/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 33 ms] (4) HASKELL (5) Narrow [SOUND, 0 ms] (6) AND (7) QDP (8) TransformationProof [EQUIVALENT, 0 ms] (9) QDP (10) DependencyGraphProof [EQUIVALENT, 0 ms] (11) QDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) QDP (14) QReductionProof [EQUIVALENT, 0 ms] (15) QDP (16) TransformationProof [EQUIVALENT, 0 ms] (17) QDP (18) UsableRulesProof [EQUIVALENT, 0 ms] (19) QDP (20) QReductionProof [EQUIVALENT, 0 ms] (21) QDP (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) QDP (29) QReductionProof [EQUIVALENT, 0 ms] (30) QDP (31) QDPSizeChangeProof [EQUIVALENT, 0 ms] (32) YES (33) QDP (34) TransformationProof [EQUIVALENT, 0 ms] (35) QDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) QDP (38) QReductionProof [EQUIVALENT, 0 ms] (39) QDP (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] (41) YES ---------------------------------------- (0) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; isPrefixOf :: Eq a => [a] -> [a] -> Bool; isPrefixOf [] _ = True; isPrefixOf _ [] = False; isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys; isSuffixOf :: Eq a => [a] -> [a] -> Bool; isSuffixOf x y = reverse x `isPrefixOf` reverse y; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; isPrefixOf :: Eq a => [a] -> [a] -> Bool; isPrefixOf [] vy = True; isPrefixOf vz [] = False; isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys; isSuffixOf :: Eq a => [a] -> [a] -> Bool; isSuffixOf x y = reverse x `isPrefixOf` reverse y; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (3) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (4) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; isPrefixOf :: Eq a => [a] -> [a] -> Bool; isPrefixOf [] vy = True; isPrefixOf vz [] = False; isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys; isSuffixOf :: Eq a => [a] -> [a] -> Bool; isSuffixOf x y = reverse x `isPrefixOf` reverse y; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (5) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="List.isSuffixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="List.isSuffixOf wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", 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Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 426[label="",style="solid", color="blue", weight=9]; 426 -> 354[label="",style="solid", color="blue", weight=3]; 427[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 427[label="",style="solid", color="blue", weight=9]; 427 -> 355[label="",style="solid", color="blue", weight=3]; 428[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 428[label="",style="solid", color="blue", weight=9]; 428 -> 356[label="",style="solid", color="blue", weight=3]; 429[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 429[label="",style="solid", color="blue", weight=9]; 429 -> 357[label="",style="solid", color="blue", weight=3]; 430[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 430[label="",style="solid", color="blue", weight=9]; 430 -> 358[label="",style="solid", color="blue", weight=3]; 431[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 431[label="",style="solid", color="blue", weight=9]; 431 -> 359[label="",style="solid", color="blue", weight=3]; 432[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 432[label="",style="solid", color="blue", weight=9]; 432 -> 360[label="",style="solid", color="blue", weight=3]; 433[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 433[label="",style="solid", color="blue", weight=9]; 433 -> 361[label="",style="solid", color="blue", weight=3]; 434[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];348 -> 434[label="",style="solid", color="blue", weight=9]; 434 -> 362[label="",style="solid", color="blue", weight=3]; 349 -> 305[label="",style="dashed", color="red", weight=0]; 349[label="wu250 == wu260",fontsize=16,color="magenta"];349 -> 363[label="",style="dashed", color="magenta", weight=3]; 349 -> 364[label="",style="dashed", color="magenta", weight=3]; 350 -> 306[label="",style="dashed", color="red", weight=0]; 350[label="wu250 == wu260",fontsize=16,color="magenta"];350 -> 365[label="",style="dashed", color="magenta", weight=3]; 350 -> 366[label="",style="dashed", color="magenta", weight=3]; 351 -> 307[label="",style="dashed", color="red", weight=0]; 351[label="wu250 == wu260",fontsize=16,color="magenta"];351 -> 367[label="",style="dashed", color="magenta", weight=3]; 351 -> 368[label="",style="dashed", color="magenta", weight=3]; 352 -> 308[label="",style="dashed", color="red", weight=0]; 352[label="wu250 == wu260",fontsize=16,color="magenta"];352 -> 369[label="",style="dashed", color="magenta", weight=3]; 352 -> 370[label="",style="dashed", color="magenta", weight=3]; 353 -> 309[label="",style="dashed", color="red", weight=0]; 353[label="wu250 == wu260",fontsize=16,color="magenta"];353 -> 371[label="",style="dashed", color="magenta", weight=3]; 353 -> 372[label="",style="dashed", color="magenta", weight=3]; 354 -> 310[label="",style="dashed", color="red", weight=0]; 354[label="wu250 == wu260",fontsize=16,color="magenta"];354 -> 373[label="",style="dashed", color="magenta", weight=3]; 354 -> 374[label="",style="dashed", color="magenta", weight=3]; 355 -> 311[label="",style="dashed", color="red", weight=0]; 355[label="wu250 == wu260",fontsize=16,color="magenta"];355 -> 375[label="",style="dashed", color="magenta", weight=3]; 355 -> 376[label="",style="dashed", color="magenta", weight=3]; 356 -> 312[label="",style="dashed", color="red", weight=0]; 356[label="wu250 == wu260",fontsize=16,color="magenta"];356 -> 377[label="",style="dashed", color="magenta", weight=3]; 356 -> 378[label="",style="dashed", color="magenta", weight=3]; 357 -> 313[label="",style="dashed", color="red", weight=0]; 357[label="wu250 == wu260",fontsize=16,color="magenta"];357 -> 379[label="",style="dashed", color="magenta", weight=3]; 357 -> 380[label="",style="dashed", color="magenta", weight=3]; 358 -> 314[label="",style="dashed", color="red", weight=0]; 358[label="wu250 == wu260",fontsize=16,color="magenta"];358 -> 381[label="",style="dashed", color="magenta", weight=3]; 358 -> 382[label="",style="dashed", color="magenta", weight=3]; 359 -> 315[label="",style="dashed", color="red", weight=0]; 359[label="wu250 == wu260",fontsize=16,color="magenta"];359 -> 383[label="",style="dashed", color="magenta", weight=3]; 359 -> 384[label="",style="dashed", color="magenta", weight=3]; 360 -> 316[label="",style="dashed", color="red", weight=0]; 360[label="wu250 == wu260",fontsize=16,color="magenta"];360 -> 385[label="",style="dashed", color="magenta", weight=3]; 360 -> 386[label="",style="dashed", color="magenta", weight=3]; 361 -> 317[label="",style="dashed", color="red", weight=0]; 361[label="wu250 == wu260",fontsize=16,color="magenta"];361 -> 387[label="",style="dashed", color="magenta", weight=3]; 361 -> 388[label="",style="dashed", color="magenta", weight=3]; 362 -> 318[label="",style="dashed", color="red", weight=0]; 362[label="wu250 == wu260",fontsize=16,color="magenta"];362 -> 389[label="",style="dashed", color="magenta", weight=3]; 362 -> 390[label="",style="dashed", color="magenta", weight=3]; 363[label="wu250",fontsize=16,color="green",shape="box"];364[label="wu260",fontsize=16,color="green",shape="box"];365[label="wu250",fontsize=16,color="green",shape="box"];366[label="wu260",fontsize=16,color="green",shape="box"];367[label="wu250",fontsize=16,color="green",shape="box"];368[label="wu260",fontsize=16,color="green",shape="box"];369[label="wu250",fontsize=16,color="green",shape="box"];370[label="wu260",fontsize=16,color="green",shape="box"];371[label="wu250",fontsize=16,color="green",shape="box"];372[label="wu260",fontsize=16,color="green",shape="box"];373[label="wu250",fontsize=16,color="green",shape="box"];374[label="wu260",fontsize=16,color="green",shape="box"];375[label="wu250",fontsize=16,color="green",shape="box"];376[label="wu260",fontsize=16,color="green",shape="box"];377[label="wu250",fontsize=16,color="green",shape="box"];378[label="wu260",fontsize=16,color="green",shape="box"];379[label="wu250",fontsize=16,color="green",shape="box"];380[label="wu260",fontsize=16,color="green",shape="box"];381[label="wu250",fontsize=16,color="green",shape="box"];382[label="wu260",fontsize=16,color="green",shape="box"];383[label="wu250",fontsize=16,color="green",shape="box"];384[label="wu260",fontsize=16,color="green",shape="box"];385[label="wu250",fontsize=16,color="green",shape="box"];386[label="wu260",fontsize=16,color="green",shape="box"];387[label="wu250",fontsize=16,color="green",shape="box"];388[label="wu260",fontsize=16,color="green",shape="box"];389[label="wu250",fontsize=16,color="green",shape="box"];390[label="wu260",fontsize=16,color="green",shape="box"];} ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(wu250, wu251), :(wu260, wu261), ba) -> new_asAs(new_esEs(wu250, wu260, ba), wu251, wu261, ba) The TRS R consists of the following rules: new_esEs3(wu16, wu190) -> error([]) new_esEs5(wu16, wu190) -> error([]) new_esEs(wu250, wu260, ty_Double) -> new_esEs1(wu250, wu260) new_esEs1(wu16, wu190) -> error([]) new_esEs(wu250, wu260, app(app(ty_@2, bh), ca)) -> new_esEs11(wu250, wu260, bh, ca) new_esEs(wu250, wu260, app(ty_[], bb)) -> new_esEs2(wu250, wu260, bb) new_esEs7(wu16, wu190) -> error([]) new_esEs6(wu16, wu190, cf, cg, da) -> error([]) new_esEs10(wu16, wu190, db, dc) -> error([]) new_esEs(wu250, wu260, ty_Ordering) -> new_esEs3(wu250, wu260) new_esEs(wu250, wu260, app(app(app(ty_@3, bc), bd), be)) -> new_esEs6(wu250, wu260, bc, bd, be) new_esEs(wu250, wu260, ty_Char) -> new_esEs7(wu250, wu260) new_esEs(wu250, wu260, app(ty_Ratio, cb)) -> new_esEs12(wu250, wu260, cb) new_esEs(wu250, wu260, ty_Int) -> new_esEs8(wu250, wu260) new_esEs4(wu16, wu190) -> error([]) new_esEs2(wu16, wu190, ce) -> error([]) new_esEs(wu250, wu260, ty_@0) -> new_esEs9(wu250, wu260) new_esEs12(wu16, wu190, cd) -> error([]) new_esEs13(wu16, wu190, dd) -> error([]) new_esEs0(wu16, wu190) -> error([]) new_esEs(wu250, wu260, app(app(ty_Either, bf), bg)) -> new_esEs10(wu250, wu260, bf, bg) new_esEs11(wu16, wu190, de, df) -> error([]) new_esEs(wu250, wu260, ty_Float) -> new_esEs0(wu250, wu260) new_esEs(wu250, wu260, ty_Bool) -> new_esEs4(wu250, wu260) new_esEs(wu250, wu260, ty_Integer) -> new_esEs5(wu250, wu260) new_esEs(wu250, wu260, app(ty_Maybe, cc)) -> new_esEs13(wu250, wu260, cc) new_esEs9(@0, @0) -> True new_esEs8(wu16, wu190) -> error([]) The set Q consists of the following terms: new_esEs12(x0, x1, x2) new_esEs(x0, x1, app(app(ty_Either, x2), x3)) new_esEs9(@0, @0) new_esEs4(x0, x1) new_esEs(x0, x1, app(app(ty_@2, x2), x3)) new_esEs11(x0, x1, x2, x3) new_esEs(x0, x1, app(ty_[], x2)) new_esEs1(x0, x1) new_esEs(x0, x1, ty_Integer) new_esEs0(x0, x1) new_esEs(x0, x1, ty_Bool) new_esEs5(x0, x1) new_esEs10(x0, x1, x2, x3) new_esEs6(x0, x1, x2, x3, x4) new_esEs2(x0, x1, x2) new_esEs(x0, x1, ty_Float) new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_esEs(x0, x1, ty_Double) new_esEs8(x0, x1) new_esEs(x0, x1, ty_Int) new_esEs(x0, x1, app(ty_Ratio, x2)) new_esEs(x0, x1, ty_Char) new_esEs(x0, x1, app(ty_Maybe, x2)) new_esEs(x0, x1, ty_Ordering) new_esEs3(x0, x1) new_esEs(x0, x1, ty_@0) new_esEs13(x0, x1, x2) new_esEs7(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule new_asAs(True, :(wu250, wu251), :(wu260, wu261), ba) -> new_asAs(new_esEs(wu250, wu260, ba), wu251, wu261, ba) at position [0] we obtained the following new rules [LPAR04]: (new_asAs(True, :(x0, y1), :(x1, y3), ty_Double) -> new_asAs(new_esEs1(x0, x1), y1, y3, ty_Double),new_asAs(True, :(x0, y1), :(x1, y3), ty_Double) -> new_asAs(new_esEs1(x0, x1), y1, y3, ty_Double)) (new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_asAs(new_esEs11(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_esEs11(x0, x1, x2, x3), y1, y3, app(app(ty_@2, x2), x3))) (new_asAs(True, :(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_asAs(new_esEs2(x0, x1, x2), y1, y3, app(ty_[], x2)),new_asAs(True, :(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_asAs(new_esEs2(x0, x1, x2), y1, y3, app(ty_[], x2))) (new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs3(x0, x1), y1, y3, ty_Ordering),new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs3(x0, x1), y1, y3, ty_Ordering)) (new_asAs(True, :(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_asAs(new_esEs6(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_esEs6(x0, x1, x2, x3, x4), y1, y3, app(app(app(ty_@3, x2), x3), x4))) (new_asAs(True, :(x0, y1), :(x1, y3), ty_Char) -> new_asAs(new_esEs7(x0, x1), y1, y3, ty_Char),new_asAs(True, :(x0, y1), :(x1, y3), ty_Char) -> new_asAs(new_esEs7(x0, x1), y1, y3, ty_Char)) (new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_asAs(new_esEs12(x0, x1, x2), y1, y3, app(ty_Ratio, x2)),new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_asAs(new_esEs12(x0, x1, x2), y1, y3, app(ty_Ratio, x2))) (new_asAs(True, :(x0, y1), :(x1, y3), ty_Int) -> new_asAs(new_esEs8(x0, x1), y1, y3, ty_Int),new_asAs(True, :(x0, y1), :(x1, y3), ty_Int) -> new_asAs(new_esEs8(x0, x1), y1, y3, ty_Int)) (new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0),new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0)) (new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_asAs(new_esEs10(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_esEs10(x0, x1, x2, x3), y1, y3, app(app(ty_Either, x2), x3))) (new_asAs(True, :(x0, y1), :(x1, y3), ty_Float) -> new_asAs(new_esEs0(x0, x1), y1, y3, ty_Float),new_asAs(True, :(x0, y1), :(x1, y3), ty_Float) -> new_asAs(new_esEs0(x0, x1), y1, y3, ty_Float)) (new_asAs(True, :(x0, y1), :(x1, y3), ty_Bool) -> new_asAs(new_esEs4(x0, x1), y1, y3, ty_Bool),new_asAs(True, :(x0, y1), :(x1, y3), ty_Bool) -> new_asAs(new_esEs4(x0, x1), y1, y3, ty_Bool)) (new_asAs(True, :(x0, y1), :(x1, y3), ty_Integer) -> new_asAs(new_esEs5(x0, x1), y1, y3, ty_Integer),new_asAs(True, :(x0, y1), :(x1, y3), ty_Integer) -> new_asAs(new_esEs5(x0, x1), y1, y3, ty_Integer)) (new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_asAs(new_esEs13(x0, x1, x2), y1, y3, app(ty_Maybe, x2)),new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_asAs(new_esEs13(x0, x1, x2), y1, y3, app(ty_Maybe, x2))) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(x0, y1), :(x1, y3), ty_Double) -> new_asAs(new_esEs1(x0, x1), y1, y3, ty_Double) new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_asAs(new_esEs11(x0, x1, x2, x3), y1, y3, app(app(ty_@2, x2), x3)) new_asAs(True, :(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_asAs(new_esEs2(x0, x1, x2), y1, y3, app(ty_[], x2)) new_asAs(True, :(x0, y1), :(x1, y3), ty_Ordering) -> new_asAs(new_esEs3(x0, x1), y1, y3, ty_Ordering) new_asAs(True, :(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_asAs(new_esEs6(x0, x1, x2, x3, x4), y1, y3, app(app(app(ty_@3, x2), x3), x4)) new_asAs(True, :(x0, y1), :(x1, y3), ty_Char) -> new_asAs(new_esEs7(x0, x1), y1, y3, ty_Char) new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_asAs(new_esEs12(x0, x1, x2), y1, y3, app(ty_Ratio, x2)) new_asAs(True, :(x0, y1), :(x1, y3), ty_Int) -> new_asAs(new_esEs8(x0, x1), y1, y3, ty_Int) new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0) new_asAs(True, :(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_asAs(new_esEs10(x0, x1, x2, x3), y1, y3, app(app(ty_Either, x2), x3)) new_asAs(True, :(x0, y1), :(x1, y3), ty_Float) -> new_asAs(new_esEs0(x0, x1), y1, y3, ty_Float) new_asAs(True, :(x0, y1), :(x1, y3), ty_Bool) -> new_asAs(new_esEs4(x0, x1), y1, y3, ty_Bool) new_asAs(True, :(x0, y1), :(x1, y3), ty_Integer) -> new_asAs(new_esEs5(x0, x1), y1, y3, ty_Integer) new_asAs(True, :(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_asAs(new_esEs13(x0, x1, x2), y1, y3, app(ty_Maybe, x2)) The TRS R consists of the following rules: new_esEs3(wu16, wu190) -> error([]) new_esEs5(wu16, wu190) -> error([]) new_esEs(wu250, wu260, ty_Double) -> new_esEs1(wu250, wu260) new_esEs1(wu16, wu190) -> error([]) new_esEs(wu250, wu260, app(app(ty_@2, bh), ca)) -> new_esEs11(wu250, wu260, bh, ca) new_esEs(wu250, wu260, app(ty_[], bb)) -> new_esEs2(wu250, wu260, bb) new_esEs7(wu16, wu190) -> error([]) new_esEs6(wu16, wu190, cf, cg, da) -> error([]) new_esEs10(wu16, wu190, db, dc) -> error([]) new_esEs(wu250, wu260, ty_Ordering) -> new_esEs3(wu250, wu260) new_esEs(wu250, wu260, app(app(app(ty_@3, bc), bd), be)) -> new_esEs6(wu250, wu260, bc, bd, be) new_esEs(wu250, wu260, ty_Char) -> new_esEs7(wu250, wu260) new_esEs(wu250, wu260, app(ty_Ratio, cb)) -> new_esEs12(wu250, wu260, cb) new_esEs(wu250, wu260, ty_Int) -> new_esEs8(wu250, wu260) new_esEs4(wu16, wu190) -> error([]) new_esEs2(wu16, wu190, ce) -> error([]) new_esEs(wu250, wu260, ty_@0) -> new_esEs9(wu250, wu260) new_esEs12(wu16, wu190, cd) -> error([]) new_esEs13(wu16, wu190, dd) -> error([]) new_esEs0(wu16, wu190) -> error([]) new_esEs(wu250, wu260, app(app(ty_Either, bf), bg)) -> new_esEs10(wu250, wu260, bf, bg) new_esEs11(wu16, wu190, de, df) -> error([]) new_esEs(wu250, wu260, ty_Float) -> new_esEs0(wu250, wu260) new_esEs(wu250, wu260, ty_Bool) -> new_esEs4(wu250, wu260) new_esEs(wu250, wu260, ty_Integer) -> new_esEs5(wu250, wu260) new_esEs(wu250, wu260, app(ty_Maybe, cc)) -> new_esEs13(wu250, wu260, cc) new_esEs9(@0, @0) -> True new_esEs8(wu16, wu190) -> error([]) The set Q consists of the following terms: new_esEs12(x0, x1, x2) new_esEs(x0, x1, app(app(ty_Either, x2), x3)) new_esEs9(@0, @0) new_esEs4(x0, x1) new_esEs(x0, x1, app(app(ty_@2, x2), x3)) new_esEs11(x0, x1, x2, x3) new_esEs(x0, x1, app(ty_[], x2)) new_esEs1(x0, x1) new_esEs(x0, x1, ty_Integer) new_esEs0(x0, x1) new_esEs(x0, x1, ty_Bool) new_esEs5(x0, x1) new_esEs10(x0, x1, x2, x3) new_esEs6(x0, x1, x2, x3, x4) new_esEs2(x0, x1, x2) new_esEs(x0, x1, ty_Float) new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_esEs(x0, x1, ty_Double) new_esEs8(x0, x1) new_esEs(x0, x1, ty_Int) new_esEs(x0, x1, app(ty_Ratio, x2)) new_esEs(x0, x1, ty_Char) new_esEs(x0, x1, app(ty_Maybe, x2)) new_esEs(x0, x1, ty_Ordering) new_esEs3(x0, x1) new_esEs(x0, x1, ty_@0) new_esEs13(x0, x1, x2) new_esEs7(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 13 less nodes. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0) The TRS R consists of the following rules: new_esEs3(wu16, wu190) -> error([]) new_esEs5(wu16, wu190) -> error([]) new_esEs(wu250, wu260, ty_Double) -> new_esEs1(wu250, wu260) new_esEs1(wu16, wu190) -> error([]) new_esEs(wu250, wu260, app(app(ty_@2, bh), ca)) -> new_esEs11(wu250, wu260, bh, ca) new_esEs(wu250, wu260, app(ty_[], bb)) -> new_esEs2(wu250, wu260, bb) new_esEs7(wu16, wu190) -> error([]) new_esEs6(wu16, wu190, cf, cg, da) -> error([]) new_esEs10(wu16, wu190, db, dc) -> error([]) new_esEs(wu250, wu260, ty_Ordering) -> new_esEs3(wu250, wu260) new_esEs(wu250, wu260, app(app(app(ty_@3, bc), bd), be)) -> new_esEs6(wu250, wu260, bc, bd, be) new_esEs(wu250, wu260, ty_Char) -> new_esEs7(wu250, wu260) new_esEs(wu250, wu260, app(ty_Ratio, cb)) -> new_esEs12(wu250, wu260, cb) new_esEs(wu250, wu260, ty_Int) -> new_esEs8(wu250, wu260) new_esEs4(wu16, wu190) -> error([]) new_esEs2(wu16, wu190, ce) -> error([]) new_esEs(wu250, wu260, ty_@0) -> new_esEs9(wu250, wu260) new_esEs12(wu16, wu190, cd) -> error([]) new_esEs13(wu16, wu190, dd) -> error([]) new_esEs0(wu16, wu190) -> error([]) new_esEs(wu250, wu260, app(app(ty_Either, bf), bg)) -> new_esEs10(wu250, wu260, bf, bg) new_esEs11(wu16, wu190, de, df) -> error([]) new_esEs(wu250, wu260, ty_Float) -> new_esEs0(wu250, wu260) new_esEs(wu250, wu260, ty_Bool) -> new_esEs4(wu250, wu260) new_esEs(wu250, wu260, ty_Integer) -> new_esEs5(wu250, wu260) new_esEs(wu250, wu260, app(ty_Maybe, cc)) -> new_esEs13(wu250, wu260, cc) new_esEs9(@0, @0) -> True new_esEs8(wu16, wu190) -> error([]) The set Q consists of the following terms: new_esEs12(x0, x1, x2) new_esEs(x0, x1, app(app(ty_Either, x2), x3)) new_esEs9(@0, @0) new_esEs4(x0, x1) new_esEs(x0, x1, app(app(ty_@2, x2), x3)) new_esEs11(x0, x1, x2, x3) new_esEs(x0, x1, app(ty_[], x2)) new_esEs1(x0, x1) new_esEs(x0, x1, ty_Integer) new_esEs0(x0, x1) new_esEs(x0, x1, ty_Bool) new_esEs5(x0, x1) new_esEs10(x0, x1, x2, x3) new_esEs6(x0, x1, x2, x3, x4) new_esEs2(x0, x1, x2) new_esEs(x0, x1, ty_Float) new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_esEs(x0, x1, ty_Double) new_esEs8(x0, x1) new_esEs(x0, x1, ty_Int) new_esEs(x0, x1, app(ty_Ratio, x2)) new_esEs(x0, x1, ty_Char) new_esEs(x0, x1, app(ty_Maybe, x2)) new_esEs(x0, x1, ty_Ordering) new_esEs3(x0, x1) new_esEs(x0, x1, ty_@0) new_esEs13(x0, x1, x2) new_esEs7(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) UsableRulesProof (EQUIVALENT) 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) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0) The TRS R consists of the following rules: new_esEs9(@0, @0) -> True The set Q consists of the following terms: new_esEs12(x0, x1, x2) new_esEs(x0, x1, app(app(ty_Either, x2), x3)) new_esEs9(@0, @0) new_esEs4(x0, x1) new_esEs(x0, x1, app(app(ty_@2, x2), x3)) new_esEs11(x0, x1, x2, x3) new_esEs(x0, x1, app(ty_[], x2)) new_esEs1(x0, x1) new_esEs(x0, x1, ty_Integer) new_esEs0(x0, x1) new_esEs(x0, x1, ty_Bool) new_esEs5(x0, x1) new_esEs10(x0, x1, x2, x3) new_esEs6(x0, x1, x2, x3, x4) new_esEs2(x0, x1, x2) new_esEs(x0, x1, ty_Float) new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_esEs(x0, x1, ty_Double) new_esEs8(x0, x1) new_esEs(x0, x1, ty_Int) new_esEs(x0, x1, app(ty_Ratio, x2)) new_esEs(x0, x1, ty_Char) new_esEs(x0, x1, app(ty_Maybe, x2)) new_esEs(x0, x1, ty_Ordering) new_esEs3(x0, x1) new_esEs(x0, x1, ty_@0) new_esEs13(x0, x1, x2) new_esEs7(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_esEs12(x0, x1, x2) new_esEs(x0, x1, app(app(ty_Either, x2), x3)) new_esEs4(x0, x1) new_esEs(x0, x1, app(app(ty_@2, x2), x3)) new_esEs11(x0, x1, x2, x3) new_esEs(x0, x1, app(ty_[], x2)) new_esEs1(x0, x1) new_esEs(x0, x1, ty_Integer) new_esEs0(x0, x1) new_esEs(x0, x1, ty_Bool) new_esEs5(x0, x1) new_esEs10(x0, x1, x2, x3) new_esEs6(x0, x1, x2, x3, x4) new_esEs2(x0, x1, x2) new_esEs(x0, x1, ty_Float) new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_esEs(x0, x1, ty_Double) new_esEs8(x0, x1) new_esEs(x0, x1, ty_Int) new_esEs(x0, x1, app(ty_Ratio, x2)) new_esEs(x0, x1, ty_Char) new_esEs(x0, x1, app(ty_Maybe, x2)) new_esEs(x0, x1, ty_Ordering) new_esEs3(x0, x1) new_esEs(x0, x1, ty_@0) new_esEs13(x0, x1, x2) new_esEs7(x0, x1) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0) The TRS R consists of the following rules: new_esEs9(@0, @0) -> True The set Q consists of the following terms: new_esEs9(@0, @0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule new_asAs(True, :(x0, y1), :(x1, y3), ty_@0) -> new_asAs(new_esEs9(x0, x1), y1, y3, ty_@0) at position [0] we obtained the following new rules [LPAR04]: (new_asAs(True, :(@0, y1), :(@0, y3), ty_@0) -> new_asAs(True, y1, y3, ty_@0),new_asAs(True, :(@0, y1), :(@0, y3), ty_@0) -> new_asAs(True, y1, y3, ty_@0)) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(@0, y1), :(@0, y3), ty_@0) -> new_asAs(True, y1, y3, ty_@0) The TRS R consists of the following rules: new_esEs9(@0, @0) -> True The set Q consists of the following terms: new_esEs9(@0, @0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(@0, y1), :(@0, y3), ty_@0) -> new_asAs(True, y1, y3, ty_@0) R is empty. The set Q consists of the following terms: new_esEs9(@0, @0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_esEs9(@0, @0) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: new_asAs(True, :(@0, y1), :(@0, y3), ty_@0) -> new_asAs(True, y1, y3, ty_@0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_asAs(True, :(@0, y1), :(@0, y3), ty_@0) -> new_asAs(True, y1, y3, ty_@0) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4 ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, new_flip(wu19, wu1810, ba), wu1811, ba) The TRS R consists of the following rules: new_flip(wu15, wu16, ba) -> :(wu16, wu15) The set Q consists of the following terms: new_flip(x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, new_flip(wu19, wu1810, ba), wu1811, ba) at position [2] we obtained the following new rules [LPAR04]: (new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, :(wu1810, wu19), wu1811, ba),new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, :(wu1810, wu19), wu1811, ba)) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, :(wu1810, wu19), wu1811, ba) The TRS R consists of the following rules: new_flip(wu15, wu16, ba) -> :(wu16, wu15) The set Q consists of the following terms: new_flip(x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, :(wu1810, wu19), wu1811, ba) R is empty. The set Q consists of the following terms: new_flip(x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_flip(x0, x1, x2) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, :(wu1810, wu19), wu1811, ba) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_isPrefixOf(wu16, wu15, wu19, :(wu1810, wu1811), ba) -> new_isPrefixOf(wu16, wu15, :(wu1810, wu19), wu1811, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 4 > 4, 5 >= 5 ---------------------------------------- (32) YES ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(new_flip(wu15, wu16, ba), wu170, wu171, wu18, ba) The TRS R consists of the following rules: new_flip(wu15, wu16, ba) -> :(wu16, wu15) The set Q consists of the following terms: new_flip(x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(new_flip(wu15, wu16, ba), wu170, wu171, wu18, ba) at position [0] we obtained the following new rules [LPAR04]: (new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(:(wu16, wu15), wu170, wu171, wu18, ba),new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(:(wu16, wu15), wu170, wu171, wu18, ba)) ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(:(wu16, wu15), wu170, wu171, wu18, ba) The TRS R consists of the following rules: new_flip(wu15, wu16, ba) -> :(wu16, wu15) The set Q consists of the following terms: new_flip(x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(:(wu16, wu15), wu170, wu171, wu18, ba) R is empty. The set Q consists of the following terms: new_flip(x0, x1, x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_flip(x0, x1, x2) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(:(wu16, wu15), wu170, wu171, wu18, ba) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (40) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_isPrefixOf0(wu15, wu16, :(wu170, wu171), wu18, ba) -> new_isPrefixOf0(:(wu16, wu15), wu170, wu171, wu18, ba) The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4, 5 >= 5 ---------------------------------------- (41) YES