Left Termination proof of /tmp/tmpZbaNbS/nat.pl

Left Termination of the query pattern factorial_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 QDPSizeChangeProof (⇔)

                ↳29 YES

            ↳30 PiDP

                ↳31 UsableRulesProof (⇔)

                ↳32 PiDP

                ↳33 PiDPToQDPProof (⇐)

                ↳34 QDP

                ↳35 QDPSizeChangeProof (⇔)

                ↳36 YES

    ↳37 PrologToPiTRSProof (⇐)

        ↳38 PiTRS

        ↳39 DependencyPairsProof (⇔)

        ↳40 PiDP

        ↳41 DependencyGraphProof (⇔)

        ↳42 AND

            ↳43 PiDP

                ↳44 UsableRulesProof (⇔)

                ↳45 PiDP

                ↳46 PiDPToQDPProof (⇐)

                ↳47 QDP

                ↳48 NonTerminationProof (⇔)

                ↳49 NO

            ↳50 PiDP

                ↳51 UsableRulesProof (⇔)

                ↳52 PiDP

                ↳53 PiDPToQDPProof (⇐)

                ↳54 QDP

                ↳55 NonTerminationProof (⇔)

                ↳56 NO

            ↳57 PiDP

                ↳58 UsableRulesProof (⇔)

                ↳59 PiDP

                ↳60 PiDPToQDPProof (⇐)

                ↳61 QDP

                ↳62 QDPSizeChangeProof (⇔)

                ↳63 YES

            ↳64 PiDP

                ↳65 UsableRulesProof (⇔)

                ↳66 PiDP

                ↳67 PiDPToQDPProof (⇐)

                ↳68 QDP

                ↳69 QDPSizeChangeProof (⇔)

                ↳70 YES

(0) Obligation:

Clauses:

isNat(s(X)) :- isNat(X).
isNat(0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).
lt(s(X), s(Y)) :- lt(X, Y).
lt(0, s(Y)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
even(s(X)) :- odd(X).
even(0).
odd(s(X)) :- even(X).
odd(s(0)).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).
mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)).
mult(0, Y, 0).
factorial(s(X), R) :- ','(factorial(X, Y), mult(s(X), Y, R)).
factorial(0, s(0)).

Queries:

factorial(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: factorial(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: factorial(T1, T2), SCOPE: 1, CLAUSE: 20 | factorial(T1, T2), SCOPE: 1, CLAUSE: 21\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(factorial(T5, X5), mult(s(T5), X5, T7)) | factorial(s(T5), T2), SCOPE: 1, CLAUSE: 21\n\nT5 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: factorial(T1, T2), SCOPE: 1, CLAUSE: 21\n\nT1 is ground\nfactorial(T1, T2) !=? factorial(s(X3), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(factorial(T5, X5), mult(s(T5), X5, T7)), SCOPE: 2, CLAUSE: 20 | ','(factorial(T5, X5), mult(s(T5), X5, T7)), SCOPE: 2, CLAUSE: 21 | ?_2 | factorial(s(T5), T2), SCOPE: 1, CLAUSE: 21\n\nT5 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(factorial(T5, X5), mult(s(T5), X5, T7)), SCOPE: 2, CLAUSE: 20\n\nT5 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(factorial(T5, X5), mult(s(T5), X5, T7)), SCOPE: 2, CLAUSE: 21 | ?_2 | factorial(s(T5), T2), SCOPE: 1, CLAUSE: 21\n\nT5 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(factorial(T12, X24), ','(mult(s(T12), X24, X25), mult(s(s(T12)), X25, T7)))\n\nT12 is ground\nX25 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: factorial(T12, X24)\n\nT12 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(mult(s(T12), T14, X25), mult(s(s(T12)), X25, T7))\n\nT12 is ground\nT14 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: factorial(T12, X24), SCOPE: 3, CLAUSE: 20 | factorial(T12, X24), SCOPE: 3, CLAUSE: 21\n\nT12 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: factorial(T12, X24), SCOPE: 3, CLAUSE: 20\n\nT12 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: factorial(T12, X24), SCOPE: 3, CLAUSE: 21\n\nT12 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(factorial(T20, X52), mult(s(T20), X52, X53))\n\nT20 is ground\nX53 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: factorial(T20, X52)\n\nT20 is ground\nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: mult(s(T20), T22, X53)\n\nT20 is ground\nT22 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: mult(s(T20), T22, X53), SCOPE: 4, CLAUSE: 18 | mult(s(T20), T22, X53), SCOPE: 4, CLAUSE: 19\n\nT20 is ground\nT22 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: mult(s(T20), T22, X53), SCOPE: 4, CLAUSE: 18\n\nT20 is ground\nT22 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: mult(s(T20), T22, X53), SCOPE: 4, CLAUSE: 19\n\nT20 is ground\nT22 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(mult(T38, T39, X100), add(T39, X100, X101))\n\nT38 is ground\nT39 is ground\nX101 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: mult(T38, T39, X100)\n\nT38 is ground\nT39 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: add(T39, T42, X101)\n\nT39 is ground\nT42 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: mult(T38, T39, X100), SCOPE: 5, CLAUSE: 18 | mult(T38, T39, X100), SCOPE: 5, CLAUSE: 19\n\nT38 is ground\nT39 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: mult(T38, T39, X100), SCOPE: 5, CLAUSE: 18\n\nT38 is ground\nT39 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: mult(T38, T39, X100), SCOPE: 5, CLAUSE: 19\n\nT38 is ground\nT39 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(mult(T53, T54, X133), add(T54, X133, X134))\n\nT53 is ground\nT54 is ground\nX134 is free; \nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: add(T39, T42, X101), SCOPE: 6, CLAUSE: 16 | add(T39, T42, X101), SCOPE: 6, CLAUSE: 17\n\nT39 is ground\nT42 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: add(T39, T42, X101), SCOPE: 6, CLAUSE: 16\n\nT39 is ground\nT42 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: add(T39, T42, X101), SCOPE: 6, CLAUSE: 17\n\nT39 is ground\nT42 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: add(T70, T71, X170)\n\nT70 is ground\nT71 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: mult(s(T12), T14, X25)\n\nT12 is ground\nT14 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: mult(s(s(T12)), T79, T7)\n\nT12 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: mult(s(s(T12)), T79, T7), SCOPE: 7, CLAUSE: 18 | mult(s(s(T12)), T79, T7), SCOPE: 7, CLAUSE: 19\n\nT12 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: mult(s(s(T12)), T79, T7), SCOPE: 7, CLAUSE: 18\n\nT12 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: mult(s(s(T12)), T79, T7), SCOPE: 7, CLAUSE: 19\n\nT12 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(mult(s(T103), T104, X224), add(T104, X224, T106))\n\nT103 is ground\nT104 is ground\nX224 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: mult(s(T103), T104, X224)\n\nT103 is ground\nT104 is ground\nX224 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: add(T104, T109, T106)\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: add(T104, T109, T106), SCOPE: 8, CLAUSE: 16 | add(T104, T109, T106), SCOPE: 8, CLAUSE: 17\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: add(T104, T109, T106), SCOPE: 8, CLAUSE: 16\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: add(T104, T109, T106), SCOPE: 8, CLAUSE: 17\n\nT104 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: add(T127, T128, T130)\n\nT127 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(factorial(T5, X5), mult(s(T5), X5, T7)), SCOPE: 2, CLAUSE: 21\n\nT5 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ?_2 | factorial(s(T5), T2), SCOPE: 1, CLAUSE: 21\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: mult(s(0), s(0), T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: mult(s(0), s(0), T7), SCOPE: 9, CLAUSE: 18 | mult(s(0), s(0), T7), SCOPE: 9, CLAUSE: 19\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: mult(s(0), s(0), T7), SCOPE: 9, CLAUSE: 18\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: mult(s(0), s(0), T7), SCOPE: 9, CLAUSE: 19\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(mult(0, s(0), X303), add(s(0), X303, T147))\n\nX303 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: mult(0, s(0), X303)\n\nX303 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: add(s(0), T149, T147)\n\nT149 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: mult(0, s(0), X303), SCOPE: 10, CLAUSE: 18 | mult(0, s(0), X303), SCOPE: 10, CLAUSE: 19\n\nX303 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: mult(0, s(0), X303), SCOPE: 10, CLAUSE: 19\n\nX303 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: add(s(0), T149, T147), SCOPE: 11, CLAUSE: 16 | add(s(0), T149, T147), SCOPE: 11, CLAUSE: 17\n\nT149 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: add(s(0), T149, T147), SCOPE: 11, CLAUSE: 16\n\nT149 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: add(s(0), T149, T147), SCOPE: 11, CLAUSE: 17\n\nT149 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: add(0, T160, T162)\n\nT160 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: add(0, T160, T162), SCOPE: 12, CLAUSE: 16 | add(0, T160, T162), SCOPE: 12, CLAUSE: 17\n\nT160 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: add(0, T160, T162), SCOPE: 12, CLAUSE: 17\n\nT160 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: factorial(s(T5), T2), SCOPE: 1, CLAUSE: 21\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 93 [arrowhead = none ];
93 -> 2;

94 [label="EVAL with clause\nfactorial(s(X3), X4) :- ','(factorial(X3, X5), mult(s(X3), X5, X4)).\nand substitution\nX3 -> T5\nT1 -> s(T5)\nT2 -> T7\nX4 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 94 [arrowhead = none ];
94 -> 3;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 95 [arrowhead = none ];
95 -> 4;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 96 [arrowhead = none ];
96 -> 5;

97 [label="EVAL with clause\nfactorial(0, s(0)).\nand substitution\nT1 -> 0\nT2 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 97 [arrowhead = none ];
97 -> 90;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 98 [arrowhead = none ];
98 -> 91;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 99 [arrowhead = none ];
99 -> 6;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 100 [arrowhead = none ];
100 -> 7;

101 [label="EVAL with clause\nfactorial(s(X22), X23) :- ','(factorial(X22, X24), mult(s(X22), X24, X23)).\nand substitution\nX22 -> T12\nT5 -> s(T12)\nX5 -> X25\nX23 -> X25", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 101 [arrowhead = none ];
101 -> 8;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 102 [arrowhead = none ];
102 -> 9;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 103 [arrowhead = none ];
103 -> 62;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 104 [arrowhead = none ];
104 -> 63;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 105 [arrowhead = none ];
105 -> 10;

106 [label="SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:\nX24 -> T14", fontsize=14, style = filled, fillcolor = violet];
8 -> 106 [arrowhead = none ];
106 -> 11;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 107 [arrowhead = none ];
107 -> 12;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 108 [arrowhead = none ];
108 -> 45;

109 [label="SPLIT 2\nnew knowledge:\nT79 is ground\nreplacements:\nX25 -> T79", fontsize=14, style = filled, fillcolor = violet];
11 -> 109 [arrowhead = none ];
109 -> 46;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 110 [arrowhead = none ];
110 -> 13;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 111 [arrowhead = none ];
111 -> 14;

112 [label="EVAL with clause\nfactorial(s(X50), X51) :- ','(factorial(X50, X52), mult(s(X50), X52, X51)).\nand substitution\nX50 -> T20\nT12 -> s(T20)\nX24 -> X53\nX51 -> X53", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 112 [arrowhead = none ];
112 -> 15;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 113 [arrowhead = none ];
113 -> 16;

114 [label="EVAL with clause\nfactorial(0, s(0)).\nand substitution\nT12 -> 0\nX24 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 114 [arrowhead = none ];
114 -> 42;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 115 [arrowhead = none ];
115 -> 43;

116 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 116 [arrowhead = none ];
116 -> 17;

117 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX52 -> T22", fontsize=14, style = filled, fillcolor = violet];
15 -> 117 [arrowhead = none ];
117 -> 18;

118 [label="INSTANCE with matching:\nT12 -> T20\nX24 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 118 [arrowhead = none , style=dashed];
118 -> 10[style=dashed];

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 119 [arrowhead = none ];
119 -> 19;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 120 [arrowhead = none ];
120 -> 20;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 121 [arrowhead = none ];
121 -> 21;

122 [label="ONLY EVAL with clause\nmult(s(X97), X98, X99) :- ','(mult(X97, X98, X100), add(X98, X100, X99)).\nand substitution\nT20 -> T38\nX97 -> T38\nT22 -> T39\nX98 -> T39\nX53 -> X101\nX99 -> X101", fontsize=14, style = filled, fillcolor = white];
20 -> 122 [arrowhead = none ];
122 -> 22;

123 [label="BACKTRACK\nfor clause: mult(0, Y, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
21 -> 123 [arrowhead = none ];
123 -> 41;

124 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 124 [arrowhead = none ];
124 -> 23;

125 [label="SPLIT 2\nnew knowledge:\nT42 is ground\nreplacements:\nX100 -> T42", fontsize=14, style = filled, fillcolor = violet];
22 -> 125 [arrowhead = none ];
125 -> 24;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 126 [arrowhead = none ];
126 -> 25;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 127 [arrowhead = none ];
127 -> 33;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 128 [arrowhead = none ];
128 -> 26;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 129 [arrowhead = none ];
129 -> 27;

130 [label="EVAL with clause\nmult(s(X130), X131, X132) :- ','(mult(X130, X131, X133), add(X131, X133, X132)).\nand substitution\nX130 -> T53\nT38 -> s(T53)\nT39 -> T54\nX131 -> T54\nX100 -> X134\nX132 -> X134", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 130 [arrowhead = none ];
130 -> 28;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 131 [arrowhead = none ];
131 -> 29;

132 [label="EVAL with clause\nmult(0, X145, 0).\nand substitution\nT38 -> 0\nT39 -> T59\nX145 -> T59\nX100 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 132 [arrowhead = none ];
132 -> 30;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 133 [arrowhead = none ];
133 -> 31;

134 [label="INSTANCE with matching:\nT38 -> T53\nT39 -> T54\nX100 -> X133\nX101 -> X134", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 134 [arrowhead = none , style=dashed];
134 -> 22[style=dashed];

135 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 135 [arrowhead = none ];
135 -> 32;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 136 [arrowhead = none ];
136 -> 34;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 137 [arrowhead = none ];
137 -> 35;

138 [label="EVAL with clause\nadd(s(X167), X168, s(X169)) :- add(X167, X168, X169).\nand substitution\nX167 -> T70\nT39 -> s(T70)\nT42 -> T71\nX168 -> T71\nX169 -> X170\nX101 -> s(X170)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 138 [arrowhead = none ];
138 -> 36;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 139 [arrowhead = none ];
139 -> 37;

140 [label="EVAL with clause\nadd(0, X180, X180).\nand substitution\nT39 -> 0\nT42 -> T76\nX180 -> T76\nX101 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 140 [arrowhead = none ];
140 -> 38;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 141 [arrowhead = none ];
141 -> 39;

142 [label="INSTANCE with matching:\nT39 -> T70\nT42 -> T71\nX101 -> X170", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 142 [arrowhead = none , style=dashed];
142 -> 24[style=dashed];

143 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 143 [arrowhead = none ];
143 -> 40;

144 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 144 [arrowhead = none ];
144 -> 44;

145 [label="INSTANCE with matching:\nT20 -> T12\nT22 -> T14\nX53 -> X25", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 145 [arrowhead = none , style=dashed];
145 -> 18[style=dashed];

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 146 [arrowhead = none ];
146 -> 47;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 147 [arrowhead = none ];
147 -> 48;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 148 [arrowhead = none ];
148 -> 49;

149 [label="ONLY EVAL with clause\nmult(s(X221), X222, X223) :- ','(mult(X221, X222, X224), add(X222, X224, X223)).\nand substitution\nT12 -> T103\nX221 -> s(T103)\nT79 -> T104\nX222 -> T104\nT7 -> T106\nX223 -> T106\nT105 -> T106", fontsize=14, style = filled, fillcolor = white];
48 -> 149 [arrowhead = none ];
149 -> 50;

150 [label="BACKTRACK\nfor clause: mult(0, Y, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 150 [arrowhead = none ];
150 -> 61;

151 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 151 [arrowhead = none ];
151 -> 51;

152 [label="SPLIT 2\nnew knowledge:\nT109 is ground\nreplacements:\nX224 -> T109", fontsize=14, style = filled, fillcolor = violet];
50 -> 152 [arrowhead = none ];
152 -> 52;

153 [label="INSTANCE with matching:\nT12 -> T103\nT14 -> T104\nX25 -> X224", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 153 [arrowhead = none , style=dashed];
153 -> 45[style=dashed];

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 154 [arrowhead = none ];
154 -> 53;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 155 [arrowhead = none ];
155 -> 54;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 156 [arrowhead = none ];
156 -> 55;

157 [label="EVAL with clause\nadd(s(X250), X251, s(X252)) :- add(X250, X251, X252).\nand substitution\nX250 -> T127\nT104 -> s(T127)\nT109 -> T128\nX251 -> T128\nX252 -> T130\nT106 -> s(T130)\nT129 -> T130", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 157 [arrowhead = none ];
157 -> 56;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 158 [arrowhead = none ];
158 -> 57;

159 [label="EVAL with clause\nadd(0, X261, X261).\nand substitution\nT104 -> 0\nT109 -> T136\nX261 -> T136\nT106 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 159 [arrowhead = none ];
159 -> 58;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 160 [arrowhead = none ];
160 -> 59;

161 [label="INSTANCE with matching:\nT104 -> T127\nT109 -> T128\nT106 -> T130", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 161 [arrowhead = none , style=dashed];
161 -> 52[style=dashed];

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 162 [arrowhead = none ];
162 -> 60;

163 [label="EVAL with clause\nfactorial(0, s(0)).\nand substitution\nT5 -> 0\nX5 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 163 [arrowhead = none ];
163 -> 64;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 164 [arrowhead = none ];
164 -> 65;

165 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 165 [arrowhead = none ];
165 -> 88;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 166 [arrowhead = none ];
166 -> 66;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 167 [arrowhead = none ];
167 -> 67;

168 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 168 [arrowhead = none ];
168 -> 68;

169 [label="ONLY EVAL with clause\nmult(s(X300), X301, X302) :- ','(mult(X300, X301, X303), add(X301, X303, X302)).\nand substitution\nX300 -> 0\nX301 -> s(0)\nT7 -> T147\nX302 -> T147\nT146 -> T147", fontsize=14, style = filled, fillcolor = white];
67 -> 169 [arrowhead = none ];
169 -> 69;

170 [label="BACKTRACK\nfor clause: mult(0, Y, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
68 -> 170 [arrowhead = none ];
170 -> 87;

171 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
69 -> 171 [arrowhead = none ];
171 -> 70;

172 [label="SPLIT 2\nnew knowledge:\nT149 is ground\nreplacements:\nX303 -> T149", fontsize=14, style = filled, fillcolor = violet];
69 -> 172 [arrowhead = none ];
172 -> 71;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 173 [arrowhead = none ];
173 -> 72;

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 174 [arrowhead = none ];
174 -> 76;

175 [label="BACKTRACK\nfor clause: mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R))because of non-unification", fontsize=14, style = filled, fillcolor = white];
72 -> 175 [arrowhead = none ];
175 -> 73;

176 [label="ONLY EVAL with clause\nmult(0, X323, 0).\nand substitution\nX323 -> s(0)\nX303 -> 0", fontsize=14, style = filled, fillcolor = white];
73 -> 176 [arrowhead = none ];
176 -> 74;

177 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 177 [arrowhead = none ];
177 -> 75;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 178 [arrowhead = none ];
178 -> 77;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 179 [arrowhead = none ];
179 -> 78;

180 [label="EVAL with clause\nadd(s(X339), X340, s(X341)) :- add(X339, X340, X341).\nand substitution\nX339 -> 0\nT149 -> T160\nX340 -> T160\nX341 -> T162\nT147 -> s(T162)\nT161 -> T162", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 180 [arrowhead = none ];
180 -> 79;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 181 [arrowhead = none ];
181 -> 80;

182 [label="BACKTRACK\nfor clause: add(0, X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
78 -> 182 [arrowhead = none ];
182 -> 86;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 183 [arrowhead = none ];
183 -> 81;

184 [label="BACKTRACK\nfor clause: add(s(X), Y, s(Z)) :- add(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
81 -> 184 [arrowhead = none ];
184 -> 82;

185 [label="EVAL with clause\nadd(0, X355, X355).\nand substitution\nT160 -> T167\nX355 -> T167\nT162 -> T167", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 185 [arrowhead = none ];
185 -> 83;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 186 [arrowhead = none ];
186 -> 84;

187 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 187 [arrowhead = none ];
187 -> 85;

188 [label="BACKTRACK\nfor clause: factorial(0, s(0))because of non-unification", fontsize=14, style = filled, fillcolor = white];
88 -> 188 [arrowhead = none ];
188 -> 89;

189 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 189 [arrowhead = none ];
189 -> 92;

}

(2) Obligation:

Clauses:

factorial10(s(T20), X53) :- factorial10(T20, X52).
factorial10(s(T20), X53) :- ','(factorial10(T20, T22), mult18(T20, T22, X53)).
factorial10(0, s(0)).
p22(T38, T39, X100, X101) :- mult23(T38, T39, X100).
p22(T38, T39, T42, X101) :- ','(mult23(T38, T39, T42), add24(T39, T42, X101)).
add24(s(T70), T71, s(X170)) :- add24(T70, T71, X170).
add24(0, T76, T76).
mult18(T38, T39, X101) :- p22(T38, T39, X100, X101).
add52(s(T127), T128, s(T130)) :- add52(T127, T128, T130).
add52(0, T136, T136).
mult23(s(T53), T54, X134) :- p22(T53, T54, X133, X134).
mult23(0, T59, 0).
mult70(0).
factorial1(s(s(T12)), T7) :- factorial10(T12, X24).
factorial1(s(s(T12)), T7) :- ','(factorial10(T12, T14), mult18(T12, T14, X25)).
factorial1(s(s(T103)), T106) :- ','(factorial10(T103, T14), ','(mult18(T103, T14, T104), mult18(T103, T104, X224))).
factorial1(s(s(T103)), T106) :- ','(factorial10(T103, T14), ','(mult18(T103, T14, T104), ','(mult18(T103, T104, T109), add52(T104, T109, T106)))).
factorial1(s(0), T147) :- mult70(X303).
factorial1(s(0), s(T167)) :- mult70(T167).
factorial1(0, s(0)).

Queries:

factorial1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
factorial1_in: (b,f)
factorial10_in: (b,f)
mult18_in: (b,f,f)
p22_in: (b,f,f,f)
mult23_in: (b,f,f)
add24_in: (f,f,f)
add52_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(5) 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:

FACTORIAL1_IN_GA(s(s(T12)), T7) → U11_GA(T12, T7, factorial10_in_ga(T12, X24))
FACTORIAL1_IN_GA(s(s(T12)), T7) → FACTORIAL10_IN_GA(T12, X24)
FACTORIAL10_IN_GA(s(T20), X53) → U1_GA(T20, X53, factorial10_in_ga(T20, X52))
FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)
FACTORIAL10_IN_GA(s(T20), X53) → U2_GA(T20, X53, factorial10_in_ga(T20, T22))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → U3_GA(T20, X53, mult18_in_gaa(T20, T22, X53))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → MULT18_IN_GAA(T20, T22, X53)
MULT18_IN_GAA(T38, T39, X101) → U8_GAA(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
MULT18_IN_GAA(T38, T39, X101) → P22_IN_GAAA(T38, T39, X100, X101)
P22_IN_GAAA(T38, T39, X100, X101) → U4_GAAA(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → U10_GAA(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)
P22_IN_GAAA(T38, T39, T42, X101) → U5_GAAA(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_GAAA(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → ADD24_IN_AAA(T39, T42, X101)
ADD24_IN_AAA(s(T70), T71, s(X170)) → U7_AAA(T70, T71, X170, add24_in_aaa(T70, T71, X170))
ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)
FACTORIAL1_IN_GA(s(s(T12)), T7) → U12_GA(T12, T7, factorial10_in_ga(T12, T14))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → U13_GA(T12, T7, mult18_in_gaa(T12, T14, X25))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → MULT18_IN_GAA(T12, T14, X25)
FACTORIAL1_IN_GA(s(s(T103)), T106) → U14_GA(T103, T106, factorial10_in_ga(T103, T14))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → U15_GA(T103, T106, mult18_in_gaa(T103, T14, T104))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → MULT18_IN_GAA(T103, T14, T104)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_GA(T103, T106, mult18_in_gaa(T103, T104, X224))
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → MULT18_IN_GAA(T103, T104, X224)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_GA(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_GA(T103, T106, add52_in_aaa(T104, T109, T106))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → ADD52_IN_AAA(T104, T109, T106)
ADD52_IN_AAA(s(T127), T128, s(T130)) → U9_AAA(T127, T128, T130, add52_in_aaa(T127, T128, T130))
ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)
FACTORIAL1_IN_GA(s(0), T147) → U19_GA(T147, mult70_in_a(X303))
FACTORIAL1_IN_GA(s(0), T147) → MULT70_IN_A(X303)
FACTORIAL1_IN_GA(s(0), s(T167)) → U20_GA(T167, mult70_in_a(T167))
FACTORIAL1_IN_GA(s(0), s(T167)) → MULT70_IN_A(T167)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

The argument filtering Pi contains the following mapping:
factorial1_in_ga(x1, x2) = factorial1_in_ga(x1)
s(x1) = s(x1)
U11_ga(x1, x2, x3) = U11_ga(x3)
factorial10_in_ga(x1, x2) = factorial10_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
0 = 0
factorial10_out_ga(x1, x2) = factorial10_out_ga
U3_ga(x1, x2, x3) = U3_ga(x3)
mult18_in_gaa(x1, x2, x3) = mult18_in_gaa(x1)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x4)
p22_in_gaaa(x1, x2, x3, x4) = p22_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5) = U4_gaaa(x5)
mult23_in_gaa(x1, x2, x3) = mult23_in_gaa(x1)
U10_gaa(x1, x2, x3, x4) = U10_gaa(x4)
U5_gaaa(x1, x2, x3, x4, x5) = U5_gaaa(x5)
mult23_out_gaa(x1, x2, x3) = mult23_out_gaa
U6_gaaa(x1, x2, x3, x4, x5) = U6_gaaa(x5)
p22_out_gaaa(x1, x2, x3, x4) = p22_out_gaaa
add24_in_aaa(x1, x2, x3) = add24_in_aaa
U7_aaa(x1, x2, x3, x4) = U7_aaa(x4)
add24_out_aaa(x1, x2, x3) = add24_out_aaa(x1)
mult18_out_gaa(x1, x2, x3) = mult18_out_gaa
factorial1_out_ga(x1, x2) = factorial1_out_ga
U12_ga(x1, x2, x3) = U12_ga(x1, x3)
U13_ga(x1, x2, x3) = U13_ga(x3)
U14_ga(x1, x2, x3) = U14_ga(x1, x3)
U15_ga(x1, x2, x3) = U15_ga(x1, x3)
U16_ga(x1, x2, x3) = U16_ga(x3)
U17_ga(x1, x2, x3, x4) = U17_ga(x4)
U18_ga(x1, x2, x3) = U18_ga(x3)
add52_in_aaa(x1, x2, x3) = add52_in_aaa
U9_aaa(x1, x2, x3, x4) = U9_aaa(x4)
add52_out_aaa(x1, x2, x3) = add52_out_aaa(x1)
U19_ga(x1, x2) = U19_ga(x2)
mult70_in_a(x1) = mult70_in_a
mult70_out_a(x1) = mult70_out_a(x1)
U20_ga(x1, x2) = U20_ga(x2)
FACTORIAL1_IN_GA(x1, x2) = FACTORIAL1_IN_GA(x1)
U11_GA(x1, x2, x3) = U11_GA(x3)
FACTORIAL10_IN_GA(x1, x2) = FACTORIAL10_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U3_GA(x1, x2, x3) = U3_GA(x3)
MULT18_IN_GAA(x1, x2, x3) = MULT18_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4) = U8_GAA(x4)
P22_IN_GAAA(x1, x2, x3, x4) = P22_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5) = U4_GAAA(x5)
MULT23_IN_GAA(x1, x2, x3) = MULT23_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4) = U10_GAA(x4)
U5_GAAA(x1, x2, x3, x4, x5) = U5_GAAA(x5)
U6_GAAA(x1, x2, x3, x4, x5) = U6_GAAA(x5)
ADD24_IN_AAA(x1, x2, x3) = ADD24_IN_AAA
U7_AAA(x1, x2, x3, x4) = U7_AAA(x4)
U12_GA(x1, x2, x3) = U12_GA(x1, x3)
U13_GA(x1, x2, x3) = U13_GA(x3)
U14_GA(x1, x2, x3) = U14_GA(x1, x3)
U15_GA(x1, x2, x3) = U15_GA(x1, x3)
U16_GA(x1, x2, x3) = U16_GA(x3)
U17_GA(x1, x2, x3, x4) = U17_GA(x4)
U18_GA(x1, x2, x3) = U18_GA(x3)
ADD52_IN_AAA(x1, x2, x3) = ADD52_IN_AAA
U9_AAA(x1, x2, x3, x4) = U9_AAA(x4)
U19_GA(x1, x2) = U19_GA(x2)
MULT70_IN_A(x1) = MULT70_IN_A
U20_GA(x1, x2) = U20_GA(x2)

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

(6) Obligation:

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

FACTORIAL1_IN_GA(s(s(T12)), T7) → U11_GA(T12, T7, factorial10_in_ga(T12, X24))
FACTORIAL1_IN_GA(s(s(T12)), T7) → FACTORIAL10_IN_GA(T12, X24)
FACTORIAL10_IN_GA(s(T20), X53) → U1_GA(T20, X53, factorial10_in_ga(T20, X52))
FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)
FACTORIAL10_IN_GA(s(T20), X53) → U2_GA(T20, X53, factorial10_in_ga(T20, T22))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → U3_GA(T20, X53, mult18_in_gaa(T20, T22, X53))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → MULT18_IN_GAA(T20, T22, X53)
MULT18_IN_GAA(T38, T39, X101) → U8_GAA(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
MULT18_IN_GAA(T38, T39, X101) → P22_IN_GAAA(T38, T39, X100, X101)
P22_IN_GAAA(T38, T39, X100, X101) → U4_GAAA(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → U10_GAA(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)
P22_IN_GAAA(T38, T39, T42, X101) → U5_GAAA(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_GAAA(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → ADD24_IN_AAA(T39, T42, X101)
ADD24_IN_AAA(s(T70), T71, s(X170)) → U7_AAA(T70, T71, X170, add24_in_aaa(T70, T71, X170))
ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)
FACTORIAL1_IN_GA(s(s(T12)), T7) → U12_GA(T12, T7, factorial10_in_ga(T12, T14))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → U13_GA(T12, T7, mult18_in_gaa(T12, T14, X25))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → MULT18_IN_GAA(T12, T14, X25)
FACTORIAL1_IN_GA(s(s(T103)), T106) → U14_GA(T103, T106, factorial10_in_ga(T103, T14))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → U15_GA(T103, T106, mult18_in_gaa(T103, T14, T104))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → MULT18_IN_GAA(T103, T14, T104)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_GA(T103, T106, mult18_in_gaa(T103, T104, X224))
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → MULT18_IN_GAA(T103, T104, X224)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_GA(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_GA(T103, T106, add52_in_aaa(T104, T109, T106))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → ADD52_IN_AAA(T104, T109, T106)
ADD52_IN_AAA(s(T127), T128, s(T130)) → U9_AAA(T127, T128, T130, add52_in_aaa(T127, T128, T130))
ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)
FACTORIAL1_IN_GA(s(0), T147) → U19_GA(T147, mult70_in_a(X303))
FACTORIAL1_IN_GA(s(0), T147) → MULT70_IN_A(X303)
FACTORIAL1_IN_GA(s(0), s(T167)) → U20_GA(T167, mult70_in_a(T167))
FACTORIAL1_IN_GA(s(0), s(T167)) → MULT70_IN_A(T167)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 30 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD52_IN_AAA(x1, x2, x3) = ADD52_IN_AAA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

ADD52_IN_AAA → ADD52_IN_AAA

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

(14) 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 = ADD52_IN_AAA evaluates to t =ADD52_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 ADD52_IN_AAA to ADD52_IN_AAA.

(15) NO

(16) Obligation:

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

ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD24_IN_AAA(x1, x2, x3) = ADD24_IN_AAA

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

ADD24_IN_AAA → ADD24_IN_AAA

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

(21) 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 = ADD24_IN_AAA evaluates to t =ADD24_IN_AAA

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ADD24_IN_AAA to ADD24_IN_AAA.

(22) NO

(23) Obligation:

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

P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
P22_IN_GAAA(x1, x2, x3, x4) = P22_IN_GAAA(x1)
MULT23_IN_GAA(x1, x2, x3) = MULT23_IN_GAA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

P22_IN_GAAA(T38) → MULT23_IN_GAA(T38)
MULT23_IN_GAA(s(T53)) → P22_IN_GAAA(T53)

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

(28) 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:

MULT23_IN_GAA(s(T53)) → P22_IN_GAAA(T53)
The graph contains the following edges 1 > 1
P22_IN_GAAA(T38) → MULT23_IN_GAA(T38)
The graph contains the following edges 1 >= 1

(29) YES

(30) Obligation:

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

FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
FACTORIAL10_IN_GA(x1, x2) = FACTORIAL10_IN_GA(x1)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

FACTORIAL10_IN_GA(s(T20)) → FACTORIAL10_IN_GA(T20)

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

(35) 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:

FACTORIAL10_IN_GA(s(T20)) → FACTORIAL10_IN_GA(T20)
The graph contains the following edges 1 > 1

(36) YES

(37) PrologToPiTRSProof (SOUND transformation)

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(38) Obligation:

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

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(39) 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:

FACTORIAL1_IN_GA(s(s(T12)), T7) → U11_GA(T12, T7, factorial10_in_ga(T12, X24))
FACTORIAL1_IN_GA(s(s(T12)), T7) → FACTORIAL10_IN_GA(T12, X24)
FACTORIAL10_IN_GA(s(T20), X53) → U1_GA(T20, X53, factorial10_in_ga(T20, X52))
FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)
FACTORIAL10_IN_GA(s(T20), X53) → U2_GA(T20, X53, factorial10_in_ga(T20, T22))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → U3_GA(T20, X53, mult18_in_gaa(T20, T22, X53))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → MULT18_IN_GAA(T20, T22, X53)
MULT18_IN_GAA(T38, T39, X101) → U8_GAA(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
MULT18_IN_GAA(T38, T39, X101) → P22_IN_GAAA(T38, T39, X100, X101)
P22_IN_GAAA(T38, T39, X100, X101) → U4_GAAA(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → U10_GAA(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)
P22_IN_GAAA(T38, T39, T42, X101) → U5_GAAA(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_GAAA(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → ADD24_IN_AAA(T39, T42, X101)
ADD24_IN_AAA(s(T70), T71, s(X170)) → U7_AAA(T70, T71, X170, add24_in_aaa(T70, T71, X170))
ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)
FACTORIAL1_IN_GA(s(s(T12)), T7) → U12_GA(T12, T7, factorial10_in_ga(T12, T14))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → U13_GA(T12, T7, mult18_in_gaa(T12, T14, X25))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → MULT18_IN_GAA(T12, T14, X25)
FACTORIAL1_IN_GA(s(s(T103)), T106) → U14_GA(T103, T106, factorial10_in_ga(T103, T14))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → U15_GA(T103, T106, mult18_in_gaa(T103, T14, T104))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → MULT18_IN_GAA(T103, T14, T104)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_GA(T103, T106, mult18_in_gaa(T103, T104, X224))
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → MULT18_IN_GAA(T103, T104, X224)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_GA(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_GA(T103, T106, add52_in_aaa(T104, T109, T106))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → ADD52_IN_AAA(T104, T109, T106)
ADD52_IN_AAA(s(T127), T128, s(T130)) → U9_AAA(T127, T128, T130, add52_in_aaa(T127, T128, T130))
ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)
FACTORIAL1_IN_GA(s(0), T147) → U19_GA(T147, mult70_in_a(X303))
FACTORIAL1_IN_GA(s(0), T147) → MULT70_IN_A(X303)
FACTORIAL1_IN_GA(s(0), s(T167)) → U20_GA(T167, mult70_in_a(T167))
FACTORIAL1_IN_GA(s(0), s(T167)) → MULT70_IN_A(T167)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

The argument filtering Pi contains the following mapping:
factorial1_in_ga(x1, x2) = factorial1_in_ga(x1)
s(x1) = s(x1)
U11_ga(x1, x2, x3) = U11_ga(x1, x3)
factorial10_in_ga(x1, x2) = factorial10_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
0 = 0
factorial10_out_ga(x1, x2) = factorial10_out_ga(x1)
U3_ga(x1, x2, x3) = U3_ga(x1, x3)
mult18_in_gaa(x1, x2, x3) = mult18_in_gaa(x1)
U8_gaa(x1, x2, x3, x4) = U8_gaa(x1, x4)
p22_in_gaaa(x1, x2, x3, x4) = p22_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5) = U4_gaaa(x1, x5)
mult23_in_gaa(x1, x2, x3) = mult23_in_gaa(x1)
U10_gaa(x1, x2, x3, x4) = U10_gaa(x1, x4)
U5_gaaa(x1, x2, x3, x4, x5) = U5_gaaa(x1, x5)
mult23_out_gaa(x1, x2, x3) = mult23_out_gaa(x1)
U6_gaaa(x1, x2, x3, x4, x5) = U6_gaaa(x1, x5)
p22_out_gaaa(x1, x2, x3, x4) = p22_out_gaaa(x1)
add24_in_aaa(x1, x2, x3) = add24_in_aaa
U7_aaa(x1, x2, x3, x4) = U7_aaa(x4)
add24_out_aaa(x1, x2, x3) = add24_out_aaa(x1)
mult18_out_gaa(x1, x2, x3) = mult18_out_gaa(x1)
factorial1_out_ga(x1, x2) = factorial1_out_ga(x1)
U12_ga(x1, x2, x3) = U12_ga(x1, x3)
U13_ga(x1, x2, x3) = U13_ga(x1, x3)
U14_ga(x1, x2, x3) = U14_ga(x1, x3)
U15_ga(x1, x2, x3) = U15_ga(x1, x3)
U16_ga(x1, x2, x3) = U16_ga(x1, x3)
U17_ga(x1, x2, x3, x4) = U17_ga(x1, x4)
U18_ga(x1, x2, x3) = U18_ga(x1, x3)
add52_in_aaa(x1, x2, x3) = add52_in_aaa
U9_aaa(x1, x2, x3, x4) = U9_aaa(x4)
add52_out_aaa(x1, x2, x3) = add52_out_aaa(x1)
U19_ga(x1, x2) = U19_ga(x2)
mult70_in_a(x1) = mult70_in_a
mult70_out_a(x1) = mult70_out_a(x1)
U20_ga(x1, x2) = U20_ga(x2)
FACTORIAL1_IN_GA(x1, x2) = FACTORIAL1_IN_GA(x1)
U11_GA(x1, x2, x3) = U11_GA(x1, x3)
FACTORIAL10_IN_GA(x1, x2) = FACTORIAL10_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
MULT18_IN_GAA(x1, x2, x3) = MULT18_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4) = U8_GAA(x1, x4)
P22_IN_GAAA(x1, x2, x3, x4) = P22_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5) = U4_GAAA(x1, x5)
MULT23_IN_GAA(x1, x2, x3) = MULT23_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4) = U10_GAA(x1, x4)
U5_GAAA(x1, x2, x3, x4, x5) = U5_GAAA(x1, x5)
U6_GAAA(x1, x2, x3, x4, x5) = U6_GAAA(x1, x5)
ADD24_IN_AAA(x1, x2, x3) = ADD24_IN_AAA
U7_AAA(x1, x2, x3, x4) = U7_AAA(x4)
U12_GA(x1, x2, x3) = U12_GA(x1, x3)
U13_GA(x1, x2, x3) = U13_GA(x1, x3)
U14_GA(x1, x2, x3) = U14_GA(x1, x3)
U15_GA(x1, x2, x3) = U15_GA(x1, x3)
U16_GA(x1, x2, x3) = U16_GA(x1, x3)
U17_GA(x1, x2, x3, x4) = U17_GA(x1, x4)
U18_GA(x1, x2, x3) = U18_GA(x1, x3)
ADD52_IN_AAA(x1, x2, x3) = ADD52_IN_AAA
U9_AAA(x1, x2, x3, x4) = U9_AAA(x4)
U19_GA(x1, x2) = U19_GA(x2)
MULT70_IN_A(x1) = MULT70_IN_A
U20_GA(x1, x2) = U20_GA(x2)

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

(40) Obligation:

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

FACTORIAL1_IN_GA(s(s(T12)), T7) → U11_GA(T12, T7, factorial10_in_ga(T12, X24))
FACTORIAL1_IN_GA(s(s(T12)), T7) → FACTORIAL10_IN_GA(T12, X24)
FACTORIAL10_IN_GA(s(T20), X53) → U1_GA(T20, X53, factorial10_in_ga(T20, X52))
FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)
FACTORIAL10_IN_GA(s(T20), X53) → U2_GA(T20, X53, factorial10_in_ga(T20, T22))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → U3_GA(T20, X53, mult18_in_gaa(T20, T22, X53))
U2_GA(T20, X53, factorial10_out_ga(T20, T22)) → MULT18_IN_GAA(T20, T22, X53)
MULT18_IN_GAA(T38, T39, X101) → U8_GAA(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
MULT18_IN_GAA(T38, T39, X101) → P22_IN_GAAA(T38, T39, X100, X101)
P22_IN_GAAA(T38, T39, X100, X101) → U4_GAAA(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → U10_GAA(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)
P22_IN_GAAA(T38, T39, T42, X101) → U5_GAAA(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_GAAA(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
U5_GAAA(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → ADD24_IN_AAA(T39, T42, X101)
ADD24_IN_AAA(s(T70), T71, s(X170)) → U7_AAA(T70, T71, X170, add24_in_aaa(T70, T71, X170))
ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)
FACTORIAL1_IN_GA(s(s(T12)), T7) → U12_GA(T12, T7, factorial10_in_ga(T12, T14))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → U13_GA(T12, T7, mult18_in_gaa(T12, T14, X25))
U12_GA(T12, T7, factorial10_out_ga(T12, T14)) → MULT18_IN_GAA(T12, T14, X25)
FACTORIAL1_IN_GA(s(s(T103)), T106) → U14_GA(T103, T106, factorial10_in_ga(T103, T14))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → U15_GA(T103, T106, mult18_in_gaa(T103, T14, T104))
U14_GA(T103, T106, factorial10_out_ga(T103, T14)) → MULT18_IN_GAA(T103, T14, T104)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_GA(T103, T106, mult18_in_gaa(T103, T104, X224))
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → MULT18_IN_GAA(T103, T104, X224)
U15_GA(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_GA(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_GA(T103, T106, add52_in_aaa(T104, T109, T106))
U17_GA(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → ADD52_IN_AAA(T104, T109, T106)
ADD52_IN_AAA(s(T127), T128, s(T130)) → U9_AAA(T127, T128, T130, add52_in_aaa(T127, T128, T130))
ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)
FACTORIAL1_IN_GA(s(0), T147) → U19_GA(T147, mult70_in_a(X303))
FACTORIAL1_IN_GA(s(0), T147) → MULT70_IN_A(X303)
FACTORIAL1_IN_GA(s(0), s(T167)) → U20_GA(T167, mult70_in_a(T167))
FACTORIAL1_IN_GA(s(0), s(T167)) → MULT70_IN_A(T167)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(41) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 30 less nodes.

(42) Complex Obligation (AND)

(43) Obligation:

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

ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

ADD52_IN_AAA(s(T127), T128, s(T130)) → ADD52_IN_AAA(T127, T128, T130)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD52_IN_AAA(x1, x2, x3) = ADD52_IN_AAA

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

ADD52_IN_AAA → ADD52_IN_AAA

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

(48) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ADD52_IN_AAA to ADD52_IN_AAA.

(49) NO

(50) Obligation:

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

ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

ADD24_IN_AAA(s(T70), T71, s(X170)) → ADD24_IN_AAA(T70, T71, X170)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
ADD24_IN_AAA(x1, x2, x3) = ADD24_IN_AAA

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

ADD24_IN_AAA → ADD24_IN_AAA

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

(55) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ADD24_IN_AAA to ADD24_IN_AAA.

(56) NO

(57) Obligation:

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

P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

P22_IN_GAAA(T38, T39, X100, X101) → MULT23_IN_GAA(T38, T39, X100)
MULT23_IN_GAA(s(T53), T54, X134) → P22_IN_GAAA(T53, T54, X133, X134)

(60) PiDPToQDPProof (SOUND transformation)

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

(61) Obligation:

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

P22_IN_GAAA(T38) → MULT23_IN_GAA(T38)
MULT23_IN_GAA(s(T53)) → P22_IN_GAAA(T53)

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

(62) 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:

MULT23_IN_GAA(s(T53)) → P22_IN_GAAA(T53)
The graph contains the following edges 1 > 1
P22_IN_GAAA(T38) → MULT23_IN_GAA(T38)
The graph contains the following edges 1 >= 1

(63) YES

(64) Obligation:

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

FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)

The TRS R consists of the following rules:

factorial1_in_ga(s(s(T12)), T7) → U11_ga(T12, T7, factorial10_in_ga(T12, X24))
factorial10_in_ga(s(T20), X53) → U1_ga(T20, X53, factorial10_in_ga(T20, X52))
factorial10_in_ga(s(T20), X53) → U2_ga(T20, X53, factorial10_in_ga(T20, T22))
factorial10_in_ga(0, s(0)) → factorial10_out_ga(0, s(0))
U2_ga(T20, X53, factorial10_out_ga(T20, T22)) → U3_ga(T20, X53, mult18_in_gaa(T20, T22, X53))
mult18_in_gaa(T38, T39, X101) → U8_gaa(T38, T39, X101, p22_in_gaaa(T38, T39, X100, X101))
p22_in_gaaa(T38, T39, X100, X101) → U4_gaaa(T38, T39, X100, X101, mult23_in_gaa(T38, T39, X100))
mult23_in_gaa(s(T53), T54, X134) → U10_gaa(T53, T54, X134, p22_in_gaaa(T53, T54, X133, X134))
p22_in_gaaa(T38, T39, T42, X101) → U5_gaaa(T38, T39, T42, X101, mult23_in_gaa(T38, T39, T42))
mult23_in_gaa(0, T59, 0) → mult23_out_gaa(0, T59, 0)
U5_gaaa(T38, T39, T42, X101, mult23_out_gaa(T38, T39, T42)) → U6_gaaa(T38, T39, T42, X101, add24_in_aaa(T39, T42, X101))
add24_in_aaa(s(T70), T71, s(X170)) → U7_aaa(T70, T71, X170, add24_in_aaa(T70, T71, X170))
add24_in_aaa(0, T76, T76) → add24_out_aaa(0, T76, T76)
U7_aaa(T70, T71, X170, add24_out_aaa(T70, T71, X170)) → add24_out_aaa(s(T70), T71, s(X170))
U6_gaaa(T38, T39, T42, X101, add24_out_aaa(T39, T42, X101)) → p22_out_gaaa(T38, T39, T42, X101)
U10_gaa(T53, T54, X134, p22_out_gaaa(T53, T54, X133, X134)) → mult23_out_gaa(s(T53), T54, X134)
U4_gaaa(T38, T39, X100, X101, mult23_out_gaa(T38, T39, X100)) → p22_out_gaaa(T38, T39, X100, X101)
U8_gaa(T38, T39, X101, p22_out_gaaa(T38, T39, X100, X101)) → mult18_out_gaa(T38, T39, X101)
U3_ga(T20, X53, mult18_out_gaa(T20, T22, X53)) → factorial10_out_ga(s(T20), X53)
U1_ga(T20, X53, factorial10_out_ga(T20, X52)) → factorial10_out_ga(s(T20), X53)
U11_ga(T12, T7, factorial10_out_ga(T12, X24)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T12)), T7) → U12_ga(T12, T7, factorial10_in_ga(T12, T14))
U12_ga(T12, T7, factorial10_out_ga(T12, T14)) → U13_ga(T12, T7, mult18_in_gaa(T12, T14, X25))
U13_ga(T12, T7, mult18_out_gaa(T12, T14, X25)) → factorial1_out_ga(s(s(T12)), T7)
factorial1_in_ga(s(s(T103)), T106) → U14_ga(T103, T106, factorial10_in_ga(T103, T14))
U14_ga(T103, T106, factorial10_out_ga(T103, T14)) → U15_ga(T103, T106, mult18_in_gaa(T103, T14, T104))
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U16_ga(T103, T106, mult18_in_gaa(T103, T104, X224))
U16_ga(T103, T106, mult18_out_gaa(T103, T104, X224)) → factorial1_out_ga(s(s(T103)), T106)
U15_ga(T103, T106, mult18_out_gaa(T103, T14, T104)) → U17_ga(T103, T106, T104, mult18_in_gaa(T103, T104, T109))
U17_ga(T103, T106, T104, mult18_out_gaa(T103, T104, T109)) → U18_ga(T103, T106, add52_in_aaa(T104, T109, T106))
add52_in_aaa(s(T127), T128, s(T130)) → U9_aaa(T127, T128, T130, add52_in_aaa(T127, T128, T130))
add52_in_aaa(0, T136, T136) → add52_out_aaa(0, T136, T136)
U9_aaa(T127, T128, T130, add52_out_aaa(T127, T128, T130)) → add52_out_aaa(s(T127), T128, s(T130))
U18_ga(T103, T106, add52_out_aaa(T104, T109, T106)) → factorial1_out_ga(s(s(T103)), T106)
factorial1_in_ga(s(0), T147) → U19_ga(T147, mult70_in_a(X303))
mult70_in_a(0) → mult70_out_a(0)
U19_ga(T147, mult70_out_a(X303)) → factorial1_out_ga(s(0), T147)
factorial1_in_ga(s(0), s(T167)) → U20_ga(T167, mult70_in_a(T167))
U20_ga(T167, mult70_out_a(T167)) → factorial1_out_ga(s(0), s(T167))
factorial1_in_ga(0, s(0)) → factorial1_out_ga(0, s(0))

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

FACTORIAL10_IN_GA(s(T20), X53) → FACTORIAL10_IN_GA(T20, X52)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
FACTORIAL10_IN_GA(x1, x2) = FACTORIAL10_IN_GA(x1)

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

(67) PiDPToQDPProof (SOUND transformation)

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

(68) Obligation:

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

FACTORIAL10_IN_GA(s(T20)) → FACTORIAL10_IN_GA(T20)

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

(69) 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:

FACTORIAL10_IN_GA(s(T20)) → FACTORIAL10_IN_GA(T20)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) NonTerminationProof (EQUIVALENT transformation)

(15) NO

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) NonTerminationProof (EQUIVALENT transformation)

(22) NO

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) PiDPToQDPProof (SOUND transformation)

(34) Obligation:

(35) QDPSizeChangeProof (EQUIVALENT transformation)

(36) YES

(37) PrologToPiTRSProof (SOUND transformation)

(38) Obligation:

(39) DependencyPairsProof (EQUIVALENT transformation)

(40) Obligation:

(41) DependencyGraphProof (EQUIVALENT transformation)

(42) Complex Obligation (AND)

(43) Obligation:

(44) UsableRulesProof (EQUIVALENT transformation)

(45) Obligation:

(46) PiDPToQDPProof (SOUND transformation)

(47) Obligation:

(48) NonTerminationProof (EQUIVALENT transformation)

(49) NO

(50) Obligation:

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

(53) PiDPToQDPProof (SOUND transformation)

(54) Obligation:

(55) NonTerminationProof (EQUIVALENT transformation)

(56) NO

(57) Obligation:

(58) UsableRulesProof (EQUIVALENT transformation)

(59) Obligation:

(60) PiDPToQDPProof (SOUND transformation)

(61) Obligation:

(62) QDPSizeChangeProof (EQUIVALENT transformation)

(63) YES

(64) Obligation:

(65) UsableRulesProof (EQUIVALENT transformation)

(66) Obligation:

(67) PiDPToQDPProof (SOUND transformation)

(68) Obligation:

(69) QDPSizeChangeProof (EQUIVALENT transformation)

(70) YES