/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 22 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isPrefixOf [] _ = True;
isPrefixOf _ [] = False;
isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isPrefixOf [] wu = True;
isPrefixOf wv [] = False;
isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isPrefixOf [] wu = True;
isPrefixOf wv [] = False;
isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
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135 -> 7[label="",style="solid", color="burlywood", weight=3];
136[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 136[label="",style="solid", color="burlywood", weight=9];
136 -> 8[label="",style="solid", color="burlywood", weight=3];
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8[label="List.isPrefixOf (ww30 : ww31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
10[label="ww30 == ww40 && List.isPrefixOf ww31 ww41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
13[label="List.isPrefixOf ww31 ww41",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
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137 -> 16[label="",style="solid", color="burlywood", weight=3];
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140[label="ww300/Neg ww3000",fontsize=10,color="white",style="solid",shape="box"];19 -> 140[label="",style="solid", color="burlywood", weight=9];
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142[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 142[label="",style="solid", color="burlywood", weight=9];
142 -> 23[label="",style="solid", color="burlywood", weight=3];
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144[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 144[label="",style="solid", color="burlywood", weight=9];
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146[label="ww400/Neg ww4000",fontsize=10,color="white",style="solid",shape="box"];22 -> 146[label="",style="solid", color="burlywood", weight=9];
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153 -> 34[label="",style="solid", color="burlywood", weight=3];
154[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 154[label="",style="solid", color="burlywood", weight=9];
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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];
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];
155 -> 37[label="",style="solid", color="burlywood", weight=3];
156[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 156[label="",style="solid", color="burlywood", weight=9];
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158[label="ww4000/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 158[label="",style="solid", color="burlywood", weight=9];
158 -> 40[label="",style="solid", color="burlywood", weight=3];
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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];
36[label="(False && ww301 == ww401) && ww5",fontsize=16,color="black",shape="triangle"];36 -> 50[label="",style="solid", color="black", weight=3];
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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];
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];
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];
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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];
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165 -> 61[label="",style="solid", color="burlywood", weight=3];
166[label="ww30000/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 166[label="",style="solid", color="burlywood", weight=9];
166 -> 62[label="",style="solid", color="burlywood", weight=3];
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53 -> 36[label="",style="dashed", color="red", weight=0];
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55 -> 66[label="",style="dashed", color="magenta", weight=3];
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167 -> 67[label="",style="solid", color="burlywood", weight=3];
168[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 168[label="",style="solid", color="burlywood", weight=9];
168 -> 68[label="",style="solid", color="burlywood", weight=3];
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];
169 -> 69[label="",style="solid", color="burlywood", weight=3];
170[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 70[label="",style="solid", color="burlywood", weight=3];
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];
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];
68[label="(primEqNat (Succ ww300000) Zero && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3];
69[label="(primEqNat Zero (Succ ww400000) && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3];
70[label="(primEqNat Zero Zero && ww301 == ww401) && ww5",fontsize=16,color="black",shape="box"];70 -> 75[label="",style="solid", color="black", weight=3];
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];
171 -> 76[label="",style="solid", color="burlywood", weight=3];
172[label="ww301/Neg ww3010",fontsize=10,color="white",style="solid",shape="box"];71 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 77[label="",style="solid", color="burlywood", weight=3];
72 -> 48[label="",style="dashed", color="red", weight=0];
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72 -> 79[label="",style="dashed", color="magenta", weight=3];
73 -> 36[label="",style="dashed", color="red", weight=0];
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74[label="(False && ww301 == ww401) && ww5",fontsize=16,color="magenta"];75 -> 52[label="",style="dashed", color="red", weight=0];
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];
173 -> 80[label="",style="solid", color="burlywood", weight=3];
174[label="ww3010/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 174[label="",style="solid", color="burlywood", weight=9];
174 -> 81[label="",style="solid", color="burlywood", weight=3];
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];
175 -> 82[label="",style="solid", color="burlywood", weight=3];
176[label="ww3010/Zero",fontsize=10,color="white",style="solid",shape="box"];77 -> 176[label="",style="solid", color="burlywood", weight=9];
176 -> 83[label="",style="solid", color="burlywood", weight=3];
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];
177 -> 84[label="",style="solid", color="burlywood", weight=3];
178[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];80 -> 178[label="",style="solid", color="burlywood", weight=9];
178 -> 85[label="",style="solid", color="burlywood", weight=3];
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];
179 -> 86[label="",style="solid", color="burlywood", weight=3];
180[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];81 -> 180[label="",style="solid", color="burlywood", weight=9];
180 -> 87[label="",style="solid", color="burlywood", weight=3];
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];
181 -> 88[label="",style="solid", color="burlywood", weight=3];
182[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];82 -> 182[label="",style="solid", color="burlywood", weight=9];
182 -> 89[label="",style="solid", color="burlywood", weight=3];
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];
183 -> 90[label="",style="solid", color="burlywood", weight=3];
184[label="ww401/Neg ww4010",fontsize=10,color="white",style="solid",shape="box"];83 -> 184[label="",style="solid", color="burlywood", weight=9];
184 -> 91[label="",style="solid", color="burlywood", weight=3];
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];
185 -> 92[label="",style="solid", color="burlywood", weight=3];
186[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];84 -> 186[label="",style="solid", color="burlywood", weight=9];
186 -> 93[label="",style="solid", color="burlywood", weight=3];
85[label="primEqInt (Pos (Succ ww30100)) (Neg ww4010) && ww5",fontsize=16,color="black",shape="box"];85 -> 94[label="",style="solid", color="black", weight=3];
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];
187 -> 95[label="",style="solid", color="burlywood", weight=3];
188[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];86 -> 188[label="",style="solid", color="burlywood", weight=9];
188 -> 96[label="",style="solid", color="burlywood", weight=3];
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];
189 -> 97[label="",style="solid", color="burlywood", weight=3];
190[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];87 -> 190[label="",style="solid", color="burlywood", weight=9];
190 -> 98[label="",style="solid", color="burlywood", weight=3];
88[label="primEqInt (Neg (Succ ww30100)) (Pos ww4010) && ww5",fontsize=16,color="black",shape="box"];88 -> 99[label="",style="solid", color="black", weight=3];
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];
191 -> 100[label="",style="solid", color="burlywood", weight=3];
192[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];89 -> 192[label="",style="solid", color="burlywood", weight=9];
192 -> 101[label="",style="solid", color="burlywood", weight=3];
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];
193 -> 102[label="",style="solid", color="burlywood", weight=3];
194[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];90 -> 194[label="",style="solid", color="burlywood", weight=9];
194 -> 103[label="",style="solid", color="burlywood", weight=3];
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];
195 -> 104[label="",style="solid", color="burlywood", weight=3];
196[label="ww4010/Zero",fontsize=10,color="white",style="solid",shape="box"];91 -> 196[label="",style="solid", color="burlywood", weight=9];
196 -> 105[label="",style="solid", color="burlywood", weight=3];
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];
93[label="primEqInt (Pos (Succ ww30100)) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];93 -> 107[label="",style="solid", color="black", weight=3];
94 -> 50[label="",style="dashed", color="red", weight=0];
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];
96[label="primEqInt (Pos Zero) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];96 -> 109[label="",style="solid", color="black", weight=3];
97[label="primEqInt (Pos Zero) (Neg (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];97 -> 110[label="",style="solid", color="black", weight=3];
98[label="primEqInt (Pos Zero) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];98 -> 111[label="",style="solid", color="black", weight=3];
99 -> 50[label="",style="dashed", color="red", weight=0];
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];
101[label="primEqInt (Neg (Succ ww30100)) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];101 -> 113[label="",style="solid", color="black", weight=3];
102[label="primEqInt (Neg Zero) (Pos (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];102 -> 114[label="",style="solid", color="black", weight=3];
103[label="primEqInt (Neg Zero) (Pos Zero) && ww5",fontsize=16,color="black",shape="box"];103 -> 115[label="",style="solid", color="black", weight=3];
104[label="primEqInt (Neg Zero) (Neg (Succ ww40100)) && ww5",fontsize=16,color="black",shape="box"];104 -> 116[label="",style="solid", color="black", weight=3];
105[label="primEqInt (Neg Zero) (Neg Zero) && ww5",fontsize=16,color="black",shape="box"];105 -> 117[label="",style="solid", color="black", weight=3];
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];
197 -> 118[label="",style="solid", color="burlywood", weight=3];
198[label="ww30100/Zero",fontsize=10,color="white",style="solid",shape="box"];106 -> 198[label="",style="solid", color="burlywood", weight=9];
198 -> 119[label="",style="solid", color="burlywood", weight=3];
107 -> 50[label="",style="dashed", color="red", weight=0];
107[label="False && ww5",fontsize=16,color="magenta"];108 -> 50[label="",style="dashed", color="red", weight=0];
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];
110 -> 50[label="",style="dashed", color="red", weight=0];
110[label="False && ww5",fontsize=16,color="magenta"];111 -> 109[label="",style="dashed", color="red", weight=0];
111[label="True && ww5",fontsize=16,color="magenta"];112 -> 106[label="",style="dashed", color="red", weight=0];
112[label="primEqNat ww30100 ww40100 && ww5",fontsize=16,color="magenta"];112 -> 121[label="",style="dashed", color="magenta", weight=3];
112 -> 122[label="",style="dashed", color="magenta", weight=3];
113 -> 50[label="",style="dashed", color="red", weight=0];
113[label="False && ww5",fontsize=16,color="magenta"];114 -> 50[label="",style="dashed", color="red", weight=0];
114[label="False && ww5",fontsize=16,color="magenta"];115 -> 109[label="",style="dashed", color="red", weight=0];
115[label="True && ww5",fontsize=16,color="magenta"];116 -> 50[label="",style="dashed", color="red", weight=0];
116[label="False && ww5",fontsize=16,color="magenta"];117 -> 109[label="",style="dashed", color="red", weight=0];
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];
199 -> 123[label="",style="solid", color="burlywood", weight=3];
200[label="ww40100/Zero",fontsize=10,color="white",style="solid",shape="box"];118 -> 200[label="",style="solid", color="burlywood", weight=9];
200 -> 124[label="",style="solid", color="burlywood", weight=3];
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];
201 -> 125[label="",style="solid", color="burlywood", weight=3];
202[label="ww40100/Zero",fontsize=10,color="white",style="solid",shape="box"];119 -> 202[label="",style="solid", color="burlywood", weight=9];
202 -> 126[label="",style="solid", color="burlywood", weight=3];
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];
124[label="primEqNat (Succ ww301000) Zero && ww5",fontsize=16,color="black",shape="box"];124 -> 128[label="",style="solid", color="black", weight=3];
125[label="primEqNat Zero (Succ ww401000) && ww5",fontsize=16,color="black",shape="box"];125 -> 129[label="",style="solid", color="black", weight=3];
126[label="primEqNat Zero Zero && ww5",fontsize=16,color="black",shape="box"];126 -> 130[label="",style="solid", color="black", weight=3];
127 -> 106[label="",style="dashed", color="red", weight=0];
127[label="primEqNat ww301000 ww401000 && ww5",fontsize=16,color="magenta"];127 -> 131[label="",style="dashed", color="magenta", weight=3];
127 -> 132[label="",style="dashed", color="magenta", weight=3];
128 -> 50[label="",style="dashed", color="red", weight=0];
128[label="False && ww5",fontsize=16,color="magenta"];129 -> 50[label="",style="dashed", color="red", weight=0];
129[label="False && ww5",fontsize=16,color="magenta"];130 -> 109[label="",style="dashed", color="red", weight=0];
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"];}

----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_asAs0(Succ(ww300000), Succ(ww400000), ww301, ww401, ww5) -> new_asAs0(ww300000, ww400000, ww301, ww401, ww5)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_asAs0(Succ(ww300000), Succ(ww400000), ww301, ww401, ww5) -> new_asAs0(ww300000, ww400000, ww301, ww401, ww5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5


----------------------------------------

(9)
YES

----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_asAs(Succ(ww301000), Succ(ww401000), ww5) -> new_asAs(ww301000, ww401000, ww5)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_asAs(Succ(ww301000), Succ(ww401000), ww5) -> new_asAs(ww301000, ww401000, ww5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


----------------------------------------

(12)
YES

----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf(:(ww30, ww31), :(ww40, ww41)) -> new_isPrefixOf(ww31, ww41)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_isPrefixOf(:(ww30, ww31), :(ww40, ww41)) -> new_isPrefixOf(ww31, ww41)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(15)
YES