/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 0 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) AND (9) QDP (10) TransformationProof [EQUIVALENT, 0 ms] (11) QDP (12) TransformationProof [EQUIVALENT, 0 ms] (13) QDP (14) TransformationProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) TransformationProof [EQUIVALENT, 0 ms] (20) QDP (21) TransformationProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 1 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) TransformationProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) QDP (36) TransformationProof [EQUIVALENT, 0 ms] (37) QDP (38) TransformationProof [EQUIVALENT, 0 ms] (39) QDP (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] (41) YES (42) QDP (43) DependencyGraphProof [EQUIVALENT, 0 ms] (44) AND (45) QDP (46) QDPSizeChangeProof [EQUIVALENT, 0 ms] (47) YES (48) QDP (49) QDPSizeChangeProof [EQUIVALENT, 0 ms] (50) YES ---------------------------------------- (0) Obligation: mainModule Main module FiniteMap where { import qualified Main; import qualified Maybe; import qualified Prelude; data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; instance (Eq a, Eq b) => Eq FiniteMap b a where { } foldFM_LE :: Ord b => (b -> c -> a -> a) -> a -> b -> FiniteMap b c -> a; foldFM_LE k z fr EmptyFM = z; foldFM_LE k z fr (Branch key elt _ fm_l fm_r) | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r | otherwise = foldFM_LE k z fr fm_l; keysFM_LE :: Ord a => FiniteMap a b -> a -> [a]; keysFM_LE fm fr = foldFM_LE (\key elt rest ->key : rest) [] fr fm; } module Maybe where { import qualified FiniteMap; import qualified Main; import qualified Prelude; } module Main where { import qualified FiniteMap; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\keyeltrest->key : rest" is transformed to "keysFM_LE0 key elt rest = key : rest; " ---------------------------------------- (2) Obligation: mainModule Main module FiniteMap where { import qualified Main; import qualified Maybe; import qualified Prelude; data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; instance (Eq a, Eq b) => Eq FiniteMap a b where { } foldFM_LE :: Ord b => (b -> c -> a -> a) -> a -> b -> FiniteMap b c -> a; foldFM_LE k z fr EmptyFM = z; foldFM_LE k z fr (Branch key elt _ fm_l fm_r) | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r | otherwise = foldFM_LE k z fr fm_l; keysFM_LE :: Ord a => FiniteMap a b -> a -> [a]; keysFM_LE fm fr = foldFM_LE keysFM_LE0 [] fr fm; keysFM_LE0 key elt rest = key : rest; } module Maybe where { import qualified FiniteMap; import qualified Main; import qualified Prelude; } module Main where { import qualified FiniteMap; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (4) Obligation: mainModule Main module FiniteMap where { import qualified Main; import qualified Maybe; import qualified Prelude; data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; instance (Eq a, Eq b) => Eq FiniteMap a b where { } foldFM_LE :: Ord b => (b -> a -> c -> c) -> c -> b -> FiniteMap b a -> c; foldFM_LE k z fr EmptyFM = z; foldFM_LE k z fr (Branch key elt vy fm_l fm_r) | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r | otherwise = foldFM_LE k z fr fm_l; keysFM_LE :: Ord b => FiniteMap b a -> b -> [b]; keysFM_LE fm fr = foldFM_LE keysFM_LE0 [] fr fm; keysFM_LE0 key elt rest = key : rest; } module Maybe where { import qualified FiniteMap; import qualified Main; import qualified Prelude; } module Main where { import qualified FiniteMap; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (5) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " The following Function with conditions "foldFM_LE k z fr EmptyFM = z; foldFM_LE k z fr (Branch key elt vy fm_l fm_r)|key <= frfoldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r|otherwisefoldFM_LE k z fr fm_l; " is transformed to "foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM; foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r); " "foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r; foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise; " "foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l; " "foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr); " "foldFM_LE3 k z fr EmptyFM = z; foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy; " ---------------------------------------- (6) Obligation: mainModule Main module FiniteMap where { import qualified Main; import qualified Maybe; import qualified Prelude; data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; instance (Eq a, Eq b) => Eq FiniteMap b a where { } foldFM_LE :: Ord c => (c -> a -> b -> b) -> b -> c -> FiniteMap c a -> b; foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM; foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r); foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l; foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r; foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise; foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr); foldFM_LE3 k z fr EmptyFM = z; foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy; keysFM_LE :: Ord b => FiniteMap b a -> b -> [b]; keysFM_LE fm fr = foldFM_LE keysFM_LE0 [] fr fm; keysFM_LE0 key elt rest = key : rest; } module Maybe where { import qualified FiniteMap; import qualified Main; import qualified Prelude; } module Main where { import qualified FiniteMap; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="FiniteMap.keysFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="FiniteMap.keysFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="FiniteMap.keysFM_LE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];2593[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 2593[label="",style="solid", color="burlywood", weight=9]; 2593 -> 6[label="",style="solid", color="burlywood", weight=3]; 2594[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 2594[label="",style="solid", color="burlywood", weight=9]; 2594 -> 7[label="",style="solid", color="burlywood", weight=3]; 6[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 8[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 9[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 10[label="[]",fontsize=16,color="green",shape="box"];11[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 <= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 12[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= GT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 13[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == GT))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 14[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (primCmpInt wz30 wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2595[label="wz30/Pos wz300",fontsize=10,color="white",style="solid",shape="box"];14 -> 2595[label="",style="solid", color="burlywood", weight=9]; 2595 -> 15[label="",style="solid", color="burlywood", weight=3]; 2596[label="wz30/Neg wz300",fontsize=10,color="white",style="solid",shape="box"];14 -> 2596[label="",style="solid", color="burlywood", weight=9]; 2596 -> 16[label="",style="solid", color="burlywood", weight=3]; 15[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 (Pos wz300) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos wz300) wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2597[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];15 -> 2597[label="",style="solid", color="burlywood", weight=9]; 2597 -> 17[label="",style="solid", color="burlywood", weight=3]; 2598[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 2598[label="",style="solid", color="burlywood", weight=9]; 2598 -> 18[label="",style="solid", color="burlywood", weight=3]; 16[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 (Neg wz300) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg wz300) wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2599[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];16 -> 2599[label="",style="solid", color="burlywood", weight=9]; 2599 -> 19[label="",style="solid", color="burlywood", weight=3]; 2600[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 2600[label="",style="solid", color="burlywood", weight=9]; 2600 -> 20[label="",style="solid", color="burlywood", weight=3]; 17[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 (Pos (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos (Succ wz3000)) wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2601[label="wz4/Pos wz40",fontsize=10,color="white",style="solid",shape="box"];17 -> 2601[label="",style="solid", color="burlywood", weight=9]; 2601 -> 21[label="",style="solid", color="burlywood", weight=3]; 2602[label="wz4/Neg wz40",fontsize=10,color="white",style="solid",shape="box"];17 -> 2602[label="",style="solid", color="burlywood", weight=9]; 2602 -> 22[label="",style="solid", color="burlywood", weight=3]; 18[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 (Pos Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos Zero) wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2603[label="wz4/Pos wz40",fontsize=10,color="white",style="solid",shape="box"];18 -> 2603[label="",style="solid", color="burlywood", weight=9]; 2603 -> 23[label="",style="solid", color="burlywood", weight=3]; 2604[label="wz4/Neg wz40",fontsize=10,color="white",style="solid",shape="box"];18 -> 2604[label="",style="solid", color="burlywood", weight=9]; 2604 -> 24[label="",style="solid", color="burlywood", weight=3]; 19[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 (Neg (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg (Succ wz3000)) wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2605[label="wz4/Pos wz40",fontsize=10,color="white",style="solid",shape="box"];19 -> 2605[label="",style="solid", color="burlywood", weight=9]; 2605 -> 25[label="",style="solid", color="burlywood", weight=3]; 2606[label="wz4/Neg wz40",fontsize=10,color="white",style="solid",shape="box"];19 -> 2606[label="",style="solid", color="burlywood", weight=9]; 2606 -> 26[label="",style="solid", color="burlywood", weight=3]; 20[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 (Neg Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg Zero) wz4 == GT))",fontsize=16,color="burlywood",shape="box"];2607[label="wz4/Pos wz40",fontsize=10,color="white",style="solid",shape="box"];20 -> 2607[label="",style="solid", color="burlywood", weight=9]; 2607 -> 27[label="",style="solid", color="burlywood", weight=3]; 2608[label="wz4/Neg wz40",fontsize=10,color="white",style="solid",shape="box"];20 -> 2608[label="",style="solid", color="burlywood", weight=9]; 2608 -> 28[label="",style="solid", color="burlywood", weight=3]; 21[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos wz40) (Pos (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos (Succ wz3000)) (Pos wz40) == GT))",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3]; 22[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Neg wz40) (Pos (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos (Succ wz3000)) (Neg wz40) == GT))",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3]; 23[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos wz40) (Pos Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos Zero) (Pos wz40) == GT))",fontsize=16,color="burlywood",shape="box"];2609[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 2609[label="",style="solid", color="burlywood", weight=9]; 2609 -> 31[label="",style="solid", color="burlywood", weight=3]; 2610[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 2610[label="",style="solid", color="burlywood", weight=9]; 2610 -> 32[label="",style="solid", color="burlywood", weight=3]; 24[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Neg wz40) (Pos Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos Zero) (Neg wz40) == GT))",fontsize=16,color="burlywood",shape="box"];2611[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 2611[label="",style="solid", color="burlywood", weight=9]; 2611 -> 33[label="",style="solid", color="burlywood", weight=3]; 2612[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 2612[label="",style="solid", color="burlywood", weight=9]; 2612 -> 34[label="",style="solid", color="burlywood", weight=3]; 25[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos wz40) (Neg (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg (Succ wz3000)) (Pos wz40) == GT))",fontsize=16,color="black",shape="box"];25 -> 35[label="",style="solid", color="black", weight=3]; 26[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Neg wz40) (Neg (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg (Succ wz3000)) (Neg wz40) == GT))",fontsize=16,color="black",shape="box"];26 -> 36[label="",style="solid", color="black", weight=3]; 27[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos wz40) (Neg Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg Zero) (Pos wz40) == GT))",fontsize=16,color="burlywood",shape="box"];2613[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];27 -> 2613[label="",style="solid", color="burlywood", weight=9]; 2613 -> 37[label="",style="solid", color="burlywood", weight=3]; 2614[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 2614[label="",style="solid", color="burlywood", weight=9]; 2614 -> 38[label="",style="solid", color="burlywood", weight=3]; 28[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Neg wz40) (Neg Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Neg Zero) (Neg wz40) == GT))",fontsize=16,color="burlywood",shape="box"];2615[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];28 -> 2615[label="",style="solid", color="burlywood", weight=9]; 2615 -> 39[label="",style="solid", color="burlywood", weight=3]; 2616[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 2616[label="",style="solid", color="burlywood", weight=9]; 2616 -> 40[label="",style="solid", color="burlywood", weight=3]; 29[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos wz40) (Pos (Succ wz3000)) wz31 wz32 wz33 wz34 (not (primCmpNat (Succ wz3000) wz40 == GT))",fontsize=16,color="burlywood",shape="box"];2617[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];29 -> 2617[label="",style="solid", color="burlywood", weight=9]; 2617 -> 41[label="",style="solid", color="burlywood", weight=3]; 2618[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 2618[label="",style="solid", color="burlywood", weight=9]; 2618 -> 42[label="",style="solid", color="burlywood", weight=3]; 30[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Neg wz40) (Pos (Succ wz3000)) wz31 wz32 wz33 wz34 (not (GT == GT))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3]; 31[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos (Succ wz400)) (Pos Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos Zero) (Pos (Succ wz400)) == GT))",fontsize=16,color="black",shape="box"];31 -> 44[label="",style="solid", color="black", weight=3]; 32[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Pos Zero) (Pos Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3]; 33[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] (Neg (Succ wz400)) (Pos Zero) wz31 wz32 wz33 wz34 (not (primCmpInt (Pos Zero) (Neg (Succ wz400)) == GT))",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3]; 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241[label="",style="solid", color="black", weight=3]; 206[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 (Neg Zero) wz31 wz8) (Pos (Succ wz400)) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];206 -> 242[label="",style="solid", color="black", weight=3]; 207[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 (Neg Zero) wz31 wz8) (Pos (Succ wz400)) (FiniteMap.Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];207 -> 243[label="",style="solid", color="black", weight=3]; 208[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 (Neg Zero) wz31 wz9) (Pos Zero) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];208 -> 244[label="",style="solid", color="black", weight=3]; 209[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 (Neg Zero) wz31 wz9) (Pos Zero) (FiniteMap.Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];209 -> 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332[label="",style="dashed", color="magenta", weight=3]; 289[label="Neg Zero : wz8",fontsize=16,color="green",shape="box"];290 -> 242[label="",style="dashed", color="red", weight=0]; 290[label="FiniteMap.keysFM_LE0 (Neg Zero) wz31 wz8",fontsize=16,color="magenta"];291[label="Succ wz400",fontsize=16,color="green",shape="box"];292[label="wz9",fontsize=16,color="green",shape="box"];293 -> 242[label="",style="dashed", color="red", weight=0]; 293[label="FiniteMap.keysFM_LE0 (Neg Zero) wz31 wz9",fontsize=16,color="magenta"];293 -> 334[label="",style="dashed", color="magenta", weight=3]; 294[label="Zero",fontsize=16,color="green",shape="box"];295[label="wz10",fontsize=16,color="green",shape="box"];296[label="wz10",fontsize=16,color="green",shape="box"];1983[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz189 (Pos (Succ wz190)) (Pos (Succ wz191)) wz192 wz193 wz194 wz195 True",fontsize=16,color="black",shape="box"];1983 -> 2001[label="",style="solid", color="black", weight=3]; 1984 -> 2002[label="",style="dashed", color="red", weight=0]; 1984[label="FiniteMap.keysFM_LE0 (Pos (Succ wz191)) wz192 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz189 (Pos (Succ wz190)) wz194)",fontsize=16,color="magenta"];1984 -> 2003[label="",style="dashed", color="magenta", weight=3]; 1985[label="Succ wz190",fontsize=16,color="green",shape="box"];1986[label="wz195",fontsize=16,color="green",shape="box"];652[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos wz40) wz343",fontsize=16,color="burlywood",shape="triangle"];2651[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];652 -> 2651[label="",style="solid", color="burlywood", weight=9]; 2651 -> 750[label="",style="solid", color="burlywood", weight=3]; 2652[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];652 -> 2652[label="",style="solid", color="burlywood", weight=9]; 2652 -> 751[label="",style="solid", color="burlywood", weight=3]; 313[label="wz11",fontsize=16,color="green",shape="box"];314 -> 224[label="",style="dashed", color="red", weight=0]; 314[label="FiniteMap.keysFM_LE0 (Pos Zero) wz31 wz11",fontsize=16,color="magenta"];314 -> 348[label="",style="dashed", color="magenta", weight=3]; 315[label="Succ wz400",fontsize=16,color="green",shape="box"];1563[label="wz147",fontsize=16,color="green",shape="box"];1564[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) wz1300 wz1301 wz1302 wz1303 wz1304 (wz1300 <= Neg (Succ wz125))",fontsize=16,color="black",shape="box"];1564 -> 1592[label="",style="solid", color="black", weight=3]; 333[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz340 wz341 wz342 wz343 wz344 (compare wz340 (Neg Zero) /= GT)",fontsize=16,color="black",shape="box"];333 -> 365[label="",style="solid", color="black", weight=3]; 316[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos wz40) wz340 wz341 wz342 wz343 wz344 (not (compare wz340 (Pos 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2582[label="wz257",fontsize=16,color="green",shape="box"];331[label="wz12",fontsize=16,color="green",shape="box"];332[label="wz12",fontsize=16,color="green",shape="box"];334[label="wz9",fontsize=16,color="green",shape="box"];2001 -> 652[label="",style="dashed", color="red", weight=0]; 2001[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz189 (Pos (Succ wz190)) wz194",fontsize=16,color="magenta"];2001 -> 2004[label="",style="dashed", color="magenta", weight=3]; 2001 -> 2005[label="",style="dashed", color="magenta", weight=3]; 2001 -> 2006[label="",style="dashed", color="magenta", weight=3]; 2003 -> 652[label="",style="dashed", color="red", weight=0]; 2003[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz189 (Pos (Succ wz190)) wz194",fontsize=16,color="magenta"];2003 -> 2007[label="",style="dashed", color="magenta", weight=3]; 2003 -> 2008[label="",style="dashed", color="magenta", weight=3]; 2003 -> 2009[label="",style="dashed", color="magenta", weight=3]; 2002[label="FiniteMap.keysFM_LE0 (Pos (Succ wz191)) wz192 wz198",fontsize=16,color="black",shape="triangle"];2002 -> 2010[label="",style="solid", color="black", weight=3]; 750[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos wz40) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];750 -> 806[label="",style="solid", color="black", weight=3]; 751[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos wz40) (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];751 -> 807[label="",style="solid", color="black", weight=3]; 348[label="wz11",fontsize=16,color="green",shape="box"];1592[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) wz1300 wz1301 wz1302 wz1303 wz1304 (compare wz1300 (Neg (Succ wz125)) /= GT)",fontsize=16,color="black",shape="box"];1592 -> 1688[label="",style="solid", color="black", weight=3]; 365[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz340 wz341 wz342 wz343 wz344 (not 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2588[label="",style="dashed", color="magenta", weight=3]; 2583 -> 2589[label="",style="dashed", color="magenta", weight=3]; 2584[label="wz258",fontsize=16,color="green",shape="box"];2585 -> 1351[label="",style="dashed", color="red", weight=0]; 2585[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz256 (Neg (Succ wz257)) wz261",fontsize=16,color="magenta"];2585 -> 2590[label="",style="dashed", color="magenta", weight=3]; 2585 -> 2591[label="",style="dashed", color="magenta", weight=3]; 2585 -> 2592[label="",style="dashed", color="magenta", weight=3]; 2586[label="wz259",fontsize=16,color="green",shape="box"];2004[label="wz189",fontsize=16,color="green",shape="box"];2005[label="Succ wz190",fontsize=16,color="green",shape="box"];2006[label="wz194",fontsize=16,color="green",shape="box"];2007[label="wz189",fontsize=16,color="green",shape="box"];2008[label="Succ wz190",fontsize=16,color="green",shape="box"];2009[label="wz194",fontsize=16,color="green",shape="box"];2010[label="Pos (Succ wz191) : wz198",fontsize=16,color="green",shape="box"];806[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz13 (Pos wz40) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];806 -> 863[label="",style="solid", color="black", weight=3]; 807[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz13 (Pos wz40) (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];807 -> 864[label="",style="solid", color="black", weight=3]; 1688[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) wz1300 wz1301 wz1302 wz1303 wz1304 (not (compare wz1300 (Neg (Succ wz125)) == GT))",fontsize=16,color="black",shape="box"];1688 -> 1709[label="",style="solid", color="black", weight=3]; 380[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz340 wz341 wz342 wz343 wz344 (not (primCmpInt wz340 (Neg Zero) == GT))",fontsize=16,color="burlywood",shape="box"];2655[label="wz340/Pos wz3400",fontsize=10,color="white",style="solid",shape="box"];380 -> 2655[label="",style="solid", color="burlywood", weight=9]; 2655 -> 416[label="",style="solid", color="burlywood", weight=3]; 2656[label="wz340/Neg wz3400",fontsize=10,color="white",style="solid",shape="box"];380 -> 2656[label="",style="solid", color="burlywood", weight=9]; 2656 -> 417[label="",style="solid", color="burlywood", weight=3]; 381[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos wz40) (Pos wz3400) wz341 wz342 wz343 wz344 (not (primCmpInt (Pos wz3400) (Pos wz40) == GT))",fontsize=16,color="burlywood",shape="box"];2657[label="wz3400/Succ wz34000",fontsize=10,color="white",style="solid",shape="box"];381 -> 2657[label="",style="solid", color="burlywood", weight=9]; 2657 -> 418[label="",style="solid", color="burlywood", weight=3]; 2658[label="wz3400/Zero",fontsize=10,color="white",style="solid",shape="box"];381 -> 2658[label="",style="solid", color="burlywood", weight=9]; 2658 -> 419[label="",style="solid", color="burlywood", weight=3]; 382[label="FiniteMap.foldFM_LE1 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2587[label="wz261",fontsize=16,color="green",shape="box"];2588[label="wz256",fontsize=16,color="green",shape="box"];2589[label="wz257",fontsize=16,color="green",shape="box"];2590[label="wz261",fontsize=16,color="green",shape="box"];2591[label="wz256",fontsize=16,color="green",shape="box"];2592[label="wz257",fontsize=16,color="green",shape="box"];863[label="wz13",fontsize=16,color="green",shape="box"];864 -> 229[label="",style="dashed", color="red", weight=0]; 864[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos wz40) wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= Pos wz40)",fontsize=16,color="magenta"];864 -> 921[label="",style="dashed", color="magenta", weight=3]; 864 -> 922[label="",style="dashed", color="magenta", weight=3]; 864 -> 923[label="",style="dashed", color="magenta", weight=3]; 864 -> 924[label="",style="dashed", color="magenta", weight=3]; 864 -> 925[label="",style="dashed", color="magenta", weight=3]; 1709[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz147 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weight=3]; 2664[label="wz3400/Zero",fontsize=10,color="white",style="solid",shape="box"];416 -> 2664[label="",style="solid", color="burlywood", weight=9]; 2664 -> 452[label="",style="solid", color="burlywood", weight=3]; 417[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) (Neg wz3400) wz341 wz342 wz343 wz344 (not (primCmpInt (Neg wz3400) (Neg Zero) == GT))",fontsize=16,color="burlywood",shape="box"];2665[label="wz3400/Succ wz34000",fontsize=10,color="white",style="solid",shape="box"];417 -> 2665[label="",style="solid", color="burlywood", weight=9]; 2665 -> 453[label="",style="solid", color="burlywood", weight=3]; 2666[label="wz3400/Zero",fontsize=10,color="white",style="solid",shape="box"];417 -> 2666[label="",style="solid", color="burlywood", weight=9]; 2666 -> 454[label="",style="solid", color="burlywood", weight=3]; 418[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos wz40) (Pos (Succ wz34000)) wz341 wz342 wz343 wz344 (not (primCmpInt (Pos (Succ wz34000)) 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weight=3]; 421[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos wz40) (Neg Zero) wz341 wz342 wz343 wz344 (not (primCmpInt (Neg Zero) (Pos wz40) == GT))",fontsize=16,color="burlywood",shape="box"];2669[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];421 -> 2669[label="",style="solid", color="burlywood", weight=9]; 2669 -> 459[label="",style="solid", color="burlywood", weight=3]; 2670[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];421 -> 2670[label="",style="solid", color="burlywood", weight=9]; 2670 -> 460[label="",style="solid", color="burlywood", weight=3]; 921[label="wz3430",fontsize=16,color="green",shape="box"];922[label="wz3432",fontsize=16,color="green",shape="box"];923[label="wz3434",fontsize=16,color="green",shape="box"];924[label="wz3433",fontsize=16,color="green",shape="box"];925[label="wz3431",fontsize=16,color="green",shape="box"];1732[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) 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color="burlywood", weight=3]; 2674[label="wz13000/Zero",fontsize=10,color="white",style="solid",shape="box"];1733 -> 2674[label="",style="solid", color="burlywood", weight=9]; 2674 -> 1758[label="",style="solid", color="burlywood", weight=3]; 451[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) (Pos (Succ wz34000)) wz341 wz342 wz343 wz344 (not (primCmpInt (Pos (Succ wz34000)) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];451 -> 491[label="",style="solid", color="black", weight=3]; 452[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) (Pos Zero) wz341 wz342 wz343 wz344 (not (primCmpInt (Pos Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];452 -> 492[label="",style="solid", color="black", weight=3]; 453[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) (Neg (Succ wz34000)) wz341 wz342 wz343 wz344 (not (primCmpInt (Neg (Succ wz34000)) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];453 -> 493[label="",style="solid", color="black", weight=3]; 454[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz20 (Neg Zero) (Neg Zero) wz341 wz342 wz343 wz344 (not (primCmpInt (Neg Zero) (Neg Zero) == GT))",fontsize=16,color="black",shape="box"];454 -> 494[label="",style="solid", color="black", weight=3]; 455[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos wz40) (Pos (Succ wz34000)) wz341 wz342 wz343 wz344 (not (primCmpNat (Succ wz34000) wz40 == GT))",fontsize=16,color="burlywood",shape="box"];2675[label="wz40/Succ wz400",fontsize=10,color="white",style="solid",shape="box"];455 -> 2675[label="",style="solid", color="burlywood", weight=9]; 2675 -> 495[label="",style="solid", color="burlywood", weight=3]; 2676[label="wz40/Zero",fontsize=10,color="white",style="solid",shape="box"];455 -> 2676[label="",style="solid", color="burlywood", weight=9]; 2676 -> 496[label="",style="solid", color="burlywood", weight=3]; 456[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 (Pos (Succ 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FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 (Neg (Succ wz34000)) wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz343)) (Neg Zero) wz344",fontsize=16,color="magenta"];737 -> 794[label="",style="dashed", color="magenta", weight=3]; 737 -> 795[label="",style="dashed", color="magenta", weight=3]; 738[label="wz344",fontsize=16,color="green",shape="box"];739[label="wz341",fontsize=16,color="green",shape="box"];740 -> 735[label="",style="dashed", color="red", weight=0]; 740[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz343",fontsize=16,color="magenta"];745[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz13 (Pos Zero) (Pos (Succ wz34000)) wz341 wz342 wz343 wz344 otherwise",fontsize=16,color="black",shape="box"];745 -> 801[label="",style="solid", color="black", weight=3]; 746 -> 652[label="",style="dashed", color="red", weight=0]; 746[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 (Pos Zero) wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos (Succ wz400)) wz343)) (Pos (Succ wz400)) wz344",fontsize=16,color="magenta"];746 -> 802[label="",style="dashed", color="magenta", weight=3]; 746 -> 803[label="",style="dashed", color="magenta", weight=3]; 746 -> 804[label="",style="dashed", color="magenta", weight=3]; 747[label="wz344",fontsize=16,color="green",shape="box"];748 -> 652[label="",style="dashed", color="red", weight=0]; 748[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos Zero) wz343",fontsize=16,color="magenta"];748 -> 805[label="",style="dashed", color="magenta", weight=3]; 749[label="wz341",fontsize=16,color="green",shape="box"];752 -> 242[label="",style="dashed", color="red", weight=0]; 752[label="FiniteMap.keysFM_LE0 (Neg Zero) wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos (Succ wz400)) wz343)",fontsize=16,color="magenta"];752 -> 808[label="",style="dashed", color="magenta", weight=3]; 752 -> 809[label="",style="dashed", color="magenta", weight=3]; 753[label="Succ wz400",fontsize=16,color="green",shape="box"];754[label="wz344",fontsize=16,color="green",shape="box"];755 -> 242[label="",style="dashed", color="red", weight=0]; 755[label="FiniteMap.keysFM_LE0 (Neg Zero) wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos Zero) wz343)",fontsize=16,color="magenta"];755 -> 810[label="",style="dashed", color="magenta", weight=3]; 755 -> 811[label="",style="dashed", color="magenta", weight=3]; 756[label="Zero",fontsize=16,color="green",shape="box"];757[label="wz344",fontsize=16,color="green",shape="box"];1925[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) (Pos (Succ wz130000)) wz1301 wz1302 wz1303 wz1304 True",fontsize=16,color="black",shape="box"];1925 -> 1936[label="",style="solid", color="black", weight=3]; 1926[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) (Pos Zero) wz1301 wz1302 wz1303 wz1304 True",fontsize=16,color="black",shape="box"];1926 -> 1937[label="",style="solid", color="black", weight=3]; 1931[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) (Neg Zero) wz1301 wz1302 wz1303 wz1304 otherwise",fontsize=16,color="black",shape="box"];1931 -> 1943[label="",style="solid", color="black", weight=3]; 791 -> 735[label="",style="dashed", color="red", weight=0]; 791[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz343",fontsize=16,color="magenta"];792[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];792 -> 848[label="",style="solid", color="black", weight=3]; 793[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];793 -> 849[label="",style="solid", color="black", weight=3]; 794[label="wz344",fontsize=16,color="green",shape="box"];795 -> 193[label="",style="dashed", color="red", weight=0]; 795[label="FiniteMap.keysFM_LE0 (Neg (Succ wz34000)) wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz343)",fontsize=16,color="magenta"];795 -> 850[label="",style="dashed", color="magenta", weight=3]; 795 -> 851[label="",style="dashed", color="magenta", weight=3]; 795 -> 852[label="",style="dashed", color="magenta", weight=3]; 801[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz13 (Pos Zero) (Pos (Succ wz34000)) wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];801 -> 860[label="",style="solid", color="black", weight=3]; 802 -> 224[label="",style="dashed", color="red", weight=0]; 802[label="FiniteMap.keysFM_LE0 (Pos Zero) wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos (Succ wz400)) wz343)",fontsize=16,color="magenta"];802 -> 861[label="",style="dashed", color="magenta", weight=3]; 802 -> 862[label="",style="dashed", color="magenta", weight=3]; 803[label="Succ wz400",fontsize=16,color="green",shape="box"];804[label="wz344",fontsize=16,color="green",shape="box"];805[label="Zero",fontsize=16,color="green",shape="box"];808[label="wz341",fontsize=16,color="green",shape="box"];809 -> 652[label="",style="dashed", color="red", weight=0]; 809[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos (Succ wz400)) wz343",fontsize=16,color="magenta"];809 -> 865[label="",style="dashed", color="magenta", weight=3]; 810[label="wz341",fontsize=16,color="green",shape="box"];811 -> 652[label="",style="dashed", color="red", weight=0]; 811[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos Zero) wz343",fontsize=16,color="magenta"];811 -> 866[label="",style="dashed", color="magenta", weight=3]; 1936 -> 1351[label="",style="dashed", color="red", weight=0]; 1936[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) wz1303",fontsize=16,color="magenta"];1936 -> 1949[label="",style="dashed", color="magenta", weight=3]; 1937 -> 1351[label="",style="dashed", color="red", weight=0]; 1937[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) wz1303",fontsize=16,color="magenta"];1937 -> 1950[label="",style="dashed", color="magenta", weight=3]; 1943[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) (Neg Zero) wz1301 wz1302 wz1303 wz1304 True",fontsize=16,color="black",shape="box"];1943 -> 1958[label="",style="solid", color="black", weight=3]; 848 -> 277[label="",style="dashed", color="red", weight=0]; 848[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz20 (Neg Zero) FiniteMap.EmptyFM",fontsize=16,color="magenta"];848 -> 1039[label="",style="dashed", color="magenta", weight=3]; 849 -> 283[label="",style="dashed", color="red", weight=0]; 849[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz20 (Neg Zero) (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="magenta"];849 -> 1040[label="",style="dashed", color="magenta", weight=3]; 849 -> 1041[label="",style="dashed", color="magenta", weight=3]; 849 -> 1042[label="",style="dashed", color="magenta", weight=3]; 849 -> 1043[label="",style="dashed", color="magenta", weight=3]; 849 -> 1044[label="",style="dashed", color="magenta", weight=3]; 850[label="wz34000",fontsize=16,color="green",shape="box"];851 -> 735[label="",style="dashed", color="red", weight=0]; 851[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz20 (Neg Zero) wz343",fontsize=16,color="magenta"];852[label="wz341",fontsize=16,color="green",shape="box"];860 -> 652[label="",style="dashed", color="red", weight=0]; 860[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos Zero) wz343",fontsize=16,color="magenta"];860 -> 1061[label="",style="dashed", color="magenta", weight=3]; 861 -> 652[label="",style="dashed", color="red", weight=0]; 861[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 (Pos (Succ wz400)) wz343",fontsize=16,color="magenta"];861 -> 1062[label="",style="dashed", color="magenta", weight=3]; 862[label="wz341",fontsize=16,color="green",shape="box"];865[label="Succ wz400",fontsize=16,color="green",shape="box"];866[label="Zero",fontsize=16,color="green",shape="box"];1949[label="wz1303",fontsize=16,color="green",shape="box"];1950[label="wz1303",fontsize=16,color="green",shape="box"];1958 -> 1351[label="",style="dashed", color="red", weight=0]; 1958[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz147 (Neg (Succ wz125)) wz1303",fontsize=16,color="magenta"];1958 -> 1968[label="",style="dashed", color="magenta", weight=3]; 1039[label="wz20",fontsize=16,color="green",shape="box"];1040[label="wz3430",fontsize=16,color="green",shape="box"];1041[label="wz3432",fontsize=16,color="green",shape="box"];1042[label="wz3434",fontsize=16,color="green",shape="box"];1043[label="wz3433",fontsize=16,color="green",shape="box"];1044[label="wz3431",fontsize=16,color="green",shape="box"];1061[label="Zero",fontsize=16,color="green",shape="box"];1062[label="Succ wz400",fontsize=16,color="green",shape="box"];1968[label="wz1303",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h) new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) The TRS R consists of the following rules: new_foldFM_LE8(wz31, wz7, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE00(wz31, wz7, h), h) new_foldFM_LE30(wz19, h) -> wz19 new_foldFM_LE20(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h) new_foldFM_LE8(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE20(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz20, wz343, h) new_keysFM_LE00(wz31, wz6, h) -> :(Pos(Zero), wz6) new_foldFM_LE5(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE20(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE7(wz31, wz10, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE01(wz31, wz10, h), h) new_foldFM_LE20(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_keysFM_LE0(wz3000, wz31, wz5, h) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE5(wz20, EmptyFM, h) -> new_foldFM_LE30(wz20, h) new_foldFM_LE7(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) new_keysFM_LE01(wz31, wz8, h) -> :(Neg(Zero), wz8) new_foldFM_LE20(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) The set Q consists of the following terms: new_keysFM_LE01(x0, x1, x2) new_foldFM_LE20(x0, Pos(Zero), x1, x2, x3, x4, x5) new_foldFM_LE8(x0, x1, EmptyFM, x2) new_foldFM_LE5(x0, EmptyFM, x1) new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE7(x0, x1, EmptyFM, x2) new_foldFM_LE8(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE20(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE5(x0, Branch(x1, x2, x3, x4, x5), x6) new_foldFM_LE20(x0, Neg(Zero), x1, x2, x3, x4, x5) new_foldFM_LE30(x0, x1) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE20(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(:(Neg(Succ(wz34000)), new_foldFM_LE5(wz20, wz343, h)), wz344, h),new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(:(Neg(Succ(wz34000)), new_foldFM_LE5(wz20, wz343, h)), wz344, h)) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(:(Neg(Succ(wz34000)), new_foldFM_LE5(wz20, wz343, h)), wz344, h) The TRS R consists of the following rules: new_foldFM_LE8(wz31, wz7, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE00(wz31, wz7, h), h) new_foldFM_LE30(wz19, h) -> wz19 new_foldFM_LE20(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h) new_foldFM_LE8(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE20(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz20, wz343, h) new_keysFM_LE00(wz31, wz6, h) -> :(Pos(Zero), wz6) new_foldFM_LE5(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE20(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE7(wz31, wz10, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE01(wz31, wz10, h), h) new_foldFM_LE20(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_keysFM_LE0(wz3000, wz31, wz5, h) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE5(wz20, EmptyFM, h) -> new_foldFM_LE30(wz20, h) new_foldFM_LE7(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) new_keysFM_LE01(wz31, wz8, h) -> :(Neg(Zero), wz8) new_foldFM_LE20(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) The set Q consists of the following terms: new_keysFM_LE01(x0, x1, x2) new_foldFM_LE20(x0, Pos(Zero), x1, x2, x3, x4, x5) new_foldFM_LE8(x0, x1, EmptyFM, x2) new_foldFM_LE5(x0, EmptyFM, x1) new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE7(x0, x1, EmptyFM, x2) new_foldFM_LE8(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE20(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE5(x0, Branch(x1, x2, x3, x4, x5), x6) new_foldFM_LE20(x0, Neg(Zero), x1, x2, x3, x4, x5) new_foldFM_LE30(x0, x1) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE20(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Neg(Zero), wz10), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Neg(Zero), wz10), wz340, wz341, wz342, wz343, wz344, h)) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(:(Neg(Succ(wz34000)), new_foldFM_LE5(wz20, wz343, h)), wz344, h) new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Neg(Zero), wz10), wz340, wz341, wz342, wz343, wz344, h) The TRS R consists of the following rules: new_foldFM_LE8(wz31, wz7, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE00(wz31, wz7, h), h) new_foldFM_LE30(wz19, h) -> wz19 new_foldFM_LE20(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h) new_foldFM_LE8(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE20(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz20, wz343, h) new_keysFM_LE00(wz31, wz6, h) -> :(Pos(Zero), wz6) new_foldFM_LE5(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE20(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE7(wz31, wz10, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE01(wz31, wz10, h), h) new_foldFM_LE20(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_keysFM_LE0(wz3000, wz31, wz5, h) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE5(wz20, EmptyFM, h) -> new_foldFM_LE30(wz20, h) new_foldFM_LE7(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) new_keysFM_LE01(wz31, wz8, h) -> :(Neg(Zero), wz8) new_foldFM_LE20(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) The set Q consists of the following terms: new_keysFM_LE01(x0, x1, x2) new_foldFM_LE20(x0, Pos(Zero), x1, x2, x3, x4, x5) new_foldFM_LE8(x0, x1, EmptyFM, x2) new_foldFM_LE5(x0, EmptyFM, x1) new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE7(x0, x1, EmptyFM, x2) new_foldFM_LE8(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE20(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE5(x0, Branch(x1, x2, x3, x4, x5), x6) new_foldFM_LE20(x0, Neg(Zero), x1, x2, x3, x4, x5) new_foldFM_LE30(x0, x1) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE20(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Pos(Zero), wz7), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Pos(Zero), wz7), wz340, wz341, wz342, wz343, wz344, h)) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(:(Neg(Succ(wz34000)), new_foldFM_LE5(wz20, wz343, h)), wz344, h) new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Neg(Zero), wz10), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Pos(Zero), wz7), wz340, wz341, wz342, wz343, wz344, h) The TRS R consists of the following rules: new_foldFM_LE8(wz31, wz7, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE00(wz31, wz7, h), h) new_foldFM_LE30(wz19, h) -> wz19 new_foldFM_LE20(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(new_keysFM_LE0(wz34000, wz341, new_foldFM_LE5(wz20, wz343, h), h), wz344, h) new_foldFM_LE8(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) new_foldFM_LE20(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz20, wz343, h) new_keysFM_LE00(wz31, wz6, h) -> :(Pos(Zero), wz6) new_foldFM_LE5(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE20(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) new_foldFM_LE7(wz31, wz10, EmptyFM, h) -> new_foldFM_LE30(new_keysFM_LE01(wz31, wz10, h), h) new_foldFM_LE20(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) new_keysFM_LE0(wz3000, wz31, wz5, h) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE5(wz20, EmptyFM, h) -> new_foldFM_LE30(wz20, h) new_foldFM_LE7(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE20(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) new_keysFM_LE01(wz31, wz8, h) -> :(Neg(Zero), wz8) new_foldFM_LE20(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) The set Q consists of the following terms: new_keysFM_LE01(x0, x1, x2) new_foldFM_LE20(x0, Pos(Zero), x1, x2, x3, x4, x5) new_foldFM_LE8(x0, x1, EmptyFM, x2) new_foldFM_LE5(x0, EmptyFM, x1) new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE7(x0, x1, EmptyFM, x2) new_foldFM_LE8(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE20(x0, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE5(x0, Branch(x1, x2, x3, x4, x5), x6) new_foldFM_LE20(x0, Neg(Zero), x1, x2, x3, x4, x5) new_foldFM_LE30(x0, x1) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE20(x0, Neg(Succ(x1)), x2, x3, x4, x5, x6) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Neg(Zero), wz10), wz340, wz341, wz342, wz343, wz344, h) The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7 *new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) The graph contains the following edges 1 >= 1, 5 > 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 7 >= 7 *new_foldFM_LE3(wz20, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE2(wz20, wz3430, wz3431, wz3432, wz3433, wz3434, h) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7 *new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE2(:(Pos(Zero), wz7), wz340, wz341, wz342, wz343, wz344, h) The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7 *new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4 *new_foldFM_LE2(wz20, Pos(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE5(wz20, wz343, h), wz344, h) The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4 *new_foldFM_LE2(wz20, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3 *new_foldFM_LE2(wz20, Neg(Zero), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3 *new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz20, wz343, h) The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3 *new_foldFM_LE2(wz20, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(:(Neg(Succ(wz34000)), new_foldFM_LE5(wz20, wz343, h)), wz344, h) The graph contains the following edges 6 >= 2, 7 >= 3 ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h),new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h)) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba),new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba)) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba),new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba)) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba),new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba)) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Pos(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba),new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Pos(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba)) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Pos(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h),new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h)) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Pos(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba),new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba)) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba) new_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Pos(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba) new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba) The TRS R consists of the following rules: new_foldFM_LE19(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE110(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) new_foldFM_LE19(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE00(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_foldFM_LE19(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(wz13, Zero, wz343, ba) new_foldFM_LE19(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE22(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) new_foldFM_LE22(wz3000, wz31, wz5, wz40, EmptyFM, ba) -> new_keysFM_LE0(wz3000, wz31, wz5, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) new_keysFM_LE02(wz191, wz192, wz198, h) -> :(Pos(Succ(wz191)), wz198) new_foldFM_LE22(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE0(wz3000, wz31, wz5, ba), wz40, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE00(wz31, wz6, ba) -> :(Pos(Zero), wz6) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, EmptyFM, ba) -> wz13 new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE14(new_keysFM_LE02(wz191, wz192, new_foldFM_LE14(wz189, Succ(wz190), wz194, h), h), Succ(wz190), wz195, h) new_foldFM_LE21(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE19(new_keysFM_LE00(wz31, wz6, ba), Zero, wz340, wz341, wz342, wz343, wz344, ba) new_keysFM_LE0(wz3000, wz31, wz5, ba) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE21(wz31, wz6, EmptyFM, ba) -> new_keysFM_LE00(wz31, wz6, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE111(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) new_foldFM_LE14(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE19(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) new_foldFM_LE19(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE21(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) new_foldFM_LE19(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Succ(wz400), wz343, ba), ba), Succ(wz400), wz344, ba) new_keysFM_LE01(wz31, wz8, ba) -> :(Neg(Zero), wz8) new_foldFM_LE19(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE14(new_keysFM_LE01(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), ba), Zero, wz344, ba) new_foldFM_LE110(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE14(wz189, Succ(wz190), wz194, h) The set Q consists of the following terms: new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE19(x0, Succ(x1), Pos(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE19(x0, Succ(x1), Neg(Zero), x2, x3, x4, x5, x6) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE14(x0, x1, EmptyFM, x2) new_foldFM_LE19(x0, Succ(x1), Pos(Zero), x2, x3, x4, x5, x6) new_foldFM_LE22(x0, x1, x2, x3, Branch(x4, x5, x6, x7, x8), x9) new_foldFM_LE22(x0, x1, x2, x3, EmptyFM, x4) new_keysFM_LE00(x0, x1, x2) new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_foldFM_LE19(x0, x1, Neg(Succ(x2)), x3, x4, x5, x6, x7) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE21(x0, x1, EmptyFM, x2) new_foldFM_LE110(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE14(x0, x1, Branch(x2, x3, x4, x5, x6), x7) new_keysFM_LE01(x0, x1, x2) new_foldFM_LE19(x0, Zero, Pos(Succ(x1)), x2, x3, x4, x5, x6) new_foldFM_LE19(x0, Zero, Neg(Zero), x1, x2, x3, x4, x5) new_keysFM_LE02(x0, x1, x2, x3) new_foldFM_LE111(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE19(x0, Zero, Pos(Zero), x1, x2, x3, x4, x5) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) 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_foldFM_LE17(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Pos(Zero), wz6), Zero, wz340, wz341, wz342, wz343, wz344, ba) The graph contains the following edges 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 4 >= 8 *new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 6 > 7, 8 >= 8 *new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Succ(wz1970), h) -> new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, wz1960, wz1970, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 > 8, 9 > 9, 10 >= 10 *new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Zero, h) -> new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8 *new_foldFM_LE16(wz13, Succ(wz400), Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE13(wz13, wz400, wz34000, wz341, wz342, wz343, wz344, wz34000, wz400, ba) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 3 > 8, 2 > 9, 8 >= 10 *new_foldFM_LE18(wz3000, wz31, wz5, wz40, Branch(wz340, wz341, wz342, wz343, wz344), ba) -> new_foldFM_LE16(:(Neg(Succ(wz3000)), wz5), wz40, wz340, wz341, wz342, wz343, wz344, ba) The graph contains the following edges 4 >= 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 5 > 7, 6 >= 8 *new_foldFM_LE9(wz13, wz40, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), ba) -> new_foldFM_LE16(wz13, wz40, wz3430, wz3431, wz3432, wz3433, wz3434, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 4 >= 8 *new_foldFM_LE16(wz13, wz40, Neg(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE18(wz34000, wz341, new_foldFM_LE14(wz13, wz40, wz343, ba), wz40, wz344, ba) The graph contains the following edges 3 > 1, 4 >= 2, 2 >= 4, 7 >= 5, 8 >= 6 *new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE17(wz341, new_foldFM_LE14(wz13, Zero, wz343, ba), wz344, ba) The graph contains the following edges 4 >= 1, 7 >= 3, 8 >= 4 *new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Succ(wz1960), Zero, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4 *new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4 *new_foldFM_LE13(wz189, wz190, wz191, wz192, wz193, wz194, wz195, Zero, Succ(wz1970), h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) The graph contains the following edges 7 >= 3, 10 >= 4 *new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 6 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Zero, Pos(Succ(wz34000)), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Zero, Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Zero, wz343, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 6 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Zero, Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Zero, wz343, ba)), Zero, wz344, ba) The graph contains the following edges 2 >= 2, 3 > 2, 7 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(wz13, Succ(wz400), wz343, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Succ(wz400), Pos(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Pos(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba) The graph contains the following edges 2 >= 2, 7 >= 3, 8 >= 4 *new_foldFM_LE16(wz13, Succ(wz400), Neg(Zero), wz341, wz342, wz343, wz344, ba) -> new_foldFM_LE9(:(Neg(Zero), new_foldFM_LE14(wz13, Succ(wz400), wz343, ba)), Succ(wz400), wz344, ba) The graph contains the following edges 2 >= 2, 7 >= 3, 8 >= 4 *new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(wz189, Succ(wz190), wz194, h) The graph contains the following edges 1 >= 1, 6 >= 3, 8 >= 4 *new_foldFM_LE15(wz189, wz190, wz191, wz192, wz193, wz194, wz195, h) -> new_foldFM_LE9(:(Pos(Succ(wz191)), new_foldFM_LE14(wz189, Succ(wz190), wz194, h)), Succ(wz190), wz195, h) The graph contains the following edges 7 >= 3, 8 >= 4 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_foldFM_LE(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE1(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) The TRS R consists of the following rules: new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE0(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_keysFM_LE0(wz3000, wz31, wz5, bb) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE0(wz147, wz125, EmptyFM, h) -> wz147 new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_foldFM_LE0(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) new_foldFM_LE0(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE11(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE0(wz256, wz257, wz261, ba) new_foldFM_LE0(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) new_foldFM_LE0(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) The set Q consists of the following terms: new_foldFM_LE0(x0, x1, Branch(Pos(Succ(x2)), x3, x4, x5, x6), x7) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE0(x0, x1, EmptyFM, x2) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE12(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE0(x0, x1, Branch(Pos(Zero), x2, x3, x4, x5), x6) new_foldFM_LE0(x0, x1, Branch(Neg(Succ(x2)), x3, x4, x5, x6), x7) new_foldFM_LE0(x0, x1, Branch(Neg(Zero), x2, x3, x4, x5), x6) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba),new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba)) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_foldFM_LE(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE1(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba) The TRS R consists of the following rules: new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE0(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_keysFM_LE0(wz3000, wz31, wz5, bb) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE0(wz147, wz125, EmptyFM, h) -> wz147 new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_foldFM_LE0(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) new_foldFM_LE0(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE11(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE0(wz256, wz257, wz261, ba) new_foldFM_LE0(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) new_foldFM_LE0(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) The set Q consists of the following terms: new_foldFM_LE0(x0, x1, Branch(Pos(Succ(x2)), x3, x4, x5, x6), x7) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE0(x0, x1, EmptyFM, x2) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE12(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE0(x0, x1, Branch(Pos(Zero), x2, x3, x4, x5), x6) new_foldFM_LE0(x0, x1, Branch(Neg(Succ(x2)), x3, x4, x5, x6), x7) new_foldFM_LE0(x0, x1, Branch(Neg(Zero), x2, x3, x4, x5), x6) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) at position [0] we obtained the following new rules [LPAR04]: (new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba),new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba)) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_foldFM_LE(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE1(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) new_foldFM_LE(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba) new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba) The TRS R consists of the following rules: new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE0(new_keysFM_LE0(wz258, wz259, new_foldFM_LE0(wz256, wz257, wz261, ba), ba), wz257, wz262, ba) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_keysFM_LE0(wz3000, wz31, wz5, bb) -> :(Neg(Succ(wz3000)), wz5) new_foldFM_LE0(wz147, wz125, EmptyFM, h) -> wz147 new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE12(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) new_foldFM_LE0(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) new_foldFM_LE0(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE11(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) new_foldFM_LE11(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE0(wz256, wz257, wz261, ba) new_foldFM_LE0(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) new_foldFM_LE0(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE0(wz147, wz125, wz1303, h) The set Q consists of the following terms: new_foldFM_LE0(x0, x1, Branch(Pos(Succ(x2)), x3, x4, x5, x6), x7) new_keysFM_LE0(x0, x1, x2, x3) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Succ(x7), x8) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Zero, Zero, x7) new_foldFM_LE0(x0, x1, EmptyFM, x2) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Succ(x8), x9) new_foldFM_LE11(x0, x1, x2, x3, x4, x5, x6, Succ(x7), Zero, x8) new_foldFM_LE12(x0, x1, x2, x3, x4, x5, x6, x7) new_foldFM_LE0(x0, x1, Branch(Pos(Zero), x2, x3, x4, x5), x6) new_foldFM_LE0(x0, x1, Branch(Neg(Succ(x2)), x3, x4, x5, x6), x7) new_foldFM_LE0(x0, x1, Branch(Neg(Zero), x2, x3, x4, x5), x6) 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_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Succ(wz2640), ba) -> new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, wz2630, wz2640, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 > 8, 9 > 9, 10 >= 10 *new_foldFM_LE(wz147, wz125, Branch(Neg(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE1(wz147, wz125, wz130000, wz1301, wz1302, wz1303, wz1304, Succ(wz125), Succ(wz130000), h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 3 > 9, 4 >= 10 *new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Zero, ba) -> new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8 *new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Succ(wz2630), Zero, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 10 >= 4 *new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 10 >= 4 *new_foldFM_LE1(wz256, wz257, wz258, wz259, wz260, wz261, wz262, Zero, Succ(wz2640), ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba) The graph contains the following edges 2 >= 2, 7 >= 3, 10 >= 4 *new_foldFM_LE(wz147, wz125, Branch(Neg(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *new_foldFM_LE(wz147, wz125, Branch(Pos(Zero), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *new_foldFM_LE(wz147, wz125, Branch(Pos(Succ(wz130000)), wz1301, wz1302, wz1303, wz1304), h) -> new_foldFM_LE(wz147, wz125, wz1303, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(wz256, wz257, wz261, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 6 >= 3, 8 >= 4 *new_foldFM_LE10(wz256, wz257, wz258, wz259, wz260, wz261, wz262, ba) -> new_foldFM_LE(:(Neg(Succ(wz258)), new_foldFM_LE0(wz256, wz257, wz261, ba)), wz257, wz262, ba) The graph contains the following edges 2 >= 2, 7 >= 3, 8 >= 4 ---------------------------------------- (41) YES ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE23(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) new_foldFM_LE23(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(wz40), wz33, h) new_foldFM_LE23(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) new_foldFM_LE23(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) new_foldFM_LE23(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) new_foldFM_LE23(Pos(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Succ(wz400)), wz33, h) new_foldFM_LE23(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(wz40), wz33, h) new_foldFM_LE23(Pos(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Succ(wz400)), wz33, h) new_foldFM_LE23(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) new_foldFM_LE23(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (44) Complex Obligation (AND) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE23(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) new_foldFM_LE23(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(wz40), wz33, h) new_foldFM_LE23(Pos(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Succ(wz400)), wz33, h) new_foldFM_LE23(Pos(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Succ(wz400)), wz33, h) new_foldFM_LE23(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) new_foldFM_LE23(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) 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_foldFM_LE23(Pos(wz40), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(wz40), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Pos(Zero), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Pos(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Pos(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Zero), wz33, h) The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Pos(Succ(wz400)), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Succ(wz400)), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Pos(Succ(wz400)), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Pos(Succ(wz400)), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 ---------------------------------------- (47) YES ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldFM_LE23(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) new_foldFM_LE23(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) new_foldFM_LE23(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) new_foldFM_LE23(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(wz40), wz33, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) 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_foldFM_LE23(Neg(Zero), Branch(Neg(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Neg(Zero), Branch(Neg(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Neg(Zero), Branch(Pos(Zero), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(Zero), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_foldFM_LE23(Neg(wz40), Branch(Pos(Succ(wz3000)), wz31, wz32, wz33, wz34), h) -> new_foldFM_LE23(Neg(wz40), wz33, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 ---------------------------------------- (50) YES