10.65/4.70 YES 12.36/5.20 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 12.36/5.20 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.36/5.20 12.36/5.20 12.36/5.20 H-Termination with start terms of the given HASKELL could be proven: 12.36/5.20 12.36/5.20 (0) HASKELL 12.36/5.20 (1) BR [EQUIVALENT, 0 ms] 12.36/5.20 (2) HASKELL 12.36/5.20 (3) COR [EQUIVALENT, 0 ms] 12.36/5.20 (4) HASKELL 12.36/5.20 (5) Narrow [SOUND, 0 ms] 12.36/5.20 (6) QDP 12.36/5.20 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 12.36/5.20 (8) AND 12.36/5.20 (9) QDP 12.36/5.20 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.36/5.20 (11) YES 12.36/5.20 (12) QDP 12.36/5.20 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.36/5.20 (14) YES 12.36/5.20 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (0) 12.36/5.20 Obligation: 12.36/5.20 mainModule Main 12.36/5.20 module FiniteMap where { 12.36/5.20 import qualified Main; 12.36/5.20 import qualified Maybe; 12.36/5.20 import qualified Prelude; 12.36/5.20 data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; 12.36/5.20 12.36/5.20 instance (Eq a, Eq b) => Eq FiniteMap a b where { 12.36/5.20 } 12.36/5.20 foldFM_LE :: Ord b => (b -> c -> a -> a) -> a -> b -> FiniteMap b c -> a; 12.36/5.20 foldFM_LE k z fr EmptyFM = z; 12.36/5.20 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 12.36/5.20 | otherwise = foldFM_LE k z fr fm_l; 12.36/5.20 12.36/5.20 } 12.36/5.20 module Maybe where { 12.36/5.20 import qualified FiniteMap; 12.36/5.20 import qualified Main; 12.36/5.20 import qualified Prelude; 12.36/5.20 } 12.36/5.20 module Main where { 12.36/5.20 import qualified FiniteMap; 12.36/5.20 import qualified Maybe; 12.36/5.20 import qualified Prelude; 12.36/5.20 } 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (1) BR (EQUIVALENT) 12.36/5.20 Replaced joker patterns by fresh variables and removed binding patterns. 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (2) 12.36/5.20 Obligation: 12.36/5.20 mainModule Main 12.36/5.20 module FiniteMap where { 12.36/5.20 import qualified Main; 12.36/5.20 import qualified Maybe; 12.36/5.20 import qualified Prelude; 12.36/5.20 data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; 12.36/5.20 12.36/5.20 instance (Eq a, Eq b) => Eq FiniteMap b a where { 12.36/5.20 } 12.36/5.20 foldFM_LE :: Ord c => (c -> a -> b -> b) -> b -> c -> FiniteMap c a -> b; 12.36/5.20 foldFM_LE k z fr EmptyFM = z; 12.36/5.20 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 12.36/5.20 | otherwise = foldFM_LE k z fr fm_l; 12.36/5.20 12.36/5.20 } 12.36/5.20 module Maybe where { 12.36/5.20 import qualified FiniteMap; 12.36/5.20 import qualified Main; 12.36/5.20 import qualified Prelude; 12.36/5.20 } 12.36/5.20 module Main where { 12.36/5.20 import qualified FiniteMap; 12.36/5.20 import qualified Maybe; 12.36/5.20 import qualified Prelude; 12.36/5.20 } 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (3) COR (EQUIVALENT) 12.36/5.20 Cond Reductions: 12.36/5.20 The following Function with conditions 12.36/5.20 "undefined |Falseundefined; 12.36/5.20 " 12.36/5.20 is transformed to 12.36/5.20 "undefined = undefined1; 12.36/5.20 " 12.36/5.20 "undefined0 True = undefined; 12.36/5.20 " 12.36/5.20 "undefined1 = undefined0 False; 12.36/5.20 " 12.36/5.20 The following Function with conditions 12.36/5.20 "foldFM_LE k z fr EmptyFM = z; 12.36/5.20 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; 12.36/5.20 " 12.36/5.20 is transformed to 12.36/5.20 "foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM; 12.36/5.20 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); 12.36/5.20 " 12.36/5.20 "foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l; 12.36/5.20 " 12.36/5.20 "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; 12.36/5.20 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; 12.36/5.20 " 12.36/5.20 "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); 12.36/5.20 " 12.36/5.20 "foldFM_LE3 k z fr EmptyFM = z; 12.36/5.20 foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy; 12.36/5.20 " 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (4) 12.36/5.20 Obligation: 12.36/5.20 mainModule Main 12.36/5.20 module FiniteMap where { 12.36/5.20 import qualified Main; 12.36/5.20 import qualified Maybe; 12.36/5.20 import qualified Prelude; 12.36/5.20 data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; 12.36/5.20 12.36/5.20 instance (Eq a, Eq b) => Eq FiniteMap a b where { 12.36/5.20 } 12.36/5.20 foldFM_LE :: Ord b => (b -> c -> a -> a) -> a -> b -> FiniteMap b c -> a; 12.36/5.20 foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM; 12.36/5.20 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); 12.36/5.20 12.36/5.20 foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l; 12.36/5.20 12.36/5.20 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; 12.36/5.20 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; 12.36/5.20 12.36/5.20 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); 12.36/5.20 12.36/5.20 foldFM_LE3 k z fr EmptyFM = z; 12.36/5.20 foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy; 12.36/5.20 12.36/5.20 } 12.36/5.20 module Maybe where { 12.36/5.20 import qualified FiniteMap; 12.36/5.20 import qualified Main; 12.36/5.20 import qualified Prelude; 12.36/5.20 } 12.36/5.20 module Main where { 12.36/5.20 import qualified FiniteMap; 12.36/5.20 import qualified Maybe; 12.36/5.20 import qualified Prelude; 12.36/5.20 } 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (5) Narrow (SOUND) 12.36/5.20 Haskell To QDPs 12.36/5.20 12.36/5.20 digraph dp_graph { 12.36/5.20 node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 12.36/5.20 3[label="FiniteMap.foldFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 12.36/5.20 4[label="FiniteMap.foldFM_LE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 12.36/5.20 5[label="FiniteMap.foldFM_LE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3]; 12.36/5.20 6[label="FiniteMap.foldFM_LE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];345[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 345[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 345 -> 7[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 346[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 346[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 346 -> 8[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 7[label="FiniteMap.foldFM_LE wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 12.36/5.20 8[label="FiniteMap.foldFM_LE wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 12.36/5.20 9[label="FiniteMap.foldFM_LE3 wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 12.36/5.20 10[label="FiniteMap.foldFM_LE2 wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 12.36/5.20 11[label="wz4",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 <= wz5)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 12.36/5.20 13[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= GT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 12.36/5.20 14[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == GT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 12.36/5.20 15[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (primCmpChar wz60 wz5 == GT))",fontsize=16,color="burlywood",shape="box"];347[label="wz60/Char wz600",fontsize=10,color="white",style="solid",shape="box"];15 -> 347[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 347 -> 16[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 16[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpChar (Char wz600) wz5 == GT))",fontsize=16,color="burlywood",shape="box"];348[label="wz5/Char wz50",fontsize=10,color="white",style="solid",shape="box"];16 -> 348[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 348 -> 17[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 17[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char wz50) (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpChar (Char wz600) (Char wz50) == GT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3]; 12.36/5.20 18[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char wz50) (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpNat wz600 wz50 == GT))",fontsize=16,color="burlywood",shape="box"];349[label="wz600/Succ wz6000",fontsize=10,color="white",style="solid",shape="box"];18 -> 349[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 349 -> 19[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 350[label="wz600/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 350[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 350 -> 20[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 19[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char wz50) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat (Succ wz6000) wz50 == GT))",fontsize=16,color="burlywood",shape="box"];351[label="wz50/Succ wz500",fontsize=10,color="white",style="solid",shape="box"];19 -> 351[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 351 -> 21[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 352[label="wz50/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 352[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 352 -> 22[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 20[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char wz50) (Char Zero) wz61 wz62 wz63 wz64 (not (primCmpNat Zero wz50 == GT))",fontsize=16,color="burlywood",shape="box"];353[label="wz50/Succ wz500",fontsize=10,color="white",style="solid",shape="box"];20 -> 353[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 353 -> 23[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 354[label="wz50/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 354[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 354 -> 24[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 21[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat (Succ wz6000) (Succ wz500) == GT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 12.36/5.20 22[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat (Succ wz6000) Zero == GT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 12.36/5.20 23[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 (not (primCmpNat Zero (Succ wz500) == GT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 12.36/5.20 24[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 12.36/5.20 25 -> 238[label="",style="dashed", color="red", weight=0]; 12.36/5.20 25[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (primCmpNat wz6000 wz500 == GT))",fontsize=16,color="magenta"];25 -> 239[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 240[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 241[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 242[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 243[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 244[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 245[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 246[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 247[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 25 -> 248[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 26[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not (GT == GT))",fontsize=16,color="black",shape="box"];26 -> 31[label="",style="solid", color="black", weight=3]; 12.36/5.20 27[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 (not (LT == GT))",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3]; 12.36/5.20 28[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 (not (EQ == GT))",fontsize=16,color="black",shape="box"];28 -> 33[label="",style="solid", color="black", weight=3]; 12.36/5.20 239[label="wz500",fontsize=16,color="green",shape="box"];240[label="wz3",fontsize=16,color="green",shape="box"];241[label="wz4",fontsize=16,color="green",shape="box"];242[label="wz500",fontsize=16,color="green",shape="box"];243[label="wz62",fontsize=16,color="green",shape="box"];244[label="wz6000",fontsize=16,color="green",shape="box"];245[label="wz61",fontsize=16,color="green",shape="box"];246[label="wz63",fontsize=16,color="green",shape="box"];247[label="wz64",fontsize=16,color="green",shape="box"];248[label="wz6000",fontsize=16,color="green",shape="box"];238[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat wz42 wz43 == GT))",fontsize=16,color="burlywood",shape="triangle"];355[label="wz42/Succ wz420",fontsize=10,color="white",style="solid",shape="box"];238 -> 355[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 355 -> 309[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 356[label="wz42/Zero",fontsize=10,color="white",style="solid",shape="box"];238 -> 356[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 356 -> 310[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 31[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 (not True)",fontsize=16,color="black",shape="box"];31 -> 38[label="",style="solid", color="black", weight=3]; 12.36/5.20 32[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3]; 12.36/5.20 33[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3]; 12.36/5.20 309[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) wz43 == GT))",fontsize=16,color="burlywood",shape="box"];357[label="wz43/Succ wz430",fontsize=10,color="white",style="solid",shape="box"];309 -> 357[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 357 -> 311[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 358[label="wz43/Zero",fontsize=10,color="white",style="solid",shape="box"];309 -> 358[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 358 -> 312[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 310[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat Zero wz43 == GT))",fontsize=16,color="burlywood",shape="box"];359[label="wz43/Succ wz430",fontsize=10,color="white",style="solid",shape="box"];310 -> 359[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 359 -> 313[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 360[label="wz43/Zero",fontsize=10,color="white",style="solid",shape="box"];310 -> 360[label="",style="solid", color="burlywood", weight=9]; 12.36/5.20 360 -> 314[label="",style="solid", color="burlywood", weight=3]; 12.36/5.20 38[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3]; 12.36/5.20 39[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3]; 12.36/5.20 40[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char Zero) (Char Zero) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3]; 12.36/5.20 311[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) (Succ wz430) == GT))",fontsize=16,color="black",shape="box"];311 -> 315[label="",style="solid", color="black", weight=3]; 12.36/5.20 312[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) Zero == GT))",fontsize=16,color="black",shape="box"];312 -> 316[label="",style="solid", color="black", weight=3]; 12.36/5.20 313[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat Zero (Succ wz430) == GT))",fontsize=16,color="black",shape="box"];313 -> 317[label="",style="solid", color="black", weight=3]; 12.36/5.20 314[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];314 -> 318[label="",style="solid", color="black", weight=3]; 12.36/5.20 45[label="FiniteMap.foldFM_LE0 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3]; 12.36/5.20 46 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 46[label="FiniteMap.foldFM_LE wz3 (wz3 (Char Zero) wz61 (FiniteMap.foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63)) (Char (Succ wz500)) wz64",fontsize=16,color="magenta"];46 -> 54[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 46 -> 55[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 46 -> 56[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 47 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 47[label="FiniteMap.foldFM_LE wz3 (wz3 (Char Zero) wz61 (FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63)) (Char Zero) wz64",fontsize=16,color="magenta"];47 -> 57[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 47 -> 58[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 47 -> 59[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 315 -> 238[label="",style="dashed", color="red", weight=0]; 12.36/5.20 315[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat wz420 wz430 == GT))",fontsize=16,color="magenta"];315 -> 319[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 315 -> 320[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 316[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (GT == GT))",fontsize=16,color="black",shape="box"];316 -> 321[label="",style="solid", color="black", weight=3]; 12.36/5.20 317[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (LT == GT))",fontsize=16,color="black",shape="box"];317 -> 322[label="",style="solid", color="black", weight=3]; 12.36/5.20 318[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (EQ == GT))",fontsize=16,color="black",shape="box"];318 -> 323[label="",style="solid", color="black", weight=3]; 12.36/5.20 53[label="FiniteMap.foldFM_LE0 wz3 wz4 (Char Zero) (Char (Succ wz6000)) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];53 -> 67[label="",style="solid", color="black", weight=3]; 12.36/5.20 54[label="wz3 (Char Zero) wz61 (FiniteMap.foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63)",fontsize=16,color="green",shape="box"];54 -> 68[label="",style="dashed", color="green", weight=3]; 12.36/5.20 54 -> 69[label="",style="dashed", color="green", weight=3]; 12.36/5.20 54 -> 70[label="",style="dashed", color="green", weight=3]; 12.36/5.20 55[label="wz64",fontsize=16,color="green",shape="box"];56[label="Char (Succ wz500)",fontsize=16,color="green",shape="box"];57[label="wz3 (Char Zero) wz61 (FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63)",fontsize=16,color="green",shape="box"];57 -> 71[label="",style="dashed", color="green", weight=3]; 12.36/5.20 57 -> 72[label="",style="dashed", color="green", weight=3]; 12.36/5.20 57 -> 73[label="",style="dashed", color="green", weight=3]; 12.36/5.20 58[label="wz64",fontsize=16,color="green",shape="box"];59[label="Char Zero",fontsize=16,color="green",shape="box"];319[label="wz430",fontsize=16,color="green",shape="box"];320[label="wz420",fontsize=16,color="green",shape="box"];321[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not True)",fontsize=16,color="black",shape="box"];321 -> 324[label="",style="solid", color="black", weight=3]; 12.36/5.20 322[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not False)",fontsize=16,color="black",shape="triangle"];322 -> 325[label="",style="solid", color="black", weight=3]; 12.36/5.20 323 -> 322[label="",style="dashed", color="red", weight=0]; 12.36/5.20 323[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not False)",fontsize=16,color="magenta"];67 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 67[label="FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63",fontsize=16,color="magenta"];67 -> 81[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 67 -> 82[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 68[label="Char Zero",fontsize=16,color="green",shape="box"];69[label="wz61",fontsize=16,color="green",shape="box"];70 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 70[label="FiniteMap.foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63",fontsize=16,color="magenta"];70 -> 83[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 70 -> 84[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 71[label="Char Zero",fontsize=16,color="green",shape="box"];72[label="wz61",fontsize=16,color="green",shape="box"];73 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 73[label="FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63",fontsize=16,color="magenta"];73 -> 85[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 73 -> 86[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 324[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 False",fontsize=16,color="black",shape="box"];324 -> 326[label="",style="solid", color="black", weight=3]; 12.36/5.20 325[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 True",fontsize=16,color="black",shape="box"];325 -> 327[label="",style="solid", color="black", weight=3]; 12.36/5.20 81[label="wz63",fontsize=16,color="green",shape="box"];82[label="Char Zero",fontsize=16,color="green",shape="box"];83[label="wz63",fontsize=16,color="green",shape="box"];84[label="Char (Succ wz500)",fontsize=16,color="green",shape="box"];85[label="wz63",fontsize=16,color="green",shape="box"];86[label="Char Zero",fontsize=16,color="green",shape="box"];326[label="FiniteMap.foldFM_LE0 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 otherwise",fontsize=16,color="black",shape="box"];326 -> 328[label="",style="solid", color="black", weight=3]; 12.36/5.20 327 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 327[label="FiniteMap.foldFM_LE wz34 (wz34 (Char (Succ wz37)) wz38 (FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40)) (Char (Succ wz36)) wz41",fontsize=16,color="magenta"];327 -> 329[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 327 -> 330[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 327 -> 331[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 327 -> 332[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 328[label="FiniteMap.foldFM_LE0 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 True",fontsize=16,color="black",shape="box"];328 -> 333[label="",style="solid", color="black", weight=3]; 12.36/5.20 329[label="wz34 (Char (Succ wz37)) wz38 (FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40)",fontsize=16,color="green",shape="box"];329 -> 334[label="",style="dashed", color="green", weight=3]; 12.36/5.20 329 -> 335[label="",style="dashed", color="green", weight=3]; 12.36/5.20 329 -> 336[label="",style="dashed", color="green", weight=3]; 12.36/5.20 330[label="wz41",fontsize=16,color="green",shape="box"];331[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];332[label="wz34",fontsize=16,color="green",shape="box"];333 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 333[label="FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40",fontsize=16,color="magenta"];333 -> 337[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 333 -> 338[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 333 -> 339[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 333 -> 340[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 334[label="Char (Succ wz37)",fontsize=16,color="green",shape="box"];335[label="wz38",fontsize=16,color="green",shape="box"];336 -> 6[label="",style="dashed", color="red", weight=0]; 12.36/5.20 336[label="FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40",fontsize=16,color="magenta"];336 -> 341[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 336 -> 342[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 336 -> 343[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 336 -> 344[label="",style="dashed", color="magenta", weight=3]; 12.36/5.20 337[label="wz35",fontsize=16,color="green",shape="box"];338[label="wz40",fontsize=16,color="green",shape="box"];339[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];340[label="wz34",fontsize=16,color="green",shape="box"];341[label="wz35",fontsize=16,color="green",shape="box"];342[label="wz40",fontsize=16,color="green",shape="box"];343[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];344[label="wz34",fontsize=16,color="green",shape="box"];} 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (6) 12.36/5.20 Obligation: 12.36/5.20 Q DP problem: 12.36/5.20 The TRS P consists of the following rules: 12.36/5.20 12.36/5.20 new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc) 12.36/5.20 new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc) 12.36/5.20 new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) -> new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) 12.36/5.20 new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) -> new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba) 12.36/5.20 new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc) 12.36/5.20 new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc) 12.36/5.20 12.36/5.20 R is empty. 12.36/5.20 Q is empty. 12.36/5.20 We have to consider all minimal (P,Q,R)-chains. 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (7) DependencyGraphProof (EQUIVALENT) 12.36/5.20 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (8) 12.36/5.20 Complex Obligation (AND) 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (9) 12.36/5.20 Obligation: 12.36/5.20 Q DP problem: 12.36/5.20 The TRS P consists of the following rules: 12.36/5.20 12.36/5.20 new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc) 12.36/5.20 new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) -> new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) 12.36/5.20 new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba) 12.36/5.20 new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) -> new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba) 12.36/5.20 12.36/5.20 R is empty. 12.36/5.20 Q is empty. 12.36/5.20 We have to consider all minimal (P,Q,R)-chains. 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (10) QDPSizeChangeProof (EQUIVALENT) 12.36/5.20 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. 12.36/5.20 12.36/5.20 From the DPs we obtained the following set of size-change graphs: 12.36/5.20 *new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 3 > 8, 2 > 9, 4 >= 10, 5 >= 11 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) -> new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba) 12.36/5.20 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, 11 >= 11 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) -> new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8, 11 >= 9 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4, 11 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4, 11 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba) 12.36/5.20 The graph contains the following edges 1 >= 1, 7 >= 3, 10 >= 4, 11 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba) 12.36/5.20 The graph contains the following edges 1 >= 1, 7 >= 3, 8 >= 4, 9 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba) 12.36/5.20 The graph contains the following edges 1 >= 1, 6 >= 3, 8 >= 4, 9 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (11) 12.36/5.20 YES 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (12) 12.36/5.20 Obligation: 12.36/5.20 Q DP problem: 12.36/5.20 The TRS P consists of the following rules: 12.36/5.20 12.36/5.20 new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc) 12.36/5.20 new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc) 12.36/5.20 new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc) 12.36/5.20 12.36/5.20 R is empty. 12.36/5.20 Q is empty. 12.36/5.20 We have to consider all minimal (P,Q,R)-chains. 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (13) QDPSizeChangeProof (EQUIVALENT) 12.36/5.20 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. 12.36/5.20 12.36/5.20 From the DPs we obtained the following set of size-change graphs: 12.36/5.20 *new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 *new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc) 12.36/5.20 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5 12.36/5.20 12.36/5.20 12.36/5.20 ---------------------------------------- 12.36/5.20 12.36/5.20 (14) 12.36/5.20 YES 12.65/5.24 EOF