Left Termination of the query pattern ways_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

nat(0).
nat(s(X)) :- nat(X).
plus(0, X, X) :- nat(X).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
ways(0, X1, s(0)).
ways(X2, [], 0).
ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N))))).
ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N))).

Queries:

ways(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

Clauses:

nat(0).
nat(s(X)) :- nat(X).
plus(0, X, X) :- nat(X).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
ways(0, X1, s(0)).
ways(X2, [], 0).
ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N))))).
ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N))).
=(X, X).

Queries:

ways(g,g,a).

(3) BuiltinConflictTransformerProof (SOUND transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(4) Obligation:

Clauses:

nat(0).
nat(s(X)) :- nat(X).
plus(0, X, X) :- nat(X).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
ways(0, X1, s(0)).
ways(X2, [], 0).
ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N))))).
ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N))).
user_defined_=(X, X).

Queries:

ways(g,g,a).

(5) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ways(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ways(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | ways(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | ways(T1, T2, T3), SCOPE: 1, CLAUSE: 6 | ways(T1, T2, T3), SCOPE: 1, CLAUSE: 7\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | ways(0, T5, T3), SCOPE: 1, CLAUSE: 5 | ways(0, T5, T3), SCOPE: 1, CLAUSE: 6 | ways(0, T5, T3), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ways(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | ways(T1, T2, T3), SCOPE: 1, CLAUSE: 6 | ways(T1, T2, T3), SCOPE: 1, CLAUSE: 7\n\nT1 is ground\nT2 is ground\nways(T1, T2, T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ways(0, T5, T3), SCOPE: 1, CLAUSE: 5 | ways(0, T5, T3), SCOPE: 1, CLAUSE: 6 | ways(0, T5, T3), SCOPE: 1, CLAUSE: 7\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | ways(0, [], T3), SCOPE: 1, CLAUSE: 6 | ways(0, [], T3), SCOPE: 1, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ways(0, T5, T3), SCOPE: 1, CLAUSE: 6 | ways(0, T5, T3), SCOPE: 1, CLAUSE: 7\n\nT5 is ground\nways(0, T5, T3) !=? ways(X10, [], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ways(0, [], T3), SCOPE: 1, CLAUSE: 6 | ways(0, [], T3), SCOPE: 1, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ways(0, [], T3), SCOPE: 1, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(user_defined_=(0, s(X28)), ','(plus(s(T9), X29, 0), ','(ways(0, T10, X30), ','(ways(X29, .(s(T9), T10), X31), plus(X30, X31, T12))))) | ways(0, .(s(T9), T10), T3), SCOPE: 1, CLAUSE: 7\n\nT9 is ground\nT10 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ways(0, T5, T3), SCOPE: 1, CLAUSE: 7\n\nT5 is ground\nways(0, T5, T3) !=? ways(X10, [], 0)\nways(0, T5, T3) !=? ways(X24, .(s(X25), X26), X27)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(user_defined_=(0, s(X28)), ','(plus(s(T9), X29, 0), ','(ways(0, T10, X30), ','(ways(X29, .(s(T9), T10), X31), plus(X30, X31, T12))))), SCOPE: 2, CLAUSE: 8 | ?_2 | ways(0, .(s(T9), T10), T3), SCOPE: 1, CLAUSE: 7\n\nT9 is ground\nT10 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ?_2 | ways(0, .(s(T9), T10), T3), SCOPE: 1, CLAUSE: 7\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ways(0, .(s(T9), T10), T3), SCOPE: 1, CLAUSE: 7\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(user_defined_=(0, s(X53)), ','(plus(0, s(X54), s(T21)), ways(0, T22, T24)))\n\nT21 is ground\nT22 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(user_defined_=(0, s(X53)), ','(plus(0, s(X54), s(T21)), ways(0, T22, T24))), SCOPE: 3, CLAUSE: 8\n\nT21 is ground\nT22 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true | ways(T30, [], T3), SCOPE: 1, CLAUSE: 6 | ways(T30, [], T3), SCOPE: 1, CLAUSE: 7\n\nT30 is ground\nways(T30, [], T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ways(T1, T2, T3), SCOPE: 1, CLAUSE: 6 | ways(T1, T2, T3), SCOPE: 1, CLAUSE: 7\n\nT1 is ground\nT2 is ground\nways(T1, T2, T3) !=? ways(0, X7, s(0))\nways(T1, T2, T3) !=? ways(X63, [], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ways(T30, [], T3), SCOPE: 1, CLAUSE: 6 | ways(T30, [], T3), SCOPE: 1, CLAUSE: 7\n\nT30 is ground\nways(T30, [], T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ways(T30, [], T3), SCOPE: 1, CLAUSE: 7\n\nT30 is ground\nways(T30, [], T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(user_defined_=(T37, s(X81)), ','(plus(s(T38), X82, T37), ','(ways(T37, T39, X83), ','(ways(X82, .(s(T38), T39), X84), plus(X83, X84, T41))))) | ways(T37, .(s(T38), T39), T3), SCOPE: 1, CLAUSE: 7\n\nT37 is ground\nT38 is ground\nT39 is ground\nX81 is free; \nX82 is free; \nX83 is free; \nX84 is free; \nways(T37, .(s(T38), T39), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ways(T1, T2, T3), SCOPE: 1, CLAUSE: 7\n\nT1 is ground\nT2 is ground\nways(T1, T2, T3) !=? ways(0, X7, s(0))\nways(T1, T2, T3) !=? ways(X63, [], 0)\nways(T1, T2, T3) !=? ways(X77, .(s(X78), X79), X80)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(user_defined_=(T37, s(X81)), ','(plus(s(T38), X82, T37), ','(ways(T37, T39, X83), ','(ways(X82, .(s(T38), T39), X84), plus(X83, X84, T41))))), SCOPE: 4, CLAUSE: 8 | ?_4 | ways(T37, .(s(T38), T39), T3), SCOPE: 1, CLAUSE: 7\n\nT37 is ground\nT38 is ground\nT39 is ground\nX81 is free; \nX82 is free; \nX83 is free; \nX84 is free; \nways(T37, .(s(T38), T39), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(user_defined_=(T37, s(X81)), ','(plus(s(T38), X82, T37), ','(ways(T37, T39, X83), ','(ways(X82, .(s(T38), T39), X84), plus(X83, X84, T41))))), SCOPE: 4, CLAUSE: 8\n\nT37 is ground\nT38 is ground\nT39 is ground\nX81 is free; \nX82 is free; \nX83 is free; \nX84 is free; \nways(T37, .(s(T38), T39), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ?_4 | ways(T37, .(s(T38), T39), T3), SCOPE: 1, CLAUSE: 7\n\nT37 is ground\nT38 is ground\nT39 is ground\nways(T37, .(s(T38), T39), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(plus(s(T38), X82, s(T51)), ','(ways(s(T51), T39, X83), ','(ways(X82, .(s(T38), T39), X84), plus(X83, X84, T41))))\n\nT38 is ground\nT39 is ground\nT51 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: plus(s(T38), X82, s(T51))\n\nT38 is ground\nT51 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(ways(s(T51), T39, X83), ','(ways(T54, .(s(T38), T39), X84), plus(X83, X84, T41)))\n\nT38 is ground\nT39 is ground\nT51 is ground\nT54 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: plus(s(T38), X82, s(T51)), SCOPE: 5, CLAUSE: 2 | plus(s(T38), X82, s(T51)), SCOPE: 5, CLAUSE: 3\n\nT38 is ground\nT51 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: plus(s(T38), X82, s(T51)), SCOPE: 5, CLAUSE: 3\n\nT38 is ground\nT51 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: plus(T62, X119, T63)\n\nT62 is ground\nT63 is ground\nX119 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: plus(T62, X119, T63), SCOPE: 6, CLAUSE: 2 | plus(T62, X119, T63), SCOPE: 6, CLAUSE: 3\n\nT62 is ground\nT63 is ground\nX119 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: plus(T62, X119, T63), SCOPE: 6, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nX119 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: plus(T62, X119, T63), SCOPE: 6, CLAUSE: 3\n\nT62 is ground\nT63 is ground\nX119 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: nat(T69)\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: nat(T69), SCOPE: 7, CLAUSE: 0 | nat(T69), SCOPE: 7, CLAUSE: 1\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: nat(T69), SCOPE: 7, CLAUSE: 0\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: nat(T69), SCOPE: 7, CLAUSE: 1\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: nat(T72)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: plus(T77, X148, T78)\n\nT77 is ground\nT78 is ground\nX148 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ways(s(T51), T39, X83)\n\nT39 is ground\nT51 is ground\nX83 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(ways(T54, .(s(T38), T39), X84), plus(T82, X84, T41))\n\nT38 is ground\nT39 is ground\nT54 is ground\nT82 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ways(T54, .(s(T38), T39), X84)\n\nT38 is ground\nT39 is ground\nT54 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: plus(T82, T89, T41)\n\nT82 is ground\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: plus(T82, T89, T41), SCOPE: 8, CLAUSE: 2 | plus(T82, T89, T41), SCOPE: 8, CLAUSE: 3\n\nT82 is ground\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: plus(T82, T89, T41), SCOPE: 8, CLAUSE: 2\n\nT82 is ground\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: plus(T82, T89, T41), SCOPE: 8, CLAUSE: 3\n\nT82 is ground\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: nat(T99)\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: plus(T106, T107, T109)\n\nT106 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ways(T37, .(s(T38), T39), T3), SCOPE: 1, CLAUSE: 7\n\nT37 is ground\nT38 is ground\nT39 is ground\nways(T37, .(s(T38), T39), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(user_defined_=(T121, s(X202)), ','(plus(T121, s(X203), s(T122)), ways(T121, T123, T125)))\n\nT121 is ground\nT122 is ground\nT123 is ground\nX202 is free; \nX203 is free; \nways(T121, .(s(T122), T123), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(user_defined_=(T121, s(X202)), ','(plus(T121, s(X203), s(T122)), ways(T121, T123, T125))), SCOPE: 9, CLAUSE: 8\n\nT121 is ground\nT122 is ground\nT123 is ground\nX202 is free; \nX203 is free; \nways(T121, .(s(T122), T123), T3) !=? ways(0, X7, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(plus(s(T132), s(X203), s(T122)), ways(s(T132), T123, T125))\n\nT122 is ground\nT123 is ground\nT132 is ground\nX203 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: plus(s(T132), s(X203), s(T122))\n\nT122 is ground\nT132 is ground\nX203 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ways(s(T132), T123, T125)\n\nT123 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: plus(s(T132), s(X203), s(T122)), SCOPE: 10, CLAUSE: 2 | plus(s(T132), s(X203), s(T122)), SCOPE: 10, CLAUSE: 3\n\nT122 is ground\nT132 is ground\nX203 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: plus(s(T132), s(X203), s(T122)), SCOPE: 10, CLAUSE: 3\n\nT122 is ground\nT132 is ground\nX203 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: plus(T142, s(X234), T143)\n\nT142 is ground\nT143 is ground\nX234 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: plus(T142, s(X234), T143), SCOPE: 11, CLAUSE: 2 | plus(T142, s(X234), T143), SCOPE: 11, CLAUSE: 3\n\nT142 is ground\nT143 is ground\nX234 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: plus(T142, s(X234), T143), SCOPE: 11, CLAUSE: 2\n\nT142 is ground\nT143 is ground\nX234 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: plus(T142, s(X234), T143), SCOPE: 11, CLAUSE: 3\n\nT142 is ground\nT143 is ground\nX234 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: nat(s(X246))\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: nat(s(X246)), SCOPE: 12, CLAUSE: 0 | nat(s(X246)), SCOPE: 12, CLAUSE: 1\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: nat(s(X246)), SCOPE: 12, CLAUSE: 1\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: nat(X255)\n\nX255 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: nat(X255), SCOPE: 13, CLAUSE: 0 | nat(X255), SCOPE: 13, CLAUSE: 1\n\nX255 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: nat(X255), SCOPE: 13, CLAUSE: 0\n\nX255 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: nat(X255), SCOPE: 13, CLAUSE: 1\n\nX255 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: nat(X261)\n\nX261 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: plus(T148, s(X273), T149)\n\nT148 is ground\nT149 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 90 [arrowhead = none ];
90 -> 2;

91 [label="EVAL with clause\nways(0, X7, s(0)).\nand substitution\nT1 -> 0\nT2 -> T5\nX7 -> T5\nT3 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 91 [arrowhead = none ];
91 -> 3;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 92 [arrowhead = none ];
92 -> 4;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 93 [arrowhead = none ];
93 -> 5;

94 [label="EVAL with clause\nways(X63, [], 0).\nand substitution\nT1 -> T30\nX63 -> T30\nT2 -> []\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 94 [arrowhead = none ];
94 -> 20;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 95 [arrowhead = none ];
95 -> 21;

96 [label="EVAL with clause\nways(X10, [], 0).\nand substitution\nX10 -> 0\nT5 -> []\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 96 [arrowhead = none ];
96 -> 6;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 97 [arrowhead = none ];
97 -> 7;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 98 [arrowhead = none ];
98 -> 8;

99 [label="EVAL with clause\nways(X24, .(s(X25), X26), X27) :- ','(user_defined_=(X24, s(X28)), ','(plus(s(X25), X29, X24), ','(ways(X24, X26, X30), ','(ways(X29, .(s(X25), X26), X31), plus(X30, X31, X27))))).\nand substitution\nX24 -> 0\nX25 -> T9\nX26 -> T10\nT5 -> .(s(T9), T10)\nT3 -> T12\nX27 -> T12\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 99 [arrowhead = none ];
99 -> 11;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 100 [arrowhead = none ];
100 -> 12;

101 [label="BACKTRACK\nfor clause: ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N)))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
8 -> 101 [arrowhead = none ];
101 -> 9;

102 [label="BACKTRACK\nfor clause: ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
9 -> 102 [arrowhead = none ];
102 -> 10;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 103 [arrowhead = none ];
103 -> 13;

104 [label="BACKTRACK\nfor clause: ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N)))\nwith clash: (ways(0, T5, T3), ways(X24, .(s(X25), X26), X27))", fontsize=14, style = filled, fillcolor = white];
12 -> 104 [arrowhead = none ];
104 -> 19;

105 [label="BACKTRACK\nfor clause: user_defined_=(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 105 [arrowhead = none ];
105 -> 14;

106 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 106 [arrowhead = none ];
106 -> 15;

107 [label="ONLY EVAL with clause\nways(X49, .(s(X50), X51), X52) :- ','(user_defined_=(X49, s(X53)), ','(plus(X49, s(X54), s(X50)), ways(X49, X51, X52))).\nand substitution\nX49 -> 0\nT9 -> T21\nX50 -> T21\nT10 -> T22\nX51 -> T22\nT3 -> T24\nX52 -> T24\nT23 -> T24", fontsize=14, style = filled, fillcolor = white];
15 -> 107 [arrowhead = none ];
107 -> 16;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 108 [arrowhead = none ];
108 -> 17;

109 [label="BACKTRACK\nfor clause: user_defined_=(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
17 -> 109 [arrowhead = none ];
109 -> 18;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 110 [arrowhead = none ];
110 -> 22;

111 [label="EVAL with clause\nways(X77, .(s(X78), X79), X80) :- ','(user_defined_=(X77, s(X81)), ','(plus(s(X78), X82, X77), ','(ways(X77, X79, X83), ','(ways(X82, .(s(X78), X79), X84), plus(X83, X84, X80))))).\nand substitution\nT1 -> T37\nX77 -> T37\nX78 -> T38\nX79 -> T39\nT2 -> .(s(T38), T39)\nT3 -> T41\nX80 -> T41\nT40 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 111 [arrowhead = none ];
111 -> 25;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 112 [arrowhead = none ];
112 -> 26;

113 [label="BACKTRACK\nfor clause: ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N)))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
22 -> 113 [arrowhead = none ];
113 -> 23;

114 [label="BACKTRACK\nfor clause: ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 114 [arrowhead = none ];
114 -> 24;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 115 [arrowhead = none ];
115 -> 27;

116 [label="BACKTRACK\nfor clause: ways(Amount, .(s(C), Coins), N) :- ','(user_defined_=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N)))\nwith clash: (ways(T1, T2, T3), ways(X77, .(s(X78), X79), X80))", fontsize=14, style = filled, fillcolor = white];
26 -> 116 [arrowhead = none ];
116 -> 89;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 117 [arrowhead = none ];
117 -> 28;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 118 [arrowhead = none ];
118 -> 29;

119 [label="EVAL with clause\nuser_defined_=(X93, X93).\nand substitution\nT37 -> s(T51)\nX93 -> s(T51)\nX81 -> T51\nT50 -> s(T51)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 119 [arrowhead = none ];
119 -> 30;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 120 [arrowhead = none ];
120 -> 31;

121 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 121 [arrowhead = none ];
121 -> 63;

122 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 122 [arrowhead = none ];
122 -> 32;

123 [label="SPLIT 2\nnew knowledge:\nT54 is ground\nreplacements:\nX82 -> T54", fontsize=14, style = filled, fillcolor = violet];
30 -> 123 [arrowhead = none ];
123 -> 33;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 124 [arrowhead = none ];
124 -> 34;

125 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
33 -> 125 [arrowhead = none ];
125 -> 52;

126 [label="SPLIT 2\nnew knowledge:\nT82 is ground\nreplacements:\nX83 -> T82", fontsize=14, style = filled, fillcolor = violet];
33 -> 126 [arrowhead = none ];
126 -> 53;

127 [label="BACKTRACK\nfor clause: plus(0, X, X) :- nat(X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
34 -> 127 [arrowhead = none ];
127 -> 35;

128 [label="ONLY EVAL with clause\nplus(s(X116), X117, s(X118)) :- plus(X116, X117, X118).\nand substitution\nT38 -> T62\nX116 -> T62\nX82 -> X119\nX117 -> X119\nT51 -> T63\nX118 -> T63", fontsize=14, style = filled, fillcolor = white];
35 -> 128 [arrowhead = none ];
128 -> 36;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 129 [arrowhead = none ];
129 -> 37;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 130 [arrowhead = none ];
130 -> 38;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 131 [arrowhead = none ];
131 -> 39;

132 [label="EVAL with clause\nplus(0, X130, X130) :- nat(X130).\nand substitution\nT62 -> 0\nX119 -> T69\nX130 -> T69\nT63 -> T69\nX131 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 132 [arrowhead = none ];
132 -> 40;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 133 [arrowhead = none ];
133 -> 41;

134 [label="EVAL with clause\nplus(s(X145), X146, s(X147)) :- plus(X145, X146, X147).\nand substitution\nX145 -> T77\nT62 -> s(T77)\nX119 -> X148\nX146 -> X148\nX147 -> T78\nT63 -> s(T78)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 134 [arrowhead = none ];
134 -> 50;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 135 [arrowhead = none ];
135 -> 51;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 136 [arrowhead = none ];
136 -> 42;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 137 [arrowhead = none ];
137 -> 43;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 138 [arrowhead = none ];
138 -> 44;

139 [label="EVAL with clause\nnat(0).\nand substitution\nT69 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 139 [arrowhead = none ];
139 -> 45;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 140 [arrowhead = none ];
140 -> 46;

141 [label="EVAL with clause\nnat(s(X135)) :- nat(X135).\nand substitution\nX135 -> T72\nT69 -> s(T72)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 141 [arrowhead = none ];
141 -> 48;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 142 [arrowhead = none ];
142 -> 49;

143 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 143 [arrowhead = none ];
143 -> 47;

144 [label="INSTANCE with matching:\nT69 -> T72", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 144 [arrowhead = none , style=dashed];
144 -> 40[style=dashed];

145 [label="INSTANCE with matching:\nT62 -> T77\nX119 -> X148\nT63 -> T78", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 145 [arrowhead = none , style=dashed];
145 -> 36[style=dashed];

146 [label="INSTANCE with matching:\nT1 -> s(T51)\nT2 -> T39\nT3 -> X83", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 146 [arrowhead = none , style=dashed];
146 -> 1[style=dashed];

147 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 147 [arrowhead = none ];
147 -> 54;

148 [label="SPLIT 2\nnew knowledge:\nT89 is ground\nreplacements:\nX84 -> T89", fontsize=14, style = filled, fillcolor = violet];
53 -> 148 [arrowhead = none ];
148 -> 55;

149 [label="INSTANCE with matching:\nT1 -> T54\nT2 -> .(s(T38), T39)\nT3 -> X84", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 149 [arrowhead = none , style=dashed];
149 -> 1[style=dashed];

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 150 [arrowhead = none ];
150 -> 56;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 151 [arrowhead = none ];
151 -> 57;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 152 [arrowhead = none ];
152 -> 58;

153 [label="EVAL with clause\nplus(0, X171, X171) :- nat(X171).\nand substitution\nT82 -> 0\nT89 -> T99\nX171 -> T99\nT41 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 153 [arrowhead = none ];
153 -> 59;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 154 [arrowhead = none ];
154 -> 60;

155 [label="EVAL with clause\nplus(s(X179), X180, s(X181)) :- plus(X179, X180, X181).\nand substitution\nX179 -> T106\nT82 -> s(T106)\nT89 -> T107\nX180 -> T107\nX181 -> T109\nT41 -> s(T109)\nT108 -> T109", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 155 [arrowhead = none ];
155 -> 61;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 156 [arrowhead = none ];
156 -> 62;

157 [label="INSTANCE with matching:\nT69 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 157 [arrowhead = none , style=dashed];
157 -> 40[style=dashed];

158 [label="INSTANCE with matching:\nT82 -> T106\nT89 -> T107\nT41 -> T109", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 158 [arrowhead = none , style=dashed];
158 -> 55[style=dashed];

159 [label="ONLY EVAL with clause\nways(X198, .(s(X199), X200), X201) :- ','(user_defined_=(X198, s(X202)), ','(plus(X198, s(X203), s(X199)), ways(X198, X200, X201))).\nand substitution\nT37 -> T121\nX198 -> T121\nT38 -> T122\nX199 -> T122\nT39 -> T123\nX200 -> T123\nT3 -> T125\nX201 -> T125\nT124 -> T125", fontsize=14, style = filled, fillcolor = white];
63 -> 159 [arrowhead = none ];
159 -> 64;

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 160 [arrowhead = none ];
160 -> 65;

161 [label="EVAL with clause\nuser_defined_=(X208, X208).\nand substitution\nT121 -> s(T132)\nX208 -> s(T132)\nX202 -> T132\nT131 -> s(T132)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 161 [arrowhead = none ];
161 -> 66;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 162 [arrowhead = none ];
162 -> 67;

163 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 163 [arrowhead = none ];
163 -> 68;

164 [label="SPLIT 2\nnew knowledge:\nT135 is ground\nreplacements:\nX203 -> T135", fontsize=14, style = filled, fillcolor = violet];
66 -> 164 [arrowhead = none ];
164 -> 69;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 165 [arrowhead = none ];
165 -> 70;

166 [label="INSTANCE with matching:\nT1 -> s(T132)\nT2 -> T123\nT3 -> T125", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 166 [arrowhead = none , style=dashed];
166 -> 1[style=dashed];

167 [label="BACKTRACK\nfor clause: plus(0, X, X) :- nat(X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 167 [arrowhead = none ];
167 -> 71;

168 [label="ONLY EVAL with clause\nplus(s(X231), X232, s(X233)) :- plus(X231, X232, X233).\nand substitution\nT132 -> T142\nX231 -> T142\nX203 -> X234\nX232 -> s(X234)\nT122 -> T143\nX233 -> T143", fontsize=14, style = filled, fillcolor = white];
71 -> 168 [arrowhead = none ];
168 -> 72;

169 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 169 [arrowhead = none ];
169 -> 73;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 170 [arrowhead = none ];
170 -> 74;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 171 [arrowhead = none ];
171 -> 75;

172 [label="EVAL with clause\nplus(0, X245, X245) :- nat(X245).\nand substitution\nT142 -> 0\nX234 -> X246\nX245 -> s(X246)\nT143 -> s(X246)", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 172 [arrowhead = none ];
172 -> 76;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 173 [arrowhead = none ];
173 -> 77;

174 [label="EVAL with clause\nplus(s(X270), X271, s(X272)) :- plus(X270, X271, X272).\nand substitution\nX270 -> T148\nT142 -> s(T148)\nX234 -> X273\nX271 -> s(X273)\nX272 -> T149\nT143 -> s(T149)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 174 [arrowhead = none ];
174 -> 87;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 175 [arrowhead = none ];
175 -> 88;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 176 [arrowhead = none ];
176 -> 78;

177 [label="BACKTRACK\nfor clause: nat(0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
78 -> 177 [arrowhead = none ];
177 -> 79;

178 [label="ONLY EVAL with clause\nnat(s(X254)) :- nat(X254).\nand substitution\nX246 -> X255\nX254 -> X255", fontsize=14, style = filled, fillcolor = white];
79 -> 178 [arrowhead = none ];
178 -> 80;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 179 [arrowhead = none ];
179 -> 81;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 180 [arrowhead = none ];
180 -> 82;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 181 [arrowhead = none ];
181 -> 83;

182 [label="ONLY EVAL with clause\nnat(0).\nand substitution\nX255 -> 0", fontsize=14, style = filled, fillcolor = white];
82 -> 182 [arrowhead = none ];
182 -> 84;

183 [label="ONLY EVAL with clause\nnat(s(X260)) :- nat(X260).\nand substitution\nX260 -> X261\nX255 -> s(X261)", fontsize=14, style = filled, fillcolor = white];
83 -> 183 [arrowhead = none ];
183 -> 86;

184 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 184 [arrowhead = none ];
184 -> 85;

185 [label="INSTANCE with matching:\nX255 -> X261", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 185 [arrowhead = none , style=dashed];
185 -> 80[style=dashed];

186 [label="INSTANCE with matching:\nT142 -> T148\nX234 -> X273\nT143 -> T149", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 186 [arrowhead = none , style=dashed];
186 -> 72[style=dashed];

}

(6) Obligation:

Clauses:

nat40(0).
nat40(s(T72)) :- nat40(T72).
plus36(0, T69, T69) :- nat40(T69).
plus36(s(T77), X148, s(T78)) :- plus36(T77, X148, T78).
plus55(0, T99, T99) :- nat40(T99).
plus55(s(T106), T107, s(T109)) :- plus55(T106, T107, T109).
nat80(0).
nat80(s(X261)) :- nat80(X261).
plus72(0, X255, s(X255)) :- nat80(X255).
plus72(s(T148), X273, s(T149)) :- plus72(T148, X273, T149).
plus32(T62, X119, T63) :- plus36(T62, X119, T63).
plus68(T142, X234, T143) :- plus72(T142, X234, T143).
ways1(0, T5, s(0)).
ways1(0, [], 0).
ways1(T30, [], 0).
ways1(s(T51), .(s(T38), T39), T41) :- plus32(T38, X82, T51).
ways1(s(T51), .(s(T38), T39), T41) :- ','(plus32(T38, T54, T51), ways1(s(T51), T39, X83)).
ways1(s(T51), .(s(T38), T39), T41) :- ','(plus32(T38, T54, T51), ','(ways1(s(T51), T39, T82), ways1(T54, .(s(T38), T39), X84))).
ways1(s(T51), .(s(T38), T39), T41) :- ','(plus32(T38, T54, T51), ','(ways1(s(T51), T39, T82), ','(ways1(T54, .(s(T38), T39), T89), plus55(T82, T89, T41)))).
ways1(s(T132), .(s(T122), T123), T125) :- plus68(T132, X203, T122).
ways1(s(T132), .(s(T122), T123), T125) :- ','(plus68(T132, T135, T122), ways1(s(T132), T123, T125)).

Queries:

ways1(g,g,a).

(7) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ways1_in: (b,b,f)
plus32_in: (b,f,b)
plus36_in: (b,f,b)
nat40_in: (b) (f)
plus68_in: (b,f,b)
plus72_in: (b,f,b)
nat80_in: (b)
plus55_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(8) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)

(9) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U11_GGA(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → PLUS32_IN_GAG(T38, X82, T51)
PLUS32_IN_GAG(T62, X119, T63) → U9_GAG(T62, X119, T63, plus36_in_gag(T62, X119, T63))
PLUS32_IN_GAG(T62, X119, T63) → PLUS36_IN_GAG(T62, X119, T63)
PLUS36_IN_GAG(0, T69, T69) → U2_GAG(T69, nat40_in_g(T69))
PLUS36_IN_GAG(0, T69, T69) → NAT40_IN_G(T69)
NAT40_IN_G(s(T72)) → U1_G(T72, nat40_in_g(T72))
NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)
PLUS36_IN_GAG(s(T77), X148, s(T78)) → U3_GAG(T77, X148, T78, plus36_in_gag(T77, X148, T78))
PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_GGA(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U18_GGA(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → PLUS68_IN_GAG(T132, X203, T122)
PLUS68_IN_GAG(T142, X234, T143) → U10_GAG(T142, X234, T143, plus72_in_gag(T142, X234, T143))
PLUS68_IN_GAG(T142, X234, T143) → PLUS72_IN_GAG(T142, X234, T143)
PLUS72_IN_GAG(0, X255, s(X255)) → U7_GAG(X255, nat80_in_g(X255))
PLUS72_IN_GAG(0, X255, s(X255)) → NAT80_IN_G(X255)
NAT80_IN_G(s(X261)) → U6_G(X261, nat80_in_g(X261))
NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)
PLUS72_IN_GAG(s(T148), X273, s(T149)) → U8_GAG(T148, X273, T149, plus72_in_gag(T148, X273, T149))
PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_GGA(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_GGA(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_GGA(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_GGA(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → PLUS55_IN_AAA(T82, T89, T41)
PLUS55_IN_AAA(0, T99, T99) → U4_AAA(T99, nat40_in_a(T99))
PLUS55_IN_AAA(0, T99, T99) → NAT40_IN_A(T99)
NAT40_IN_A(s(T72)) → U1_A(T72, nat40_in_a(T72))
NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)
PLUS55_IN_AAA(s(T106), T107, s(T109)) → U5_AAA(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x5)
PLUS32_IN_GAG(x1, x2, x3)  =  PLUS32_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x4)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)
U2_GAG(x1, x2)  =  U2_GAG(x1, x2)
NAT40_IN_G(x1)  =  NAT40_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x4)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x5)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x5)
PLUS68_IN_GAG(x1, x2, x3)  =  PLUS68_IN_GAG(x1, x3)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x4)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)
U7_GAG(x1, x2)  =  U7_GAG(x1, x2)
NAT80_IN_G(x1)  =  NAT80_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x2)
U8_GAG(x1, x2, x3, x4)  =  U8_GAG(x4)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x2, x3, x5, x6)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x5)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x6)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x5)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA
U4_AAA(x1, x2)  =  U4_AAA(x2)
NAT40_IN_A(x1)  =  NAT40_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U5_AAA(x1, x2, x3, x4)  =  U5_AAA(x4)

We have to consider all (P,R,Pi)-chains

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U11_GGA(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → PLUS32_IN_GAG(T38, X82, T51)
PLUS32_IN_GAG(T62, X119, T63) → U9_GAG(T62, X119, T63, plus36_in_gag(T62, X119, T63))
PLUS32_IN_GAG(T62, X119, T63) → PLUS36_IN_GAG(T62, X119, T63)
PLUS36_IN_GAG(0, T69, T69) → U2_GAG(T69, nat40_in_g(T69))
PLUS36_IN_GAG(0, T69, T69) → NAT40_IN_G(T69)
NAT40_IN_G(s(T72)) → U1_G(T72, nat40_in_g(T72))
NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)
PLUS36_IN_GAG(s(T77), X148, s(T78)) → U3_GAG(T77, X148, T78, plus36_in_gag(T77, X148, T78))
PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_GGA(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U18_GGA(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → PLUS68_IN_GAG(T132, X203, T122)
PLUS68_IN_GAG(T142, X234, T143) → U10_GAG(T142, X234, T143, plus72_in_gag(T142, X234, T143))
PLUS68_IN_GAG(T142, X234, T143) → PLUS72_IN_GAG(T142, X234, T143)
PLUS72_IN_GAG(0, X255, s(X255)) → U7_GAG(X255, nat80_in_g(X255))
PLUS72_IN_GAG(0, X255, s(X255)) → NAT80_IN_G(X255)
NAT80_IN_G(s(X261)) → U6_G(X261, nat80_in_g(X261))
NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)
PLUS72_IN_GAG(s(T148), X273, s(T149)) → U8_GAG(T148, X273, T149, plus72_in_gag(T148, X273, T149))
PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_GGA(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_GGA(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_GGA(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_GGA(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → PLUS55_IN_AAA(T82, T89, T41)
PLUS55_IN_AAA(0, T99, T99) → U4_AAA(T99, nat40_in_a(T99))
PLUS55_IN_AAA(0, T99, T99) → NAT40_IN_A(T99)
NAT40_IN_A(s(T72)) → U1_A(T72, nat40_in_a(T72))
NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)
PLUS55_IN_AAA(s(T106), T107, s(T109)) → U5_AAA(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x5)
PLUS32_IN_GAG(x1, x2, x3)  =  PLUS32_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x4)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)
U2_GAG(x1, x2)  =  U2_GAG(x1, x2)
NAT40_IN_G(x1)  =  NAT40_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x4)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x5)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x5)
PLUS68_IN_GAG(x1, x2, x3)  =  PLUS68_IN_GAG(x1, x3)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x4)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)
U7_GAG(x1, x2)  =  U7_GAG(x1, x2)
NAT80_IN_G(x1)  =  NAT80_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x2)
U8_GAG(x1, x2, x3, x4)  =  U8_GAG(x4)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x2, x3, x5, x6)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x5)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x6)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x5)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA
U4_AAA(x1, x2)  =  U4_AAA(x2)
NAT40_IN_A(x1)  =  NAT40_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U5_AAA(x1, x2, x3, x4)  =  U5_AAA(x4)

We have to consider all (P,R,Pi)-chains

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 26 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
NAT40_IN_A(x1)  =  NAT40_IN_A

We have to consider all (P,R,Pi)-chains

(14) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(15) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NAT40_IN_A(x1)  =  NAT40_IN_A

We have to consider all (P,R,Pi)-chains

(16) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT40_IN_A → NAT40_IN_A

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

(18) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = NAT40_IN_A evaluates to t =NAT40_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from NAT40_IN_A to NAT40_IN_A.

(19) NO

(20) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA

We have to consider all (P,R,Pi)-chains

(21) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(22) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA

We have to consider all (P,R,Pi)-chains

(23) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS55_IN_AAA → PLUS55_IN_AAA

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

(25) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = PLUS55_IN_AAA evaluates to t =PLUS55_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PLUS55_IN_AAA to PLUS55_IN_AAA.

(26) NO

(27) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
NAT80_IN_G(x1)  =  NAT80_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(28) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(29) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)

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

(30) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)

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

(32) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)
  The graph contains the following edges 1 > 1

(33) YES

(34) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(35) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(36) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(37) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS72_IN_GAG(s(T148), s(T149)) → PLUS72_IN_GAG(T148, T149)

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

(39) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• PLUS72_IN_GAG(s(T148), s(T149)) → PLUS72_IN_GAG(T148, T149)
  The graph contains the following edges 1 > 1, 2 > 2

(40) YES

(41) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
NAT40_IN_G(x1)  =  NAT40_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(42) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(43) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)

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

(44) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(45) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)

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

(46) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)
  The graph contains the following edges 1 > 1

(47) YES

(48) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(49) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(50) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(51) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(52) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS36_IN_GAG(s(T77), s(T78)) → PLUS36_IN_GAG(T77, T78)

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

(53) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• PLUS36_IN_GAG(s(T77), s(T78)) → PLUS36_IN_GAG(T77, T78)
  The graph contains the following edges 1 > 1, 2 > 2

(54) YES

(55) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x2, x3, x5, x6)

We have to consider all (P,R,Pi)-chains

(56) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(57) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)

The TRS R consists of the following rules:

plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g
U1_g(x1, x2)  =  U1_g(x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x2)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g
U6_g(x1, x2)  =  U6_g(x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x2)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x2)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x2, x3, x5, x6)

We have to consider all (P,R,Pi)-chains

(58) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(59) Obligation:

Q DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, plus32_in_gag(T38, T51))
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → WAYS1_IN_GGA(s(T51), T39)
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T123, plus68_in_gag(T132, T122))
U19_GGA(T132, T123, plus68_out_gag(T135)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(60) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, plus32_in_gag(T38, T51)) at position [3] we obtained the following new rules [LPAR04]:

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))

(61) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → WAYS1_IN_GGA(s(T51), T39)
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T123, plus68_in_gag(T132, T122))
U19_GGA(T132, T123, plus68_out_gag(T135)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(62) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T123, plus68_in_gag(T132, T122)) at position [2] we obtained the following new rules [LPAR04]:

WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T123, U10_gag(plus72_in_gag(T132, T122)))

(63) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → WAYS1_IN_GGA(s(T51), T39)
U19_GGA(T132, T123, plus68_out_gag(T135)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T123, U10_gag(plus72_in_gag(T132, T122)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U19_GGA(T132, T123, plus68_out_gag(T135)) → WAYS1_IN_GGA(s(T132), T123)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = 1   
POL(U10_gag(x₁)) = x₁   
POL(U11_gga(x₁)) = 0   
POL(U12_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃   
POL(U12_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U13_gga(x₁)) = 0   
POL(U14_GGA(x₁, x₂, x₃, x₄)) = x₁ + x₂   
POL(U14_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U15_gga(x₁)) = 0   
POL(U16_gga(x₁)) = 0   
POL(U17_gga(x₁)) = 0   
POL(U18_gga(x₁)) = 0   
POL(U19_GGA(x₁, x₂, x₃)) = x₂ + x₃   
POL(U19_gga(x₁, x₂, x₃)) = 0   
POL(U1_a(x₁)) = 0   
POL(U1_g(x₁)) = 0   
POL(U20_gga(x₁)) = 0   
POL(U2_gag(x₁, x₂)) = 0   
POL(U3_gag(x₁)) = 0   
POL(U4_aaa(x₁)) = 0   
POL(U5_aaa(x₁)) = 0   
POL(U6_g(x₁)) = x₁   
POL(U7_gag(x₁, x₂)) = x₂   
POL(U8_gag(x₁)) = x₁   
POL(U9_gag(x₁)) = 0   
POL(WAYS1_IN_GGA(x₁, x₂)) = x₂   
POL([]) = 0   
POL(nat40_in_a) = 0   
POL(nat40_in_g(x₁)) = 0   
POL(nat40_out_a(x₁)) = 0   
POL(nat40_out_g) = 0   
POL(nat80_in_g(x₁)) = x₁   
POL(nat80_out_g) = 1   
POL(plus32_in_gag(x₁, x₂)) = 0   
POL(plus32_out_gag(x₁)) = 0   
POL(plus36_in_gag(x₁, x₂)) = 0   
POL(plus36_out_gag(x₁)) = 0   
POL(plus55_in_aaa) = 0   
POL(plus55_out_aaa(x₁, x₂, x₃)) = 0   
POL(plus68_in_gag(x₁, x₂)) = x₂   
POL(plus68_out_gag(x₁)) = 1   
POL(plus72_in_gag(x₁, x₂)) = x₂   
POL(plus72_out_gag(x₁)) = 1   
POL(s(x₁)) = x₁   
POL(ways1_in_gga(x₁, x₂)) = 0   
POL(ways1_out_gga) = 0   

The following usable rules [FROCOS05] were oriented:

plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U6_g(nat80_out_g) → nat80_out_g

(65) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → WAYS1_IN_GGA(s(T51), T39)
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T123, U10_gag(plus72_in_gag(T132, T122)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(66) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(67) Obligation:

Q DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → WAYS1_IN_GGA(s(T51), T39)
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → WAYS1_IN_GGA(s(T51), T39)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = 0   
POL(U10_gag(x₁)) = x₁   
POL(U11_gga(x₁)) = 0   
POL(U12_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U12_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U13_gga(x₁)) = 0   
POL(U14_GGA(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U14_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U15_gga(x₁)) = 0   
POL(U16_gga(x₁)) = 0   
POL(U17_gga(x₁)) = 0   
POL(U18_gga(x₁)) = 0   
POL(U19_gga(x₁, x₂, x₃)) = 0   
POL(U1_a(x₁)) = 0   
POL(U1_g(x₁)) = 0   
POL(U20_gga(x₁)) = 0   
POL(U2_gag(x₁, x₂)) = 0   
POL(U3_gag(x₁)) = 0   
POL(U4_aaa(x₁)) = 0   
POL(U5_aaa(x₁)) = 0   
POL(U6_g(x₁)) = 0   
POL(U7_gag(x₁, x₂)) = 0   
POL(U8_gag(x₁)) = 0   
POL(U9_gag(x₁)) = 0   
POL(WAYS1_IN_GGA(x₁, x₂)) = x₂   
POL([]) = 0   
POL(nat40_in_a) = 0   
POL(nat40_in_g(x₁)) = 0   
POL(nat40_out_a(x₁)) = 0   
POL(nat40_out_g) = 0   
POL(nat80_in_g(x₁)) = 0   
POL(nat80_out_g) = 0   
POL(plus32_in_gag(x₁, x₂)) = x₁ + x₂   
POL(plus32_out_gag(x₁)) = 0   
POL(plus36_in_gag(x₁, x₂)) = 0   
POL(plus36_out_gag(x₁)) = 0   
POL(plus55_in_aaa) = 0   
POL(plus55_out_aaa(x₁, x₂, x₃)) = 0   
POL(plus68_in_gag(x₁, x₂)) = x₁   
POL(plus68_out_gag(x₁)) = 0   
POL(plus72_in_gag(x₁, x₂)) = 0   
POL(plus72_out_gag(x₁)) = x₁   
POL(s(x₁)) = 1   
POL(ways1_in_gga(x₁, x₂)) = 0   
POL(ways1_out_gga) = 0   

The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

Q DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))
U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(plus36_in_gag(T38, T51)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 0   
POL(0) = 0   
POL(U10_gag(x₁)) = 1 + x₁   
POL(U11_gga(x₁)) = 0   
POL(U12_GGA(x₁, x₂, x₃, x₄)) = x₄   
POL(U12_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U13_gga(x₁)) = 0   
POL(U14_GGA(x₁, x₂, x₃, x₄)) = x₃   
POL(U14_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U15_gga(x₁)) = 0   
POL(U16_gga(x₁)) = 0   
POL(U17_gga(x₁)) = 0   
POL(U18_gga(x₁)) = 0   
POL(U19_gga(x₁, x₂, x₃)) = 0   
POL(U1_a(x₁)) = 0   
POL(U1_g(x₁)) = 0   
POL(U20_gga(x₁)) = 0   
POL(U2_gag(x₁, x₂)) = x₁   
POL(U3_gag(x₁)) = x₁   
POL(U4_aaa(x₁)) = x₁   
POL(U5_aaa(x₁)) = 0   
POL(U6_g(x₁)) = x₁   
POL(U7_gag(x₁, x₂)) = 0   
POL(U8_gag(x₁)) = 0   
POL(U9_gag(x₁)) = x₁   
POL(WAYS1_IN_GGA(x₁, x₂)) = x₁   
POL([]) = 0   
POL(nat40_in_a) = 0   
POL(nat40_in_g(x₁)) = 0   
POL(nat40_out_a(x₁)) = x₁   
POL(nat40_out_g) = 0   
POL(nat80_in_g(x₁)) = 1 + x₁   
POL(nat80_out_g) = 0   
POL(plus32_in_gag(x₁, x₂)) = 0   
POL(plus32_out_gag(x₁)) = x₁   
POL(plus36_in_gag(x₁, x₂)) = x₂   
POL(plus36_out_gag(x₁)) = x₁   
POL(plus55_in_aaa) = 0   
POL(plus55_out_aaa(x₁, x₂, x₃)) = 0   
POL(plus68_in_gag(x₁, x₂)) = 1 + x₁   
POL(plus68_out_gag(x₁)) = 1   
POL(plus72_in_gag(x₁, x₂)) = x₁   
POL(plus72_out_gag(x₁)) = 1   
POL(s(x₁)) = 1 + x₁   
POL(ways1_in_gga(x₁, x₂)) = 0   
POL(ways1_out_gga) = 0   

The following usable rules [FROCOS05] were oriented:

plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T51, T38, T39, plus32_out_gag(T54)) → U14_GGA(T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T38, T39, T54, ways1_out_gga) → WAYS1_IN_GGA(T54, .(s(T38), T39))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T123, plus68_in_gag(T132, T122))
U9_gag(plus36_out_gag(X119)) → plus32_out_gag(X119)
U10_gag(plus72_out_gag(X234)) → plus68_out_gag(X234)
U11_gga(plus32_out_gag(X82)) → ways1_out_gga
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U13_gga(ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T54)) → U14_gga(T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(plus68_out_gag(X203)) → ways1_out_gga
U19_gga(T132, T123, plus68_out_gag(T135)) → U20_gga(ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(plus72_in_gag(T148, T149))
U13_gga(ways1_out_gga) → ways1_out_gga
U14_gga(T38, T39, T54, ways1_out_gga) → U15_gga(ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T38, T39, T54, ways1_out_gga) → U16_gga(ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(ways1_out_gga) → ways1_out_gga
U2_gag(T69, nat40_out_g) → plus36_out_gag(T69)
U3_gag(plus36_out_gag(X148)) → plus36_out_gag(X148)
U7_gag(X255, nat80_out_g) → plus72_out_gag(X255)
U8_gag(plus72_out_gag(X273)) → plus72_out_gag(X273)
U15_gga(ways1_out_gga) → ways1_out_gga
U16_gga(ways1_out_gga) → U17_gga(plus55_in_aaa)
nat40_in_g(0) → nat40_out_g
nat40_in_g(s(T72)) → U1_g(nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g
nat80_in_g(s(X261)) → U6_g(nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga
U17_gga(plus55_out_aaa(T82, T89, T41)) → ways1_out_gga
U1_g(nat40_out_g) → nat40_out_g
U6_g(nat80_out_g) → nat80_out_g
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0)
U10_gag(x0)
U11_gga(x0)
U12_gga(x0, x1, x2, x3)
U18_gga(x0)
U19_gga(x0, x1, x2)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0)
U14_gga(x0, x1, x2, x3)
U20_gga(x0)
U2_gag(x0, x1)
U3_gag(x0)
U7_gag(x0, x1)
U8_gag(x0)
U15_gga(x0)
U16_gga(x0)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0)
U1_g(x0)
U6_g(x0)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(72) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(73) TRUE

(74) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ways1_in: (b,b,f)
plus32_in: (b,f,b)
plus36_in: (b,f,b)
nat40_in: (b) (f)
plus68_in: (b,f,b)
plus72_in: (b,f,b)
nat80_in: (b)
plus55_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(75) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)

(76) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U11_GGA(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → PLUS32_IN_GAG(T38, X82, T51)
PLUS32_IN_GAG(T62, X119, T63) → U9_GAG(T62, X119, T63, plus36_in_gag(T62, X119, T63))
PLUS32_IN_GAG(T62, X119, T63) → PLUS36_IN_GAG(T62, X119, T63)
PLUS36_IN_GAG(0, T69, T69) → U2_GAG(T69, nat40_in_g(T69))
PLUS36_IN_GAG(0, T69, T69) → NAT40_IN_G(T69)
NAT40_IN_G(s(T72)) → U1_G(T72, nat40_in_g(T72))
NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)
PLUS36_IN_GAG(s(T77), X148, s(T78)) → U3_GAG(T77, X148, T78, plus36_in_gag(T77, X148, T78))
PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_GGA(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U18_GGA(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → PLUS68_IN_GAG(T132, X203, T122)
PLUS68_IN_GAG(T142, X234, T143) → U10_GAG(T142, X234, T143, plus72_in_gag(T142, X234, T143))
PLUS68_IN_GAG(T142, X234, T143) → PLUS72_IN_GAG(T142, X234, T143)
PLUS72_IN_GAG(0, X255, s(X255)) → U7_GAG(X255, nat80_in_g(X255))
PLUS72_IN_GAG(0, X255, s(X255)) → NAT80_IN_G(X255)
NAT80_IN_G(s(X261)) → U6_G(X261, nat80_in_g(X261))
NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)
PLUS72_IN_GAG(s(T148), X273, s(T149)) → U8_GAG(T148, X273, T149, plus72_in_gag(T148, X273, T149))
PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_GGA(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_GGA(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_GGA(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_GGA(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → PLUS55_IN_AAA(T82, T89, T41)
PLUS55_IN_AAA(0, T99, T99) → U4_AAA(T99, nat40_in_a(T99))
PLUS55_IN_AAA(0, T99, T99) → NAT40_IN_A(T99)
NAT40_IN_A(s(T72)) → U1_A(T72, nat40_in_a(T72))
NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)
PLUS55_IN_AAA(s(T106), T107, s(T109)) → U5_AAA(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x2, x3, x5)
PLUS32_IN_GAG(x1, x2, x3)  =  PLUS32_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x1, x3, x4)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)
U2_GAG(x1, x2)  =  U2_GAG(x1, x2)
NAT40_IN_G(x1)  =  NAT40_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x1, x3, x4)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x1, x2, x3, x5)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
PLUS68_IN_GAG(x1, x2, x3)  =  PLUS68_IN_GAG(x1, x3)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x1, x3, x4)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)
U7_GAG(x1, x2)  =  U7_GAG(x1, x2)
NAT80_IN_G(x1)  =  NAT80_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x1, x2)
U8_GAG(x1, x2, x3, x4)  =  U8_GAG(x1, x3, x4)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x5, x6)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x1, x2, x3, x5)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x1, x2, x3, x6)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x1, x2, x3, x5)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA
U4_AAA(x1, x2)  =  U4_AAA(x2)
NAT40_IN_A(x1)  =  NAT40_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U5_AAA(x1, x2, x3, x4)  =  U5_AAA(x4)

We have to consider all (P,R,Pi)-chains

(77) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U11_GGA(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → PLUS32_IN_GAG(T38, X82, T51)
PLUS32_IN_GAG(T62, X119, T63) → U9_GAG(T62, X119, T63, plus36_in_gag(T62, X119, T63))
PLUS32_IN_GAG(T62, X119, T63) → PLUS36_IN_GAG(T62, X119, T63)
PLUS36_IN_GAG(0, T69, T69) → U2_GAG(T69, nat40_in_g(T69))
PLUS36_IN_GAG(0, T69, T69) → NAT40_IN_G(T69)
NAT40_IN_G(s(T72)) → U1_G(T72, nat40_in_g(T72))
NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)
PLUS36_IN_GAG(s(T77), X148, s(T78)) → U3_GAG(T77, X148, T78, plus36_in_gag(T77, X148, T78))
PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)
WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_GGA(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U18_GGA(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → PLUS68_IN_GAG(T132, X203, T122)
PLUS68_IN_GAG(T142, X234, T143) → U10_GAG(T142, X234, T143, plus72_in_gag(T142, X234, T143))
PLUS68_IN_GAG(T142, X234, T143) → PLUS72_IN_GAG(T142, X234, T143)
PLUS72_IN_GAG(0, X255, s(X255)) → U7_GAG(X255, nat80_in_g(X255))
PLUS72_IN_GAG(0, X255, s(X255)) → NAT80_IN_G(X255)
NAT80_IN_G(s(X261)) → U6_G(X261, nat80_in_g(X261))
NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)
PLUS72_IN_GAG(s(T148), X273, s(T149)) → U8_GAG(T148, X273, T149, plus72_in_gag(T148, X273, T149))
PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_GGA(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_GGA(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_GGA(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_GGA(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
U16_GGA(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → PLUS55_IN_AAA(T82, T89, T41)
PLUS55_IN_AAA(0, T99, T99) → U4_AAA(T99, nat40_in_a(T99))
PLUS55_IN_AAA(0, T99, T99) → NAT40_IN_A(T99)
NAT40_IN_A(s(T72)) → U1_A(T72, nat40_in_a(T72))
NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)
PLUS55_IN_AAA(s(T106), T107, s(T109)) → U5_AAA(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x1, x2, x3, x5)
PLUS32_IN_GAG(x1, x2, x3)  =  PLUS32_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x1, x3, x4)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)
U2_GAG(x1, x2)  =  U2_GAG(x1, x2)
NAT40_IN_G(x1)  =  NAT40_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x1, x3, x4)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U13_GGA(x1, x2, x3, x4, x5)  =  U13_GGA(x1, x2, x3, x5)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
PLUS68_IN_GAG(x1, x2, x3)  =  PLUS68_IN_GAG(x1, x3)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x1, x3, x4)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)
U7_GAG(x1, x2)  =  U7_GAG(x1, x2)
NAT80_IN_G(x1)  =  NAT80_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x1, x2)
U8_GAG(x1, x2, x3, x4)  =  U8_GAG(x1, x3, x4)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x5, x6)
U15_GGA(x1, x2, x3, x4, x5)  =  U15_GGA(x1, x2, x3, x5)
U16_GGA(x1, x2, x3, x4, x5, x6)  =  U16_GGA(x1, x2, x3, x6)
U17_GGA(x1, x2, x3, x4, x5)  =  U17_GGA(x1, x2, x3, x5)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA
U4_AAA(x1, x2)  =  U4_AAA(x2)
NAT40_IN_A(x1)  =  NAT40_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U5_AAA(x1, x2, x3, x4)  =  U5_AAA(x4)

We have to consider all (P,R,Pi)-chains

(78) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 26 less nodes.

(79) Complex Obligation (AND)

(80) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
NAT40_IN_A(x1)  =  NAT40_IN_A

We have to consider all (P,R,Pi)-chains

(81) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(82) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_A(s(T72)) → NAT40_IN_A(T72)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NAT40_IN_A(x1)  =  NAT40_IN_A

We have to consider all (P,R,Pi)-chains

(83) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(84) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT40_IN_A → NAT40_IN_A

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

(85) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = NAT40_IN_A evaluates to t =NAT40_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from NAT40_IN_A to NAT40_IN_A.

(86) NO

(87) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA

We have to consider all (P,R,Pi)-chains

(88) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(89) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS55_IN_AAA(s(T106), T107, s(T109)) → PLUS55_IN_AAA(T106, T107, T109)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS55_IN_AAA(x1, x2, x3)  =  PLUS55_IN_AAA

We have to consider all (P,R,Pi)-chains

(90) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(91) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS55_IN_AAA → PLUS55_IN_AAA

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

(92) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = PLUS55_IN_AAA evaluates to t =PLUS55_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from PLUS55_IN_AAA to PLUS55_IN_AAA.

(93) NO

(94) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
NAT80_IN_G(x1)  =  NAT80_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(95) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(96) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)

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

(97) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(98) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)

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

(99) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• NAT80_IN_G(s(X261)) → NAT80_IN_G(X261)
  The graph contains the following edges 1 > 1

(100) YES

(101) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(102) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(103) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS72_IN_GAG(s(T148), X273, s(T149)) → PLUS72_IN_GAG(T148, X273, T149)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS72_IN_GAG(x1, x2, x3)  =  PLUS72_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(104) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS72_IN_GAG(s(T148), s(T149)) → PLUS72_IN_GAG(T148, T149)

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

(106) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• PLUS72_IN_GAG(s(T148), s(T149)) → PLUS72_IN_GAG(T148, T149)
  The graph contains the following edges 1 > 1, 2 > 2

(107) YES

(108) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
NAT40_IN_G(x1)  =  NAT40_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(109) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(110) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)

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

(111) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)

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

(113) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• NAT40_IN_G(s(T72)) → NAT40_IN_G(T72)
  The graph contains the following edges 1 > 1

(114) YES

(115) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(116) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(117) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS36_IN_GAG(s(T77), X148, s(T78)) → PLUS36_IN_GAG(T77, X148, T78)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS36_IN_GAG(x1, x2, x3)  =  PLUS36_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(118) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(119) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS36_IN_GAG(s(T77), s(T78)) → PLUS36_IN_GAG(T77, T78)

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

(120) QDPSizeChangeProof (EQUIVALENT transformation)

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:

• PLUS36_IN_GAG(s(T77), s(T78)) → PLUS36_IN_GAG(T77, T78)
  The graph contains the following edges 1 > 1, 2 > 2

(121) YES

(122) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)

The TRS R consists of the following rules:

ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
ways1_in_gga(0, [], 0) → ways1_out_gga(0, [], 0)
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x5, x6)

We have to consider all (P,R,Pi)-chains

(123) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(124) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39), T41) → U12_GGA(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39, X83)
WAYS1_IN_GGA(s(T132), .(s(T122), T123), T125) → U19_GGA(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U19_GGA(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123, T125)
U12_GGA(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U14_GGA(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → WAYS1_IN_GGA(T54, .(s(T38), T39), X84)

The TRS R consists of the following rules:

plus32_in_gag(T62, X119, T63) → U9_gag(T62, X119, T63, plus36_in_gag(T62, X119, T63))
plus68_in_gag(T142, X234, T143) → U10_gag(T142, X234, T143, plus72_in_gag(T142, X234, T143))
ways1_in_gga(T30, [], 0) → ways1_out_gga(T30, [], 0)
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U11_gga(T51, T38, T39, T41, plus32_in_gag(T38, X82, T51))
ways1_in_gga(s(T51), .(s(T38), T39), T41) → U12_gga(T51, T38, T39, T41, plus32_in_gag(T38, T54, T51))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U18_gga(T132, T122, T123, T125, plus68_in_gag(T132, X203, T122))
ways1_in_gga(s(T132), .(s(T122), T123), T125) → U19_gga(T132, T122, T123, T125, plus68_in_gag(T132, T135, T122))
U9_gag(T62, X119, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, X234, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, T41, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, T41, ways1_in_gga(s(T51), T39, X83))
U12_gga(T51, T38, T39, T41, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T41, T54, ways1_in_gga(s(T51), T39, T82))
U18_gga(T132, T122, T123, T125, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U19_gga(T132, T122, T123, T125, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, T125, ways1_in_gga(s(T132), T123, T125))
plus36_in_gag(0, T69, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), X148, s(T78)) → U3_gag(T77, X148, T78, plus36_in_gag(T77, X148, T78))
plus72_in_gag(0, X255, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), X273, s(T149)) → U8_gag(T148, X273, T149, plus72_in_gag(T148, X273, T149))
U13_gga(T51, T38, T39, T41, ways1_out_gga(s(T51), T39, X83)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U15_gga(T51, T38, T39, T41, ways1_in_gga(T54, .(s(T38), T39), X84))
U14_gga(T51, T38, T39, T41, T54, ways1_out_gga(s(T51), T39, T82)) → U16_gga(T51, T38, T39, T41, T82, ways1_in_gga(T54, .(s(T38), T39), T89))
U20_gga(T132, T122, T123, T125, ways1_out_gga(s(T132), T123, T125)) → ways1_out_gga(s(T132), .(s(T122), T123), T125)
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, X148, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, X273, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, T41, ways1_out_gga(T54, .(s(T38), T39), X84)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U16_gga(T51, T38, T39, T41, T82, ways1_out_gga(T54, .(s(T38), T39), T89)) → U17_gga(T51, T38, T39, T41, plus55_in_aaa(T82, T89, T41))
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5, s(0)) → ways1_out_gga(0, T5, s(0))
U17_gga(T51, T38, T39, T41, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39), T41)
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa(0, T99, T99) → U4_aaa(T99, nat40_in_a(T99))
plus55_in_aaa(s(T106), T107, s(T109)) → U5_aaa(T106, T107, T109, plus55_in_aaa(T106, T107, T109))
U4_aaa(T99, nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(T106, T107, T109, plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a(0) → nat40_out_a(0)
nat40_in_a(s(T72)) → U1_a(T72, nat40_in_a(T72))
U1_a(T72, nat40_out_a(T72)) → nat40_out_a(s(T72))

The argument filtering Pi contains the following mapping:
ways1_in_gga(x1, x2, x3)  =  ways1_in_gga(x1, x2)
0  =  0
ways1_out_gga(x1, x2, x3)  =  ways1_out_gga(x1, x2)
[]  =  []
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x1, x2, x3, x5)
plus32_in_gag(x1, x2, x3)  =  plus32_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
plus36_in_gag(x1, x2, x3)  =  plus36_in_gag(x1, x3)
U2_gag(x1, x2)  =  U2_gag(x1, x2)
nat40_in_g(x1)  =  nat40_in_g(x1)
nat40_out_g(x1)  =  nat40_out_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
plus36_out_gag(x1, x2, x3)  =  plus36_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x1, x3, x4)
plus32_out_gag(x1, x2, x3)  =  plus32_out_gag(x1, x2, x3)
U12_gga(x1, x2, x3, x4, x5)  =  U12_gga(x1, x2, x3, x5)
U13_gga(x1, x2, x3, x4, x5)  =  U13_gga(x1, x2, x3, x5)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
plus68_in_gag(x1, x2, x3)  =  plus68_in_gag(x1, x3)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x3, x4)
plus72_in_gag(x1, x2, x3)  =  plus72_in_gag(x1, x3)
U7_gag(x1, x2)  =  U7_gag(x1, x2)
nat80_in_g(x1)  =  nat80_in_g(x1)
nat80_out_g(x1)  =  nat80_out_g(x1)
U6_g(x1, x2)  =  U6_g(x1, x2)
plus72_out_gag(x1, x2, x3)  =  plus72_out_gag(x1, x2, x3)
U8_gag(x1, x2, x3, x4)  =  U8_gag(x1, x3, x4)
plus68_out_gag(x1, x2, x3)  =  plus68_out_gag(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4, x5, x6)  =  U14_gga(x1, x2, x3, x5, x6)
U15_gga(x1, x2, x3, x4, x5)  =  U15_gga(x1, x2, x3, x5)
U16_gga(x1, x2, x3, x4, x5, x6)  =  U16_gga(x1, x2, x3, x6)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
plus55_in_aaa(x1, x2, x3)  =  plus55_in_aaa
U4_aaa(x1, x2)  =  U4_aaa(x2)
nat40_in_a(x1)  =  nat40_in_a
nat40_out_a(x1)  =  nat40_out_a(x1)
U1_a(x1, x2)  =  U1_a(x2)
plus55_out_aaa(x1, x2, x3)  =  plus55_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4)  =  U5_aaa(x4)
WAYS1_IN_GGA(x1, x2, x3)  =  WAYS1_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4, x5)  =  U12_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x5, x6)

We have to consider all (P,R,Pi)-chains

(125) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(126) Obligation:

Q DP problem:
The TRS P consists of the following rules:

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, plus32_in_gag(T38, T51))
U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39)
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T122, T123, plus68_in_gag(T132, T122))
U19_GGA(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → WAYS1_IN_GGA(T54, .(s(T38), T39))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(T62, T63, plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(T142, T143, plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga(T30, [])
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(T132, T122, T123, plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T122, T123, plus68_in_gag(T132, T122))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(T132, T122, T123, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123))
U19_gga(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(T148, T149, plus72_in_gag(T148, T149))
U13_gga(T51, T38, T39, ways1_out_gga(s(T51), T39)) → ways1_out_gga(s(T51), .(s(T38), T39))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U15_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U16_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(T132, T122, T123, ways1_out_gga(s(T132), T123)) → ways1_out_gga(s(T132), .(s(T122), T123))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → ways1_out_gga(s(T51), .(s(T38), T39))
U16_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → U17_gga(T51, T38, T39, plus55_in_aaa)
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga(0, T5)
U17_gga(T51, T38, T39, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0, x1, x2)
U10_gag(x0, x1, x2)
U11_gga(x0, x1, x2, x3)
U12_gga(x0, x1, x2, x3)
U18_gga(x0, x1, x2, x3)
U19_gga(x0, x1, x2, x3)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0, x1, x2, x3)
U14_gga(x0, x1, x2, x3, x4)
U20_gga(x0, x1, x2, x3)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U7_gag(x0, x1)
U8_gag(x0, x1, x2)
U15_gga(x0, x1, x2, x3)
U16_gga(x0, x1, x2, x3)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0, x1, x2, x3)
U1_g(x0, x1)
U6_g(x0, x1)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(127) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, plus32_in_gag(T38, T51)) at position [3] we obtained the following new rules [LPAR04]:

WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(T38, T51, plus36_in_gag(T38, T51)))

(128) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39)
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T122, T123, plus68_in_gag(T132, T122))
U19_GGA(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(T38, T51, plus36_in_gag(T38, T51)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(T62, T63, plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(T142, T143, plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga(T30, [])
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(T132, T122, T123, plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T122, T123, plus68_in_gag(T132, T122))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(T132, T122, T123, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123))
U19_gga(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(T148, T149, plus72_in_gag(T148, T149))
U13_gga(T51, T38, T39, ways1_out_gga(s(T51), T39)) → ways1_out_gga(s(T51), .(s(T38), T39))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U15_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U16_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(T132, T122, T123, ways1_out_gga(s(T132), T123)) → ways1_out_gga(s(T132), .(s(T122), T123))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → ways1_out_gga(s(T51), .(s(T38), T39))
U16_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → U17_gga(T51, T38, T39, plus55_in_aaa)
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga(0, T5)
U17_gga(T51, T38, T39, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0, x1, x2)
U10_gag(x0, x1, x2)
U11_gga(x0, x1, x2, x3)
U12_gga(x0, x1, x2, x3)
U18_gga(x0, x1, x2, x3)
U19_gga(x0, x1, x2, x3)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0, x1, x2, x3)
U14_gga(x0, x1, x2, x3, x4)
U20_gga(x0, x1, x2, x3)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U7_gag(x0, x1)
U8_gag(x0, x1, x2)
U15_gga(x0, x1, x2, x3)
U16_gga(x0, x1, x2, x3)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0, x1, x2, x3)
U1_g(x0, x1)
U6_g(x0, x1)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(129) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T122, T123, plus68_in_gag(T132, T122)) at position [3] we obtained the following new rules [LPAR04]:

WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T122, T123, U10_gag(T132, T122, plus72_in_gag(T132, T122)))

(130) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39)
U19_GGA(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(T38, T51, plus36_in_gag(T38, T51)))
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T122, T123, U10_gag(T132, T122, plus72_in_gag(T132, T122)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(T62, T63, plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(T142, T143, plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga(T30, [])
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(T132, T122, T123, plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T122, T123, plus68_in_gag(T132, T122))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(T132, T122, T123, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123))
U19_gga(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(T148, T149, plus72_in_gag(T148, T149))
U13_gga(T51, T38, T39, ways1_out_gga(s(T51), T39)) → ways1_out_gga(s(T51), .(s(T38), T39))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U15_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U16_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(T132, T122, T123, ways1_out_gga(s(T132), T123)) → ways1_out_gga(s(T132), .(s(T122), T123))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → ways1_out_gga(s(T51), .(s(T38), T39))
U16_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → U17_gga(T51, T38, T39, plus55_in_aaa)
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga(0, T5)
U17_gga(T51, T38, T39, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0, x1, x2)
U10_gag(x0, x1, x2)
U11_gga(x0, x1, x2, x3)
U12_gga(x0, x1, x2, x3)
U18_gga(x0, x1, x2, x3)
U19_gga(x0, x1, x2, x3)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0, x1, x2, x3)
U14_gga(x0, x1, x2, x3, x4)
U20_gga(x0, x1, x2, x3)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U7_gag(x0, x1)
U8_gag(x0, x1, x2)
U15_gga(x0, x1, x2, x3)
U16_gga(x0, x1, x2, x3)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0, x1, x2, x3)
U1_g(x0, x1)
U6_g(x0, x1)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → WAYS1_IN_GGA(s(T51), T39)
WAYS1_IN_GGA(s(T132), .(s(T122), T123)) → U19_GGA(T132, T122, T123, U10_gag(T132, T122, plus72_in_gag(T132, T122)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(U10_gag(x₁, x₂, x₃)) = 0   
POL(U11_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U12_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U12_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U13_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U14_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃   
POL(U14_gga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U15_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U16_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U17_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U18_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U19_GGA(x₁, x₂, x₃, x₄)) = x₃   
POL(U19_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U1_a(x₁)) = 1   
POL(U1_g(x₁, x₂)) = 0   
POL(U20_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U2_gag(x₁, x₂)) = 0   
POL(U3_gag(x₁, x₂, x₃)) = 0   
POL(U4_aaa(x₁)) = 1   
POL(U5_aaa(x₁)) = 0   
POL(U6_g(x₁, x₂)) = 0   
POL(U7_gag(x₁, x₂)) = 0   
POL(U8_gag(x₁, x₂, x₃)) = 0   
POL(U9_gag(x₁, x₂, x₃)) = 0   
POL(WAYS1_IN_GGA(x₁, x₂)) = x₂   
POL([]) = 0   
POL(nat40_in_a) = 1   
POL(nat40_in_g(x₁)) = 0   
POL(nat40_out_a(x₁)) = 0   
POL(nat40_out_g(x₁)) = 0   
POL(nat80_in_g(x₁)) = 0   
POL(nat80_out_g(x₁)) = 0   
POL(plus32_in_gag(x₁, x₂)) = 0   
POL(plus32_out_gag(x₁, x₂, x₃)) = 0   
POL(plus36_in_gag(x₁, x₂)) = 0   
POL(plus36_out_gag(x₁, x₂, x₃)) = 0   
POL(plus55_in_aaa) = 1   
POL(plus55_out_aaa(x₁, x₂, x₃)) = 0   
POL(plus68_in_gag(x₁, x₂)) = x₂   
POL(plus68_out_gag(x₁, x₂, x₃)) = 0   
POL(plus72_in_gag(x₁, x₂)) = 0   
POL(plus72_out_gag(x₁, x₂, x₃)) = 0   
POL(s(x₁)) = 0   
POL(ways1_in_gga(x₁, x₂)) = 0   
POL(ways1_out_gga(x₁, x₂)) = 0   

The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U19_GGA(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → WAYS1_IN_GGA(s(T132), T123)
U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U14_GGA(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(T38, T51, plus36_in_gag(T38, T51)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(T62, T63, plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(T142, T143, plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga(T30, [])
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(T132, T122, T123, plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T122, T123, plus68_in_gag(T132, T122))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(T132, T122, T123, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123))
U19_gga(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(T148, T149, plus72_in_gag(T148, T149))
U13_gga(T51, T38, T39, ways1_out_gga(s(T51), T39)) → ways1_out_gga(s(T51), .(s(T38), T39))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U15_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U16_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(T132, T122, T123, ways1_out_gga(s(T132), T123)) → ways1_out_gga(s(T132), .(s(T122), T123))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → ways1_out_gga(s(T51), .(s(T38), T39))
U16_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → U17_gga(T51, T38, T39, plus55_in_aaa)
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga(0, T5)
U17_gga(T51, T38, T39, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0, x1, x2)
U10_gag(x0, x1, x2)
U11_gga(x0, x1, x2, x3)
U12_gga(x0, x1, x2, x3)
U18_gga(x0, x1, x2, x3)
U19_gga(x0, x1, x2, x3)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0, x1, x2, x3)
U14_gga(x0, x1, x2, x3, x4)
U20_gga(x0, x1, x2, x3)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U7_gag(x0, x1)
U8_gag(x0, x1, x2)
U15_gga(x0, x1, x2, x3)
U16_gga(x0, x1, x2, x3)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0, x1, x2, x3)
U1_g(x0, x1)
U6_g(x0, x1)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(133) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(134) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U14_GGA(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(T38, T51, plus36_in_gag(T38, T51)))
U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(T62, T63, plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(T142, T143, plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga(T30, [])
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(T132, T122, T123, plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T122, T123, plus68_in_gag(T132, T122))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(T132, T122, T123, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123))
U19_gga(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(T148, T149, plus72_in_gag(T148, T149))
U13_gga(T51, T38, T39, ways1_out_gga(s(T51), T39)) → ways1_out_gga(s(T51), .(s(T38), T39))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U15_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U16_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(T132, T122, T123, ways1_out_gga(s(T132), T123)) → ways1_out_gga(s(T132), .(s(T122), T123))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → ways1_out_gga(s(T51), .(s(T38), T39))
U16_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → U17_gga(T51, T38, T39, plus55_in_aaa)
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga(0, T5)
U17_gga(T51, T38, T39, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0, x1, x2)
U10_gag(x0, x1, x2)
U11_gga(x0, x1, x2, x3)
U12_gga(x0, x1, x2, x3)
U18_gga(x0, x1, x2, x3)
U19_gga(x0, x1, x2, x3)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0, x1, x2, x3)
U14_gga(x0, x1, x2, x3, x4)
U20_gga(x0, x1, x2, x3)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U7_gag(x0, x1)
U8_gag(x0, x1, x2)
U15_gga(x0, x1, x2, x3)
U16_gga(x0, x1, x2, x3)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0, x1, x2, x3)
U1_g(x0, x1)
U6_g(x0, x1)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(135) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12_GGA(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_GGA(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₂   
POL(0) = 0   
POL(U10_gag(x₁, x₂, x₃)) = 0   
POL(U11_gga(x₁, x₂, x₃, x₄)) = x₃   
POL(U12_GGA(x₁, x₂, x₃, x₄)) = x₄   
POL(U12_gga(x₁, x₂, x₃, x₄)) = x₃   
POL(U13_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U14_GGA(x₁, x₂, x₃, x₄, x₅)) = x₄   
POL(U14_gga(x₁, x₂, x₃, x₄, x₅)) = x₅   
POL(U15_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U16_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U17_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U18_gga(x₁, x₂, x₃, x₄)) = x₃   
POL(U19_gga(x₁, x₂, x₃, x₄)) = x₃   
POL(U1_a(x₁)) = 0   
POL(U1_g(x₁, x₂)) = 0   
POL(U20_gga(x₁, x₂, x₃, x₄)) = 0   
POL(U2_gag(x₁, x₂)) = x₁   
POL(U3_gag(x₁, x₂, x₃)) = 1 + x₃   
POL(U4_aaa(x₁)) = 0   
POL(U5_aaa(x₁)) = 0   
POL(U6_g(x₁, x₂)) = 0   
POL(U7_gag(x₁, x₂)) = 0   
POL(U8_gag(x₁, x₂, x₃)) = 0   
POL(U9_gag(x₁, x₂, x₃)) = 1 + x₃   
POL(WAYS1_IN_GGA(x₁, x₂)) = x₁   
POL([]) = 0   
POL(nat40_in_a) = 0   
POL(nat40_in_g(x₁)) = 0   
POL(nat40_out_a(x₁)) = 0   
POL(nat40_out_g(x₁)) = 0   
POL(nat80_in_g(x₁)) = 0   
POL(nat80_out_g(x₁)) = 0   
POL(plus32_in_gag(x₁, x₂)) = 0   
POL(plus32_out_gag(x₁, x₂, x₃)) = 1 + x₂   
POL(plus36_in_gag(x₁, x₂)) = x₂   
POL(plus36_out_gag(x₁, x₂, x₃)) = x₂   
POL(plus55_in_aaa) = 0   
POL(plus55_out_aaa(x₁, x₂, x₃)) = 0   
POL(plus68_in_gag(x₁, x₂)) = 0   
POL(plus68_out_gag(x₁, x₂, x₃)) = 0   
POL(plus72_in_gag(x₁, x₂)) = 0   
POL(plus72_out_gag(x₁, x₂, x₃)) = 0   
POL(s(x₁)) = 1 + x₁   
POL(ways1_in_gga(x₁, x₂)) = x₂   
POL(ways1_out_gga(x₁, x₂)) = 0   

The following usable rules [FROCOS05] were oriented:

plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)

(136) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U14_GGA(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → WAYS1_IN_GGA(T54, .(s(T38), T39))
WAYS1_IN_GGA(s(T51), .(s(T38), T39)) → U12_GGA(T51, T38, T39, U9_gag(T38, T51, plus36_in_gag(T38, T51)))

The TRS R consists of the following rules:

plus32_in_gag(T62, T63) → U9_gag(T62, T63, plus36_in_gag(T62, T63))
plus68_in_gag(T142, T143) → U10_gag(T142, T143, plus72_in_gag(T142, T143))
ways1_in_gga(T30, []) → ways1_out_gga(T30, [])
ways1_in_gga(s(T51), .(s(T38), T39)) → U11_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T51), .(s(T38), T39)) → U12_gga(T51, T38, T39, plus32_in_gag(T38, T51))
ways1_in_gga(s(T132), .(s(T122), T123)) → U18_gga(T132, T122, T123, plus68_in_gag(T132, T122))
ways1_in_gga(s(T132), .(s(T122), T123)) → U19_gga(T132, T122, T123, plus68_in_gag(T132, T122))
U9_gag(T62, T63, plus36_out_gag(T62, X119, T63)) → plus32_out_gag(T62, X119, T63)
U10_gag(T142, T143, plus72_out_gag(T142, X234, T143)) → plus68_out_gag(T142, X234, T143)
U11_gga(T51, T38, T39, plus32_out_gag(T38, X82, T51)) → ways1_out_gga(s(T51), .(s(T38), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U13_gga(T51, T38, T39, ways1_in_gga(s(T51), T39))
U12_gga(T51, T38, T39, plus32_out_gag(T38, T54, T51)) → U14_gga(T51, T38, T39, T54, ways1_in_gga(s(T51), T39))
U18_gga(T132, T122, T123, plus68_out_gag(T132, X203, T122)) → ways1_out_gga(s(T132), .(s(T122), T123))
U19_gga(T132, T122, T123, plus68_out_gag(T132, T135, T122)) → U20_gga(T132, T122, T123, ways1_in_gga(s(T132), T123))
plus36_in_gag(0, T69) → U2_gag(T69, nat40_in_g(T69))
plus36_in_gag(s(T77), s(T78)) → U3_gag(T77, T78, plus36_in_gag(T77, T78))
plus72_in_gag(0, s(X255)) → U7_gag(X255, nat80_in_g(X255))
plus72_in_gag(s(T148), s(T149)) → U8_gag(T148, T149, plus72_in_gag(T148, T149))
U13_gga(T51, T38, T39, ways1_out_gga(s(T51), T39)) → ways1_out_gga(s(T51), .(s(T38), T39))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U15_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U14_gga(T51, T38, T39, T54, ways1_out_gga(s(T51), T39)) → U16_gga(T51, T38, T39, ways1_in_gga(T54, .(s(T38), T39)))
U20_gga(T132, T122, T123, ways1_out_gga(s(T132), T123)) → ways1_out_gga(s(T132), .(s(T122), T123))
U2_gag(T69, nat40_out_g(T69)) → plus36_out_gag(0, T69, T69)
U3_gag(T77, T78, plus36_out_gag(T77, X148, T78)) → plus36_out_gag(s(T77), X148, s(T78))
U7_gag(X255, nat80_out_g(X255)) → plus72_out_gag(0, X255, s(X255))
U8_gag(T148, T149, plus72_out_gag(T148, X273, T149)) → plus72_out_gag(s(T148), X273, s(T149))
U15_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → ways1_out_gga(s(T51), .(s(T38), T39))
U16_gga(T51, T38, T39, ways1_out_gga(T54, .(s(T38), T39))) → U17_gga(T51, T38, T39, plus55_in_aaa)
nat40_in_g(0) → nat40_out_g(0)
nat40_in_g(s(T72)) → U1_g(T72, nat40_in_g(T72))
nat80_in_g(0) → nat80_out_g(0)
nat80_in_g(s(X261)) → U6_g(X261, nat80_in_g(X261))
ways1_in_gga(0, T5) → ways1_out_gga(0, T5)
U17_gga(T51, T38, T39, plus55_out_aaa(T82, T89, T41)) → ways1_out_gga(s(T51), .(s(T38), T39))
U1_g(T72, nat40_out_g(T72)) → nat40_out_g(s(T72))
U6_g(X261, nat80_out_g(X261)) → nat80_out_g(s(X261))
plus55_in_aaa → U4_aaa(nat40_in_a)
plus55_in_aaa → U5_aaa(plus55_in_aaa)
U4_aaa(nat40_out_a(T99)) → plus55_out_aaa(0, T99, T99)
U5_aaa(plus55_out_aaa(T106, T107, T109)) → plus55_out_aaa(s(T106), T107, s(T109))
nat40_in_a → nat40_out_a(0)
nat40_in_a → U1_a(nat40_in_a)
U1_a(nat40_out_a(T72)) → nat40_out_a(s(T72))

The set Q consists of the following terms:

plus32_in_gag(x0, x1)
plus68_in_gag(x0, x1)
ways1_in_gga(x0, x1)
U9_gag(x0, x1, x2)
U10_gag(x0, x1, x2)
U11_gga(x0, x1, x2, x3)
U12_gga(x0, x1, x2, x3)
U18_gga(x0, x1, x2, x3)
U19_gga(x0, x1, x2, x3)
plus36_in_gag(x0, x1)
plus72_in_gag(x0, x1)
U13_gga(x0, x1, x2, x3)
U14_gga(x0, x1, x2, x3, x4)
U20_gga(x0, x1, x2, x3)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U7_gag(x0, x1)
U8_gag(x0, x1, x2)
U15_gga(x0, x1, x2, x3)
U16_gga(x0, x1, x2, x3)
nat40_in_g(x0)
nat80_in_g(x0)
U17_gga(x0, x1, x2, x3)
U1_g(x0, x1)
U6_g(x0, x1)
plus55_in_aaa
U4_aaa(x0)
U5_aaa(x0)
nat40_in_a
U1_a(x0)

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

(137) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(138) TRUE