];
10.84/4.41	}
10.84/4.41	msum :: MonadPlus b => [b a]  ->  b a;
10.84/4.41	msum = foldr mplus mzero;
10.84/4.41	
10.84/4.41	}
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(1) BR (EQUIVALENT)
10.84/4.41	Replaced joker patterns by fresh variables and removed binding patterns.
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(2)
10.84/4.41	Obligation:
10.84/4.41	mainModule Main 
10.84/4.41	module Maybe where {
10.84/4.41	import qualified Main;
10.84/4.41	import qualified Monad;
10.84/4.41	import qualified Prelude;
10.84/4.41	}
10.84/4.41	module Main where {
10.84/4.41	import qualified Maybe;
10.84/4.41	import qualified Monad;
10.84/4.41	import qualified Prelude;
10.84/4.41	}
10.84/4.41	module Monad where {
10.84/4.41	import qualified Main;
10.84/4.41	import qualified Maybe;
10.84/4.41	import qualified Prelude;
10.84/4.41	class Monad a => MonadPlus a where {
10.84/4.41	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.84/4.41	mzero :: MonadPlus a => a b;
10.84/4.41	}
10.84/4.41	instance MonadPlus Maybe where { 
10.84/4.41	mplus Nothing ys = ys;
10.84/4.41	mplus xs _ys = xs;
10.84/4.41	mzero = Nothing;
10.84/4.41	}
10.84/4.41	instance MonadPlus [] where { 
10.84/4.41	mplus = (++);
10.84/4.41	mzero = [];
10.84/4.41	}
10.84/4.41	msum :: MonadPlus b => [b a]  ->  b a;
10.84/4.41	msum = foldr mplus mzero;
10.84/4.41	
10.84/4.41	}
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(3) COR (EQUIVALENT)
10.84/4.41	Cond Reductions:
10.84/4.41	The following Function with conditions
10.84/4.41	"undefined |Falseundefined;
10.84/4.41	"
10.84/4.41	is transformed to
10.84/4.41	"undefined  = undefined1;
10.84/4.41	"
10.84/4.41	"undefined0 True = undefined;
10.84/4.41	"
10.84/4.41	"undefined1  = undefined0 False;
10.84/4.41	"
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(4)
10.84/4.41	Obligation:
10.84/4.41	mainModule Main 
10.84/4.41	module Maybe where {
10.84/4.41	import qualified Main;
10.84/4.41	import qualified Monad;
10.84/4.41	import qualified Prelude;
10.84/4.41	}
10.84/4.41	module Main where {
10.84/4.41	import qualified Maybe;
10.84/4.41	import qualified Monad;
10.84/4.41	import qualified Prelude;
10.84/4.41	}
10.84/4.41	module Monad where {
10.84/4.41	import qualified Main;
10.84/4.41	import qualified Maybe;
10.84/4.41	import qualified Prelude;
10.84/4.41	class Monad a => MonadPlus a where {
10.84/4.41	mplus :: MonadPlus a => a b  ->  a b  ->  a b;
10.84/4.41	mzero :: MonadPlus a => a b;
10.84/4.41	}
10.84/4.41	instance MonadPlus Maybe where { 
10.84/4.41	mplus Nothing ys = ys;
10.84/4.41	mplus xs _ys = xs;
10.84/4.41	mzero = Nothing;
10.84/4.41	}
10.84/4.41	instance MonadPlus [] where { 
10.84/4.41	mplus = (++);
10.84/4.41	mzero = [];
10.84/4.41	}
10.84/4.41	msum :: MonadPlus a => [a b]  ->  a b;
10.84/4.41	msum = foldr mplus mzero;
10.84/4.41	
10.84/4.41	}
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(5) Narrow (SOUND)
10.84/4.41	Haskell To QDPs
10.84/4.41	
10.84/4.41	digraph dp_graph {
10.84/4.41	node [outthreshold=100, inthreshold=100];1[label="Monad.msum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.84/4.41	3[label="Monad.msum vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.84/4.41	4[label="foldr Monad.mplus Monad.mzero vy3",fontsize=16,color="burlywood",shape="triangle"];35[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
10.84/4.41	35 -> 5[label="",style="solid", color="burlywood", weight=3];
10.84/4.41	36[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 36[label="",style="solid", color="burlywood", weight=9];
10.84/4.41	36 -> 6[label="",style="solid", color="burlywood", weight=3];
10.84/4.41	5[label="foldr Monad.mplus Monad.mzero (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.84/4.41	6[label="foldr Monad.mplus Monad.mzero []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.84/4.41	7[label="Monad.mplus vy30 (foldr Monad.mplus Monad.mzero vy31)",fontsize=16,color="blue",shape="box"];37[label="Monad.mplus :: ([] a) -> ([] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="blue", weight=9];
10.84/4.41	37 -> 18[label="",style="solid", color="blue", weight=3];
10.84/4.41	38[label="Monad.mplus :: (Maybe a) -> (Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];7 -> 38[label="",style="solid", color="blue", weight=9];
10.84/4.41	38 -> 19[label="",style="solid", color="blue", weight=3];
10.84/4.41	8[label="Monad.mzero",fontsize=16,color="blue",shape="box"];39[label="Monad.mzero :: [] a",fontsize=10,color="white",style="solid",shape="box"];8 -> 39[label="",style="solid", color="blue", weight=9];
10.84/4.41	39 -> 11[label="",style="solid", color="blue", weight=3];
10.84/4.41	40[label="Monad.mzero :: Maybe a",fontsize=10,color="white",style="solid",shape="box"];8 -> 40[label="",style="solid", color="blue", weight=9];
10.84/4.41	40 -> 12[label="",style="solid", color="blue", weight=3];
10.84/4.41	18 -> 14[label="",style="dashed", color="red", weight=0];
10.84/4.41	18[label="Monad.mplus vy30 (foldr Monad.mplus Monad.mzero vy31)",fontsize=16,color="magenta"];18 -> 23[label="",style="dashed", color="magenta", weight=3];
10.84/4.41	19 -> 15[label="",style="dashed", color="red", weight=0];
10.84/4.41	19[label="Monad.mplus vy30 (foldr Monad.mplus Monad.mzero vy31)",fontsize=16,color="magenta"];19 -> 24[label="",style="dashed", color="magenta", weight=3];
10.84/4.41	11[label="Monad.mzero",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
10.84/4.41	12[label="Monad.mzero",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
10.84/4.41	23 -> 4[label="",style="dashed", color="red", weight=0];
10.84/4.41	23[label="foldr Monad.mplus Monad.mzero vy31",fontsize=16,color="magenta"];23 -> 29[label="",style="dashed", color="magenta", weight=3];
10.84/4.41	14[label="Monad.mplus vy30 vy4",fontsize=16,color="black",shape="triangle"];14 -> 20[label="",style="solid", color="black", weight=3];
10.84/4.41	24 -> 4[label="",style="dashed", color="red", weight=0];
10.84/4.41	24[label="foldr Monad.mplus Monad.mzero vy31",fontsize=16,color="magenta"];24 -> 30[label="",style="dashed", color="magenta", weight=3];
10.84/4.41	15[label="Monad.mplus vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];41[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];15 -> 41[label="",style="solid", color="burlywood", weight=9];
10.84/4.41	41 -> 21[label="",style="solid", color="burlywood", weight=3];
10.84/4.41	42[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];15 -> 42[label="",style="solid", color="burlywood", weight=9];
10.84/4.41	42 -> 22[label="",style="solid", color="burlywood", weight=3];
10.84/4.41	16[label="[]",fontsize=16,color="green",shape="box"];17[label="Nothing",fontsize=16,color="green",shape="box"];29[label="vy31",fontsize=16,color="green",shape="box"];20[label="(++) vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];43[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];20 -> 43[label="",style="solid", color="burlywood", weight=9];
10.84/4.41	43 -> 25[label="",style="solid", color="burlywood", weight=3];
10.84/4.41	44[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];20 -> 44[label="",style="solid", color="burlywood", weight=9];
10.84/4.41	44 -> 26[label="",style="solid", color="burlywood", weight=3];
10.84/4.41	30[label="vy31",fontsize=16,color="green",shape="box"];21[label="Monad.mplus Nothing vy4",fontsize=16,color="black",shape="box"];21 -> 27[label="",style="solid", color="black", weight=3];
10.84/4.41	22[label="Monad.mplus (Just vy300) vy4",fontsize=16,color="black",shape="box"];22 -> 28[label="",style="solid", color="black", weight=3];
10.84/4.41	25[label="(++) (vy300 : vy301) vy4",fontsize=16,color="black",shape="box"];25 -> 31[label="",style="solid", color="black", weight=3];
10.84/4.41	26[label="(++) [] vy4",fontsize=16,color="black",shape="box"];26 -> 32[label="",style="solid", color="black", weight=3];
10.84/4.41	27[label="vy4",fontsize=16,color="green",shape="box"];28[label="Just vy300",fontsize=16,color="green",shape="box"];31[label="vy300 : vy301 ++ vy4",fontsize=16,color="green",shape="box"];31 -> 33[label="",style="dashed", color="green", weight=3];
10.84/4.41	32[label="vy4",fontsize=16,color="green",shape="box"];33 -> 20[label="",style="dashed", color="red", weight=0];
10.84/4.41	33[label="vy301 ++ vy4",fontsize=16,color="magenta"];33 -> 34[label="",style="dashed", color="magenta", weight=3];
10.84/4.41	34[label="vy301",fontsize=16,color="green",shape="box"];}
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(6)
10.84/4.41	Complex Obligation (AND)
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(7)
10.84/4.41	Obligation:
10.84/4.41	Q DP problem:
10.84/4.41	The TRS P consists of the following rules:
10.84/4.41	
10.84/4.41	   new_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], h)
10.84/4.41	   new_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, h)
10.84/4.41	
10.84/4.41	R is empty.
10.84/4.41	Q is empty.
10.84/4.41	We have to consider all minimal (P,Q,R)-chains.
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(8) DependencyGraphProof (EQUIVALENT)
10.84/4.41	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(9)
10.84/4.41	Complex Obligation (AND)
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(10)
10.84/4.41	Obligation:
10.84/4.41	Q DP problem:
10.84/4.41	The TRS P consists of the following rules:
10.84/4.41	
10.84/4.41	   new_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, h)
10.84/4.41	
10.84/4.41	R is empty.
10.84/4.41	Q is empty.
10.84/4.41	We have to consider all minimal (P,Q,R)-chains.
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(11) QDPSizeChangeProof (EQUIVALENT)
10.84/4.41	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. 
10.84/4.41	
10.84/4.41	From the DPs we obtained the following set of size-change graphs:
10.84/4.41	*new_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, h)
10.84/4.41	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.84/4.41	
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(12)
10.84/4.41	YES
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(13)
10.84/4.41	Obligation:
10.84/4.41	Q DP problem:
10.84/4.41	The TRS P consists of the following rules:
10.84/4.41	
10.84/4.41	   new_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], h)
10.84/4.41	
10.84/4.41	R is empty.
10.84/4.41	Q is empty.
10.84/4.41	We have to consider all minimal (P,Q,R)-chains.
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(14) QDPSizeChangeProof (EQUIVALENT)
10.84/4.41	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. 
10.84/4.41	
10.84/4.41	From the DPs we obtained the following set of size-change graphs:
10.84/4.41	*new_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], h)
10.84/4.41	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.84/4.41	
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(15)
10.84/4.41	YES
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(16)
10.84/4.41	Obligation:
10.84/4.41	Q DP problem:
10.84/4.41	The TRS P consists of the following rules:
10.84/4.41	
10.84/4.41	   new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
10.84/4.41	
10.84/4.41	R is empty.
10.84/4.41	Q is empty.
10.84/4.41	We have to consider all minimal (P,Q,R)-chains.
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(17) QDPSizeChangeProof (EQUIVALENT)
10.84/4.41	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. 
10.84/4.41	
10.84/4.41	From the DPs we obtained the following set of size-change graphs:
10.84/4.41	*new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
10.84/4.41	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
10.84/4.41	
10.84/4.41	
10.84/4.41	----------------------------------------
10.84/4.41	
10.84/4.41	(18)
10.84/4.41	YES
11.11/4.44	EOF