10.37/4.38 YES 12.15/4.88 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 12.15/4.88 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.15/4.88 12.15/4.88 12.15/4.88 H-Termination with start terms of the given HASKELL could be proven: 12.15/4.88 12.15/4.88 (0) HASKELL 12.15/4.88 (1) BR [EQUIVALENT, 0 ms] 12.15/4.88 (2) HASKELL 12.15/4.88 (3) COR [EQUIVALENT, 25 ms] 12.15/4.88 (4) HASKELL 12.15/4.88 (5) Narrow [SOUND, 0 ms] 12.15/4.88 (6) AND 12.15/4.88 (7) QDP 12.15/4.88 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.15/4.88 (9) YES 12.15/4.88 (10) QDP 12.15/4.88 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.15/4.88 (12) YES 12.15/4.88 (13) QDP 12.15/4.88 (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.15/4.88 (15) YES 12.15/4.88 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (0) 12.15/4.88 Obligation: 12.15/4.88 mainModule Main 12.15/4.88 module Maybe where { 12.15/4.88 import qualified List; 12.15/4.88 import qualified Main; 12.15/4.88 import qualified Prelude; 12.15/4.88 } 12.15/4.88 module List where { 12.15/4.88 import qualified Main; 12.15/4.88 import qualified Maybe; 12.15/4.88 import qualified Prelude; 12.15/4.88 isPrefixOf :: Eq a => [a] -> [a] -> Bool; 12.15/4.88 isPrefixOf [] _ = True; 12.15/4.88 isPrefixOf _ [] = False; 12.15/4.88 isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys; 12.15/4.88 12.15/4.88 } 12.15/4.88 module Main where { 12.15/4.88 import qualified List; 12.15/4.88 import qualified Maybe; 12.15/4.88 import qualified Prelude; 12.15/4.88 } 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (1) BR (EQUIVALENT) 12.15/4.88 Replaced joker patterns by fresh variables and removed binding patterns. 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (2) 12.15/4.88 Obligation: 12.15/4.88 mainModule Main 12.15/4.88 module Maybe where { 12.15/4.88 import qualified List; 12.15/4.88 import qualified Main; 12.15/4.88 import qualified Prelude; 12.15/4.88 } 12.15/4.88 module List where { 12.15/4.88 import qualified Main; 12.15/4.88 import qualified Maybe; 12.15/4.88 import qualified Prelude; 12.15/4.88 isPrefixOf :: Eq a => [a] -> [a] -> Bool; 12.15/4.88 isPrefixOf [] wu = True; 12.15/4.88 isPrefixOf wv [] = False; 12.15/4.88 isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys; 12.15/4.88 12.15/4.88 } 12.15/4.88 module Main where { 12.15/4.88 import qualified List; 12.15/4.88 import qualified Maybe; 12.15/4.88 import qualified Prelude; 12.15/4.88 } 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (3) COR (EQUIVALENT) 12.15/4.88 Cond Reductions: 12.15/4.88 The following Function with conditions 12.15/4.88 "undefined |Falseundefined; 12.15/4.88 " 12.15/4.88 is transformed to 12.15/4.88 "undefined = undefined1; 12.15/4.88 " 12.15/4.88 "undefined0 True = undefined; 12.15/4.88 " 12.15/4.88 "undefined1 = undefined0 False; 12.15/4.88 " 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (4) 12.15/4.88 Obligation: 12.15/4.88 mainModule Main 12.15/4.88 module Maybe where { 12.15/4.88 import qualified List; 12.15/4.88 import qualified Main; 12.15/4.88 import qualified Prelude; 12.15/4.88 } 12.15/4.88 module List where { 12.15/4.88 import qualified Main; 12.15/4.88 import qualified Maybe; 12.15/4.88 import qualified Prelude; 12.15/4.88 isPrefixOf :: Eq a => [a] -> [a] -> Bool; 12.15/4.88 isPrefixOf [] wu = True; 12.15/4.88 isPrefixOf wv [] = False; 12.15/4.88 isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys; 12.15/4.88 12.15/4.88 } 12.15/4.88 module Main where { 12.15/4.88 import qualified List; 12.15/4.88 import qualified Maybe; 12.15/4.88 import qualified Prelude; 12.15/4.88 } 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (5) Narrow (SOUND) 12.15/4.88 Haskell To QDPs 12.15/4.88 12.15/4.88 digraph dp_graph { 12.15/4.88 node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 12.15/4.88 3[label="List.isPrefixOf ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 12.15/4.88 4[label="List.isPrefixOf ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];133[label="ww3/ww30 : ww31",fontsize=10,color="white",style="solid",shape="box"];4 -> 133[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 133 -> 5[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 134[label="ww3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 134[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 134 -> 6[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 5[label="List.isPrefixOf (ww30 : ww31) ww4",fontsize=16,color="burlywood",shape="box"];135[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 135[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 135 -> 7[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 136[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 136[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 136 -> 8[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 6[label="List.isPrefixOf [] ww4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3]; 12.15/4.88 7[label="List.isPrefixOf (ww30 : ww31) (ww40 : ww41)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 12.15/4.88 8[label="List.isPrefixOf (ww30 : ww31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 12.15/4.88 9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0]; 12.15/4.88 10[label="ww30 == ww40 && List.isPrefixOf ww31 ww41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0]; 12.15/4.88 13[label="List.isPrefixOf ww31 ww41",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 13 -> 15[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 12[label="ww30 == ww40 && ww5",fontsize=16,color="burlywood",shape="triangle"];137[label="ww30/ww300 :% ww301",fontsize=10,color="white",style="solid",shape="box"];12 -> 137[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 137 -> 16[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 14[label="ww31",fontsize=16,color="green",shape="box"];15[label="ww41",fontsize=16,color="green",shape="box"];16[label="ww300 :% ww301 == ww40 && ww5",fontsize=16,color="burlywood",shape="box"];138[label="ww40/ww400 :% ww401",fontsize=10,color="white",style="solid",shape="box"];16 -> 138[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 138 -> 17[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 17[label="ww300 :% ww301 == ww400 :% ww401 && ww5",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3]; 12.15/4.88 18[label="(ww300 == ww400 && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3]; 12.15/4.88 19[label="(primEqInt ww300 ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];139[label="ww300/Pos ww3000",fontsize=10,color="white",style="solid",shape="box"];19 -> 139[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 139 -> 20[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 140[label="ww300/Neg ww3000",fontsize=10,color="white",style="solid",shape="box"];19 -> 140[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 140 -> 21[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 20[label="(primEqInt (Pos ww3000) ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];141[label="ww3000/Succ ww30000",fontsize=10,color="white",style="solid",shape="box"];20 -> 141[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 141 -> 22[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 142[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 142[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 142 -> 23[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 21[label="(primEqInt (Neg ww3000) ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];143[label="ww3000/Succ ww30000",fontsize=10,color="white",style="solid",shape="box"];21 -> 143[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 143 -> 24[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 144[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 144[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 144 -> 25[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 22[label="(primEqInt (Pos (Succ ww30000)) ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];145[label="ww400/Pos ww4000",fontsize=10,color="white",style="solid",shape="box"];22 -> 145[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 145 -> 26[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 146[label="ww400/Neg ww4000",fontsize=10,color="white",style="solid",shape="box"];22 -> 146[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 146 -> 27[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 23[label="(primEqInt (Pos Zero) ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];147[label="ww400/Pos ww4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 147[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 147 -> 28[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 148[label="ww400/Neg ww4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 148[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 148 -> 29[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 24[label="(primEqInt (Neg (Succ ww30000)) ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];149[label="ww400/Pos ww4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 149[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 149 -> 30[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 150[label="ww400/Neg ww4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 150[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 150 -> 31[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 25[label="(primEqInt (Neg Zero) ww400 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];151[label="ww400/Pos ww4000",fontsize=10,color="white",style="solid",shape="box"];25 -> 151[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 151 -> 32[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 152[label="ww400/Neg ww4000",fontsize=10,color="white",style="solid",shape="box"];25 -> 152[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 152 -> 33[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 26[label="(primEqInt (Pos (Succ ww30000)) (Pos ww4000) && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];153[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];26 -> 153[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 153 -> 34[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 154[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 154[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 154 -> 35[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 27[label="(primEqInt (Pos (Succ ww30000)) (Neg ww4000) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3]; 12.15/4.88 28[label="(primEqInt (Pos Zero) (Pos ww4000) && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];155[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];28 -> 155[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 155 -> 37[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 156[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 156[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 156 -> 38[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 29[label="(primEqInt (Pos Zero) (Neg ww4000) && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];157[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];29 -> 157[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 157 -> 39[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 158[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 158[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 158 -> 40[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 30[label="(primEqInt (Neg (Succ ww30000)) (Pos ww4000) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];30 -> 41[label="",style="solid", color="black", weight=3]; 12.15/4.88 31[label="(primEqInt (Neg (Succ ww30000)) (Neg ww4000) && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];159[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];31 -> 159[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 159 -> 42[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 160[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 160[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 160 -> 43[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 32[label="(primEqInt (Neg Zero) (Pos ww4000) && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];161[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];32 -> 161[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 161 -> 44[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 162[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 162[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 162 -> 45[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 33[label="(primEqInt (Neg Zero) (Neg ww4000) && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];163[label="ww4000/Succ ww40000",fontsize=10,color="white",style="solid",shape="box"];33 -> 163[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 163 -> 46[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 164[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 164[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 164 -> 47[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 34[label="(primEqInt (Pos (Succ ww30000)) (Pos (Succ ww40000)) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3]; 12.15/4.88 35[label="(primEqInt (Pos (Succ ww30000)) (Pos Zero) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3]; 12.15/4.88 36[label="(False && ww301 == ww401) && ww5",fontsize=16,color="black",shape="triangle"];36 -> 50[label="",style="solid", color="black", weight=3]; 12.15/4.88 37[label="(primEqInt (Pos Zero) (Pos (Succ ww40000)) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3]; 12.15/4.88 38[label="(primEqInt (Pos Zero) (Pos Zero) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3]; 12.15/4.88 39[label="(primEqInt (Pos Zero) (Neg (Succ ww40000)) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3]; 12.15/4.88 40[label="(primEqInt (Pos Zero) (Neg Zero) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3]; 12.15/4.88 41 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 41[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];42[label="(primEqInt (Neg (Succ ww30000)) (Neg (Succ ww40000)) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3]; 12.15/4.88 43[label="(primEqInt (Neg (Succ ww30000)) (Neg Zero) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3]; 12.15/4.88 44[label="(primEqInt (Neg Zero) (Pos (Succ ww40000)) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3]; 12.15/4.88 45[label="(primEqInt (Neg Zero) (Pos Zero) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3]; 12.15/4.88 46[label="(primEqInt (Neg Zero) (Neg (Succ ww40000)) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3]; 12.15/4.88 47[label="(primEqInt (Neg Zero) (Neg Zero) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3]; 12.15/4.88 48[label="(primEqNat ww30000 ww40000 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="triangle"];165[label="ww30000/Succ ww300000",fontsize=10,color="white",style="solid",shape="box"];48 -> 165[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 165 -> 61[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 166[label="ww30000/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 166[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 166 -> 62[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 49 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 49[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];50[label="False && ww5",fontsize=16,color="black",shape="triangle"];50 -> 63[label="",style="solid", color="black", weight=3]; 12.15/4.88 51 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 51[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];52[label="(True && ww301 == ww401) && ww5",fontsize=16,color="black",shape="triangle"];52 -> 64[label="",style="solid", color="black", weight=3]; 12.15/4.88 53 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 53[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];54 -> 52[label="",style="dashed", color="red", weight=0]; 12.15/4.88 54[label="(True && ww301 == ww401) && ww5",fontsize=16,color="magenta"];55 -> 48[label="",style="dashed", color="red", weight=0]; 12.15/4.88 55[label="(primEqNat ww30000 ww40000 && ww301 == ww401) && ww5",fontsize=16,color="magenta"];55 -> 65[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 55 -> 66[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 56 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 56[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];57 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 57[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];58 -> 52[label="",style="dashed", color="red", weight=0]; 12.15/4.88 58[label="(True && ww301 == ww401) && ww5",fontsize=16,color="magenta"];59 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 59[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];60 -> 52[label="",style="dashed", color="red", weight=0]; 12.15/4.88 60[label="(True && ww301 == ww401) && ww5",fontsize=16,color="magenta"];61[label="(primEqNat (Succ ww300000) ww40000 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];167[label="ww40000/Succ ww400000",fontsize=10,color="white",style="solid",shape="box"];61 -> 167[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 167 -> 67[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 168[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 168[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 168 -> 68[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 62[label="(primEqNat Zero ww40000 && ww301 == ww401) && ww5",fontsize=16,color="burlywood",shape="box"];169[label="ww40000/Succ ww400000",fontsize=10,color="white",style="solid",shape="box"];62 -> 169[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 169 -> 69[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 170[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 170[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 170 -> 70[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 63[label="False",fontsize=16,color="green",shape="box"];64[label="ww301 == ww401 && ww5",fontsize=16,color="black",shape="box"];64 -> 71[label="",style="solid", color="black", weight=3]; 12.15/4.88 65[label="ww30000",fontsize=16,color="green",shape="box"];66[label="ww40000",fontsize=16,color="green",shape="box"];67[label="(primEqNat (Succ ww300000) (Succ ww400000) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];67 -> 72[label="",style="solid", color="black", weight=3]; 12.15/4.88 68[label="(primEqNat (Succ ww300000) Zero && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3]; 12.15/4.88 69[label="(primEqNat Zero (Succ ww400000) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3]; 12.15/4.88 70[label="(primEqNat Zero Zero && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];70 -> 75[label="",style="solid", color="black", weight=3]; 12.15/4.88 71[label="primEqInt ww301 ww401 && ww5",fontsize=16,color="burlywood",shape="box"];171[label="ww301/Pos ww3010",fontsize=10,color="white",style="solid",shape="box"];71 -> 171[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 171 -> 76[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 172[label="ww301/Neg ww3010",fontsize=10,color="white",style="solid",shape="box"];71 -> 172[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 172 -> 77[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 72 -> 48[label="",style="dashed", color="red", weight=0]; 12.15/4.88 72[label="(primEqNat ww300000 ww400000 && ww301 == ww401) && ww5",fontsize=16,color="magenta"];72 -> 78[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 72 -> 79[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 73 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 73[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];74 -> 36[label="",style="dashed", color="red", weight=0]; 12.15/4.88 74[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];75 -> 52[label="",style="dashed", color="red", weight=0]; 12.15/4.88 75[label="(True && ww301 == ww401) && ww5",fontsize=16,color="magenta"];76[label="primEqInt (Pos ww3010) ww401 && ww5",fontsize=16,color="burlywood",shape="box"];173[label="ww3010/Succ ww30100",fontsize=10,color="white",style="solid",shape="box"];76 -> 173[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 173 -> 80[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 174[label="ww3010/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 174[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 174 -> 81[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 77[label="primEqInt (Neg ww3010) ww401 && ww5",fontsize=16,color="burlywood",shape="box"];175[label="ww3010/Succ ww30100",fontsize=10,color="white",style="solid",shape="box"];77 -> 175[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 175 -> 82[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 176[label="ww3010/Zero",fontsize=10,color="white",style="solid",shape="box"];77 -> 176[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 176 -> 83[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 78[label="ww300000",fontsize=16,color="green",shape="box"];79[label="ww400000",fontsize=16,color="green",shape="box"];80[label="primEqInt (Pos (Succ ww30100)) ww401 && ww5",fontsize=16,color="burlywood",shape="box"];177[label="ww401/Pos ww4010",fontsize=10,color="white",style="solid",shape="box"];80 -> 177[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 177 -> 84[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 178[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];80 -> 178[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 178 -> 85[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 81[label="primEqInt (Pos Zero) ww401 && ww5",fontsize=16,color="burlywood",shape="box"];179[label="ww401/Pos ww4010",fontsize=10,color="white",style="solid",shape="box"];81 -> 179[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 179 -> 86[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 180[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];81 -> 180[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 180 -> 87[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 82[label="primEqInt (Neg (Succ ww30100)) ww401 && ww5",fontsize=16,color="burlywood",shape="box"];181[label="ww401/Pos ww4010",fontsize=10,color="white",style="solid",shape="box"];82 -> 181[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 181 -> 88[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 182[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];82 -> 182[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 182 -> 89[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 83[label="primEqInt (Neg Zero) ww401 && ww5",fontsize=16,color="burlywood",shape="box"];183[label="ww401/Pos ww4010",fontsize=10,color="white",style="solid",shape="box"];83 -> 183[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 183 -> 90[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 184[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];83 -> 184[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 184 -> 91[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 84[label="primEqInt (Pos (Succ ww30100)) (Pos ww4010) && ww5",fontsize=16,color="burlywood",shape="box"];185[label="ww4010/Succ ww40100",fontsize=10,color="white",style="solid",shape="box"];84 -> 185[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 185 -> 92[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 186[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];84 -> 186[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 186 -> 93[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 85[label="primEqInt (Pos (Succ ww30100)) (Neg ww4010) && ww5",fontsize=16,color="black",shape="box"];85 -> 94[label="",style="solid", color="black", weight=3]; 12.15/4.88 86[label="primEqInt (Pos Zero) (Pos ww4010) && ww5",fontsize=16,color="burlywood",shape="box"];187[label="ww4010/Succ ww40100",fontsize=10,color="white",style="solid",shape="box"];86 -> 187[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 187 -> 95[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 188[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];86 -> 188[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 188 -> 96[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 87[label="primEqInt (Pos Zero) (Neg ww4010) && ww5",fontsize=16,color="burlywood",shape="box"];189[label="ww4010/Succ ww40100",fontsize=10,color="white",style="solid",shape="box"];87 -> 189[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 189 -> 97[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 190[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];87 -> 190[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 190 -> 98[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 88[label="primEqInt (Neg (Succ ww30100)) (Pos ww4010) && ww5",fontsize=16,color="black",shape="box"];88 -> 99[label="",style="solid", color="black", weight=3]; 12.15/4.88 89[label="primEqInt (Neg (Succ ww30100)) (Neg ww4010) && ww5",fontsize=16,color="burlywood",shape="box"];191[label="ww4010/Succ ww40100",fontsize=10,color="white",style="solid",shape="box"];89 -> 191[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 191 -> 100[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 192[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];89 -> 192[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 192 -> 101[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 90[label="primEqInt (Neg Zero) (Pos ww4010) && ww5",fontsize=16,color="burlywood",shape="box"];193[label="ww4010/Succ ww40100",fontsize=10,color="white",style="solid",shape="box"];90 -> 193[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 193 -> 102[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 194[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];90 -> 194[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 194 -> 103[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 91[label="primEqInt (Neg Zero) (Neg ww4010) && ww5",fontsize=16,color="burlywood",shape="box"];195[label="ww4010/Succ ww40100",fontsize=10,color="white",style="solid",shape="box"];91 -> 195[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 195 -> 104[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 196[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];91 -> 196[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 196 -> 105[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 92[label="primEqInt (Pos (Succ ww30100)) (Pos (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];92 -> 106[label="",style="solid", color="black", weight=3]; 12.15/4.88 93[label="primEqInt (Pos (Succ ww30100)) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];93 -> 107[label="",style="solid", color="black", weight=3]; 12.15/4.88 94 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 94[label="False && ww5",fontsize=16,color="magenta"];95[label="primEqInt (Pos Zero) (Pos (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];95 -> 108[label="",style="solid", color="black", weight=3]; 12.15/4.88 96[label="primEqInt (Pos Zero) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];96 -> 109[label="",style="solid", color="black", weight=3]; 12.15/4.88 97[label="primEqInt (Pos Zero) (Neg (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];97 -> 110[label="",style="solid", color="black", weight=3]; 12.15/4.88 98[label="primEqInt (Pos Zero) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];98 -> 111[label="",style="solid", color="black", weight=3]; 12.15/4.88 99 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 99[label="False && ww5",fontsize=16,color="magenta"];100[label="primEqInt (Neg (Succ ww30100)) (Neg (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];100 -> 112[label="",style="solid", color="black", weight=3]; 12.15/4.88 101[label="primEqInt (Neg (Succ ww30100)) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];101 -> 113[label="",style="solid", color="black", weight=3]; 12.15/4.88 102[label="primEqInt (Neg Zero) (Pos (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];102 -> 114[label="",style="solid", color="black", weight=3]; 12.15/4.88 103[label="primEqInt (Neg Zero) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];103 -> 115[label="",style="solid", color="black", weight=3]; 12.15/4.88 104[label="primEqInt (Neg Zero) (Neg (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];104 -> 116[label="",style="solid", color="black", weight=3]; 12.15/4.88 105[label="primEqInt (Neg Zero) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];105 -> 117[label="",style="solid", color="black", weight=3]; 12.15/4.88 106[label="primEqNat ww30100 ww40100 && ww5",fontsize=16,color="burlywood",shape="triangle"];197[label="ww30100/Succ ww301000",fontsize=10,color="white",style="solid",shape="box"];106 -> 197[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 197 -> 118[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 198[label="ww30100/Zero",fontsize=10,color="white",style="solid",shape="box"];106 -> 198[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 198 -> 119[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 107 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 107[label="False && ww5",fontsize=16,color="magenta"];108 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 108[label="False && ww5",fontsize=16,color="magenta"];109[label="True && ww5",fontsize=16,color="black",shape="triangle"];109 -> 120[label="",style="solid", color="black", weight=3]; 12.15/4.88 110 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 110[label="False && ww5",fontsize=16,color="magenta"];111 -> 109[label="",style="dashed", color="red", weight=0]; 12.15/4.88 111[label="True && ww5",fontsize=16,color="magenta"];112 -> 106[label="",style="dashed", color="red", weight=0]; 12.15/4.88 112[label="primEqNat ww30100 ww40100 && ww5",fontsize=16,color="magenta"];112 -> 121[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 112 -> 122[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 113 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 113[label="False && ww5",fontsize=16,color="magenta"];114 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 114[label="False && ww5",fontsize=16,color="magenta"];115 -> 109[label="",style="dashed", color="red", weight=0]; 12.15/4.88 115[label="True && ww5",fontsize=16,color="magenta"];116 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 116[label="False && ww5",fontsize=16,color="magenta"];117 -> 109[label="",style="dashed", color="red", weight=0]; 12.15/4.88 117[label="True && ww5",fontsize=16,color="magenta"];118[label="primEqNat (Succ ww301000) ww40100 && ww5",fontsize=16,color="burlywood",shape="box"];199[label="ww40100/Succ ww401000",fontsize=10,color="white",style="solid",shape="box"];118 -> 199[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 199 -> 123[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 200[label="ww40100/Zero",fontsize=10,color="white",style="solid",shape="box"];118 -> 200[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 200 -> 124[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 119[label="primEqNat Zero ww40100 && ww5",fontsize=16,color="burlywood",shape="box"];201[label="ww40100/Succ ww401000",fontsize=10,color="white",style="solid",shape="box"];119 -> 201[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 201 -> 125[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 202[label="ww40100/Zero",fontsize=10,color="white",style="solid",shape="box"];119 -> 202[label="",style="solid", color="burlywood", weight=9]; 12.15/4.88 202 -> 126[label="",style="solid", color="burlywood", weight=3]; 12.15/4.88 120[label="ww5",fontsize=16,color="green",shape="box"];121[label="ww40100",fontsize=16,color="green",shape="box"];122[label="ww30100",fontsize=16,color="green",shape="box"];123[label="primEqNat (Succ ww301000) (Succ ww401000) && ww5",fontsize=16,color="black",shape="box"];123 -> 127[label="",style="solid", color="black", weight=3]; 12.15/4.88 124[label="primEqNat (Succ ww301000) Zero && ww5",fontsize=16,color="black",shape="box"];124 -> 128[label="",style="solid", color="black", weight=3]; 12.15/4.88 125[label="primEqNat Zero (Succ ww401000) && ww5",fontsize=16,color="black",shape="box"];125 -> 129[label="",style="solid", color="black", weight=3]; 12.15/4.88 126[label="primEqNat Zero Zero && ww5",fontsize=16,color="black",shape="box"];126 -> 130[label="",style="solid", color="black", weight=3]; 12.15/4.88 127 -> 106[label="",style="dashed", color="red", weight=0]; 12.15/4.88 127[label="primEqNat ww301000 ww401000 && ww5",fontsize=16,color="magenta"];127 -> 131[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 127 -> 132[label="",style="dashed", color="magenta", weight=3]; 12.15/4.88 128 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 128[label="False && ww5",fontsize=16,color="magenta"];129 -> 50[label="",style="dashed", color="red", weight=0]; 12.15/4.88 129[label="False && ww5",fontsize=16,color="magenta"];130 -> 109[label="",style="dashed", color="red", weight=0]; 12.15/4.88 130[label="True && ww5",fontsize=16,color="magenta"];131[label="ww401000",fontsize=16,color="green",shape="box"];132[label="ww301000",fontsize=16,color="green",shape="box"];} 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (6) 12.15/4.88 Complex Obligation (AND) 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (7) 12.15/4.88 Obligation: 12.15/4.88 Q DP problem: 12.15/4.88 The TRS P consists of the following rules: 12.15/4.88 12.15/4.88 new_asAs0(Succ(ww300000), Succ(ww400000), ww301, ww401, ww5) -> new_asAs0(ww300000, ww400000, ww301, ww401, ww5) 12.15/4.88 12.15/4.88 R is empty. 12.15/4.88 Q is empty. 12.15/4.88 We have to consider all minimal (P,Q,R)-chains. 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (8) QDPSizeChangeProof (EQUIVALENT) 12.15/4.88 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. 12.15/4.88 12.15/4.88 From the DPs we obtained the following set of size-change graphs: 12.15/4.88 *new_asAs0(Succ(ww300000), Succ(ww400000), ww301, ww401, ww5) -> new_asAs0(ww300000, ww400000, ww301, ww401, ww5) 12.15/4.88 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5 12.15/4.88 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (9) 12.15/4.88 YES 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (10) 12.15/4.88 Obligation: 12.15/4.88 Q DP problem: 12.15/4.88 The TRS P consists of the following rules: 12.15/4.88 12.15/4.88 new_asAs(Succ(ww301000), Succ(ww401000), ww5) -> new_asAs(ww301000, ww401000, ww5) 12.15/4.88 12.15/4.88 R is empty. 12.15/4.88 Q is empty. 12.15/4.88 We have to consider all minimal (P,Q,R)-chains. 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (11) QDPSizeChangeProof (EQUIVALENT) 12.15/4.88 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. 12.15/4.88 12.15/4.88 From the DPs we obtained the following set of size-change graphs: 12.15/4.88 *new_asAs(Succ(ww301000), Succ(ww401000), ww5) -> new_asAs(ww301000, ww401000, ww5) 12.15/4.88 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 12.15/4.88 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (12) 12.15/4.88 YES 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (13) 12.15/4.88 Obligation: 12.15/4.88 Q DP problem: 12.15/4.88 The TRS P consists of the following rules: 12.15/4.88 12.15/4.88 new_isPrefixOf(:(ww30, ww31), :(ww40, ww41)) -> new_isPrefixOf(ww31, ww41) 12.15/4.88 12.15/4.88 R is empty. 12.15/4.88 Q is empty. 12.15/4.88 We have to consider all minimal (P,Q,R)-chains. 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (14) QDPSizeChangeProof (EQUIVALENT) 12.15/4.88 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. 12.15/4.88 12.15/4.88 From the DPs we obtained the following set of size-change graphs: 12.15/4.88 *new_isPrefixOf(:(ww30, ww31), :(ww40, ww41)) -> new_isPrefixOf(ww31, ww41) 12.15/4.88 The graph contains the following edges 1 > 1, 2 > 2 12.15/4.88 12.15/4.88 12.15/4.88 ---------------------------------------- 12.15/4.88 12.15/4.88 (15) 12.15/4.88 YES 12.30/4.97 EOF