Left Termination proof of /tmp/tmp_UwWZp/divminuslinear.pl

Left Termination of the query pattern
div(g,g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 BuiltinConflictTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToPrologProblemTransformerProof (⇒, 184 ms)

↳4 Prolog

↳5 PrologToPiTRSProof (⇒, 48 ms)

↳6 PiTRS

↳7 DependencyPairsProof (⇔, 84 ms)

↳8 PiDP

↳9 DependencyGraphProof (⇔, 0 ms)

↳10 AND

    ↳11 PiDP

        ↳12 UsableRulesProof (⇔, 0 ms)

        ↳13 PiDP

        ↳14 PiDPToQDPProof (⇒, 1 ms)

        ↳15 QDP

        ↳16 QDPSizeChangeProof (⇔, 0 ms)

        ↳17 YES

    ↳18 PiDP

        ↳19 UsableRulesProof (⇔, 0 ms)

        ↳20 PiDP

        ↳21 PiDPToQDPProof (⇒, 0 ms)

        ↳22 QDP

        ↳23 QDPOrderProof (⇔, 44 ms)

        ↳24 QDP

        ↳25 DependencyGraphProof (⇔, 0 ms)

        ↳26 TRUE

(0) Obligation:

Clauses:

minus(X, Y, Z) :- ','(=(X, 0), ','(!, =(Z, 0))).
minus(X, Y, Z) :- ','(=(Y, 0), ','(!, =(Z, X))).
minus(X, Y, Z) :- ','(=(X, s(A)), ','(=(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=(Y, 0), ','(!, fail)).
div(X, Y, Z) :- ','(=(X, 0), ','(!, =(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =(Z, s(V)))).
=(X, X).

Query: div(g,g,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

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

(2) Obligation:

Clauses:

minus(X, Y, Z) :- ','(user_defined_=(X, 0), ','(!, user_defined_=(Z, 0))).
minus(X, Y, Z) :- ','(user_defined_=(Y, 0), ','(!, user_defined_=(Z, X))).
minus(X, Y, Z) :- ','(user_defined_=(X, s(A)), ','(user_defined_=(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(user_defined_=(Y, 0), ','(!, fail)).
div(X, Y, Z) :- ','(user_defined_=(X, 0), ','(!, user_defined_=(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), user_defined_=(Z, s(V)))).
user_defined_=(X, X).

Query: div(g,g,a)

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: div(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(user_defined_=(T8, 0), ','(!_1, fail)) | div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(user_defined_=(T8, 0), ','(!_1, fail)), SCOPE: 2, CLAUSE: 6 | ?_2 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, fail) | ?_2 | div(T7, 0, T3), SCOPE: 1, CLAUSE: 4 | div(T7, 0, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ?_2 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground\nX9 is free\nuser_defined_=(T8, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: div(T7, T8, T3), SCOPE: 1, CLAUSE: 4 | div(T7, T8, T3), SCOPE: 1, CLAUSE: 5\n\nT7 is ground\nT8 is ground\nX9 is free\nuser_defined_=(T8, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(user_defined_=(T20, 0), ','(!_1, user_defined_=(T23, 0))) | div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\nX9 is free\nuser_defined_=(T21, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(user_defined_=(T20, 0), ','(!_1, user_defined_=(T23, 0))), SCOPE: 3, CLAUSE: 6 | ?_3 | div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\nX9 is free\nuser_defined_=(T21, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_1, user_defined_=(T23, 0)) | ?_3 | div(0, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT21 is ground\nX9 is free\nuser_defined_=(T21, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ?_3 | div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\nX9 is free\nX21 is free\nuser_defined_=(T21, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T20, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: user_defined_=(T23, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: user_defined_=(T23, 0), SCOPE: 4, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: div(T20, T21, T3), SCOPE: 1, CLAUSE: 5\n\nT20 is ground\nT21 is ground\nX9 is free\nX21 is free\nuser_defined_=(T21, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T20, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(minus(T39, T40, X40), ','(div(X40, T40, X41), user_defined_=(T42, s(X41))))\n\nT39 is ground\nT40 is ground\nX9 is free\nX21 is free\nX40 is free\nX41 is free\nuser_defined_=(T40, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T39, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: minus(T39, T40, X40)\n\nT39 is ground\nT40 is ground\nX9 is free\nX21 is free\nX40 is free\nuser_defined_=(T40, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T39, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(div(T45, T40, X41), user_defined_=(T42, s(X41)))\n\nT40 is ground\nT45 is ground\nX9 is free\nX41 is free\nuser_defined_=(T40, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: minus(T39, T40, X40), SCOPE: 5, CLAUSE: 0 | minus(T39, T40, X40), SCOPE: 5, CLAUSE: 1 | minus(T39, T40, X40), SCOPE: 5, CLAUSE: 2\n\nT39 is ground\nT40 is ground\nX9 is free\nX21 is free\nX40 is free\nuser_defined_=(T40, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T39, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(user_defined_=(T52, 0), ','(!_5, user_defined_=(X61, 0))) | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 1 | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX9 is free\nX21 is free\nX40 is free\nX61 is free\nuser_defined_=(T53, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T52, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(user_defined_=(T52, 0), ','(!_5, user_defined_=(X61, 0))), SCOPE: 6, CLAUSE: 6 | ?_6 | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 1 | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX9 is free\nX21 is free\nX40 is free\nX61 is free\nuser_defined_=(T53, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T52, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ?_6 | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 1 | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX9 is free\nX21 is free\nX40 is free\nuser_defined_=(T53, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T52, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: minus(T52, T53, X40), SCOPE: 5, CLAUSE: 1 | minus(T52, T53, X40), SCOPE: 5, CLAUSE: 2\n\nT52 is ground\nT53 is ground\nX9 is free\nX21 is free\nX40 is free\nuser_defined_=(T53, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T52, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(user_defined_=(T60, 0), ','(!_5, user_defined_=(X74, T59))) | minus(T59, T60, X40), SCOPE: 5, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX9 is free\nX21 is free\nX40 is free\nX74 is free\nuser_defined_=(T60, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T59, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(user_defined_=(T60, 0), ','(!_5, user_defined_=(X74, T59))), SCOPE: 7, CLAUSE: 6 | ?_7 | minus(T59, T60, X40), SCOPE: 5, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX9 is free\nX21 is free\nX40 is free\nX74 is free\nuser_defined_=(T60, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T59, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ?_7 | minus(T59, T60, X40), SCOPE: 5, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX9 is free\nX21 is free\nX40 is free\nuser_defined_=(T60, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T59, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: minus(T59, T60, X40), SCOPE: 5, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nX9 is free\nX21 is free\nX40 is free\nuser_defined_=(T60, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T59, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(user_defined_=(T66, s(X95)), ','(user_defined_=(T67, s(X96)), minus(X95, X96, X97)))\n\nT66 is ground\nT67 is ground\nX9 is free\nX21 is free\nX97 is free\nX95 is free\nX96 is free\nuser_defined_=(T67, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T66, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(user_defined_=(T66, s(X95)), ','(user_defined_=(T67, s(X96)), minus(X95, X96, X97))), SCOPE: 8, CLAUSE: 6\n\nT66 is ground\nT67 is ground\nX9 is free\nX21 is free\nX97 is free\nX95 is free\nX96 is free\nuser_defined_=(T67, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T66, 0) !=? user_defined_=(X21, X21)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(user_defined_=(T67, s(X96)), minus(T73, X96, X97))\n\nT67 is ground\nT73 is ground\nX9 is free\nX97 is free\nX96 is free\nuser_defined_=(T67, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(user_defined_=(T67, s(X96)), minus(T73, X96, X97)), SCOPE: 9, CLAUSE: 6\n\nT67 is ground\nT73 is ground\nX9 is free\nX97 is free\nX96 is free\nuser_defined_=(T67, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: minus(T73, T80, X97)\n\nT73 is ground\nT80 is ground\nX97 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: minus(T73, T80, X97), SCOPE: 10, CLAUSE: 0 | minus(T73, T80, X97), SCOPE: 10, CLAUSE: 1 | minus(T73, T80, X97), SCOPE: 10, CLAUSE: 2\n\nT73 is ground\nT80 is ground\nX97 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(user_defined_=(T87, 0), ','(!_10, user_defined_=(X130, 0))) | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 1 | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX97 is free\nX130 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(user_defined_=(T87, 0), ','(!_10, user_defined_=(X130, 0))), SCOPE: 11, CLAUSE: 6 | ?_11 | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 1 | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX97 is free\nX130 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(!_10, user_defined_=(X130, 0)) | ?_11 | minus(0, T88, X97), SCOPE: 10, CLAUSE: 1 | minus(0, T88, X97), SCOPE: 10, CLAUSE: 2\n\nT88 is ground\nX97 is free\nX130 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ?_11 | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 1 | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX97 is free\nX133 is free\nuser_defined_=(T87, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: user_defined_=(X130, 0)\n\nX130 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: user_defined_=(X130, 0), SCOPE: 12, CLAUSE: 6\n\nX130 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: minus(T87, T88, X97), SCOPE: 10, CLAUSE: 1 | minus(T87, T88, X97), SCOPE: 10, CLAUSE: 2\n\nT87 is ground\nT88 is ground\nX97 is free\nX133 is free\nuser_defined_=(T87, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(user_defined_=(T97, 0), ','(!_10, user_defined_=(X151, T96))) | minus(T96, T97, X97), SCOPE: 10, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX97 is free\nX133 is free\nX151 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(user_defined_=(T97, 0), ','(!_10, user_defined_=(X151, T96))), SCOPE: 13, CLAUSE: 6 | ?_13 | minus(T96, T97, X97), SCOPE: 10, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX97 is free\nX133 is free\nX151 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(!_10, user_defined_=(X151, T96)) | ?_13 | minus(T96, 0, X97), SCOPE: 10, CLAUSE: 2\n\nT96 is ground\nX97 is free\nX133 is free\nX151 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ?_13 | minus(T96, T97, X97), SCOPE: 10, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX97 is free\nX133 is free\nX154 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)\nuser_defined_=(T97, 0) !=? user_defined_=(X154, X154)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: user_defined_=(X151, T96)\n\nT96 is ground\nX133 is free\nX151 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: user_defined_=(X151, T96), SCOPE: 14, CLAUSE: 6\n\nT96 is ground\nX133 is free\nX151 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: minus(T96, T97, X97), SCOPE: 10, CLAUSE: 2\n\nT96 is ground\nT97 is ground\nX97 is free\nX133 is free\nX154 is free\nuser_defined_=(T96, 0) !=? user_defined_=(X133, X133)\nuser_defined_=(T97, 0) !=? user_defined_=(X154, X154)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(user_defined_=(T108, s(X180)), ','(user_defined_=(T109, s(X181)), minus(X180, X181, X182)))\n\nT108 is ground\nT109 is ground\nX133 is free\nX154 is free\nX182 is free\nX180 is free\nX181 is free\nuser_defined_=(T108, 0) !=? user_defined_=(X133, X133)\nuser_defined_=(T109, 0) !=? user_defined_=(X154, X154)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(user_defined_=(T108, s(X180)), ','(user_defined_=(T109, s(X181)), minus(X180, X181, X182))), SCOPE: 15, CLAUSE: 6\n\nT108 is ground\nT109 is ground\nX133 is free\nX154 is free\nX182 is free\nX180 is free\nX181 is free\nuser_defined_=(T108, 0) !=? user_defined_=(X133, X133)\nuser_defined_=(T109, 0) !=? user_defined_=(X154, X154)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(user_defined_=(T109, s(X181)), minus(T115, X181, X182))\n\nT109 is ground\nT115 is ground\nX154 is free\nX182 is free\nX181 is free\nuser_defined_=(T109, 0) !=? user_defined_=(X154, X154)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(user_defined_=(T109, s(X181)), minus(T115, X181, X182)), SCOPE: 16, CLAUSE: 6\n\nT109 is ground\nT115 is ground\nX154 is free\nX182 is free\nX181 is free\nuser_defined_=(T109, 0) !=? user_defined_=(X154, X154)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: minus(T115, T122, X182)\n\nT115 is ground\nT122 is ground\nX182 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: div(T45, T40, X41)\n\nT40 is ground\nT45 is ground\nX9 is free\nX41 is free\nuser_defined_=(T40, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: user_defined_=(T42, s(T127))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: div(T45, T40, X41), SCOPE: 17, CLAUSE: 3 | div(T45, T40, X41), SCOPE: 17, CLAUSE: 4 | div(T45, T40, X41), SCOPE: 17, CLAUSE: 5\n\nT40 is ground\nT45 is ground\nX9 is free\nX41 is free\nuser_defined_=(T40, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(user_defined_=(T135, 0), ','(!_17, fail)) | div(T134, T135, X41), SCOPE: 17, CLAUSE: 4 | div(T134, T135, X41), SCOPE: 17, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX9 is free\nX41 is free\nuser_defined_=(T135, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(user_defined_=(T135, 0), ','(!_17, fail)), SCOPE: 18, CLAUSE: 6 | ?_18 | div(T134, T135, X41), SCOPE: 17, CLAUSE: 4 | div(T134, T135, X41), SCOPE: 17, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX9 is free\nX41 is free\nuser_defined_=(T135, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ?_18 | div(T134, T135, X41), SCOPE: 17, CLAUSE: 4 | div(T134, T135, X41), SCOPE: 17, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX9 is free\nX41 is free\nuser_defined_=(T135, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: div(T134, T135, X41), SCOPE: 17, CLAUSE: 4 | div(T134, T135, X41), SCOPE: 17, CLAUSE: 5\n\nT134 is ground\nT135 is ground\nX9 is free\nX41 is free\nuser_defined_=(T135, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(user_defined_=(T141, 0), ','(!_17, user_defined_=(X236, 0))) | div(T141, T142, X41), SCOPE: 17, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX9 is free\nX41 is free\nX236 is free\nuser_defined_=(T142, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(user_defined_=(T141, 0), ','(!_17, user_defined_=(X236, 0))), SCOPE: 19, CLAUSE: 6 | ?_19 | div(T141, T142, X41), SCOPE: 17, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX9 is free\nX41 is free\nX236 is free\nuser_defined_=(T142, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(!_17, user_defined_=(X236, 0)) | ?_19 | div(0, T142, X41), SCOPE: 17, CLAUSE: 5\n\nT142 is ground\nX9 is free\nX41 is free\nX236 is free\nuser_defined_=(T142, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ?_19 | div(T141, T142, X41), SCOPE: 17, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX9 is free\nX41 is free\nX239 is free\nuser_defined_=(T142, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T141, 0) !=? user_defined_=(X239, X239)", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: user_defined_=(X236, 0)\n\nX236 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: user_defined_=(X236, 0), SCOPE: 20, CLAUSE: 6\n\nX236 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: div(T141, T142, X41), SCOPE: 17, CLAUSE: 5\n\nT141 is ground\nT142 is ground\nX9 is free\nX41 is free\nX239 is free\nuser_defined_=(T142, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T141, 0) !=? user_defined_=(X239, X239)", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(minus(T152, T153, X263), ','(div(X263, T153, X264), user_defined_=(X265, s(X264))))\n\nT152 is ground\nT153 is ground\nX9 is free\nX239 is free\nX265 is free\nX263 is free\nX264 is free\nuser_defined_=(T153, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T152, 0) !=? user_defined_=(X239, X239)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: minus(T152, T153, X263)\n\nT152 is ground\nT153 is ground\nX9 is free\nX239 is free\nX263 is free\nuser_defined_=(T153, 0) !=? user_defined_=(X9, X9)\nuser_defined_=(T152, 0) !=? user_defined_=(X239, X239)", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(div(T156, T153, X264), user_defined_=(X265, s(X264)))\n\nT153 is ground\nT156 is ground\nX9 is free\nX265 is free\nX264 is free\nuser_defined_=(T153, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: div(T156, T153, X264)\n\nT153 is ground\nT156 is ground\nX9 is free\nX264 is free\nuser_defined_=(T153, 0) !=? user_defined_=(X9, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: user_defined_=(X265, s(T161))\n\nX265 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: user_defined_=(X265, s(T161)), SCOPE: 21, CLAUSE: 6\n\nX265 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: user_defined_=(T42, s(T127)), SCOPE: 22, CLAUSE: 6\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="ONLY EVAL with clause\ndiv(X4, X5, X6) :- ','(user_defined_=(X5, 0), ','(!_1, fail)).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T9,\nX6 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 94 [arrowhead = none ];
94 -> 3;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 95 [arrowhead = none ];
95 -> 4;

96 [label="EVAL with clause\nuser_defined_=(X9, X9).\nand substitutionT8 -> 0,\nX9 -> 0,\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 96 [arrowhead = none ];
96 -> 5;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 97 [arrowhead = none ];
97 -> 6;

98 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 98 [arrowhead = none ];
98 -> 7;

99 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 99 [arrowhead = none ];
99 -> 9;

100 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 100 [arrowhead = none ];
100 -> 8;

101 [label="ONLY EVAL with clause\ndiv(X16, X17, X18) :- ','(user_defined_=(X16, 0), ','(!_1, user_defined_=(X18, 0))).\nand substitutionT7 -> T20,\nX16 -> T20,\nT8 -> T21,\nX17 -> T21,\nT3 -> T23,\nX18 -> T23,\nT22 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 101 [arrowhead = none ];
101 -> 10;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 102 [arrowhead = none ];
102 -> 11;

103 [label="EVAL with clause\nuser_defined_=(X21, X21).\nand substitutionT20 -> 0,\nX21 -> 0,\nT26 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 103 [arrowhead = none ];
103 -> 12;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 104 [arrowhead = none ];
104 -> 13;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 105 [arrowhead = none ];
105 -> 14;

106 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 106 [arrowhead = none ];
106 -> 19;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 107 [arrowhead = none ];
107 -> 15;

108 [label="EVAL with clause\nuser_defined_=(X24, X24).\nand substitutionT23 -> 0,\nX24 -> 0,\nT29 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 108 [arrowhead = none ];
108 -> 16;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 109 [arrowhead = none ];
109 -> 17;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 110 [arrowhead = none ];
110 -> 18;

111 [label="ONLY EVAL with clause\ndiv(X37, X38, X39) :- ','(minus(X37, X38, X40), ','(div(X40, X38, X41), user_defined_=(X39, s(X41)))).\nand substitutionT20 -> T39,\nX37 -> T39,\nT21 -> T40,\nX38 -> T40,\nT3 -> T42,\nX39 -> T42,\nT41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 111 [arrowhead = none ];
111 -> 20;

112 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 112 [arrowhead = none ];
112 -> 21;

113 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nT40 is ground\nT45 is ground\nreplacements:X40 -> T45", fontsize=14, style = filled, fillcolor = violet];
20 -> 113 [arrowhead = none ];
113 -> 22;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 114 [arrowhead = none ];
114 -> 23;

115 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 115 [arrowhead = none ];
115 -> 65;

116 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT40 is ground\nreplacements:X41 -> T127", fontsize=14, style = filled, fillcolor = violet];
22 -> 116 [arrowhead = none ];
116 -> 66;

117 [label="ONLY EVAL with clause\nminus(X58, X59, X60) :- ','(user_defined_=(X58, 0), ','(!_5, user_defined_=(X60, 0))).\nand substitutionT39 -> T52,\nX58 -> T52,\nT40 -> T53,\nX59 -> T53,\nX40 -> X61,\nX60 -> X61", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 117 [arrowhead = none ];
117 -> 24;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 118 [arrowhead = none ];
118 -> 25;

119 [label="BACKTRACK\nfor clause: user_defined_=(X, X)\nwith clash: (user_defined_=(T52, 0), user_defined_=(X21, X21))", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 119 [arrowhead = none ];
119 -> 26;

120 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 120 [arrowhead = none ];
120 -> 27;

121 [label="ONLY EVAL with clause\nminus(X71, X72, X73) :- ','(user_defined_=(X72, 0), ','(!_5, user_defined_=(X73, X71))).\nand substitutionT52 -> T59,\nX71 -> T59,\nT53 -> T60,\nX72 -> T60,\nX40 -> X74,\nX73 -> X74", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 121 [arrowhead = none ];
121 -> 28;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 122 [arrowhead = none ];
122 -> 29;

123 [label="BACKTRACK\nfor clause: user_defined_=(X, X)\nwith clash: (user_defined_=(T60, 0), user_defined_=(X9, X9))", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 123 [arrowhead = none ];
123 -> 30;

124 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 124 [arrowhead = none ];
124 -> 31;

125 [label="ONLY EVAL with clause\nminus(X92, X93, X94) :- ','(user_defined_=(X92, s(X95)), ','(user_defined_=(X93, s(X96)), minus(X95, X96, X94))).\nand substitutionT59 -> T66,\nX92 -> T66,\nT60 -> T67,\nX93 -> T67,\nX40 -> X97,\nX94 -> X97", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 125 [arrowhead = none ];
125 -> 32;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 126 [arrowhead = none ];
126 -> 33;

127 [label="EVAL with clause\nuser_defined_=(X106, X106).\nand substitutionT66 -> s(T73),\nX106 -> s(T73),\nX95 -> T73,\nT72 -> s(T73)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 127 [arrowhead = none ];
127 -> 34;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 128 [arrowhead = none ];
128 -> 35;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 129 [arrowhead = none ];
129 -> 36;

130 [label="EVAL with clause\nuser_defined_=(X114, X114).\nand substitutionT67 -> s(T80),\nX114 -> s(T80),\nX96 -> T80,\nT79 -> s(T80)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 130 [arrowhead = none ];
130 -> 37;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 131 [arrowhead = none ];
131 -> 38;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 132 [arrowhead = none ];
132 -> 39;

133 [label="ONLY EVAL with clause\nminus(X127, X128, X129) :- ','(user_defined_=(X127, 0), ','(!_10, user_defined_=(X129, 0))).\nand substitutionT73 -> T87,\nX127 -> T87,\nT80 -> T88,\nX128 -> T88,\nX97 -> X130,\nX129 -> X130", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 133 [arrowhead = none ];
133 -> 40;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 134 [arrowhead = none ];
134 -> 41;

135 [label="EVAL with clause\nuser_defined_=(X133, X133).\nand substitutionT87 -> 0,\nX133 -> 0,\nT91 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 135 [arrowhead = none ];
135 -> 42;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 136 [arrowhead = none ];
136 -> 43;

137 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 137 [arrowhead = none ];
137 -> 44;

138 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 138 [arrowhead = none ];
138 -> 48;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 139 [arrowhead = none ];
139 -> 45;

140 [label="ONLY EVAL with clause\nuser_defined_=(X138, X138).\nand substitutionX130 -> 0,\nX138 -> 0,\nX139 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 140 [arrowhead = none ];
140 -> 46;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 141 [arrowhead = none ];
141 -> 47;

142 [label="ONLY EVAL with clause\nminus(X148, X149, X150) :- ','(user_defined_=(X149, 0), ','(!_10, user_defined_=(X150, X148))).\nand substitutionT87 -> T96,\nX148 -> T96,\nT88 -> T97,\nX149 -> T97,\nX97 -> X151,\nX150 -> X151", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 142 [arrowhead = none ];
142 -> 49;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 143 [arrowhead = none ];
143 -> 50;

144 [label="EVAL with clause\nuser_defined_=(X154, X154).\nand substitutionT97 -> 0,\nX154 -> 0,\nT100 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 144 [arrowhead = none ];
144 -> 51;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 145 [arrowhead = none ];
145 -> 52;

146 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 146 [arrowhead = none ];
146 -> 53;

147 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 147 [arrowhead = none ];
147 -> 57;

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 148 [arrowhead = none ];
148 -> 54;

149 [label="ONLY EVAL with clause\nuser_defined_=(X159, X159).\nand substitutionX151 -> T103,\nX159 -> T103,\nT96 -> T103,\nX160 -> T103", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 149 [arrowhead = none ];
149 -> 55;

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

151 [label="ONLY EVAL with clause\nminus(X177, X178, X179) :- ','(user_defined_=(X177, s(X180)), ','(user_defined_=(X178, s(X181)), minus(X180, X181, X179))).\nand substitutionT96 -> T108,\nX177 -> T108,\nT97 -> T109,\nX178 -> T109,\nX97 -> X182,\nX179 -> X182", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 151 [arrowhead = none ];
151 -> 58;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 152 [arrowhead = none ];
152 -> 59;

153 [label="EVAL with clause\nuser_defined_=(X191, X191).\nand substitutionT108 -> s(T115),\nX191 -> s(T115),\nX180 -> T115,\nT114 -> s(T115)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 153 [arrowhead = none ];
153 -> 60;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 154 [arrowhead = none ];
154 -> 61;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 155 [arrowhead = none ];
155 -> 62;

156 [label="EVAL with clause\nuser_defined_=(X199, X199).\nand substitutionT109 -> s(T122),\nX199 -> s(T122),\nX181 -> T122,\nT121 -> s(T122)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 156 [arrowhead = none ];
156 -> 63;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 157 [arrowhead = none ];
157 -> 64;

158 [label="INSTANCE with matching:\nT73 -> T115\nT80 -> T122\nX97 -> X182", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 158 [arrowhead = none , style=dashed];
158 -> 37[style=dashed];

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 159 [arrowhead = none ];
159 -> 67;

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 160 [arrowhead = none ];
160 -> 89;

161 [label="ONLY EVAL with clause\ndiv(X220, X221, X222) :- ','(user_defined_=(X221, 0), ','(!_17, fail)).\nand substitutionT45 -> T134,\nX220 -> T134,\nT40 -> T135,\nX221 -> T135,\nX41 -> X223,\nX222 -> X223", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 161 [arrowhead = none ];
161 -> 68;

162 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 162 [arrowhead = none ];
162 -> 69;

163 [label="BACKTRACK\nfor clause: user_defined_=(X, X)\nwith clash: (user_defined_=(T135, 0), user_defined_=(X9, X9))", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 163 [arrowhead = none ];
163 -> 70;

164 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 164 [arrowhead = none ];
164 -> 71;

165 [label="ONLY EVAL with clause\ndiv(X233, X234, X235) :- ','(user_defined_=(X233, 0), ','(!_17, user_defined_=(X235, 0))).\nand substitutionT134 -> T141,\nX233 -> T141,\nT135 -> T142,\nX234 -> T142,\nX41 -> X236,\nX235 -> X236", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 165 [arrowhead = none ];
165 -> 72;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 166 [arrowhead = none ];
166 -> 73;

167 [label="EVAL with clause\nuser_defined_=(X239, X239).\nand substitutionT141 -> 0,\nX239 -> 0,\nT145 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 167 [arrowhead = none ];
167 -> 74;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 168 [arrowhead = none ];
168 -> 75;

169 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 169 [arrowhead = none ];
169 -> 76;

170 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 170 [arrowhead = none ];
170 -> 80;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 171 [arrowhead = none ];
171 -> 77;

172 [label="ONLY EVAL with clause\nuser_defined_=(X244, X244).\nand substitutionX236 -> 0,\nX244 -> 0,\nX245 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 172 [arrowhead = none ];
172 -> 78;

173 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 173 [arrowhead = none ];
173 -> 79;

174 [label="ONLY EVAL with clause\ndiv(X260, X261, X262) :- ','(minus(X260, X261, X263), ','(div(X263, X261, X264), user_defined_=(X262, s(X264)))).\nand substitutionT141 -> T152,\nX260 -> T152,\nT142 -> T153,\nX261 -> T153,\nX41 -> X265,\nX262 -> X265", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 174 [arrowhead = none ];
174 -> 81;

175 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
81 -> 175 [arrowhead = none ];
175 -> 82;

176 [label="SPLIT 2\nnew knowledge:\nT152 is ground\nT153 is ground\nT156 is ground\nreplacements:X263 -> T156", fontsize=14, style = filled, fillcolor = violet];
81 -> 176 [arrowhead = none ];
176 -> 83;

177 [label="INSTANCE with matching:\nT39 -> T152\nT40 -> T153\nX40 -> X263\nX21 -> X239", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 177 [arrowhead = none , style=dashed];
177 -> 21[style=dashed];

178 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
83 -> 178 [arrowhead = none ];
178 -> 84;

179 [label="SPLIT 2\nnew knowledge:\nT156 is ground\nT153 is ground\nreplacements:X264 -> T161", fontsize=14, style = filled, fillcolor = violet];
83 -> 179 [arrowhead = none ];
179 -> 85;

180 [label="INSTANCE with matching:\nT45 -> T156\nT40 -> T153\nX41 -> X264", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 180 [arrowhead = none , style=dashed];
180 -> 65[style=dashed];

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 181 [arrowhead = none ];
181 -> 86;

182 [label="ONLY EVAL with clause\nuser_defined_=(X286, X286).\nand substitutionX265 -> s(T166),\nX286 -> s(T166),\nT161 -> T166,\nX287 -> s(T166)", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 182 [arrowhead = none ];
182 -> 87;

183 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 183 [arrowhead = none ];
183 -> 88;

184 [label="EVAL with clause\nuser_defined_=(X290, X290).\nand substitutionT42 -> s(T172),\nX290 -> s(T172),\nT127 -> T172,\nT171 -> s(T172)", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 184 [arrowhead = none ];
184 -> 90;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 185 [arrowhead = none ];
185 -> 91;

186 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 186 [arrowhead = none ];
186 -> 92;

}

(4) Obligation:

Clauses:

minusA(0, T88, 0).
minusA(T103, 0, T103).
minusA(s(T115), s(T122), X182) :- minusA(T115, T122, X182).
minusB(s(T73), s(T80), X97) :- minusA(T73, T80, X97).
divC(0, T142, 0).
divC(T152, T153, X265) :- minusB(T152, T153, X263).
divC(T152, T153, X265) :- ','(minusB(T152, T153, T156), divC(T156, T153, X264)).
divC(T152, T153, s(T166)) :- ','(minusB(T152, T153, T156), divC(T156, T153, T166)).
divD(0, T21, 0).
divD(T39, T40, T42) :- minusB(T39, T40, X40).
divD(T39, T40, T42) :- ','(minusB(T39, T40, T45), divC(T45, T40, X41)).
divD(T39, T40, s(T172)) :- ','(minusB(T39, T40, T45), divC(T45, T40, T172)).

Query: divD(g,g,a)

(5) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
divD_in: (b,b,f)
minusB_in: (b,b,f)
minusA_in: (b,b,f)
divC_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

divD_in_gga(0, T21, 0) → divD_out_gga(0, T21, 0)
divD_in_gga(T39, T40, T42) → U8_gga(T39, T40, T42, minusB_in_gga(T39, T40, X40))
minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
U8_gga(T39, T40, T42, minusB_out_gga(T39, T40, X40)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, T42) → U9_gga(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_gga(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_gga(T39, T40, T42, divC_in_gga(T45, T40, X41))
divC_in_gga(0, T142, 0) → divC_out_gga(0, T142, 0)
divC_in_gga(T152, T153, X265) → U3_gga(T152, T153, X265, minusB_in_gga(T152, T153, X263))
U3_gga(T152, T153, X265, minusB_out_gga(T152, T153, X263)) → divC_out_gga(T152, T153, X265)
divC_in_gga(T152, T153, X265) → U4_gga(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_gga(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_gga(T152, T153, X265, divC_in_gga(T156, T153, X264))
divC_in_gga(T152, T153, s(T166)) → U6_gga(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_gga(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_gga(T152, T153, T166, divC_in_gga(T156, T153, T166))
U7_gga(T152, T153, T166, divC_out_gga(T156, T153, T166)) → divC_out_gga(T152, T153, s(T166))
U5_gga(T152, T153, X265, divC_out_gga(T156, T153, X264)) → divC_out_gga(T152, T153, X265)
U10_gga(T39, T40, T42, divC_out_gga(T45, T40, X41)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, s(T172)) → U11_gga(T39, T40, T172, minusB_in_gga(T39, T40, T45))
U11_gga(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_gga(T39, T40, T172, divC_in_gga(T45, T40, T172))
U12_gga(T39, T40, T172, divC_out_gga(T45, T40, T172)) → divD_out_gga(T39, T40, s(T172))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

divD_in_gga(0, T21, 0) → divD_out_gga(0, T21, 0)
divD_in_gga(T39, T40, T42) → U8_gga(T39, T40, T42, minusB_in_gga(T39, T40, X40))
minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
U8_gga(T39, T40, T42, minusB_out_gga(T39, T40, X40)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, T42) → U9_gga(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_gga(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_gga(T39, T40, T42, divC_in_gga(T45, T40, X41))
divC_in_gga(0, T142, 0) → divC_out_gga(0, T142, 0)
divC_in_gga(T152, T153, X265) → U3_gga(T152, T153, X265, minusB_in_gga(T152, T153, X263))
U3_gga(T152, T153, X265, minusB_out_gga(T152, T153, X263)) → divC_out_gga(T152, T153, X265)
divC_in_gga(T152, T153, X265) → U4_gga(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_gga(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_gga(T152, T153, X265, divC_in_gga(T156, T153, X264))
divC_in_gga(T152, T153, s(T166)) → U6_gga(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_gga(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_gga(T152, T153, T166, divC_in_gga(T156, T153, T166))
U7_gga(T152, T153, T166, divC_out_gga(T156, T153, T166)) → divC_out_gga(T152, T153, s(T166))
U5_gga(T152, T153, X265, divC_out_gga(T156, T153, X264)) → divC_out_gga(T152, T153, X265)
U10_gga(T39, T40, T42, divC_out_gga(T45, T40, X41)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, s(T172)) → U11_gga(T39, T40, T172, minusB_in_gga(T39, T40, T45))
U11_gga(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_gga(T39, T40, T172, divC_in_gga(T45, T40, T172))
U12_gga(T39, T40, T172, divC_out_gga(T45, T40, T172)) → divD_out_gga(T39, T40, s(T172))

(7) 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:

DIVD_IN_GGA(T39, T40, T42) → U8_GGA(T39, T40, T42, minusB_in_gga(T39, T40, X40))
DIVD_IN_GGA(T39, T40, T42) → MINUSB_IN_GGA(T39, T40, X40)
MINUSB_IN_GGA(s(T73), s(T80), X97) → U2_GGA(T73, T80, X97, minusA_in_gga(T73, T80, X97))
MINUSB_IN_GGA(s(T73), s(T80), X97) → MINUSA_IN_GGA(T73, T80, X97)
MINUSA_IN_GGA(s(T115), s(T122), X182) → U1_GGA(T115, T122, X182, minusA_in_gga(T115, T122, X182))
MINUSA_IN_GGA(s(T115), s(T122), X182) → MINUSA_IN_GGA(T115, T122, X182)
DIVD_IN_GGA(T39, T40, T42) → U9_GGA(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_GGA(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_GGA(T39, T40, T42, divC_in_gga(T45, T40, X41))
U9_GGA(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → DIVC_IN_GGA(T45, T40, X41)
DIVC_IN_GGA(T152, T153, X265) → U3_GGA(T152, T153, X265, minusB_in_gga(T152, T153, X263))
DIVC_IN_GGA(T152, T153, X265) → MINUSB_IN_GGA(T152, T153, X263)
DIVC_IN_GGA(T152, T153, X265) → U4_GGA(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_GGA(T152, T153, X265, divC_in_gga(T156, T153, X264))
U4_GGA(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, X264)
DIVC_IN_GGA(T152, T153, s(T166)) → U6_GGA(T152, T153, T166, minusB_in_gga(T152, T153, T156))
DIVC_IN_GGA(T152, T153, s(T166)) → MINUSB_IN_GGA(T152, T153, T156)
U6_GGA(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_GGA(T152, T153, T166, divC_in_gga(T156, T153, T166))
U6_GGA(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, T166)
DIVD_IN_GGA(T39, T40, s(T172)) → U11_GGA(T39, T40, T172, minusB_in_gga(T39, T40, T45))
DIVD_IN_GGA(T39, T40, s(T172)) → MINUSB_IN_GGA(T39, T40, T45)
U11_GGA(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_GGA(T39, T40, T172, divC_in_gga(T45, T40, T172))
U11_GGA(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → DIVC_IN_GGA(T45, T40, T172)

The TRS R consists of the following rules:

divD_in_gga(0, T21, 0) → divD_out_gga(0, T21, 0)
divD_in_gga(T39, T40, T42) → U8_gga(T39, T40, T42, minusB_in_gga(T39, T40, X40))
minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
U8_gga(T39, T40, T42, minusB_out_gga(T39, T40, X40)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, T42) → U9_gga(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_gga(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_gga(T39, T40, T42, divC_in_gga(T45, T40, X41))
divC_in_gga(0, T142, 0) → divC_out_gga(0, T142, 0)
divC_in_gga(T152, T153, X265) → U3_gga(T152, T153, X265, minusB_in_gga(T152, T153, X263))
U3_gga(T152, T153, X265, minusB_out_gga(T152, T153, X263)) → divC_out_gga(T152, T153, X265)
divC_in_gga(T152, T153, X265) → U4_gga(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_gga(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_gga(T152, T153, X265, divC_in_gga(T156, T153, X264))
divC_in_gga(T152, T153, s(T166)) → U6_gga(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_gga(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_gga(T152, T153, T166, divC_in_gga(T156, T153, T166))
U7_gga(T152, T153, T166, divC_out_gga(T156, T153, T166)) → divC_out_gga(T152, T153, s(T166))
U5_gga(T152, T153, X265, divC_out_gga(T156, T153, X264)) → divC_out_gga(T152, T153, X265)
U10_gga(T39, T40, T42, divC_out_gga(T45, T40, X41)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, s(T172)) → U11_gga(T39, T40, T172, minusB_in_gga(T39, T40, T45))
U11_gga(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_gga(T39, T40, T172, divC_in_gga(T45, T40, T172))
U12_gga(T39, T40, T172, divC_out_gga(T45, T40, T172)) → divD_out_gga(T39, T40, s(T172))

The argument filtering Pi contains the following mapping:
divD_in_gga(x1, x2, x3) = divD_in_gga(x1, x2)
0 = 0
divD_out_gga(x1, x2, x3) = divD_out_gga
U8_gga(x1, x2, x3, x4) = U8_gga(x4)
minusB_in_gga(x1, x2, x3) = minusB_in_gga(x1, x2)
s(x1) = s(x1)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3) = minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
minusB_out_gga(x1, x2, x3) = minusB_out_gga(x3)
U9_gga(x1, x2, x3, x4) = U9_gga(x2, x4)
U10_gga(x1, x2, x3, x4) = U10_gga(x4)
divC_in_gga(x1, x2, x3) = divC_in_gga(x1, x2)
divC_out_gga(x1, x2, x3) = divC_out_gga
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
U6_gga(x1, x2, x3, x4) = U6_gga(x2, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x4)
U11_gga(x1, x2, x3, x4) = U11_gga(x2, x4)
U12_gga(x1, x2, x3, x4) = U12_gga(x4)
DIVD_IN_GGA(x1, x2, x3) = DIVD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
MINUSB_IN_GGA(x1, x2, x3) = MINUSB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x2, x4)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x4)
DIVC_IN_GGA(x1, x2, x3) = DIVC_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x4)
U11_GGA(x1, x2, x3, x4) = U11_GGA(x2, x4)
U12_GGA(x1, x2, x3, x4) = U12_GGA(x4)

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

(8) Obligation:

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

DIVD_IN_GGA(T39, T40, T42) → U8_GGA(T39, T40, T42, minusB_in_gga(T39, T40, X40))
DIVD_IN_GGA(T39, T40, T42) → MINUSB_IN_GGA(T39, T40, X40)
MINUSB_IN_GGA(s(T73), s(T80), X97) → U2_GGA(T73, T80, X97, minusA_in_gga(T73, T80, X97))
MINUSB_IN_GGA(s(T73), s(T80), X97) → MINUSA_IN_GGA(T73, T80, X97)
MINUSA_IN_GGA(s(T115), s(T122), X182) → U1_GGA(T115, T122, X182, minusA_in_gga(T115, T122, X182))
MINUSA_IN_GGA(s(T115), s(T122), X182) → MINUSA_IN_GGA(T115, T122, X182)
DIVD_IN_GGA(T39, T40, T42) → U9_GGA(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_GGA(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_GGA(T39, T40, T42, divC_in_gga(T45, T40, X41))
U9_GGA(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → DIVC_IN_GGA(T45, T40, X41)
DIVC_IN_GGA(T152, T153, X265) → U3_GGA(T152, T153, X265, minusB_in_gga(T152, T153, X263))
DIVC_IN_GGA(T152, T153, X265) → MINUSB_IN_GGA(T152, T153, X263)
DIVC_IN_GGA(T152, T153, X265) → U4_GGA(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_GGA(T152, T153, X265, divC_in_gga(T156, T153, X264))
U4_GGA(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, X264)
DIVC_IN_GGA(T152, T153, s(T166)) → U6_GGA(T152, T153, T166, minusB_in_gga(T152, T153, T156))
DIVC_IN_GGA(T152, T153, s(T166)) → MINUSB_IN_GGA(T152, T153, T156)
U6_GGA(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_GGA(T152, T153, T166, divC_in_gga(T156, T153, T166))
U6_GGA(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, T166)
DIVD_IN_GGA(T39, T40, s(T172)) → U11_GGA(T39, T40, T172, minusB_in_gga(T39, T40, T45))
DIVD_IN_GGA(T39, T40, s(T172)) → MINUSB_IN_GGA(T39, T40, T45)
U11_GGA(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_GGA(T39, T40, T172, divC_in_gga(T45, T40, T172))
U11_GGA(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → DIVC_IN_GGA(T45, T40, T172)

The TRS R consists of the following rules:

divD_in_gga(0, T21, 0) → divD_out_gga(0, T21, 0)
divD_in_gga(T39, T40, T42) → U8_gga(T39, T40, T42, minusB_in_gga(T39, T40, X40))
minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
U8_gga(T39, T40, T42, minusB_out_gga(T39, T40, X40)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, T42) → U9_gga(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_gga(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_gga(T39, T40, T42, divC_in_gga(T45, T40, X41))
divC_in_gga(0, T142, 0) → divC_out_gga(0, T142, 0)
divC_in_gga(T152, T153, X265) → U3_gga(T152, T153, X265, minusB_in_gga(T152, T153, X263))
U3_gga(T152, T153, X265, minusB_out_gga(T152, T153, X263)) → divC_out_gga(T152, T153, X265)
divC_in_gga(T152, T153, X265) → U4_gga(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_gga(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_gga(T152, T153, X265, divC_in_gga(T156, T153, X264))
divC_in_gga(T152, T153, s(T166)) → U6_gga(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_gga(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_gga(T152, T153, T166, divC_in_gga(T156, T153, T166))
U7_gga(T152, T153, T166, divC_out_gga(T156, T153, T166)) → divC_out_gga(T152, T153, s(T166))
U5_gga(T152, T153, X265, divC_out_gga(T156, T153, X264)) → divC_out_gga(T152, T153, X265)
U10_gga(T39, T40, T42, divC_out_gga(T45, T40, X41)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, s(T172)) → U11_gga(T39, T40, T172, minusB_in_gga(T39, T40, T45))
U11_gga(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_gga(T39, T40, T172, divC_in_gga(T45, T40, T172))
U12_gga(T39, T40, T172, divC_out_gga(T45, T40, T172)) → divD_out_gga(T39, T40, s(T172))

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 17 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MINUSA_IN_GGA(s(T115), s(T122), X182) → MINUSA_IN_GGA(T115, T122, X182)

The TRS R consists of the following rules:

divD_in_gga(0, T21, 0) → divD_out_gga(0, T21, 0)
divD_in_gga(T39, T40, T42) → U8_gga(T39, T40, T42, minusB_in_gga(T39, T40, X40))
minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
U8_gga(T39, T40, T42, minusB_out_gga(T39, T40, X40)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, T42) → U9_gga(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_gga(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_gga(T39, T40, T42, divC_in_gga(T45, T40, X41))
divC_in_gga(0, T142, 0) → divC_out_gga(0, T142, 0)
divC_in_gga(T152, T153, X265) → U3_gga(T152, T153, X265, minusB_in_gga(T152, T153, X263))
U3_gga(T152, T153, X265, minusB_out_gga(T152, T153, X263)) → divC_out_gga(T152, T153, X265)
divC_in_gga(T152, T153, X265) → U4_gga(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_gga(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_gga(T152, T153, X265, divC_in_gga(T156, T153, X264))
divC_in_gga(T152, T153, s(T166)) → U6_gga(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_gga(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_gga(T152, T153, T166, divC_in_gga(T156, T153, T166))
U7_gga(T152, T153, T166, divC_out_gga(T156, T153, T166)) → divC_out_gga(T152, T153, s(T166))
U5_gga(T152, T153, X265, divC_out_gga(T156, T153, X264)) → divC_out_gga(T152, T153, X265)
U10_gga(T39, T40, T42, divC_out_gga(T45, T40, X41)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, s(T172)) → U11_gga(T39, T40, T172, minusB_in_gga(T39, T40, T45))
U11_gga(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_gga(T39, T40, T172, divC_in_gga(T45, T40, T172))
U12_gga(T39, T40, T172, divC_out_gga(T45, T40, T172)) → divD_out_gga(T39, T40, s(T172))

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MINUSA_IN_GGA(s(T115), s(T122), X182) → MINUSA_IN_GGA(T115, T122, X182)

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

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

MINUSA_IN_GGA(s(T115), s(T122)) → MINUSA_IN_GGA(T115, T122)

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

(16) 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:

MINUSA_IN_GGA(s(T115), s(T122)) → MINUSA_IN_GGA(T115, T122)
The graph contains the following edges 1 > 1, 2 > 2

(17) YES

(18) Obligation:

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

DIVC_IN_GGA(T152, T153, X265) → U4_GGA(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, X264)
DIVC_IN_GGA(T152, T153, s(T166)) → U6_GGA(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_GGA(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, T166)

The TRS R consists of the following rules:

divD_in_gga(0, T21, 0) → divD_out_gga(0, T21, 0)
divD_in_gga(T39, T40, T42) → U8_gga(T39, T40, T42, minusB_in_gga(T39, T40, X40))
minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
U8_gga(T39, T40, T42, minusB_out_gga(T39, T40, X40)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, T42) → U9_gga(T39, T40, T42, minusB_in_gga(T39, T40, T45))
U9_gga(T39, T40, T42, minusB_out_gga(T39, T40, T45)) → U10_gga(T39, T40, T42, divC_in_gga(T45, T40, X41))
divC_in_gga(0, T142, 0) → divC_out_gga(0, T142, 0)
divC_in_gga(T152, T153, X265) → U3_gga(T152, T153, X265, minusB_in_gga(T152, T153, X263))
U3_gga(T152, T153, X265, minusB_out_gga(T152, T153, X263)) → divC_out_gga(T152, T153, X265)
divC_in_gga(T152, T153, X265) → U4_gga(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_gga(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → U5_gga(T152, T153, X265, divC_in_gga(T156, T153, X264))
divC_in_gga(T152, T153, s(T166)) → U6_gga(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_gga(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → U7_gga(T152, T153, T166, divC_in_gga(T156, T153, T166))
U7_gga(T152, T153, T166, divC_out_gga(T156, T153, T166)) → divC_out_gga(T152, T153, s(T166))
U5_gga(T152, T153, X265, divC_out_gga(T156, T153, X264)) → divC_out_gga(T152, T153, X265)
U10_gga(T39, T40, T42, divC_out_gga(T45, T40, X41)) → divD_out_gga(T39, T40, T42)
divD_in_gga(T39, T40, s(T172)) → U11_gga(T39, T40, T172, minusB_in_gga(T39, T40, T45))
U11_gga(T39, T40, T172, minusB_out_gga(T39, T40, T45)) → U12_gga(T39, T40, T172, divC_in_gga(T45, T40, T172))
U12_gga(T39, T40, T172, divC_out_gga(T45, T40, T172)) → divD_out_gga(T39, T40, s(T172))

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

DIVC_IN_GGA(T152, T153, X265) → U4_GGA(T152, T153, X265, minusB_in_gga(T152, T153, T156))
U4_GGA(T152, T153, X265, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, X264)
DIVC_IN_GGA(T152, T153, s(T166)) → U6_GGA(T152, T153, T166, minusB_in_gga(T152, T153, T156))
U6_GGA(T152, T153, T166, minusB_out_gga(T152, T153, T156)) → DIVC_IN_GGA(T156, T153, T166)

The TRS R consists of the following rules:

minusB_in_gga(s(T73), s(T80), X97) → U2_gga(T73, T80, X97, minusA_in_gga(T73, T80, X97))
U2_gga(T73, T80, X97, minusA_out_gga(T73, T80, X97)) → minusB_out_gga(s(T73), s(T80), X97)
minusA_in_gga(0, T88, 0) → minusA_out_gga(0, T88, 0)
minusA_in_gga(T103, 0, T103) → minusA_out_gga(T103, 0, T103)
minusA_in_gga(s(T115), s(T122), X182) → U1_gga(T115, T122, X182, minusA_in_gga(T115, T122, X182))
U1_gga(T115, T122, X182, minusA_out_gga(T115, T122, X182)) → minusA_out_gga(s(T115), s(T122), X182)

The argument filtering Pi contains the following mapping:
0 = 0
minusB_in_gga(x1, x2, x3) = minusB_in_gga(x1, x2)
s(x1) = s(x1)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3) = minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
minusB_out_gga(x1, x2, x3) = minusB_out_gga(x3)
DIVC_IN_GGA(x1, x2, x3) = DIVC_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x2, x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x2, x4)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

DIVC_IN_GGA(T152, T153) → U4_GGA(T153, minusB_in_gga(T152, T153))
U4_GGA(T153, minusB_out_gga(T156)) → DIVC_IN_GGA(T156, T153)
DIVC_IN_GGA(T152, T153) → U6_GGA(T153, minusB_in_gga(T152, T153))
U6_GGA(T153, minusB_out_gga(T156)) → DIVC_IN_GGA(T156, T153)

The TRS R consists of the following rules:

minusB_in_gga(s(T73), s(T80)) → U2_gga(minusA_in_gga(T73, T80))
U2_gga(minusA_out_gga(X97)) → minusB_out_gga(X97)
minusA_in_gga(0, T88) → minusA_out_gga(0)
minusA_in_gga(T103, 0) → minusA_out_gga(T103)
minusA_in_gga(s(T115), s(T122)) → U1_gga(minusA_in_gga(T115, T122))
U1_gga(minusA_out_gga(X182)) → minusA_out_gga(X182)

The set Q consists of the following terms:

minusB_in_gga(x0, x1)
U2_gga(x0)
minusA_in_gga(x0, x1)
U1_gga(x0)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

U4_GGA(T153, minusB_out_gga(T156)) → DIVC_IN_GGA(T156, T153)
U6_GGA(T153, minusB_out_gga(T156)) → DIVC_IN_GGA(T156, T153)

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

POL(0) = 0
POL(DIVC_IN_GGA(x₁, x₂)) = x₁
POL(U1_gga(x₁)) = x₁
POL(U2_gga(x₁)) = x₁
POL(U4_GGA(x₁, x₂)) = x₂
POL(U6_GGA(x₁, x₂)) = x₂
POL(minusA_in_gga(x₁, x₂)) = 1 + x₁
POL(minusA_out_gga(x₁)) = 1 + x₁
POL(minusB_in_gga(x₁, x₂)) = x₁
POL(minusB_out_gga(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

minusB_in_gga(s(T73), s(T80)) → U2_gga(minusA_in_gga(T73, T80))
minusA_in_gga(0, T88) → minusA_out_gga(0)
minusA_in_gga(T103, 0) → minusA_out_gga(T103)
minusA_in_gga(s(T115), s(T122)) → U1_gga(minusA_in_gga(T115, T122))
U2_gga(minusA_out_gga(X97)) → minusB_out_gga(X97)
U1_gga(minusA_out_gga(X182)) → minusA_out_gga(X182)

(24) Obligation:

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

DIVC_IN_GGA(T152, T153) → U4_GGA(T153, minusB_in_gga(T152, T153))
DIVC_IN_GGA(T152, T153) → U6_GGA(T153, minusB_in_gga(T152, T153))

The TRS R consists of the following rules:

minusB_in_gga(s(T73), s(T80)) → U2_gga(minusA_in_gga(T73, T80))
U2_gga(minusA_out_gga(X97)) → minusB_out_gga(X97)
minusA_in_gga(0, T88) → minusA_out_gga(0)
minusA_in_gga(T103, 0) → minusA_out_gga(T103)
minusA_in_gga(s(T115), s(T122)) → U1_gga(minusA_in_gga(T115, T122))
U1_gga(minusA_out_gga(X182)) → minusA_out_gga(X182)

The set Q consists of the following terms:

minusB_in_gga(x0, x1)
U2_gga(x0)
minusA_in_gga(x0, x1)
U1_gga(x0)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

(2) Obligation:

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

(4) Obligation:

(5) PrologToPiTRSProof (SOUND transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Complex Obligation (AND)

(11) Obligation:

(12) UsableRulesProof (EQUIVALENT transformation)

(13) Obligation:

(14) PiDPToQDPProof (SOUND transformation)

(15) Obligation:

(16) QDPSizeChangeProof (EQUIVALENT transformation)

(17) YES

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) PiDPToQDPProof (SOUND transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) DependencyGraphProof (EQUIVALENT transformation)

(26) TRUE