9.78/4.44 YES 12.26/5.05 proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs 12.26/5.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.26/5.05 12.26/5.05 12.26/5.05 H-Termination with start terms of the given HASKELL could be proven: 12.26/5.05 12.26/5.05 (0) HASKELL 12.26/5.05 (1) LR [EQUIVALENT, 0 ms] 12.26/5.05 (2) HASKELL 12.26/5.05 (3) BR [EQUIVALENT, 0 ms] 12.26/5.05 (4) HASKELL 12.26/5.05 (5) COR [EQUIVALENT, 0 ms] 12.26/5.05 (6) HASKELL 12.26/5.05 (7) Narrow [SOUND, 0 ms] 12.26/5.05 (8) AND 12.26/5.05 (9) QDP 12.26/5.05 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.26/5.05 (11) YES 12.26/5.05 (12) QDP 12.26/5.05 (13) TransformationProof [EQUIVALENT, 0 ms] 12.26/5.05 (14) QDP 12.26/5.05 (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.26/5.05 (16) YES 12.26/5.05 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (0) 12.26/5.05 Obligation: 12.26/5.05 mainModule Main 12.26/5.05 module FiniteMap where { 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 data FiniteMap a b = EmptyFM | Branch a b Int (FiniteMap a b) (FiniteMap a b) ; 12.26/5.05 12.26/5.05 instance (Eq a, Eq b) => Eq FiniteMap b a where { 12.26/5.05 } 12.26/5.05 foldFM_GE :: Ord b => (b -> a -> c -> c) -> c -> b -> FiniteMap b a -> c; 12.26/5.05 foldFM_GE k z fr EmptyFM = z; 12.26/5.05 foldFM_GE k z fr (Branch key elt _ fm_l fm_r) | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l 12.26/5.05 | otherwise = foldFM_GE k z fr fm_r; 12.26/5.05 12.26/5.05 keysFM_GE :: Ord a => FiniteMap a b -> a -> [a]; 12.26/5.05 keysFM_GE fm fr = foldFM_GE (\key elt rest ->key : rest) [] fr fm; 12.26/5.05 12.26/5.05 } 12.26/5.05 module Maybe where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 module Main where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (1) LR (EQUIVALENT) 12.26/5.05 Lambda Reductions: 12.26/5.05 The following Lambda expression 12.26/5.05 "\keyeltrest->key : rest" 12.26/5.05 is transformed to 12.26/5.05 "keysFM_GE0 key elt rest = key : rest; 12.26/5.05 " 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (2) 12.26/5.05 Obligation: 12.26/5.05 mainModule Main 12.26/5.05 module FiniteMap where { 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 data FiniteMap b a = EmptyFM | Branch b a Int (FiniteMap b a) (FiniteMap b a) ; 12.26/5.05 12.26/5.05 instance (Eq a, Eq b) => Eq FiniteMap a b where { 12.26/5.05 } 12.26/5.05 foldFM_GE :: Ord a => (a -> b -> c -> c) -> c -> a -> FiniteMap a b -> c; 12.26/5.05 foldFM_GE k z fr EmptyFM = z; 12.26/5.05 foldFM_GE k z fr (Branch key elt _ fm_l fm_r) | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l 12.26/5.05 | otherwise = foldFM_GE k z fr fm_r; 12.26/5.05 12.26/5.05 keysFM_GE :: Ord b => FiniteMap b a -> b -> [b]; 12.26/5.05 keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm; 12.26/5.05 12.26/5.05 keysFM_GE0 key elt rest = key : rest; 12.26/5.05 12.26/5.05 } 12.26/5.05 module Maybe where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 module Main where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (3) BR (EQUIVALENT) 12.26/5.05 Replaced joker patterns by fresh variables and removed binding patterns. 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (4) 12.26/5.05 Obligation: 12.26/5.05 mainModule Main 12.26/5.05 module FiniteMap where { 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 data FiniteMap b a = EmptyFM | Branch b a Int (FiniteMap b a) (FiniteMap b a) ; 12.26/5.05 12.26/5.05 instance (Eq a, Eq b) => Eq FiniteMap b a where { 12.26/5.05 } 12.26/5.05 foldFM_GE :: Ord a => (a -> b -> c -> c) -> c -> a -> FiniteMap a b -> c; 12.26/5.05 foldFM_GE k z fr EmptyFM = z; 12.26/5.05 foldFM_GE k z fr (Branch key elt vy fm_l fm_r) | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l 12.26/5.05 | otherwise = foldFM_GE k z fr fm_r; 12.26/5.05 12.26/5.05 keysFM_GE :: Ord a => FiniteMap a b -> a -> [a]; 12.26/5.05 keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm; 12.26/5.05 12.26/5.05 keysFM_GE0 key elt rest = key : rest; 12.26/5.05 12.26/5.05 } 12.26/5.05 module Maybe where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 module Main where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (5) COR (EQUIVALENT) 12.26/5.05 Cond Reductions: 12.26/5.05 The following Function with conditions 12.26/5.05 "undefined |Falseundefined; 12.26/5.05 " 12.26/5.05 is transformed to 12.26/5.05 "undefined = undefined1; 12.26/5.05 " 12.26/5.05 "undefined0 True = undefined; 12.26/5.05 " 12.26/5.05 "undefined1 = undefined0 False; 12.26/5.05 " 12.26/5.05 The following Function with conditions 12.26/5.05 "foldFM_GE k z fr EmptyFM = z; 12.26/5.05 foldFM_GE k z fr (Branch key elt vy fm_l fm_r)|key >= frfoldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l|otherwisefoldFM_GE k z fr fm_r; 12.26/5.05 " 12.26/5.05 is transformed to 12.26/5.05 "foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM; 12.26/5.05 foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r); 12.26/5.05 " 12.26/5.05 "foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r; 12.26/5.05 " 12.26/5.05 "foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l; 12.26/5.05 foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise; 12.26/5.05 " 12.26/5.05 "foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr); 12.26/5.05 " 12.26/5.05 "foldFM_GE3 k z fr EmptyFM = z; 12.26/5.05 foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy; 12.26/5.05 " 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (6) 12.26/5.05 Obligation: 12.26/5.05 mainModule Main 12.26/5.05 module FiniteMap where { 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 data FiniteMap b a = EmptyFM | Branch b a Int (FiniteMap b a) (FiniteMap b a) ; 12.26/5.05 12.26/5.05 instance (Eq a, Eq b) => Eq FiniteMap a b where { 12.26/5.05 } 12.26/5.05 foldFM_GE :: Ord c => (c -> a -> b -> b) -> b -> c -> FiniteMap c a -> b; 12.26/5.05 foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM; 12.26/5.05 foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r); 12.26/5.05 12.26/5.05 foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r; 12.26/5.05 12.26/5.05 foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l; 12.26/5.05 foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise; 12.26/5.05 12.26/5.05 foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr); 12.26/5.05 12.26/5.05 foldFM_GE3 k z fr EmptyFM = z; 12.26/5.05 foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy; 12.26/5.05 12.26/5.05 keysFM_GE :: Ord a => FiniteMap a b -> a -> [a]; 12.26/5.05 keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm; 12.26/5.05 12.26/5.05 keysFM_GE0 key elt rest = key : rest; 12.26/5.05 12.26/5.05 } 12.26/5.05 module Maybe where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Main; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 module Main where { 12.26/5.05 import qualified FiniteMap; 12.26/5.05 import qualified Maybe; 12.26/5.05 import qualified Prelude; 12.26/5.05 } 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (7) Narrow (SOUND) 12.26/5.05 Haskell To QDPs 12.26/5.05 12.26/5.05 digraph dp_graph { 12.26/5.05 node [outthreshold=100, inthreshold=100];1[label="FiniteMap.keysFM_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 12.26/5.05 3[label="FiniteMap.keysFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 12.26/5.05 4[label="FiniteMap.keysFM_GE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 12.26/5.05 5[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];54[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 54[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 54 -> 6[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 55[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 55[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 55 -> 7[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 6[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 12.26/5.05 7[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 12.26/5.05 8[label="FiniteMap.foldFM_GE3 FiniteMap.keysFM_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 12.26/5.05 9[label="FiniteMap.foldFM_GE2 FiniteMap.keysFM_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 12.26/5.05 10[label="[]",fontsize=16,color="green",shape="box"];11[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 >= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 12.26/5.05 12[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= LT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 12.26/5.05 13[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == LT))",fontsize=16,color="burlywood",shape="box"];56[label="wz30/()",fontsize=10,color="white",style="solid",shape="box"];13 -> 56[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 56 -> 14[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 14[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == LT))",fontsize=16,color="burlywood",shape="box"];57[label="wz4/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 57[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 57 -> 15[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 15[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 12.26/5.05 16[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3]; 12.26/5.05 17[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] () () wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3]; 12.26/5.05 18[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] () () wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3]; 12.26/5.05 19 -> 20[label="",style="dashed", color="red", weight=0]; 12.26/5.05 19[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 (FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] () wz34)) () wz33",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 21 -> 5[label="",style="dashed", color="red", weight=0]; 12.26/5.05 21[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] () wz34",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 21 -> 23[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 20[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 wz5) () wz33",fontsize=16,color="burlywood",shape="triangle"];58[label="wz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];20 -> 58[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 58 -> 24[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 59[label="wz33/FiniteMap.Branch wz330 wz331 wz332 wz333 wz334",fontsize=10,color="white",style="solid",shape="box"];20 -> 59[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 59 -> 25[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 22[label="wz34",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3]; 12.26/5.05 25[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 wz5) () (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3]; 12.26/5.05 26[label="FiniteMap.foldFM_GE3 FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3]; 12.26/5.05 27[label="FiniteMap.foldFM_GE2 FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 wz5) () (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3]; 12.26/5.05 28[label="FiniteMap.keysFM_GE0 () wz31 wz5",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3]; 12.26/5.05 29 -> 31[label="",style="dashed", color="red", weight=0]; 12.26/5.05 29[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz31 wz5) () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 30[label="() : wz5",fontsize=16,color="green",shape="box"];32 -> 28[label="",style="dashed", color="red", weight=0]; 12.26/5.05 32[label="FiniteMap.keysFM_GE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3]; 12.26/5.05 33[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (compare wz330 () /= LT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3]; 12.26/5.05 34[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (not (compare wz330 () == LT))",fontsize=16,color="burlywood",shape="box"];60[label="wz330/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 60[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 60 -> 35[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 35[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3]; 12.26/5.05 36[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (EQ == LT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3]; 12.26/5.05 37[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3]; 12.26/5.05 38[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () () wz331 wz332 wz333 wz334 True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3]; 12.26/5.05 39 -> 20[label="",style="dashed", color="red", weight=0]; 12.26/5.05 39[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 () wz331 (FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz6 () wz334)) () wz333",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 39 -> 41[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 39 -> 42[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 40[label="wz333",fontsize=16,color="green",shape="box"];41[label="wz331",fontsize=16,color="green",shape="box"];42[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz6 () wz334",fontsize=16,color="burlywood",shape="box"];61[label="wz334/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];42 -> 61[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 61 -> 43[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 62[label="wz334/FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344",fontsize=10,color="white",style="solid",shape="box"];42 -> 62[label="",style="solid", color="burlywood", weight=9]; 12.26/5.05 62 -> 44[label="",style="solid", color="burlywood", weight=3]; 12.26/5.05 43[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3]; 12.26/5.05 44[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz6 () (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 12.26/5.05 45[label="FiniteMap.foldFM_GE3 FiniteMap.keysFM_GE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3]; 12.26/5.05 46[label="FiniteMap.foldFM_GE2 FiniteMap.keysFM_GE0 wz6 () (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3]; 12.26/5.05 47[label="wz6",fontsize=16,color="green",shape="box"];48 -> 31[label="",style="dashed", color="red", weight=0]; 12.26/5.05 48[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz6 () wz3340 wz3341 wz3342 wz3343 wz3344 (wz3340 >= ())",fontsize=16,color="magenta"];48 -> 49[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 48 -> 50[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 48 -> 51[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 48 -> 52[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 48 -> 53[label="",style="dashed", color="magenta", weight=3]; 12.26/5.05 49[label="wz3340",fontsize=16,color="green",shape="box"];50[label="wz3344",fontsize=16,color="green",shape="box"];51[label="wz3343",fontsize=16,color="green",shape="box"];52[label="wz3342",fontsize=16,color="green",shape="box"];53[label="wz3341",fontsize=16,color="green",shape="box"];} 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (8) 12.26/5.05 Complex Obligation (AND) 12.26/5.05 12.26/5.05 ---------------------------------------- 12.26/5.05 12.26/5.05 (9) 12.26/5.05 Obligation: 12.26/5.05 Q DP problem: 12.26/5.05 The TRS P consists of the following rules: 12.26/5.05 12.26/5.05 new_foldFM_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE3(@0, wz34, h) 12.26/5.05 12.26/5.05 R is empty. 12.26/5.05 Q is empty. 12.26/5.05 We have to consider all minimal (P,Q,R)-chains. 12.26/5.05 ---------------------------------------- 12.26/5.06 12.26/5.06 (10) QDPSizeChangeProof (EQUIVALENT) 12.26/5.06 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. 12.26/5.06 12.26/5.06 From the DPs we obtained the following set of size-change graphs: 12.26/5.06 *new_foldFM_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE3(@0, wz34, h) 12.26/5.06 The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3 12.26/5.06 12.26/5.06 12.26/5.06 ---------------------------------------- 12.26/5.06 12.26/5.06 (11) 12.26/5.06 YES 12.26/5.06 12.26/5.06 ---------------------------------------- 12.26/5.06 12.26/5.06 (12) 12.26/5.06 Obligation: 12.26/5.06 Q DP problem: 12.26/5.06 The TRS P consists of the following rules: 12.26/5.06 12.26/5.06 new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h) 12.26/5.06 new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h) 12.26/5.06 new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h) 12.26/5.06 12.26/5.06 The TRS R consists of the following rules: 12.26/5.06 12.26/5.06 new_foldFM_GE0(wz6, EmptyFM, h) -> wz6 12.26/5.06 new_keysFM_GE0(wz31, wz5, h) -> :(@0, wz5) 12.26/5.06 new_foldFM_GE2(wz31, wz5, EmptyFM, h) -> new_keysFM_GE0(wz31, wz5, h) 12.26/5.06 new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h) 12.26/5.06 new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h) 12.26/5.06 new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h) 12.26/5.06 12.26/5.06 The set Q consists of the following terms: 12.26/5.06 12.26/5.06 new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6) 12.26/5.06 new_foldFM_GE2(x0, x1, EmptyFM, x2) 12.26/5.06 new_keysFM_GE0(x0, x1, x2) 12.26/5.06 new_foldFM_GE0(x0, EmptyFM, x1) 12.26/5.06 new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5) 12.26/5.06 new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7) 12.26/5.06 12.26/5.06 We have to consider all minimal (P,Q,R)-chains. 12.26/5.06 ---------------------------------------- 12.26/5.06 12.26/5.06 (13) TransformationProof (EQUIVALENT) 12.26/5.06 By rewriting [LPAR04] the rule new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h) at position [0] we obtained the following new rules [LPAR04]: 12.26/5.06 12.26/5.06 (new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@0, wz5), wz330, wz331, wz332, wz333, wz334, h),new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@0, wz5), wz330, wz331, wz332, wz333, wz334, h)) 12.26/5.06 12.26/5.06 12.26/5.06 ---------------------------------------- 12.26/5.06 12.26/5.06 (14) 12.26/5.06 Obligation: 12.26/5.06 Q DP problem: 12.26/5.06 The TRS P consists of the following rules: 12.26/5.06 12.26/5.06 new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h) 12.26/5.06 new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h) 12.26/5.06 new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@0, wz5), wz330, wz331, wz332, wz333, wz334, h) 12.26/5.06 12.26/5.06 The TRS R consists of the following rules: 12.26/5.06 12.26/5.06 new_foldFM_GE0(wz6, EmptyFM, h) -> wz6 12.26/5.06 new_keysFM_GE0(wz31, wz5, h) -> :(@0, wz5) 12.26/5.06 new_foldFM_GE2(wz31, wz5, EmptyFM, h) -> new_keysFM_GE0(wz31, wz5, h) 12.26/5.06 new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h) 12.26/5.06 new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h) 12.26/5.06 new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_keysFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h) 12.26/5.06 12.26/5.06 The set Q consists of the following terms: 12.26/5.06 12.26/5.06 new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6) 12.26/5.06 new_foldFM_GE2(x0, x1, EmptyFM, x2) 12.26/5.06 new_keysFM_GE0(x0, x1, x2) 12.26/5.06 new_foldFM_GE0(x0, EmptyFM, x1) 12.26/5.06 new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5) 12.26/5.06 new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7) 12.26/5.06 12.26/5.06 We have to consider all minimal (P,Q,R)-chains. 12.26/5.06 ---------------------------------------- 12.26/5.06 12.26/5.06 (15) QDPSizeChangeProof (EQUIVALENT) 12.26/5.06 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. 12.26/5.06 12.26/5.06 From the DPs we obtained the following set of size-change graphs: 12.26/5.06 *new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h) 12.26/5.06 The graph contains the following edges 1 >= 1, 6 > 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 7 >= 7 12.26/5.06 12.26/5.06 12.26/5.06 *new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h) 12.26/5.06 The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4 12.26/5.06 12.26/5.06 12.26/5.06 *new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@0, wz5), wz330, wz331, wz332, wz333, wz334, h) 12.26/5.06 The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7 12.26/5.06 12.26/5.06 12.26/5.06 ---------------------------------------- 12.26/5.06 12.26/5.06 (16) 12.26/5.06 YES 12.30/5.10 EOF