11.51/4.62	YES
13.18/5.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
13.18/5.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.18/5.10	
13.18/5.10	
13.18/5.10	H-Termination with start terms of the given HASKELL could be proven:
13.18/5.10	
13.18/5.10	(0) HASKELL
13.18/5.10	(1) BR [EQUIVALENT, 0 ms]
13.18/5.10	(2) HASKELL
13.18/5.10	(3) COR [EQUIVALENT, 0 ms]
13.18/5.10	(4) HASKELL
13.18/5.10	(5) Narrow [SOUND, 0 ms]
13.18/5.10	(6) AND
13.18/5.10	    (7) QDP
13.18/5.10	        (8) TransformationProof [EQUIVALENT, 6 ms]
13.18/5.10	        (9) QDP
13.18/5.10	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
13.18/5.10	        (11) QDP
13.18/5.10	        (12) QReductionProof [EQUIVALENT, 0 ms]
13.18/5.10	        (13) QDP
13.18/5.10	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.18/5.10	        (15) YES
13.18/5.10	    (16) QDP
13.18/5.10	        (17) TransformationProof [EQUIVALENT, 0 ms]
13.18/5.10	        (18) QDP
13.18/5.10	        (19) DependencyGraphProof [EQUIVALENT, 0 ms]
13.18/5.10	        (20) QDP
13.18/5.10	        (21) UsableRulesProof [EQUIVALENT, 0 ms]
13.18/5.10	        (22) QDP
13.18/5.10	        (23) QReductionProof [EQUIVALENT, 0 ms]
13.18/5.10	        (24) QDP
13.18/5.10	        (25) TransformationProof [EQUIVALENT, 0 ms]
13.18/5.10	        (26) QDP
13.18/5.10	        (27) DependencyGraphProof [EQUIVALENT, 0 ms]
13.18/5.10	        (28) QDP
13.18/5.10	        (29) UsableRulesProof [EQUIVALENT, 0 ms]
13.18/5.10	        (30) QDP
13.18/5.10	        (31) QReductionProof [EQUIVALENT, 0 ms]
13.18/5.10	        (32) QDP
13.18/5.10	        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.18/5.10	        (34) YES
13.18/5.10	    (35) QDP
13.18/5.10	        (36) TransformationProof [EQUIVALENT, 0 ms]
13.18/5.10	        (37) QDP
13.18/5.10	        (38) UsableRulesProof [EQUIVALENT, 0 ms]
13.18/5.10	        (39) QDP
13.18/5.10	        (40) QReductionProof [EQUIVALENT, 0 ms]
13.18/5.10	        (41) QDP
13.18/5.10	        (42) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.18/5.10	        (43) YES
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(0)
13.18/5.10	Obligation:
13.18/5.10	mainModule Main 
13.18/5.10	module Maybe where {
13.18/5.10	import qualified List;
13.18/5.10	import qualified Main;
13.18/5.10	import qualified Prelude;
13.18/5.10	}
13.18/5.10	module List where {
13.18/5.10	import qualified Main;
13.18/5.10	import qualified Maybe;
13.18/5.10	import qualified Prelude;
13.18/5.10	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
13.18/5.10	isPrefixOf [] _ = True;
13.18/5.10	isPrefixOf _ [] = False;
13.18/5.10	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
13.18/5.10	
13.18/5.10	isSuffixOf :: Eq a => [a]  ->  [a]  ->  Bool;
13.18/5.10	isSuffixOf x y = reverse x `isPrefixOf` reverse y;
13.18/5.10	
13.18/5.10	}
13.18/5.10	module Main where {
13.18/5.10	import qualified List;
13.18/5.10	import qualified Maybe;
13.18/5.10	import qualified Prelude;
13.18/5.10	}
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(1) BR (EQUIVALENT)
13.18/5.10	Replaced joker patterns by fresh variables and removed binding patterns.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(2)
13.18/5.10	Obligation:
13.18/5.10	mainModule Main 
13.18/5.10	module Maybe where {
13.18/5.10	import qualified List;
13.18/5.10	import qualified Main;
13.18/5.10	import qualified Prelude;
13.18/5.10	}
13.18/5.10	module List where {
13.18/5.10	import qualified Main;
13.18/5.10	import qualified Maybe;
13.18/5.10	import qualified Prelude;
13.18/5.10	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
13.18/5.10	isPrefixOf [] vy = True;
13.18/5.10	isPrefixOf vz [] = False;
13.18/5.10	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
13.18/5.10	
13.18/5.10	isSuffixOf :: Eq a => [a]  ->  [a]  ->  Bool;
13.18/5.10	isSuffixOf x y = reverse x `isPrefixOf` reverse y;
13.18/5.10	
13.18/5.10	}
13.18/5.10	module Main where {
13.18/5.10	import qualified List;
13.18/5.10	import qualified Maybe;
13.18/5.10	import qualified Prelude;
13.18/5.10	}
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(3) COR (EQUIVALENT)
13.18/5.10	Cond Reductions:
13.18/5.10	The following Function with conditions
13.18/5.10	"undefined |Falseundefined;
13.18/5.10	"
13.18/5.10	is transformed to
13.18/5.10	"undefined  = undefined1;
13.18/5.10	"
13.18/5.10	"undefined0 True = undefined;
13.18/5.10	"
13.18/5.10	"undefined1  = undefined0 False;
13.18/5.10	"
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(4)
13.18/5.10	Obligation:
13.18/5.10	mainModule Main 
13.18/5.10	module Maybe where {
13.18/5.10	import qualified List;
13.18/5.10	import qualified Main;
13.18/5.10	import qualified Prelude;
13.18/5.10	}
13.18/5.10	module List where {
13.18/5.10	import qualified Main;
13.18/5.10	import qualified Maybe;
13.18/5.10	import qualified Prelude;
13.18/5.10	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
13.18/5.10	isPrefixOf [] vy = True;
13.18/5.10	isPrefixOf vz [] = False;
13.18/5.10	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
13.18/5.10	
13.18/5.10	isSuffixOf :: Eq a => [a]  ->  [a]  ->  Bool;
13.18/5.10	isSuffixOf x y = reverse x `isPrefixOf` reverse y;
13.18/5.10	
13.18/5.10	}
13.18/5.10	module Main where {
13.18/5.10	import qualified List;
13.18/5.10	import qualified Maybe;
13.18/5.10	import qualified Prelude;
13.18/5.10	}
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(5) Narrow (SOUND)
13.18/5.10	Haskell To QDPs
13.18/5.10	
13.18/5.10	digraph dp_graph {
13.18/5.10	node [outthreshold=100, inthreshold=100];1[label="List.isSuffixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
13.18/5.10	3[label="List.isSuffixOf wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
13.18/5.10	4[label="List.isSuffixOf wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
13.18/5.10	5[label="List.isPrefixOf (reverse wu3) (reverse wu4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
13.18/5.10	6[label="List.isPrefixOf (foldl (flip (:)) [] wu3) (reverse wu4)",fontsize=16,color="burlywood",shape="box"];510[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];6 -> 510[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	510 -> 7[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	511[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 511[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	511 -> 8[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	7[label="List.isPrefixOf (foldl (flip (:)) [] (wu30 : wu31)) (reverse wu4)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
13.18/5.10	8[label="List.isPrefixOf (foldl (flip (:)) [] []) (reverse wu4)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
13.18/5.10	9 -> 335[label="",style="dashed", color="red", weight=0];
13.18/5.10	9[label="List.isPrefixOf (foldl (flip (:)) (flip (:) [] wu30) wu31) (reverse wu4)",fontsize=16,color="magenta"];9 -> 336[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	9 -> 337[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	9 -> 338[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	9 -> 339[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	10[label="List.isPrefixOf [] (reverse wu4)",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
13.18/5.10	336[label="wu4",fontsize=16,color="green",shape="box"];337[label="[]",fontsize=16,color="green",shape="box"];338[label="wu31",fontsize=16,color="green",shape="box"];339[label="wu30",fontsize=16,color="green",shape="box"];335[label="List.isPrefixOf (foldl (flip (:)) (flip (:) wu16 wu17) wu18) (reverse wu19)",fontsize=16,color="burlywood",shape="triangle"];512[label="wu18/wu180 : wu181",fontsize=10,color="white",style="solid",shape="box"];335 -> 512[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	512 -> 372[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	513[label="wu18/[]",fontsize=10,color="white",style="solid",shape="box"];335 -> 513[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	513 -> 373[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	13[label="True",fontsize=16,color="green",shape="box"];372[label="List.isPrefixOf (foldl (flip (:)) (flip (:) wu16 wu17) (wu180 : wu181)) (reverse wu19)",fontsize=16,color="black",shape="box"];372 -> 374[label="",style="solid", color="black", weight=3];
13.18/5.10	373[label="List.isPrefixOf (foldl (flip (:)) (flip (:) wu16 wu17) []) (reverse wu19)",fontsize=16,color="black",shape="box"];373 -> 375[label="",style="solid", color="black", weight=3];
13.18/5.10	374 -> 335[label="",style="dashed", color="red", weight=0];
13.18/5.10	374[label="List.isPrefixOf (foldl (flip (:)) (flip (:) (flip (:) wu16 wu17) wu180) wu181) (reverse wu19)",fontsize=16,color="magenta"];374 -> 376[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	374 -> 377[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	374 -> 378[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	375[label="List.isPrefixOf (flip (:) wu16 wu17) (reverse wu19)",fontsize=16,color="black",shape="box"];375 -> 379[label="",style="solid", color="black", weight=3];
13.18/5.10	376[label="flip (:) wu16 wu17",fontsize=16,color="black",shape="triangle"];376 -> 380[label="",style="solid", color="black", weight=3];
13.18/5.10	377[label="wu181",fontsize=16,color="green",shape="box"];378[label="wu180",fontsize=16,color="green",shape="box"];379[label="List.isPrefixOf ((:) wu17 wu16) (reverse wu19)",fontsize=16,color="black",shape="box"];379 -> 381[label="",style="solid", color="black", weight=3];
13.18/5.10	380[label="(:) wu17 wu16",fontsize=16,color="green",shape="box"];381 -> 386[label="",style="dashed", color="red", weight=0];
13.18/5.10	381[label="List.isPrefixOf ((:) wu17 wu16) (foldl (flip (:)) [] wu19)",fontsize=16,color="magenta"];381 -> 387[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	381 -> 388[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	387[label="[]",fontsize=16,color="green",shape="box"];388[label="wu19",fontsize=16,color="green",shape="box"];386[label="List.isPrefixOf ((:) wu17 wu16) (foldl (flip (:)) wu20 wu191)",fontsize=16,color="burlywood",shape="triangle"];514[label="wu191/wu1910 : wu1911",fontsize=10,color="white",style="solid",shape="box"];386 -> 514[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	514 -> 390[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	515[label="wu191/[]",fontsize=10,color="white",style="solid",shape="box"];386 -> 515[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	515 -> 391[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	390[label="List.isPrefixOf ((:) wu17 wu16) (foldl (flip (:)) wu20 (wu1910 : wu1911))",fontsize=16,color="black",shape="box"];390 -> 392[label="",style="solid", color="black", weight=3];
13.18/5.10	391[label="List.isPrefixOf ((:) wu17 wu16) (foldl (flip (:)) wu20 [])",fontsize=16,color="black",shape="box"];391 -> 393[label="",style="solid", color="black", weight=3];
13.18/5.10	392 -> 386[label="",style="dashed", color="red", weight=0];
13.18/5.10	392[label="List.isPrefixOf ((:) wu17 wu16) (foldl (flip (:)) (flip (:) wu20 wu1910) wu1911)",fontsize=16,color="magenta"];392 -> 394[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	392 -> 395[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	393[label="List.isPrefixOf ((:) wu17 wu16) wu20",fontsize=16,color="burlywood",shape="box"];516[label="wu20/wu200 : wu201",fontsize=10,color="white",style="solid",shape="box"];393 -> 516[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	516 -> 396[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	517[label="wu20/[]",fontsize=10,color="white",style="solid",shape="box"];393 -> 517[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	517 -> 397[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	394 -> 376[label="",style="dashed", color="red", weight=0];
13.18/5.10	394[label="flip (:) wu20 wu1910",fontsize=16,color="magenta"];394 -> 398[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	394 -> 399[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	395[label="wu1911",fontsize=16,color="green",shape="box"];396[label="List.isPrefixOf ((:) wu17 wu16) (wu200 : wu201)",fontsize=16,color="black",shape="box"];396 -> 400[label="",style="solid", color="black", weight=3];
13.18/5.10	397[label="List.isPrefixOf ((:) wu17 wu16) []",fontsize=16,color="black",shape="box"];397 -> 401[label="",style="solid", color="black", weight=3];
13.18/5.10	398[label="wu20",fontsize=16,color="green",shape="box"];399[label="wu1910",fontsize=16,color="green",shape="box"];400 -> 402[label="",style="dashed", color="red", weight=0];
13.18/5.10	400[label="wu17 == wu200 && List.isPrefixOf wu16 wu201",fontsize=16,color="magenta"];400 -> 403[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	400 -> 404[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	400 -> 405[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	401[label="False",fontsize=16,color="green",shape="box"];403[label="wu17 == wu200",fontsize=16,color="blue",shape="box"];518[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 518[label="",style="solid", color="blue", weight=9];
13.18/5.10	518 -> 406[label="",style="solid", color="blue", weight=3];
13.18/5.10	519[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 519[label="",style="solid", color="blue", weight=9];
13.18/5.10	519 -> 407[label="",style="solid", color="blue", weight=3];
13.18/5.10	520[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 520[label="",style="solid", color="blue", weight=9];
13.18/5.10	520 -> 408[label="",style="solid", color="blue", weight=3];
13.18/5.10	521[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 521[label="",style="solid", color="blue", weight=9];
13.18/5.10	521 -> 409[label="",style="solid", color="blue", weight=3];
13.18/5.10	522[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 522[label="",style="solid", color="blue", weight=9];
13.18/5.10	522 -> 410[label="",style="solid", color="blue", weight=3];
13.18/5.10	523[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 523[label="",style="solid", color="blue", weight=9];
13.18/5.10	523 -> 411[label="",style="solid", color="blue", weight=3];
13.18/5.10	524[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 524[label="",style="solid", color="blue", weight=9];
13.18/5.10	524 -> 412[label="",style="solid", color="blue", weight=3];
13.18/5.10	525[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 525[label="",style="solid", color="blue", weight=9];
13.18/5.10	525 -> 413[label="",style="solid", color="blue", weight=3];
13.18/5.10	526[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 526[label="",style="solid", color="blue", weight=9];
13.18/5.10	526 -> 414[label="",style="solid", color="blue", weight=3];
13.18/5.10	527[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 527[label="",style="solid", color="blue", weight=9];
13.18/5.10	527 -> 415[label="",style="solid", color="blue", weight=3];
13.18/5.10	528[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 528[label="",style="solid", color="blue", weight=9];
13.18/5.10	528 -> 416[label="",style="solid", color="blue", weight=3];
13.18/5.10	529[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 529[label="",style="solid", color="blue", weight=9];
13.18/5.10	529 -> 417[label="",style="solid", color="blue", weight=3];
13.18/5.10	530[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 530[label="",style="solid", color="blue", weight=9];
13.18/5.10	530 -> 418[label="",style="solid", color="blue", weight=3];
13.18/5.10	531[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];403 -> 531[label="",style="solid", color="blue", weight=9];
13.18/5.10	531 -> 419[label="",style="solid", color="blue", weight=3];
13.18/5.10	404[label="wu16",fontsize=16,color="green",shape="box"];405[label="wu201",fontsize=16,color="green",shape="box"];402[label="wu25 && List.isPrefixOf wu26 wu27",fontsize=16,color="burlywood",shape="triangle"];532[label="wu25/False",fontsize=10,color="white",style="solid",shape="box"];402 -> 532[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	532 -> 420[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	533[label="wu25/True",fontsize=10,color="white",style="solid",shape="box"];402 -> 533[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	533 -> 421[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	406[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];406 -> 422[label="",style="solid", color="black", weight=3];
13.18/5.10	407[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];407 -> 423[label="",style="solid", color="black", weight=3];
13.18/5.10	408[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];408 -> 424[label="",style="solid", color="black", weight=3];
13.18/5.10	409[label="wu17 == wu200",fontsize=16,color="burlywood",shape="triangle"];534[label="wu17/LT",fontsize=10,color="white",style="solid",shape="box"];409 -> 534[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	534 -> 425[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	535[label="wu17/EQ",fontsize=10,color="white",style="solid",shape="box"];409 -> 535[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	535 -> 426[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	536[label="wu17/GT",fontsize=10,color="white",style="solid",shape="box"];409 -> 536[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	536 -> 427[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	410[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];410 -> 428[label="",style="solid", color="black", weight=3];
13.18/5.10	411[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];411 -> 429[label="",style="solid", color="black", weight=3];
13.18/5.10	412[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];412 -> 430[label="",style="solid", color="black", weight=3];
13.18/5.10	413[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];413 -> 431[label="",style="solid", color="black", weight=3];
13.18/5.10	414[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];414 -> 432[label="",style="solid", color="black", weight=3];
13.18/5.10	415[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];415 -> 433[label="",style="solid", color="black", weight=3];
13.18/5.10	416[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];416 -> 434[label="",style="solid", color="black", weight=3];
13.18/5.10	417[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];417 -> 435[label="",style="solid", color="black", weight=3];
13.18/5.10	418[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];418 -> 436[label="",style="solid", color="black", weight=3];
13.18/5.10	419[label="wu17 == wu200",fontsize=16,color="black",shape="triangle"];419 -> 437[label="",style="solid", color="black", weight=3];
13.18/5.10	420[label="False && List.isPrefixOf wu26 wu27",fontsize=16,color="black",shape="box"];420 -> 438[label="",style="solid", color="black", weight=3];
13.18/5.10	421[label="True && List.isPrefixOf wu26 wu27",fontsize=16,color="black",shape="box"];421 -> 439[label="",style="solid", color="black", weight=3];
13.18/5.10	422[label="error []",fontsize=16,color="red",shape="box"];423[label="error []",fontsize=16,color="red",shape="box"];424[label="error []",fontsize=16,color="red",shape="box"];425[label="LT == wu200",fontsize=16,color="burlywood",shape="box"];537[label="wu200/LT",fontsize=10,color="white",style="solid",shape="box"];425 -> 537[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	537 -> 440[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	538[label="wu200/EQ",fontsize=10,color="white",style="solid",shape="box"];425 -> 538[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	538 -> 441[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	539[label="wu200/GT",fontsize=10,color="white",style="solid",shape="box"];425 -> 539[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	539 -> 442[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	426[label="EQ == wu200",fontsize=16,color="burlywood",shape="box"];540[label="wu200/LT",fontsize=10,color="white",style="solid",shape="box"];426 -> 540[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	540 -> 443[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	541[label="wu200/EQ",fontsize=10,color="white",style="solid",shape="box"];426 -> 541[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	541 -> 444[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	542[label="wu200/GT",fontsize=10,color="white",style="solid",shape="box"];426 -> 542[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	542 -> 445[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	427[label="GT == wu200",fontsize=16,color="burlywood",shape="box"];543[label="wu200/LT",fontsize=10,color="white",style="solid",shape="box"];427 -> 543[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	543 -> 446[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	544[label="wu200/EQ",fontsize=10,color="white",style="solid",shape="box"];427 -> 544[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	544 -> 447[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	545[label="wu200/GT",fontsize=10,color="white",style="solid",shape="box"];427 -> 545[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	545 -> 448[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	428[label="error []",fontsize=16,color="red",shape="box"];429[label="error []",fontsize=16,color="red",shape="box"];430[label="error []",fontsize=16,color="red",shape="box"];431[label="error []",fontsize=16,color="red",shape="box"];432[label="error []",fontsize=16,color="red",shape="box"];433[label="error []",fontsize=16,color="red",shape="box"];434[label="error []",fontsize=16,color="red",shape="box"];435[label="error []",fontsize=16,color="red",shape="box"];436[label="error []",fontsize=16,color="red",shape="box"];437[label="error []",fontsize=16,color="red",shape="box"];438[label="False",fontsize=16,color="green",shape="box"];439[label="List.isPrefixOf wu26 wu27",fontsize=16,color="burlywood",shape="box"];546[label="wu26/wu260 : wu261",fontsize=10,color="white",style="solid",shape="box"];439 -> 546[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	546 -> 449[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	547[label="wu26/[]",fontsize=10,color="white",style="solid",shape="box"];439 -> 547[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	547 -> 450[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	440[label="LT == LT",fontsize=16,color="black",shape="box"];440 -> 451[label="",style="solid", color="black", weight=3];
13.18/5.10	441[label="LT == EQ",fontsize=16,color="black",shape="box"];441 -> 452[label="",style="solid", color="black", weight=3];
13.18/5.10	442[label="LT == GT",fontsize=16,color="black",shape="box"];442 -> 453[label="",style="solid", color="black", weight=3];
13.18/5.10	443[label="EQ == LT",fontsize=16,color="black",shape="box"];443 -> 454[label="",style="solid", color="black", weight=3];
13.18/5.10	444[label="EQ == EQ",fontsize=16,color="black",shape="box"];444 -> 455[label="",style="solid", color="black", weight=3];
13.18/5.10	445[label="EQ == GT",fontsize=16,color="black",shape="box"];445 -> 456[label="",style="solid", color="black", weight=3];
13.18/5.10	446[label="GT == LT",fontsize=16,color="black",shape="box"];446 -> 457[label="",style="solid", color="black", weight=3];
13.18/5.10	447[label="GT == EQ",fontsize=16,color="black",shape="box"];447 -> 458[label="",style="solid", color="black", weight=3];
13.18/5.10	448[label="GT == GT",fontsize=16,color="black",shape="box"];448 -> 459[label="",style="solid", color="black", weight=3];
13.18/5.10	449[label="List.isPrefixOf (wu260 : wu261) wu27",fontsize=16,color="burlywood",shape="box"];548[label="wu27/wu270 : wu271",fontsize=10,color="white",style="solid",shape="box"];449 -> 548[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	548 -> 460[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	549[label="wu27/[]",fontsize=10,color="white",style="solid",shape="box"];449 -> 549[label="",style="solid", color="burlywood", weight=9];
13.18/5.10	549 -> 461[label="",style="solid", color="burlywood", weight=3];
13.18/5.10	450[label="List.isPrefixOf [] wu27",fontsize=16,color="black",shape="box"];450 -> 462[label="",style="solid", color="black", weight=3];
13.18/5.10	451[label="True",fontsize=16,color="green",shape="box"];452[label="False",fontsize=16,color="green",shape="box"];453[label="False",fontsize=16,color="green",shape="box"];454[label="False",fontsize=16,color="green",shape="box"];455[label="True",fontsize=16,color="green",shape="box"];456[label="False",fontsize=16,color="green",shape="box"];457[label="False",fontsize=16,color="green",shape="box"];458[label="False",fontsize=16,color="green",shape="box"];459[label="True",fontsize=16,color="green",shape="box"];460[label="List.isPrefixOf (wu260 : wu261) (wu270 : wu271)",fontsize=16,color="black",shape="box"];460 -> 463[label="",style="solid", color="black", weight=3];
13.18/5.10	461[label="List.isPrefixOf (wu260 : wu261) []",fontsize=16,color="black",shape="box"];461 -> 464[label="",style="solid", color="black", weight=3];
13.18/5.10	462[label="True",fontsize=16,color="green",shape="box"];463 -> 402[label="",style="dashed", color="red", weight=0];
13.18/5.10	463[label="wu260 == wu270 && List.isPrefixOf wu261 wu271",fontsize=16,color="magenta"];463 -> 465[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	463 -> 466[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	463 -> 467[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	464[label="False",fontsize=16,color="green",shape="box"];465[label="wu260 == wu270",fontsize=16,color="blue",shape="box"];550[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 550[label="",style="solid", color="blue", weight=9];
13.18/5.10	550 -> 468[label="",style="solid", color="blue", weight=3];
13.18/5.10	551[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 551[label="",style="solid", color="blue", weight=9];
13.18/5.10	551 -> 469[label="",style="solid", color="blue", weight=3];
13.18/5.10	552[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 552[label="",style="solid", color="blue", weight=9];
13.18/5.10	552 -> 470[label="",style="solid", color="blue", weight=3];
13.18/5.10	553[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 553[label="",style="solid", color="blue", weight=9];
13.18/5.10	553 -> 471[label="",style="solid", color="blue", weight=3];
13.18/5.10	554[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 554[label="",style="solid", color="blue", weight=9];
13.18/5.10	554 -> 472[label="",style="solid", color="blue", weight=3];
13.18/5.10	555[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 555[label="",style="solid", color="blue", weight=9];
13.18/5.10	555 -> 473[label="",style="solid", color="blue", weight=3];
13.18/5.10	556[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 556[label="",style="solid", color="blue", weight=9];
13.18/5.10	556 -> 474[label="",style="solid", color="blue", weight=3];
13.18/5.10	557[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 557[label="",style="solid", color="blue", weight=9];
13.18/5.10	557 -> 475[label="",style="solid", color="blue", weight=3];
13.18/5.10	558[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 558[label="",style="solid", color="blue", weight=9];
13.18/5.10	558 -> 476[label="",style="solid", color="blue", weight=3];
13.18/5.10	559[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 559[label="",style="solid", color="blue", weight=9];
13.18/5.10	559 -> 477[label="",style="solid", color="blue", weight=3];
13.18/5.10	560[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 560[label="",style="solid", color="blue", weight=9];
13.18/5.10	560 -> 478[label="",style="solid", color="blue", weight=3];
13.18/5.10	561[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 561[label="",style="solid", color="blue", weight=9];
13.18/5.10	561 -> 479[label="",style="solid", color="blue", weight=3];
13.18/5.10	562[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 562[label="",style="solid", color="blue", weight=9];
13.18/5.10	562 -> 480[label="",style="solid", color="blue", weight=3];
13.18/5.10	563[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];465 -> 563[label="",style="solid", color="blue", weight=9];
13.18/5.10	563 -> 481[label="",style="solid", color="blue", weight=3];
13.18/5.10	466[label="wu261",fontsize=16,color="green",shape="box"];467[label="wu271",fontsize=16,color="green",shape="box"];468 -> 406[label="",style="dashed", color="red", weight=0];
13.18/5.10	468[label="wu260 == wu270",fontsize=16,color="magenta"];468 -> 482[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	468 -> 483[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	469 -> 407[label="",style="dashed", color="red", weight=0];
13.18/5.10	469[label="wu260 == wu270",fontsize=16,color="magenta"];469 -> 484[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	469 -> 485[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	470 -> 408[label="",style="dashed", color="red", weight=0];
13.18/5.10	470[label="wu260 == wu270",fontsize=16,color="magenta"];470 -> 486[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	470 -> 487[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	471 -> 409[label="",style="dashed", color="red", weight=0];
13.18/5.10	471[label="wu260 == wu270",fontsize=16,color="magenta"];471 -> 488[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	471 -> 489[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	472 -> 410[label="",style="dashed", color="red", weight=0];
13.18/5.10	472[label="wu260 == wu270",fontsize=16,color="magenta"];472 -> 490[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	472 -> 491[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	473 -> 411[label="",style="dashed", color="red", weight=0];
13.18/5.10	473[label="wu260 == wu270",fontsize=16,color="magenta"];473 -> 492[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	473 -> 493[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	474 -> 412[label="",style="dashed", color="red", weight=0];
13.18/5.10	474[label="wu260 == wu270",fontsize=16,color="magenta"];474 -> 494[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	474 -> 495[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	475 -> 413[label="",style="dashed", color="red", weight=0];
13.18/5.10	475[label="wu260 == wu270",fontsize=16,color="magenta"];475 -> 496[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	475 -> 497[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	476 -> 414[label="",style="dashed", color="red", weight=0];
13.18/5.10	476[label="wu260 == wu270",fontsize=16,color="magenta"];476 -> 498[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	476 -> 499[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	477 -> 415[label="",style="dashed", color="red", weight=0];
13.18/5.10	477[label="wu260 == wu270",fontsize=16,color="magenta"];477 -> 500[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	477 -> 501[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	478 -> 416[label="",style="dashed", color="red", weight=0];
13.18/5.10	478[label="wu260 == wu270",fontsize=16,color="magenta"];478 -> 502[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	478 -> 503[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	479 -> 417[label="",style="dashed", color="red", weight=0];
13.18/5.10	479[label="wu260 == wu270",fontsize=16,color="magenta"];479 -> 504[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	479 -> 505[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	480 -> 418[label="",style="dashed", color="red", weight=0];
13.18/5.10	480[label="wu260 == wu270",fontsize=16,color="magenta"];480 -> 506[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	480 -> 507[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	481 -> 419[label="",style="dashed", color="red", weight=0];
13.18/5.10	481[label="wu260 == wu270",fontsize=16,color="magenta"];481 -> 508[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	481 -> 509[label="",style="dashed", color="magenta", weight=3];
13.18/5.10	482[label="wu270",fontsize=16,color="green",shape="box"];483[label="wu260",fontsize=16,color="green",shape="box"];484[label="wu270",fontsize=16,color="green",shape="box"];485[label="wu260",fontsize=16,color="green",shape="box"];486[label="wu270",fontsize=16,color="green",shape="box"];487[label="wu260",fontsize=16,color="green",shape="box"];488[label="wu270",fontsize=16,color="green",shape="box"];489[label="wu260",fontsize=16,color="green",shape="box"];490[label="wu270",fontsize=16,color="green",shape="box"];491[label="wu260",fontsize=16,color="green",shape="box"];492[label="wu270",fontsize=16,color="green",shape="box"];493[label="wu260",fontsize=16,color="green",shape="box"];494[label="wu270",fontsize=16,color="green",shape="box"];495[label="wu260",fontsize=16,color="green",shape="box"];496[label="wu270",fontsize=16,color="green",shape="box"];497[label="wu260",fontsize=16,color="green",shape="box"];498[label="wu270",fontsize=16,color="green",shape="box"];499[label="wu260",fontsize=16,color="green",shape="box"];500[label="wu270",fontsize=16,color="green",shape="box"];501[label="wu260",fontsize=16,color="green",shape="box"];502[label="wu270",fontsize=16,color="green",shape="box"];503[label="wu260",fontsize=16,color="green",shape="box"];504[label="wu270",fontsize=16,color="green",shape="box"];505[label="wu260",fontsize=16,color="green",shape="box"];506[label="wu270",fontsize=16,color="green",shape="box"];507[label="wu260",fontsize=16,color="green",shape="box"];508[label="wu270",fontsize=16,color="green",shape="box"];509[label="wu260",fontsize=16,color="green",shape="box"];}
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(6)
13.18/5.10	Complex Obligation (AND)
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(7)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, new_flip(wu20, wu1910, ba), wu1911, ba)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_flip(wu16, wu17, ba) -> :(wu17, wu16)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(8) TransformationProof (EQUIVALENT)
13.18/5.10	By rewriting [LPAR04] the rule new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, new_flip(wu20, wu1910, ba), wu1911, ba) at position [2] we obtained the following new rules [LPAR04]:
13.18/5.10	
13.18/5.10	   (new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, :(wu1910, wu20), wu1911, ba),new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, :(wu1910, wu20), wu1911, ba))
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(9)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, :(wu1910, wu20), wu1911, ba)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_flip(wu16, wu17, ba) -> :(wu17, wu16)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(10) UsableRulesProof (EQUIVALENT)
13.18/5.10	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(11)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, :(wu1910, wu20), wu1911, ba)
13.18/5.10	
13.18/5.10	R is empty.
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(12) QReductionProof (EQUIVALENT)
13.18/5.10	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(13)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, :(wu1910, wu20), wu1911, ba)
13.18/5.10	
13.18/5.10	R is empty.
13.18/5.10	Q is empty.
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(14) QDPSizeChangeProof (EQUIVALENT)
13.18/5.10	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. 
13.18/5.10	
13.18/5.10	From the DPs we obtained the following set of size-change graphs:
13.18/5.10	*new_isPrefixOf(wu17, wu16, wu20, :(wu1910, wu1911), ba) -> new_isPrefixOf(wu17, wu16, :(wu1910, wu20), wu1911, ba)
13.18/5.10	The graph contains the following edges 1 >= 1, 2 >= 2, 4 > 4, 5 >= 5
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(15)
13.18/5.10	YES
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(16)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(wu260, wu261), :(wu270, wu271), ba) -> new_asAs(new_esEs(wu260, wu270, ba), wu261, wu271, ba)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs(wu260, wu270, ty_Double) -> new_esEs3(wu260, wu270)
13.18/5.10	   new_esEs3(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, ty_Float) -> new_esEs7(wu260, wu270)
13.18/5.10	   new_esEs(wu260, wu270, ty_Int) -> new_esEs2(wu260, wu270)
13.18/5.10	   new_esEs(wu260, wu270, ty_@0) -> new_esEs1(wu260, wu270)
13.18/5.10	   new_esEs11(wu17, wu200, ce) -> error([])
13.18/5.10	   new_esEs5(wu17, wu200, db, dc) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, ty_Char) -> new_esEs8(wu260, wu270)
13.18/5.10	   new_esEs1(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(ty_@2, bg), bh)) -> new_esEs9(wu260, wu270, bg, bh)
13.18/5.10	   new_esEs(wu260, wu270, ty_Bool) -> new_esEs0(wu260, wu270)
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs(wu260, wu270, app(ty_[], cb)) -> new_esEs11(wu260, wu270, cb)
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs4(wu17, wu200, dd, de, df) -> error([])
13.18/5.10	   new_esEs7(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(ty_Maybe, ca)) -> new_esEs10(wu260, wu270, ca)
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs(wu260, wu270, ty_Integer) -> new_esEs12(wu260, wu270)
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs12(wu17, wu200) -> error([])
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	   new_esEs10(wu17, wu200, da) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(ty_Ratio, cc)) -> new_esEs13(wu260, wu270, cc)
13.18/5.10	   new_esEs2(wu17, wu200) -> error([])
13.18/5.10	   new_esEs13(wu17, wu200, cd) -> error([])
13.18/5.10	   new_esEs0(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(ty_Either, be), bf)) -> new_esEs5(wu260, wu270, be, bf)
13.18/5.10	   new_esEs(wu260, wu270, ty_Ordering) -> new_esEs6(wu260, wu270)
13.18/5.10	   new_esEs8(wu17, wu200) -> error([])
13.18/5.10	   new_esEs9(wu17, wu200, cf, cg) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(app(ty_@3, bb), bc), bd)) -> new_esEs4(wu260, wu270, bb, bc, bd)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs(x0, x1, ty_Int)
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs(x0, x1, ty_Float)
13.18/5.10	   new_esEs5(x0, x1, x2, x3)
13.18/5.10	   new_esEs(x0, x1, ty_Integer)
13.18/5.10	   new_esEs9(x0, x1, x2, x3)
13.18/5.10	   new_esEs8(x0, x1)
13.18/5.10	   new_esEs3(x0, x1)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs7(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_Ratio, x2))
13.18/5.10	   new_esEs(x0, x1, ty_Bool)
13.18/5.10	   new_esEs(x0, x1, app(ty_Maybe, x2))
13.18/5.10	   new_esEs10(x0, x1, x2)
13.18/5.10	   new_esEs12(x0, x1)
13.18/5.10	   new_esEs1(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
13.18/5.10	   new_esEs2(x0, x1)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs(x0, x1, ty_Char)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
13.18/5.10	   new_esEs(x0, x1, ty_Ordering)
13.18/5.10	   new_esEs(x0, x1, ty_Double)
13.18/5.10	   new_esEs11(x0, x1, x2)
13.18/5.10	   new_esEs13(x0, x1, x2)
13.18/5.10	   new_esEs0(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_[], x2))
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	   new_esEs(x0, x1, ty_@0)
13.18/5.10	   new_esEs4(x0, x1, x2, x3, x4)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(17) TransformationProof (EQUIVALENT)
13.18/5.10	By narrowing [LPAR04] the rule new_asAs(True, :(wu260, wu261), :(wu270, wu271), ba) -> new_asAs(new_esEs(wu260, wu270, ba), wu261, wu271, ba) at position [0] we obtained the following new rules [LPAR04]:
13.18/5.10	
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Double) -> new_asAs(new_esEs3(x0, x1), y1, y3, ty_Double),new_asAs(True, :(x0, y1), :(x1, y3), ty_Double) -> new_asAs(new_esEs3(x0, x1), y1, y3, ty_Double))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Float) -> new_asAs(new_esEs7(x0, x1), y1, y3, ty_Float),new_asAs(True, :(x0, y1), :(x1, y3), ty_Float) -> new_asAs(new_esEs7(x0, x1), y1, y3, ty_Float))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Int) -> new_asAs(new_esEs2(x0, x1), y1, y3, ty_Int),new_asAs(True, :(x0, y1), :(x1, y3), ty_Int) -> new_asAs(new_esEs2(x0, x1), y1, y3, ty_Int))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs1(x0, x1), y1, y3, ty_@0),new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs1(x0, x1), y1, y3, ty_@0))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Char) -> new_asAs(new_esEs8(x0, x1), y1, y3, ty_Char),new_asAs(True, :(x0, y1), :(x1, y3), ty_Char) -> new_asAs(new_esEs8(x0, x1), y1, y3, ty_Char))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_asAs(new_esEs9(x0, x1, x2, x3), y1, y3, app(app(ty_@2, x2), x3)),new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_asAs(new_esEs9(x0, x1, x2, x3), y1, y3, app(app(ty_@2, x2), x3)))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Bool) -> new_asAs(new_esEs0(x0, x1), y1, y3, ty_Bool),new_asAs(True, :(x0, y1), :(x1, y3), ty_Bool) -> new_asAs(new_esEs0(x0, x1), y1, y3, ty_Bool))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_asAs(new_esEs11(x0, x1, x2), y1, y3, app(ty_[], x2)),new_asAs(True, :(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_asAs(new_esEs11(x0, x1, x2), y1, y3, app(ty_[], x2)))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_asAs(new_esEs10(x0, x1, x2), y1, y3, app(ty_Maybe, x2)),new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_asAs(new_esEs10(x0, x1, x2), y1, y3, app(ty_Maybe, x2)))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Integer) -> new_asAs(new_esEs12(x0, x1), y1, y3, ty_Integer),new_asAs(True, :(x0, y1), :(x1, y3), ty_Integer) -> new_asAs(new_esEs12(x0, x1), y1, y3, ty_Integer))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_asAs(new_esEs13(x0, x1, x2), y1, y3, app(ty_Ratio, x2)),new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_asAs(new_esEs13(x0, x1, x2), y1, y3, app(ty_Ratio, x2)))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_asAs(new_esEs5(x0, x1, x2, x3), y1, y3, app(app(ty_Either, x2), x3)),new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_asAs(new_esEs5(x0, x1, x2, x3), y1, y3, app(app(ty_Either, x2), x3)))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering),new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_asAs(new_esEs4(x0, x1, x2, x3, x4), y1, y3, app(app(app(ty_@3, x2), x3), x4)),new_asAs(True, :(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_asAs(new_esEs4(x0, x1, x2, x3, x4), y1, y3, app(app(app(ty_@3, x2), x3), x4)))
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(18)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Double) -> new_asAs(new_esEs3(x0, x1), y1, y3, ty_Double)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Float) -> new_asAs(new_esEs7(x0, x1), y1, y3, ty_Float)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Int) -> new_asAs(new_esEs2(x0, x1), y1, y3, ty_Int)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs1(x0, x1), y1, y3, ty_@0)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Char) -> new_asAs(new_esEs8(x0, x1), y1, y3, ty_Char)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_asAs(new_esEs9(x0, x1, x2, x3), y1, y3, app(app(ty_@2, x2), x3))
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Bool) -> new_asAs(new_esEs0(x0, x1), y1, y3, ty_Bool)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_asAs(new_esEs11(x0, x1, x2), y1, y3, app(ty_[], x2))
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_asAs(new_esEs10(x0, x1, x2), y1, y3, app(ty_Maybe, x2))
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Integer) -> new_asAs(new_esEs12(x0, x1), y1, y3, ty_Integer)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_asAs(new_esEs13(x0, x1, x2), y1, y3, app(ty_Ratio, x2))
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_asAs(new_esEs5(x0, x1, x2, x3), y1, y3, app(app(ty_Either, x2), x3))
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_asAs(new_esEs4(x0, x1, x2, x3, x4), y1, y3, app(app(app(ty_@3, x2), x3), x4))
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs(wu260, wu270, ty_Double) -> new_esEs3(wu260, wu270)
13.18/5.10	   new_esEs3(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, ty_Float) -> new_esEs7(wu260, wu270)
13.18/5.10	   new_esEs(wu260, wu270, ty_Int) -> new_esEs2(wu260, wu270)
13.18/5.10	   new_esEs(wu260, wu270, ty_@0) -> new_esEs1(wu260, wu270)
13.18/5.10	   new_esEs11(wu17, wu200, ce) -> error([])
13.18/5.10	   new_esEs5(wu17, wu200, db, dc) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, ty_Char) -> new_esEs8(wu260, wu270)
13.18/5.10	   new_esEs1(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(ty_@2, bg), bh)) -> new_esEs9(wu260, wu270, bg, bh)
13.18/5.10	   new_esEs(wu260, wu270, ty_Bool) -> new_esEs0(wu260, wu270)
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs(wu260, wu270, app(ty_[], cb)) -> new_esEs11(wu260, wu270, cb)
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs4(wu17, wu200, dd, de, df) -> error([])
13.18/5.10	   new_esEs7(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(ty_Maybe, ca)) -> new_esEs10(wu260, wu270, ca)
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs(wu260, wu270, ty_Integer) -> new_esEs12(wu260, wu270)
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs12(wu17, wu200) -> error([])
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	   new_esEs10(wu17, wu200, da) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(ty_Ratio, cc)) -> new_esEs13(wu260, wu270, cc)
13.18/5.10	   new_esEs2(wu17, wu200) -> error([])
13.18/5.10	   new_esEs13(wu17, wu200, cd) -> error([])
13.18/5.10	   new_esEs0(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(ty_Either, be), bf)) -> new_esEs5(wu260, wu270, be, bf)
13.18/5.10	   new_esEs(wu260, wu270, ty_Ordering) -> new_esEs6(wu260, wu270)
13.18/5.10	   new_esEs8(wu17, wu200) -> error([])
13.18/5.10	   new_esEs9(wu17, wu200, cf, cg) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(app(ty_@3, bb), bc), bd)) -> new_esEs4(wu260, wu270, bb, bc, bd)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs(x0, x1, ty_Int)
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs(x0, x1, ty_Float)
13.18/5.10	   new_esEs5(x0, x1, x2, x3)
13.18/5.10	   new_esEs(x0, x1, ty_Integer)
13.18/5.10	   new_esEs9(x0, x1, x2, x3)
13.18/5.10	   new_esEs8(x0, x1)
13.18/5.10	   new_esEs3(x0, x1)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs7(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_Ratio, x2))
13.18/5.10	   new_esEs(x0, x1, ty_Bool)
13.18/5.10	   new_esEs(x0, x1, app(ty_Maybe, x2))
13.18/5.10	   new_esEs10(x0, x1, x2)
13.18/5.10	   new_esEs12(x0, x1)
13.18/5.10	   new_esEs1(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
13.18/5.10	   new_esEs2(x0, x1)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs(x0, x1, ty_Char)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
13.18/5.10	   new_esEs(x0, x1, ty_Ordering)
13.18/5.10	   new_esEs(x0, x1, ty_Double)
13.18/5.10	   new_esEs11(x0, x1, x2)
13.18/5.10	   new_esEs13(x0, x1, x2)
13.18/5.10	   new_esEs0(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_[], x2))
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	   new_esEs(x0, x1, ty_@0)
13.18/5.10	   new_esEs4(x0, x1, x2, x3, x4)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(19) DependencyGraphProof (EQUIVALENT)
13.18/5.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 13 less nodes.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(20)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs(wu260, wu270, ty_Double) -> new_esEs3(wu260, wu270)
13.18/5.10	   new_esEs3(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, ty_Float) -> new_esEs7(wu260, wu270)
13.18/5.10	   new_esEs(wu260, wu270, ty_Int) -> new_esEs2(wu260, wu270)
13.18/5.10	   new_esEs(wu260, wu270, ty_@0) -> new_esEs1(wu260, wu270)
13.18/5.10	   new_esEs11(wu17, wu200, ce) -> error([])
13.18/5.10	   new_esEs5(wu17, wu200, db, dc) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, ty_Char) -> new_esEs8(wu260, wu270)
13.18/5.10	   new_esEs1(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(ty_@2, bg), bh)) -> new_esEs9(wu260, wu270, bg, bh)
13.18/5.10	   new_esEs(wu260, wu270, ty_Bool) -> new_esEs0(wu260, wu270)
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs(wu260, wu270, app(ty_[], cb)) -> new_esEs11(wu260, wu270, cb)
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs4(wu17, wu200, dd, de, df) -> error([])
13.18/5.10	   new_esEs7(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(ty_Maybe, ca)) -> new_esEs10(wu260, wu270, ca)
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs(wu260, wu270, ty_Integer) -> new_esEs12(wu260, wu270)
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs12(wu17, wu200) -> error([])
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	   new_esEs10(wu17, wu200, da) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(ty_Ratio, cc)) -> new_esEs13(wu260, wu270, cc)
13.18/5.10	   new_esEs2(wu17, wu200) -> error([])
13.18/5.10	   new_esEs13(wu17, wu200, cd) -> error([])
13.18/5.10	   new_esEs0(wu17, wu200) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(ty_Either, be), bf)) -> new_esEs5(wu260, wu270, be, bf)
13.18/5.10	   new_esEs(wu260, wu270, ty_Ordering) -> new_esEs6(wu260, wu270)
13.18/5.10	   new_esEs8(wu17, wu200) -> error([])
13.18/5.10	   new_esEs9(wu17, wu200, cf, cg) -> error([])
13.18/5.10	   new_esEs(wu260, wu270, app(app(app(ty_@3, bb), bc), bd)) -> new_esEs4(wu260, wu270, bb, bc, bd)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs(x0, x1, ty_Int)
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs(x0, x1, ty_Float)
13.18/5.10	   new_esEs5(x0, x1, x2, x3)
13.18/5.10	   new_esEs(x0, x1, ty_Integer)
13.18/5.10	   new_esEs9(x0, x1, x2, x3)
13.18/5.10	   new_esEs8(x0, x1)
13.18/5.10	   new_esEs3(x0, x1)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs7(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_Ratio, x2))
13.18/5.10	   new_esEs(x0, x1, ty_Bool)
13.18/5.10	   new_esEs(x0, x1, app(ty_Maybe, x2))
13.18/5.10	   new_esEs10(x0, x1, x2)
13.18/5.10	   new_esEs12(x0, x1)
13.18/5.10	   new_esEs1(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
13.18/5.10	   new_esEs2(x0, x1)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs(x0, x1, ty_Char)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
13.18/5.10	   new_esEs(x0, x1, ty_Ordering)
13.18/5.10	   new_esEs(x0, x1, ty_Double)
13.18/5.10	   new_esEs11(x0, x1, x2)
13.18/5.10	   new_esEs13(x0, x1, x2)
13.18/5.10	   new_esEs0(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_[], x2))
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	   new_esEs(x0, x1, ty_@0)
13.18/5.10	   new_esEs4(x0, x1, x2, x3, x4)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(21) UsableRulesProof (EQUIVALENT)
13.18/5.10	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(22)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs(x0, x1, ty_Int)
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs(x0, x1, ty_Float)
13.18/5.10	   new_esEs5(x0, x1, x2, x3)
13.18/5.10	   new_esEs(x0, x1, ty_Integer)
13.18/5.10	   new_esEs9(x0, x1, x2, x3)
13.18/5.10	   new_esEs8(x0, x1)
13.18/5.10	   new_esEs3(x0, x1)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs7(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_Ratio, x2))
13.18/5.10	   new_esEs(x0, x1, ty_Bool)
13.18/5.10	   new_esEs(x0, x1, app(ty_Maybe, x2))
13.18/5.10	   new_esEs10(x0, x1, x2)
13.18/5.10	   new_esEs12(x0, x1)
13.18/5.10	   new_esEs1(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
13.18/5.10	   new_esEs2(x0, x1)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs(x0, x1, ty_Char)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
13.18/5.10	   new_esEs(x0, x1, ty_Ordering)
13.18/5.10	   new_esEs(x0, x1, ty_Double)
13.18/5.10	   new_esEs11(x0, x1, x2)
13.18/5.10	   new_esEs13(x0, x1, x2)
13.18/5.10	   new_esEs0(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_[], x2))
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	   new_esEs(x0, x1, ty_@0)
13.18/5.10	   new_esEs4(x0, x1, x2, x3, x4)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(23) QReductionProof (EQUIVALENT)
13.18/5.10	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
13.18/5.10	
13.18/5.10	   new_esEs(x0, x1, ty_Int)
13.18/5.10	   new_esEs(x0, x1, ty_Float)
13.18/5.10	   new_esEs5(x0, x1, x2, x3)
13.18/5.10	   new_esEs(x0, x1, ty_Integer)
13.18/5.10	   new_esEs9(x0, x1, x2, x3)
13.18/5.10	   new_esEs8(x0, x1)
13.18/5.10	   new_esEs3(x0, x1)
13.18/5.10	   new_esEs7(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_Ratio, x2))
13.18/5.10	   new_esEs(x0, x1, ty_Bool)
13.18/5.10	   new_esEs(x0, x1, app(ty_Maybe, x2))
13.18/5.10	   new_esEs10(x0, x1, x2)
13.18/5.10	   new_esEs12(x0, x1)
13.18/5.10	   new_esEs1(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
13.18/5.10	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
13.18/5.10	   new_esEs2(x0, x1)
13.18/5.10	   new_esEs(x0, x1, ty_Char)
13.18/5.10	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
13.18/5.10	   new_esEs(x0, x1, ty_Ordering)
13.18/5.10	   new_esEs(x0, x1, ty_Double)
13.18/5.10	   new_esEs11(x0, x1, x2)
13.18/5.10	   new_esEs13(x0, x1, x2)
13.18/5.10	   new_esEs0(x0, x1)
13.18/5.10	   new_esEs(x0, x1, app(ty_[], x2))
13.18/5.10	   new_esEs(x0, x1, ty_@0)
13.18/5.10	   new_esEs4(x0, x1, x2, x3, x4)
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(24)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(25) TransformationProof (EQUIVALENT)
13.18/5.10	By narrowing [LPAR04] the rule new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs6(x0, x1), y1, y3, ty_Ordering) at position [0] we obtained the following new rules [LPAR04]:
13.18/5.10	
13.18/5.10	   (new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering),new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(EQ, y1), :(GT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering),new_asAs(True, :(EQ, y1), :(GT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(GT, y1), :(EQ, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering),new_asAs(True, :(GT, y1), :(EQ, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering),new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering),new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(LT, y1), :(GT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering),new_asAs(True, :(LT, y1), :(GT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(GT, y1), :(LT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering),new_asAs(True, :(GT, y1), :(LT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(LT, y1), :(EQ, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering),new_asAs(True, :(LT, y1), :(EQ, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering))
13.18/5.10	   (new_asAs(True, :(EQ, y1), :(LT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering),new_asAs(True, :(EQ, y1), :(LT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering))
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(26)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(EQ, y1), :(GT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(GT, y1), :(EQ, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(LT, y1), :(GT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(GT, y1), :(LT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(LT, y1), :(EQ, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(EQ, y1), :(LT, y3), ty_Ordering) -> new_asAs(False, y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(27) DependencyGraphProof (EQUIVALENT)
13.18/5.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(28)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, EQ) -> True
13.18/5.10	   new_esEs6(EQ, GT) -> False
13.18/5.10	   new_esEs6(GT, EQ) -> False
13.18/5.10	   new_esEs6(GT, GT) -> True
13.18/5.10	   new_esEs6(LT, LT) -> True
13.18/5.10	   new_esEs6(LT, GT) -> False
13.18/5.10	   new_esEs6(GT, LT) -> False
13.18/5.10	   new_esEs6(LT, EQ) -> False
13.18/5.10	   new_esEs6(EQ, LT) -> False
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(29) UsableRulesProof (EQUIVALENT)
13.18/5.10	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(30)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	R is empty.
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(31) QReductionProof (EQUIVALENT)
13.18/5.10	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
13.18/5.10	
13.18/5.10	   new_esEs6(EQ, GT)
13.18/5.10	   new_esEs6(GT, EQ)
13.18/5.10	   new_esEs6(LT, GT)
13.18/5.10	   new_esEs6(GT, LT)
13.18/5.10	   new_esEs6(EQ, EQ)
13.18/5.10	   new_esEs6(LT, EQ)
13.18/5.10	   new_esEs6(EQ, LT)
13.18/5.10	   new_esEs6(GT, GT)
13.18/5.10	   new_esEs6(LT, LT)
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(32)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	   new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	
13.18/5.10	R is empty.
13.18/5.10	Q is empty.
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(33) QDPSizeChangeProof (EQUIVALENT)
13.18/5.10	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. 
13.18/5.10	
13.18/5.10	From the DPs we obtained the following set of size-change graphs:
13.18/5.10	*new_asAs(True, :(EQ, y1), :(EQ, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4
13.18/5.10	
13.18/5.10	
13.18/5.10	*new_asAs(True, :(GT, y1), :(GT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4
13.18/5.10	
13.18/5.10	
13.18/5.10	*new_asAs(True, :(LT, y1), :(LT, y3), ty_Ordering) -> new_asAs(True, y1, y3, ty_Ordering)
13.18/5.10	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(34)
13.18/5.10	YES
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(35)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(new_flip(wu16, wu17, ba), wu180, wu181, wu19, ba)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_flip(wu16, wu17, ba) -> :(wu17, wu16)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(36) TransformationProof (EQUIVALENT)
13.18/5.10	By rewriting [LPAR04] the rule new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(new_flip(wu16, wu17, ba), wu180, wu181, wu19, ba) at position [0] we obtained the following new rules [LPAR04]:
13.18/5.10	
13.18/5.10	   (new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(:(wu17, wu16), wu180, wu181, wu19, ba),new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(:(wu17, wu16), wu180, wu181, wu19, ba))
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(37)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(:(wu17, wu16), wu180, wu181, wu19, ba)
13.18/5.10	
13.18/5.10	The TRS R consists of the following rules:
13.18/5.10	
13.18/5.10	   new_flip(wu16, wu17, ba) -> :(wu17, wu16)
13.18/5.10	
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(38) UsableRulesProof (EQUIVALENT)
13.18/5.10	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(39)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(:(wu17, wu16), wu180, wu181, wu19, ba)
13.18/5.10	
13.18/5.10	R is empty.
13.18/5.10	The set Q consists of the following terms:
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(40) QReductionProof (EQUIVALENT)
13.18/5.10	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
13.18/5.10	
13.18/5.10	   new_flip(x0, x1, x2)
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(41)
13.18/5.10	Obligation:
13.18/5.10	Q DP problem:
13.18/5.10	The TRS P consists of the following rules:
13.18/5.10	
13.18/5.10	   new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(:(wu17, wu16), wu180, wu181, wu19, ba)
13.18/5.10	
13.18/5.10	R is empty.
13.18/5.10	Q is empty.
13.18/5.10	We have to consider all minimal (P,Q,R)-chains.
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(42) QDPSizeChangeProof (EQUIVALENT)
13.18/5.10	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. 
13.18/5.10	
13.18/5.10	From the DPs we obtained the following set of size-change graphs:
13.18/5.10	*new_isPrefixOf0(wu16, wu17, :(wu180, wu181), wu19, ba) -> new_isPrefixOf0(:(wu17, wu16), wu180, wu181, wu19, ba)
13.18/5.10	The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
13.18/5.10	
13.18/5.10	
13.18/5.10	----------------------------------------
13.18/5.10	
13.18/5.10	(43)
13.18/5.10	YES
13.18/5.14	EOF