Left Termination proof of /tmp/tmp8Dj0mO/divminuslinear.pl

Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PredefinedPredicateTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPrologProblemTransformerProof (⇐)

↳4 Prolog

↳5 PrologToPiTRSProof (⇐)

↳6 PiTRS

↳7 DependencyPairsProof (⇔)

↳8 PiDP

↳9 DependencyGraphProof (⇔)

↳10 AND

    ↳11 PiDP

        ↳12 UsableRulesProof (⇔)

            ↳13 PiDP

                ↳14 PiDPToQDPProof (⇐)

                    ↳15 QDP

                        ↳16 QDPSizeChangeProof (⇔)

                            ↳17 TRUE

    ↳18 PiDP

        ↳19 UsableRulesProof (⇔)

            ↳20 PiDP

                ↳21 PiDPToQDPProof (⇐)

                    ↳22 QDP

                        ↳23 QDPOrderProof (⇔)

                            ↳24 QDP

                                ↳25 DependencyGraphProof (⇔)

                                    ↳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).

Queries:

div(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

Clauses:

minus(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
minus(X, Y, Z) :- ','(=1(Y, 0), ','(!, =1(Z, X))).
minus(X, Y, Z) :- ','(=1(X, s(A)), ','(=1(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=1(Y, 0), ','(!, fail)).
div(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =1(Z, s(V)))).
=1(X, X).
fail :- fail(b).
fail(a).

Queries:

div(g,g,a).

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(=1(T5, 0), ','(!_1, fail)) | div(T4, T5, T3), SCOPE: 1, CLAUSE: 4 | div(T4, T5, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(=1(T5, 0), ','(!_1, fail)), SCOPE: 2, CLAUSE: 6 | div(T4, T5, T3), SCOPE: 1, CLAUSE: 4 | div(T4, T5, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, fail) | div(T4, 0, T3), SCOPE: 1, CLAUSE: 4 | div(T4, 0, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: div(T4, T5, T3), SCOPE: 1, CLAUSE: 4 | div(T4, T5, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT4 is ground\nT5 is ground\nX8 is free; \n=1(T5, 0) !=? =1(X8, X8)", 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: ','(=1(T8, 0), ','(!_1, =1(T11, 0))) | div(T8, T9, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT8 is ground\nT9 is ground\nX8 is free; \n=1(T9, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(=1(T8, 0), ','(!_1, =1(T11, 0))), SCOPE: 3, CLAUSE: 6 | div(T8, T9, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT8 is ground\nT9 is ground\nX8 is free; \n=1(T9, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(!_1, =1(T11, 0)) | div(0, T9, T3), SCOPE: 1, CLAUSE: 5 | ?_1\n\nT9 is ground\nX8 is free; \n=1(T9, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: div(T8, T9, T3), SCOPE: 1, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nX8 is free; \nX16 is free; \n=1(T9, 0) !=? =1(X8, X8)\n=1(T8, 0) !=? =1(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: =1(T11, 0)\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: =1(T11, 0), SCOPE: 4, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(minus(T14, T15, X25), ','(div(X25, T15, X26), =1(T17, s(X26))))\n\nT14 is ground\nT15 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX26 is free; \n=1(T15, 0) !=? =1(X8, X8)\n=1(T14, 0) !=? =1(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: minus(T14, T15, X25)\n\nT14 is ground\nT15 is ground\nX8 is free; \nX16 is free; \nX25 is free; \n=1(T15, 0) !=? =1(X8, X8)\n=1(T14, 0) !=? =1(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(div(T18, T15, X26), =1(T17, s(X26)))\n\nT15 is ground\nT18 is ground\nX8 is free; \nX16 is free; \nX26 is free; \n=1(T15, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: minus(T14, T15, X25), SCOPE: 5, CLAUSE: 0 | minus(T14, T15, X25), SCOPE: 5, CLAUSE: 1 | minus(T14, T15, X25), SCOPE: 5, CLAUSE: 2 | ?_5\n\nT14 is ground\nT15 is ground\nX8 is free; \nX16 is free; \nX25 is free; \n=1(T15, 0) !=? =1(X8, X8)\n=1(T14, 0) !=? =1(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(=1(T19, 0), ','(!_5, =1(X33, 0))) | minus(T19, T20, X25), SCOPE: 5, CLAUSE: 1 | minus(T19, T20, X25), SCOPE: 5, CLAUSE: 2 | ?_5\n\nT19 is ground\nT20 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX33 is free; \n=1(T20, 0) !=? =1(X8, X8)\n=1(T19, 0) !=? =1(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(=1(T19, 0), ','(!_5, =1(X33, 0))), SCOPE: 6, CLAUSE: 6 | minus(T19, T20, X25), SCOPE: 5, CLAUSE: 1 | minus(T19, T20, X25), SCOPE: 5, CLAUSE: 2 | ?_5\n\nT19 is ground\nT20 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX33 is free; \n=1(T20, 0) !=? =1(X8, X8)\n=1(T19, 0) !=? =1(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: minus(T19, T20, X25), SCOPE: 5, CLAUSE: 1 | minus(T19, T20, X25), SCOPE: 5, CLAUSE: 2 | ?_5\n\nT19 is ground\nT20 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX35 is free; \n=1(T20, 0) !=? =1(X8, X8)\n=1(T19, 0) !=? =1(X16, X16)\n=1(T19, 0) !=? =1(X35, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(=1(T22, 0), ','(!_5, =1(X42, T21))) | minus(T21, T22, X25), SCOPE: 5, CLAUSE: 2 | ?_5\n\nT21 is ground\nT22 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX35 is free; \nX42 is free; \n=1(T22, 0) !=? =1(X8, X8)\n=1(T21, 0) !=? =1(X16, X16)\n=1(T21, 0) !=? =1(X35, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(=1(T22, 0), ','(!_5, =1(X42, T21))), SCOPE: 7, CLAUSE: 6 | minus(T21, T22, X25), SCOPE: 5, CLAUSE: 2 | ?_5\n\nT21 is ground\nT22 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX35 is free; \nX42 is free; \n=1(T22, 0) !=? =1(X8, X8)\n=1(T21, 0) !=? =1(X16, X16)\n=1(T21, 0) !=? =1(X35, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: minus(T21, T22, X25), SCOPE: 5, CLAUSE: 2\n\nT21 is ground\nT22 is ground\nX8 is free; \nX16 is free; \nX25 is free; \nX44 is free; \n=1(T22, 0) !=? =1(X8, X8)\n=1(T21, 0) !=? =1(X16, X16)\n=1(T22, 0) !=? =1(X44, X44)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(=1(T23, s(X51)), ','(=1(T24, s(X52)), minus(X51, X52, X53)))\n\nT23 is ground\nT24 is ground\nX8 is free; \nX16 is free; \nX44 is free; \nX53 is free; \nX51 is free; \nX52 is free; \n=1(T24, 0) !=? =1(X8, X8)\n=1(T23, 0) !=? =1(X16, X16)\n=1(T24, 0) !=? =1(X44, X44)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(=1(T23, s(X51)), ','(=1(T24, s(X52)), minus(X51, X52, X53))), SCOPE: 8, CLAUSE: 6\n\nT23 is ground\nT24 is ground\nX8 is free; \nX16 is free; \nX44 is free; \nX53 is free; \nX51 is free; \nX52 is free; \n=1(T24, 0) !=? =1(X8, X8)\n=1(T23, 0) !=? =1(X16, X16)\n=1(T24, 0) !=? =1(X44, X44)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(=1(T24, s(X52)), minus(T26, X52, X53))\n\nT24 is ground\nT26 is ground\nX8 is free; \nX16 is free; \nX44 is free; \nX53 is free; \nX52 is free; \n=1(T24, 0) !=? =1(X8, X8)\n=1(T24, 0) !=? =1(X44, X44)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\nX8 is free\nX16 is free; \nX51 is free; \nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(=1(T24, s(X52)), minus(T26, X52, X53)), SCOPE: 9, CLAUSE: 6\n\nT24 is ground\nT26 is ground\nX8 is free; \nX44 is free; \nX53 is free; \nX52 is free; \n=1(T24, 0) !=? =1(X8, X8)\n=1(T24, 0) !=? =1(X44, X44)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: minus(T26, T28, X53)\n\nT26 is ground\nT28 is ground\nX8 is free; \nX44 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\nX8 is free\nX52 is free; \nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: minus(T26, T28, X53), SCOPE: 10, CLAUSE: 0 | minus(T26, T28, X53), SCOPE: 10, CLAUSE: 1 | minus(T26, T28, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT26 is ground\nT28 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(=1(T29, 0), ','(!_10, =1(X64, 0))) | minus(T29, T30, X53), SCOPE: 10, CLAUSE: 1 | minus(T29, T30, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT29 is ground\nT30 is ground\nX53 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(=1(T29, 0), ','(!_10, =1(X64, 0))), SCOPE: 11, CLAUSE: 6 | minus(T29, T30, X53), SCOPE: 10, CLAUSE: 1 | minus(T29, T30, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT29 is ground\nT30 is ground\nX53 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(!_10, =1(X64, 0)) | minus(0, T30, X53), SCOPE: 10, CLAUSE: 1 | minus(0, T30, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT30 is ground\nX53 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: minus(T29, T30, X53), SCOPE: 10, CLAUSE: 1 | minus(T29, T30, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT29 is ground\nT30 is ground\nX53 is free; \nX66 is free; \n=1(T29, 0) !=? =1(X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: =1(X64, 0)\n\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: =1(X64, 0), SCOPE: 12, CLAUSE: 6\n\nX64 is free", 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: ','(=1(T33, 0), ','(!_10, =1(X76, T32))) | minus(T32, T33, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT32 is ground\nT33 is ground\nX53 is free; \nX66 is free; \nX76 is free; \n=1(T32, 0) !=? =1(X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(=1(T33, 0), ','(!_10, =1(X76, T32))), SCOPE: 13, CLAUSE: 6 | minus(T32, T33, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT32 is ground\nT33 is ground\nX53 is free; \nX66 is free; \nX76 is free; \n=1(T32, 0) !=? =1(X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_10, =1(X76, T32)) | minus(T32, 0, X53), SCOPE: 10, CLAUSE: 2 | ?_10\n\nT32 is ground\nX53 is free; \nX66 is free; \nX76 is free; \n=1(T32, 0) !=? =1(X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: minus(T32, T33, X53), SCOPE: 10, CLAUSE: 2\n\nT32 is ground\nT33 is ground\nX53 is free; \nX66 is free; \nX78 is free; \n=1(T32, 0) !=? =1(X66, X66)\n=1(T33, 0) !=? =1(X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: =1(X76, T32)\n\nT32 is ground\nX66 is free; \nX76 is free; \n=1(T32, 0) !=? =1(X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: =1(X76, T32), SCOPE: 14, CLAUSE: 6\n\nT32 is ground\nX66 is free; \nX76 is free; \n=1(T32, 0) !=? =1(X66, X66)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(=1(T36, s(X88)), ','(=1(T37, s(X89)), minus(X88, X89, X90)))\n\nT36 is ground\nT37 is ground\nX66 is free; \nX78 is free; \nX90 is free; \nX88 is free; \nX89 is free; \n=1(T36, 0) !=? =1(X66, X66)\n=1(T37, 0) !=? =1(X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(=1(T36, s(X88)), ','(=1(T37, s(X89)), minus(X88, X89, X90))), SCOPE: 15, CLAUSE: 6\n\nT36 is ground\nT37 is ground\nX66 is free; \nX78 is free; \nX90 is free; \nX88 is free; \nX89 is free; \n=1(T36, 0) !=? =1(X66, X66)\n=1(T37, 0) !=? =1(X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(=1(T37, s(X89)), minus(T39, X89, X90))\n\nT37 is ground\nT39 is ground\nX66 is free; \nX78 is free; \nX90 is free; \nX89 is free; \n=1(T37, 0) !=? =1(X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\nX66 is free\nX78 is free; \nX88 is free; \nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(=1(T37, s(X89)), minus(T39, X89, X90)), SCOPE: 16, CLAUSE: 6\n\nT37 is ground\nT39 is ground\nX78 is free; \nX90 is free; \nX89 is free; \n=1(T37, 0) !=? =1(X78, X78)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: minus(T39, T41, X90)\n\nT39 is ground\nT41 is ground\nX78 is free; \nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\nX78 is free\nX89 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: div(T18, T15, X26)\n\nT15 is ground\nT18 is ground\nX8 is free; \nX26 is free; \n=1(T15, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: =1(T17, s(T42))\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: div(T18, T15, X26), SCOPE: 17, CLAUSE: 3 | div(T18, T15, X26), SCOPE: 17, CLAUSE: 4 | div(T18, T15, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT15 is ground\nT18 is ground\nX8 is free; \nX26 is free; \n=1(T15, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(=1(T44, 0), ','(!_17, fail)) | div(T43, T44, X26), SCOPE: 17, CLAUSE: 4 | div(T43, T44, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT43 is ground\nT44 is ground\nX8 is free; \nX26 is free; \n=1(T44, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(=1(T44, 0), ','(!_17, fail)), SCOPE: 18, CLAUSE: 6 | div(T43, T44, X26), SCOPE: 17, CLAUSE: 4 | div(T43, T44, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT43 is ground\nT44 is ground\nX8 is free; \nX26 is free; \n=1(T44, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: div(T43, T44, X26), SCOPE: 17, CLAUSE: 4 | div(T43, T44, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT43 is ground\nT44 is ground\nX8 is free; \nX26 is free; \nX103 is free; \n=1(T44, 0) !=? =1(X8, X8)\n=1(T44, 0) !=? =1(X103, X103)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(=1(T45, 0), ','(!_17, =1(X110, 0))) | div(T45, T46, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT45 is ground\nT46 is ground\nX8 is free; \nX26 is free; \nX103 is free; \nX110 is free; \n=1(T46, 0) !=? =1(X8, X8)\n=1(T46, 0) !=? =1(X103, X103)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(=1(T45, 0), ','(!_17, =1(X110, 0))), SCOPE: 19, CLAUSE: 6 | div(T45, T46, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT45 is ground\nT46 is ground\nX8 is free; \nX26 is free; \nX103 is free; \nX110 is free; \n=1(T46, 0) !=? =1(X8, X8)\n=1(T46, 0) !=? =1(X103, X103)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(!_17, =1(X110, 0)) | div(0, T46, X26), SCOPE: 17, CLAUSE: 5 | ?_17\n\nT46 is ground\nX8 is free; \nX26 is free; \nX103 is free; \nX110 is free; \n=1(T46, 0) !=? =1(X8, X8)\n=1(T46, 0) !=? =1(X103, X103)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: div(T45, T46, X26), SCOPE: 17, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX8 is free; \nX26 is free; \nX112 is free; \n=1(T46, 0) !=? =1(X8, X8)\n=1(T45, 0) !=? =1(X112, X112)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: =1(X110, 0)\n\nX8 is free\nX103 is free; \nX110 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: =1(X110, 0), SCOPE: 20, CLAUSE: 6\n\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(minus(T48, T49, X122), ','(div(X122, T49, X123), =1(X124, s(X123))))\n\nT48 is ground\nT49 is ground\nX8 is free; \nX112 is free; \nX124 is free; \nX122 is free; \nX123 is free; \n=1(T49, 0) !=? =1(X8, X8)\n=1(T48, 0) !=? =1(X112, X112)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: minus(T48, T49, X122)\n\nT48 is ground\nT49 is ground\nX8 is free; \nX112 is free; \nX122 is free; \n=1(T49, 0) !=? =1(X8, X8)\n=1(T48, 0) !=? =1(X112, X112)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(div(T50, T49, X123), =1(X124, s(X123)))\n\nT49 is ground\nT50 is ground\nX8 is free; \nX112 is free; \nX124 is free; \nX123 is free; \n=1(T49, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: div(T50, T49, X123)\n\nT49 is ground\nT50 is ground\nX8 is free; \nX123 is free; \n=1(T49, 0) !=? =1(X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: =1(X124, s(T51))\n\nX8 is free\nX124 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: =1(X124, s(T51)), SCOPE: 21, CLAUSE: 6\n\nX124 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: =1(T17, s(T42)), SCOPE: 22, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\nX129 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 85 [arrowhead = none ];
85 -> 2;

86 [label="EVAL with clause\ndiv(X4, X5, X6) :- ','(=1(X5, 0), ','(!, fail)).\nand substitution\nT1 -> T4\nX4 -> T4\nT2 -> T5\nX5 -> T5\nT3 -> T6\nX6 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 86 [arrowhead = none ];
86 -> 3;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 87 [arrowhead = none ];
87 -> 4;

88 [label="EVAL with clause\n=1(X8, X8).\nand substitution\nT5 -> 0\nX8 -> 0\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 88 [arrowhead = none ];
88 -> 5;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 89 [arrowhead = none ];
89 -> 6;

90 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 90 [arrowhead = none ];
90 -> 7;

91 [label="EVAL with clause\ndiv(X12, X13, X14) :- ','(=1(X12, 0), ','(!, =1(X14, 0))).\nand substitution\nT4 -> T8\nX12 -> T8\nT5 -> T9\nX13 -> T9\nT3 -> T11\nX14 -> T11\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 91 [arrowhead = none ];
91 -> 9;

92 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 92 [arrowhead = none ];
92 -> 8;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 93 [arrowhead = none ];
93 -> 10;

94 [label="EVAL with clause\n=1(X16, X16).\nand substitution\nT8 -> 0\nX16 -> 0\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 94 [arrowhead = none ];
94 -> 11;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 95 [arrowhead = none ];
95 -> 12;

96 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 96 [arrowhead = none ];
96 -> 13;

97 [label="EVAL with clause\ndiv(X22, X23, X24) :- ','(minus(X22, X23, X25), ','(div(X25, X23, X26), =1(X24, s(X26)))).\nand substitution\nT8 -> T14\nX22 -> T14\nT9 -> T15\nX23 -> T15\nT3 -> T17\nX24 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 97 [arrowhead = none ];
97 -> 18;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 98 [arrowhead = none ];
98 -> 14;

99 [label="EVAL with clause\n=1(X18, X18).\nand substitution\nT11 -> 0\nX18 -> 0\nT13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 99 [arrowhead = none ];
99 -> 15;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 100 [arrowhead = none ];
100 -> 16;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 101 [arrowhead = none ];
101 -> 17;

102 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 102 [arrowhead = none ];
102 -> 19;

103 [label="SPLIT 2\nnew knowledge:\nT18 is ground\nreplacements:\nX25 -> T18", fontsize=14, style = filled, fillcolor = violet];
18 -> 103 [arrowhead = none ];
103 -> 20;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 104 [arrowhead = none ];
104 -> 21;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 105 [arrowhead = none ];
105 -> 59;

106 [label="SPLIT 2\nreplacements:\nX26 -> T42", fontsize=14, style = filled, fillcolor = violet];
20 -> 106 [arrowhead = none ];
106 -> 60;

107 [label="EVAL with clause\nminus(X30, X31, X32) :- ','(=1(X30, 0), ','(!, =1(X32, 0))).\nand substitution\nT14 -> T19\nX30 -> T19\nT15 -> T20\nX31 -> T20\nX25 -> X33\nX32 -> X33", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 107 [arrowhead = none ];
107 -> 22;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 108 [arrowhead = none ];
108 -> 23;

109 [label="BACKTRACK\nfor clause: =1(X, X)\nwith clash: (=1(T19, 0), =1(X16, X16))", fontsize=14, style = filled, fillcolor = white];
23 -> 109 [arrowhead = none ];
109 -> 24;

110 [label="EVAL with clause\nminus(X39, X40, X41) :- ','(=1(X40, 0), ','(!, =1(X41, X39))).\nand substitution\nT19 -> T21\nX39 -> T21\nT20 -> T22\nX40 -> T22\nX25 -> X42\nX41 -> X42", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 110 [arrowhead = none ];
110 -> 25;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 111 [arrowhead = none ];
111 -> 26;

112 [label="BACKTRACK\nfor clause: =1(X, X)\nwith clash: (=1(T22, 0), =1(X8, X8))", fontsize=14, style = filled, fillcolor = white];
26 -> 112 [arrowhead = none ];
112 -> 27;

113 [label="EVAL with clause\nminus(X48, X49, X50) :- ','(=1(X48, s(X51)), ','(=1(X49, s(X52)), minus(X51, X52, X50))).\nand substitution\nT21 -> T23\nX48 -> T23\nT22 -> T24\nX49 -> T24\nX25 -> X53\nX50 -> X53", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 113 [arrowhead = none ];
113 -> 28;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 114 [arrowhead = none ];
114 -> 29;

115 [label="EVAL with clause\n=1(X55, X55).\nand substitution\nT23 -> s(T26)\nX55 -> s(T26)\nX51 -> T26\nT25 -> s(T26)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 115 [arrowhead = none ];
115 -> 30;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 116 [arrowhead = none ];
116 -> 31;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 117 [arrowhead = none ];
117 -> 32;

118 [label="EVAL with clause\n=1(X57, X57).\nand substitution\nT24 -> s(T28)\nX57 -> s(T28)\nX52 -> T28\nT27 -> s(T28)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 118 [arrowhead = none ];
118 -> 33;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 119 [arrowhead = none ];
119 -> 34;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 120 [arrowhead = none ];
120 -> 35;

121 [label="EVAL with clause\nminus(X61, X62, X63) :- ','(=1(X61, 0), ','(!, =1(X63, 0))).\nand substitution\nT26 -> T29\nX61 -> T29\nT28 -> T30\nX62 -> T30\nX53 -> X64\nX63 -> X64", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 121 [arrowhead = none ];
121 -> 36;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 122 [arrowhead = none ];
122 -> 37;

123 [label="EVAL with clause\n=1(X66, X66).\nand substitution\nT29 -> 0\nX66 -> 0\nT31 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 123 [arrowhead = none ];
123 -> 38;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 124 [arrowhead = none ];
124 -> 39;

125 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 125 [arrowhead = none ];
125 -> 40;

126 [label="EVAL with clause\nminus(X73, X74, X75) :- ','(=1(X74, 0), ','(!, =1(X75, X73))).\nand substitution\nT29 -> T32\nX73 -> T32\nT30 -> T33\nX74 -> T33\nX53 -> X76\nX75 -> X76", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 126 [arrowhead = none ];
126 -> 44;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 127 [arrowhead = none ];
127 -> 41;

128 [label="EVAL with clause\n=1(X68, X68).\nand substitution\nX64 -> 0\nX68 -> 0\nX69 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 128 [arrowhead = none ];
128 -> 42;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 129 [arrowhead = none ];
129 -> 43;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 130 [arrowhead = none ];
130 -> 45;

131 [label="EVAL with clause\n=1(X78, X78).\nand substitution\nT33 -> 0\nX78 -> 0\nT34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 131 [arrowhead = none ];
131 -> 46;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 132 [arrowhead = none ];
132 -> 47;

133 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 133 [arrowhead = none ];
133 -> 48;

134 [label="EVAL with clause\nminus(X85, X86, X87) :- ','(=1(X85, s(X88)), ','(=1(X86, s(X89)), minus(X88, X89, X87))).\nand substitution\nT32 -> T36\nX85 -> T36\nT33 -> T37\nX86 -> T37\nX53 -> X90\nX87 -> X90", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 134 [arrowhead = none ];
134 -> 52;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 135 [arrowhead = none ];
135 -> 49;

136 [label="EVAL with clause\n=1(X80, X80).\nand substitution\nX76 -> T35\nX80 -> T35\nT32 -> T35\nX81 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 136 [arrowhead = none ];
136 -> 50;

137 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 137 [arrowhead = none ];
137 -> 51;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 138 [arrowhead = none ];
138 -> 53;

139 [label="EVAL with clause\n=1(X92, X92).\nand substitution\nT36 -> s(T39)\nX92 -> s(T39)\nX88 -> T39\nT38 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 139 [arrowhead = none ];
139 -> 54;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 140 [arrowhead = none ];
140 -> 55;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 141 [arrowhead = none ];
141 -> 56;

142 [label="EVAL with clause\n=1(X94, X94).\nand substitution\nT37 -> s(T41)\nX94 -> s(T41)\nX89 -> T41\nT40 -> s(T41)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 142 [arrowhead = none ];
142 -> 57;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 143 [arrowhead = none ];
143 -> 58;

144 [label="INSTANCE with matching:\nT26 -> T39\nT28 -> T41\nX53 -> X90\nX8 -> X78\nX44 -> X90", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 144 [arrowhead = none , style=dashed];
144 -> 33[style=dashed];

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 145 [arrowhead = none ];
145 -> 61;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 146 [arrowhead = none ];
146 -> 81;

147 [label="EVAL with clause\ndiv(X98, X99, X100) :- ','(=1(X99, 0), ','(!, fail)).\nand substitution\nT18 -> T43\nX98 -> T43\nT15 -> T44\nX99 -> T44\nX26 -> X101\nX100 -> X101", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 147 [arrowhead = none ];
147 -> 62;

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 148 [arrowhead = none ];
148 -> 63;

149 [label="BACKTRACK\nfor clause: =1(X, X)\nwith clash: (=1(T44, 0), =1(X8, X8))", fontsize=14, style = filled, fillcolor = white];
63 -> 149 [arrowhead = none ];
149 -> 64;

150 [label="EVAL with clause\ndiv(X107, X108, X109) :- ','(=1(X107, 0), ','(!, =1(X109, 0))).\nand substitution\nT43 -> T45\nX107 -> T45\nT44 -> T46\nX108 -> T46\nX26 -> X110\nX109 -> X110", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 150 [arrowhead = none ];
150 -> 65;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 151 [arrowhead = none ];
151 -> 66;

152 [label="EVAL with clause\n=1(X112, X112).\nand substitution\nT45 -> 0\nX112 -> 0\nT47 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 152 [arrowhead = none ];
152 -> 67;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 153 [arrowhead = none ];
153 -> 68;

154 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 154 [arrowhead = none ];
154 -> 69;

155 [label="EVAL with clause\ndiv(X119, X120, X121) :- ','(minus(X119, X120, X122), ','(div(X122, X120, X123), =1(X121, s(X123)))).\nand substitution\nT45 -> T48\nX119 -> T48\nT46 -> T49\nX120 -> T49\nX26 -> X124\nX121 -> X124", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 155 [arrowhead = none ];
155 -> 73;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 156 [arrowhead = none ];
156 -> 70;

157 [label="EVAL with clause\n=1(X114, X114).\nand substitution\nX110 -> 0\nX114 -> 0\nX115 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 157 [arrowhead = none ];
157 -> 71;

158 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 158 [arrowhead = none ];
158 -> 72;

159 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
73 -> 159 [arrowhead = none ];
159 -> 74;

160 [label="SPLIT 2\nnew knowledge:\nT50 is ground\nreplacements:\nX122 -> T50", fontsize=14, style = filled, fillcolor = violet];
73 -> 160 [arrowhead = none ];
160 -> 75;

161 [label="INSTANCE with matching:\nT14 -> T48\nT15 -> T49\nX25 -> X122\nX16 -> X112", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 161 [arrowhead = none , style=dashed];
161 -> 19[style=dashed];

162 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
75 -> 162 [arrowhead = none ];
162 -> 76;

163 [label="SPLIT 2\nreplacements:\nX123 -> T51", fontsize=14, style = filled, fillcolor = violet];
75 -> 163 [arrowhead = none ];
163 -> 77;

164 [label="INSTANCE with matching:\nT18 -> T50\nT15 -> T49\nX26 -> X123", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 164 [arrowhead = none , style=dashed];
164 -> 59[style=dashed];

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 165 [arrowhead = none ];
165 -> 78;

166 [label="EVAL with clause\n=1(X126, X126).\nand substitution\nX124 -> s(T52)\nX126 -> s(T52)\nT51 -> T52\nX127 -> s(T52)", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 166 [arrowhead = none ];
166 -> 79;

167 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 167 [arrowhead = none ];
167 -> 80;

168 [label="EVAL with clause\n=1(X129, X129).\nand substitution\nT17 -> s(T54)\nX129 -> s(T54)\nT42 -> T54\nT53 -> s(T54)", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 168 [arrowhead = none ];
168 -> 82;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 169 [arrowhead = none ];
169 -> 83;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 170 [arrowhead = none ];
170 -> 84;

}

(4) Obligation:

Clauses:

minus33(0, T30, 0).
minus33(T35, 0, T35).
minus33(s(T39), s(T41), X90) :- minus33(T39, T41, X90).
minus19(s(0), s(T30), 0).
minus19(s(T35), s(0), T35).
minus19(s(s(T39)), s(s(T41)), X90) :- minus33(T39, T41, X90).
div59(0, T46, 0).
div59(T48, T49, X124) :- minus19(T48, T49, X122).
div59(T48, T49, X124) :- ','(minus19(T48, T49, T50), div59(T50, T49, X123)).
div59(T48, T49, s(T52)) :- ','(minus19(T48, T49, T50), div59(T50, T49, T52)).
div1(0, T9, 0).
div1(T14, T15, T17) :- minus19(T14, T15, X25).
div1(T14, T15, T17) :- ','(minus19(T14, T15, T18), div59(T18, T15, X26)).
div1(T14, T15, s(T54)) :- ','(minus19(T14, T15, T18), div59(T18, T15, T54)).

Queries:

div1(g,g,a).

(5) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
div1_in: (b,b,f)
minus19_in: (b,b,f)
minus33_in: (b,b,f)
div59_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

div1_in_gga(0, T9, 0) → div1_out_gga(0, T9, 0)
div1_in_gga(T14, T15, T17) → U8_gga(T14, T15, T17, minus19_in_gga(T14, T15, X25))
minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
U8_gga(T14, T15, T17, minus19_out_gga(T14, T15, X25)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, T17) → U9_gga(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_gga(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_gga(T14, T15, T17, div59_in_gga(T18, T15, X26))
div59_in_gga(0, T46, 0) → div59_out_gga(0, T46, 0)
div59_in_gga(T48, T49, X124) → U3_gga(T48, T49, X124, minus19_in_gga(T48, T49, X122))
U3_gga(T48, T49, X124, minus19_out_gga(T48, T49, X122)) → div59_out_gga(T48, T49, X124)
div59_in_gga(T48, T49, X124) → U4_gga(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_gga(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_gga(T48, T49, X124, div59_in_gga(T50, T49, X123))
div59_in_gga(T48, T49, s(T52)) → U6_gga(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_gga(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_gga(T48, T49, T52, div59_in_gga(T50, T49, T52))
U7_gga(T48, T49, T52, div59_out_gga(T50, T49, T52)) → div59_out_gga(T48, T49, s(T52))
U5_gga(T48, T49, X124, div59_out_gga(T50, T49, X123)) → div59_out_gga(T48, T49, X124)
U10_gga(T14, T15, T17, div59_out_gga(T18, T15, X26)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, s(T54)) → U11_gga(T14, T15, T54, minus19_in_gga(T14, T15, T18))
U11_gga(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_gga(T14, T15, T54, div59_in_gga(T18, T15, T54))
U12_gga(T14, T15, T54, div59_out_gga(T18, T15, T54)) → div1_out_gga(T14, T15, s(T54))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

div1_in_gga(0, T9, 0) → div1_out_gga(0, T9, 0)
div1_in_gga(T14, T15, T17) → U8_gga(T14, T15, T17, minus19_in_gga(T14, T15, X25))
minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
U8_gga(T14, T15, T17, minus19_out_gga(T14, T15, X25)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, T17) → U9_gga(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_gga(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_gga(T14, T15, T17, div59_in_gga(T18, T15, X26))
div59_in_gga(0, T46, 0) → div59_out_gga(0, T46, 0)
div59_in_gga(T48, T49, X124) → U3_gga(T48, T49, X124, minus19_in_gga(T48, T49, X122))
U3_gga(T48, T49, X124, minus19_out_gga(T48, T49, X122)) → div59_out_gga(T48, T49, X124)
div59_in_gga(T48, T49, X124) → U4_gga(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_gga(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_gga(T48, T49, X124, div59_in_gga(T50, T49, X123))
div59_in_gga(T48, T49, s(T52)) → U6_gga(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_gga(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_gga(T48, T49, T52, div59_in_gga(T50, T49, T52))
U7_gga(T48, T49, T52, div59_out_gga(T50, T49, T52)) → div59_out_gga(T48, T49, s(T52))
U5_gga(T48, T49, X124, div59_out_gga(T50, T49, X123)) → div59_out_gga(T48, T49, X124)
U10_gga(T14, T15, T17, div59_out_gga(T18, T15, X26)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, s(T54)) → U11_gga(T14, T15, T54, minus19_in_gga(T14, T15, T18))
U11_gga(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_gga(T14, T15, T54, div59_in_gga(T18, T15, T54))
U12_gga(T14, T15, T54, div59_out_gga(T18, T15, T54)) → div1_out_gga(T14, T15, s(T54))

(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:

DIV1_IN_GGA(T14, T15, T17) → U8_GGA(T14, T15, T17, minus19_in_gga(T14, T15, X25))
DIV1_IN_GGA(T14, T15, T17) → MINUS19_IN_GGA(T14, T15, X25)
MINUS19_IN_GGA(s(s(T39)), s(s(T41)), X90) → U2_GGA(T39, T41, X90, minus33_in_gga(T39, T41, X90))
MINUS19_IN_GGA(s(s(T39)), s(s(T41)), X90) → MINUS33_IN_GGA(T39, T41, X90)
MINUS33_IN_GGA(s(T39), s(T41), X90) → U1_GGA(T39, T41, X90, minus33_in_gga(T39, T41, X90))
MINUS33_IN_GGA(s(T39), s(T41), X90) → MINUS33_IN_GGA(T39, T41, X90)
DIV1_IN_GGA(T14, T15, T17) → U9_GGA(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_GGA(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_GGA(T14, T15, T17, div59_in_gga(T18, T15, X26))
U9_GGA(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → DIV59_IN_GGA(T18, T15, X26)
DIV59_IN_GGA(T48, T49, X124) → U3_GGA(T48, T49, X124, minus19_in_gga(T48, T49, X122))
DIV59_IN_GGA(T48, T49, X124) → MINUS19_IN_GGA(T48, T49, X122)
DIV59_IN_GGA(T48, T49, X124) → U4_GGA(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_GGA(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_GGA(T48, T49, X124, div59_in_gga(T50, T49, X123))
U4_GGA(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, X123)
DIV59_IN_GGA(T48, T49, s(T52)) → U6_GGA(T48, T49, T52, minus19_in_gga(T48, T49, T50))
DIV59_IN_GGA(T48, T49, s(T52)) → MINUS19_IN_GGA(T48, T49, T50)
U6_GGA(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_GGA(T48, T49, T52, div59_in_gga(T50, T49, T52))
U6_GGA(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, T52)
DIV1_IN_GGA(T14, T15, s(T54)) → U11_GGA(T14, T15, T54, minus19_in_gga(T14, T15, T18))
DIV1_IN_GGA(T14, T15, s(T54)) → MINUS19_IN_GGA(T14, T15, T18)
U11_GGA(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_GGA(T14, T15, T54, div59_in_gga(T18, T15, T54))
U11_GGA(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → DIV59_IN_GGA(T18, T15, T54)

The TRS R consists of the following rules:

div1_in_gga(0, T9, 0) → div1_out_gga(0, T9, 0)
div1_in_gga(T14, T15, T17) → U8_gga(T14, T15, T17, minus19_in_gga(T14, T15, X25))
minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
U8_gga(T14, T15, T17, minus19_out_gga(T14, T15, X25)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, T17) → U9_gga(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_gga(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_gga(T14, T15, T17, div59_in_gga(T18, T15, X26))
div59_in_gga(0, T46, 0) → div59_out_gga(0, T46, 0)
div59_in_gga(T48, T49, X124) → U3_gga(T48, T49, X124, minus19_in_gga(T48, T49, X122))
U3_gga(T48, T49, X124, minus19_out_gga(T48, T49, X122)) → div59_out_gga(T48, T49, X124)
div59_in_gga(T48, T49, X124) → U4_gga(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_gga(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_gga(T48, T49, X124, div59_in_gga(T50, T49, X123))
div59_in_gga(T48, T49, s(T52)) → U6_gga(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_gga(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_gga(T48, T49, T52, div59_in_gga(T50, T49, T52))
U7_gga(T48, T49, T52, div59_out_gga(T50, T49, T52)) → div59_out_gga(T48, T49, s(T52))
U5_gga(T48, T49, X124, div59_out_gga(T50, T49, X123)) → div59_out_gga(T48, T49, X124)
U10_gga(T14, T15, T17, div59_out_gga(T18, T15, X26)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, s(T54)) → U11_gga(T14, T15, T54, minus19_in_gga(T14, T15, T18))
U11_gga(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_gga(T14, T15, T54, div59_in_gga(T18, T15, T54))
U12_gga(T14, T15, T54, div59_out_gga(T18, T15, T54)) → div1_out_gga(T14, T15, s(T54))

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3) = div1_in_gga(x1, x2)
0 = 0
div1_out_gga(x1, x2, x3) = div1_out_gga
U8_gga(x1, x2, x3, x4) = U8_gga(x4)
minus19_in_gga(x1, x2, x3) = minus19_in_gga(x1, x2)
s(x1) = s(x1)
minus19_out_gga(x1, x2, x3) = minus19_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
minus33_in_gga(x1, x2, x3) = minus33_in_gga(x1, x2)
minus33_out_gga(x1, x2, x3) = minus33_out_gga(x3)
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)
div59_in_gga(x1, x2, x3) = div59_in_gga(x1, x2)
div59_out_gga(x1, x2, x3) = div59_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)
DIV1_IN_GGA(x1, x2, x3) = DIV1_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
MINUS19_IN_GGA(x1, x2, x3) = MINUS19_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
MINUS33_IN_GGA(x1, x2, x3) = MINUS33_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)
DIV59_IN_GGA(x1, x2, x3) = DIV59_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:

DIV1_IN_GGA(T14, T15, T17) → U8_GGA(T14, T15, T17, minus19_in_gga(T14, T15, X25))
DIV1_IN_GGA(T14, T15, T17) → MINUS19_IN_GGA(T14, T15, X25)
MINUS19_IN_GGA(s(s(T39)), s(s(T41)), X90) → U2_GGA(T39, T41, X90, minus33_in_gga(T39, T41, X90))
MINUS19_IN_GGA(s(s(T39)), s(s(T41)), X90) → MINUS33_IN_GGA(T39, T41, X90)
MINUS33_IN_GGA(s(T39), s(T41), X90) → U1_GGA(T39, T41, X90, minus33_in_gga(T39, T41, X90))
MINUS33_IN_GGA(s(T39), s(T41), X90) → MINUS33_IN_GGA(T39, T41, X90)
DIV1_IN_GGA(T14, T15, T17) → U9_GGA(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_GGA(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_GGA(T14, T15, T17, div59_in_gga(T18, T15, X26))
U9_GGA(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → DIV59_IN_GGA(T18, T15, X26)
DIV59_IN_GGA(T48, T49, X124) → U3_GGA(T48, T49, X124, minus19_in_gga(T48, T49, X122))
DIV59_IN_GGA(T48, T49, X124) → MINUS19_IN_GGA(T48, T49, X122)
DIV59_IN_GGA(T48, T49, X124) → U4_GGA(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_GGA(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_GGA(T48, T49, X124, div59_in_gga(T50, T49, X123))
U4_GGA(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, X123)
DIV59_IN_GGA(T48, T49, s(T52)) → U6_GGA(T48, T49, T52, minus19_in_gga(T48, T49, T50))
DIV59_IN_GGA(T48, T49, s(T52)) → MINUS19_IN_GGA(T48, T49, T50)
U6_GGA(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_GGA(T48, T49, T52, div59_in_gga(T50, T49, T52))
U6_GGA(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, T52)
DIV1_IN_GGA(T14, T15, s(T54)) → U11_GGA(T14, T15, T54, minus19_in_gga(T14, T15, T18))
DIV1_IN_GGA(T14, T15, s(T54)) → MINUS19_IN_GGA(T14, T15, T18)
U11_GGA(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_GGA(T14, T15, T54, div59_in_gga(T18, T15, T54))
U11_GGA(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → DIV59_IN_GGA(T18, T15, T54)

The TRS R consists of the following rules:

div1_in_gga(0, T9, 0) → div1_out_gga(0, T9, 0)
div1_in_gga(T14, T15, T17) → U8_gga(T14, T15, T17, minus19_in_gga(T14, T15, X25))
minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
U8_gga(T14, T15, T17, minus19_out_gga(T14, T15, X25)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, T17) → U9_gga(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_gga(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_gga(T14, T15, T17, div59_in_gga(T18, T15, X26))
div59_in_gga(0, T46, 0) → div59_out_gga(0, T46, 0)
div59_in_gga(T48, T49, X124) → U3_gga(T48, T49, X124, minus19_in_gga(T48, T49, X122))
U3_gga(T48, T49, X124, minus19_out_gga(T48, T49, X122)) → div59_out_gga(T48, T49, X124)
div59_in_gga(T48, T49, X124) → U4_gga(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_gga(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_gga(T48, T49, X124, div59_in_gga(T50, T49, X123))
div59_in_gga(T48, T49, s(T52)) → U6_gga(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_gga(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_gga(T48, T49, T52, div59_in_gga(T50, T49, T52))
U7_gga(T48, T49, T52, div59_out_gga(T50, T49, T52)) → div59_out_gga(T48, T49, s(T52))
U5_gga(T48, T49, X124, div59_out_gga(T50, T49, X123)) → div59_out_gga(T48, T49, X124)
U10_gga(T14, T15, T17, div59_out_gga(T18, T15, X26)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, s(T54)) → U11_gga(T14, T15, T54, minus19_in_gga(T14, T15, T18))
U11_gga(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_gga(T14, T15, T54, div59_in_gga(T18, T15, T54))
U12_gga(T14, T15, T54, div59_out_gga(T18, T15, T54)) → div1_out_gga(T14, T15, s(T54))

(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:

MINUS33_IN_GGA(s(T39), s(T41), X90) → MINUS33_IN_GGA(T39, T41, X90)

The TRS R consists of the following rules:

div1_in_gga(0, T9, 0) → div1_out_gga(0, T9, 0)
div1_in_gga(T14, T15, T17) → U8_gga(T14, T15, T17, minus19_in_gga(T14, T15, X25))
minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
U8_gga(T14, T15, T17, minus19_out_gga(T14, T15, X25)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, T17) → U9_gga(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_gga(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_gga(T14, T15, T17, div59_in_gga(T18, T15, X26))
div59_in_gga(0, T46, 0) → div59_out_gga(0, T46, 0)
div59_in_gga(T48, T49, X124) → U3_gga(T48, T49, X124, minus19_in_gga(T48, T49, X122))
U3_gga(T48, T49, X124, minus19_out_gga(T48, T49, X122)) → div59_out_gga(T48, T49, X124)
div59_in_gga(T48, T49, X124) → U4_gga(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_gga(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_gga(T48, T49, X124, div59_in_gga(T50, T49, X123))
div59_in_gga(T48, T49, s(T52)) → U6_gga(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_gga(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_gga(T48, T49, T52, div59_in_gga(T50, T49, T52))
U7_gga(T48, T49, T52, div59_out_gga(T50, T49, T52)) → div59_out_gga(T48, T49, s(T52))
U5_gga(T48, T49, X124, div59_out_gga(T50, T49, X123)) → div59_out_gga(T48, T49, X124)
U10_gga(T14, T15, T17, div59_out_gga(T18, T15, X26)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, s(T54)) → U11_gga(T14, T15, T54, minus19_in_gga(T14, T15, T18))
U11_gga(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_gga(T14, T15, T54, div59_in_gga(T18, T15, T54))
U12_gga(T14, T15, T54, div59_out_gga(T18, T15, T54)) → div1_out_gga(T14, T15, s(T54))

(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:

MINUS33_IN_GGA(s(T39), s(T41), X90) → MINUS33_IN_GGA(T39, T41, X90)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
MINUS33_IN_GGA(x1, x2, x3) = MINUS33_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:

MINUS33_IN_GGA(s(T39), s(T41)) → MINUS33_IN_GGA(T39, T41)

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:

MINUS33_IN_GGA(s(T39), s(T41)) → MINUS33_IN_GGA(T39, T41)
The graph contains the following edges 1 > 1, 2 > 2

(17) TRUE

(18) Obligation:

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

DIV59_IN_GGA(T48, T49, X124) → U4_GGA(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_GGA(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, X123)
DIV59_IN_GGA(T48, T49, s(T52)) → U6_GGA(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_GGA(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, T52)

The TRS R consists of the following rules:

div1_in_gga(0, T9, 0) → div1_out_gga(0, T9, 0)
div1_in_gga(T14, T15, T17) → U8_gga(T14, T15, T17, minus19_in_gga(T14, T15, X25))
minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
U8_gga(T14, T15, T17, minus19_out_gga(T14, T15, X25)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, T17) → U9_gga(T14, T15, T17, minus19_in_gga(T14, T15, T18))
U9_gga(T14, T15, T17, minus19_out_gga(T14, T15, T18)) → U10_gga(T14, T15, T17, div59_in_gga(T18, T15, X26))
div59_in_gga(0, T46, 0) → div59_out_gga(0, T46, 0)
div59_in_gga(T48, T49, X124) → U3_gga(T48, T49, X124, minus19_in_gga(T48, T49, X122))
U3_gga(T48, T49, X124, minus19_out_gga(T48, T49, X122)) → div59_out_gga(T48, T49, X124)
div59_in_gga(T48, T49, X124) → U4_gga(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_gga(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → U5_gga(T48, T49, X124, div59_in_gga(T50, T49, X123))
div59_in_gga(T48, T49, s(T52)) → U6_gga(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_gga(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → U7_gga(T48, T49, T52, div59_in_gga(T50, T49, T52))
U7_gga(T48, T49, T52, div59_out_gga(T50, T49, T52)) → div59_out_gga(T48, T49, s(T52))
U5_gga(T48, T49, X124, div59_out_gga(T50, T49, X123)) → div59_out_gga(T48, T49, X124)
U10_gga(T14, T15, T17, div59_out_gga(T18, T15, X26)) → div1_out_gga(T14, T15, T17)
div1_in_gga(T14, T15, s(T54)) → U11_gga(T14, T15, T54, minus19_in_gga(T14, T15, T18))
U11_gga(T14, T15, T54, minus19_out_gga(T14, T15, T18)) → U12_gga(T14, T15, T54, div59_in_gga(T18, T15, T54))
U12_gga(T14, T15, T54, div59_out_gga(T18, T15, T54)) → div1_out_gga(T14, T15, s(T54))

(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:

DIV59_IN_GGA(T48, T49, X124) → U4_GGA(T48, T49, X124, minus19_in_gga(T48, T49, T50))
U4_GGA(T48, T49, X124, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, X123)
DIV59_IN_GGA(T48, T49, s(T52)) → U6_GGA(T48, T49, T52, minus19_in_gga(T48, T49, T50))
U6_GGA(T48, T49, T52, minus19_out_gga(T48, T49, T50)) → DIV59_IN_GGA(T50, T49, T52)

The TRS R consists of the following rules:

minus19_in_gga(s(0), s(T30), 0) → minus19_out_gga(s(0), s(T30), 0)
minus19_in_gga(s(T35), s(0), T35) → minus19_out_gga(s(T35), s(0), T35)
minus19_in_gga(s(s(T39)), s(s(T41)), X90) → U2_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U2_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus19_out_gga(s(s(T39)), s(s(T41)), X90)
minus33_in_gga(0, T30, 0) → minus33_out_gga(0, T30, 0)
minus33_in_gga(T35, 0, T35) → minus33_out_gga(T35, 0, T35)
minus33_in_gga(s(T39), s(T41), X90) → U1_gga(T39, T41, X90, minus33_in_gga(T39, T41, X90))
U1_gga(T39, T41, X90, minus33_out_gga(T39, T41, X90)) → minus33_out_gga(s(T39), s(T41), X90)

The argument filtering Pi contains the following mapping:
0 = 0
minus19_in_gga(x1, x2, x3) = minus19_in_gga(x1, x2)
s(x1) = s(x1)
minus19_out_gga(x1, x2, x3) = minus19_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
minus33_in_gga(x1, x2, x3) = minus33_in_gga(x1, x2)
minus33_out_gga(x1, x2, x3) = minus33_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
DIV59_IN_GGA(x1, x2, x3) = DIV59_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:

DIV59_IN_GGA(T48, T49) → U4_GGA(T49, minus19_in_gga(T48, T49))
U4_GGA(T49, minus19_out_gga(T50)) → DIV59_IN_GGA(T50, T49)
DIV59_IN_GGA(T48, T49) → U6_GGA(T49, minus19_in_gga(T48, T49))
U6_GGA(T49, minus19_out_gga(T50)) → DIV59_IN_GGA(T50, T49)

The TRS R consists of the following rules:

minus19_in_gga(s(0), s(T30)) → minus19_out_gga(0)
minus19_in_gga(s(T35), s(0)) → minus19_out_gga(T35)
minus19_in_gga(s(s(T39)), s(s(T41))) → U2_gga(minus33_in_gga(T39, T41))
U2_gga(minus33_out_gga(X90)) → minus19_out_gga(X90)
minus33_in_gga(0, T30) → minus33_out_gga(0)
minus33_in_gga(T35, 0) → minus33_out_gga(T35)
minus33_in_gga(s(T39), s(T41)) → U1_gga(minus33_in_gga(T39, T41))
U1_gga(minus33_out_gga(X90)) → minus33_out_gga(X90)

The set Q consists of the following terms:

minus19_in_gga(x0, x1)
U2_gga(x0)
minus33_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].

The following pairs can be oriented strictly and are deleted.

U4_GGA(T49, minus19_out_gga(T50)) → DIV59_IN_GGA(T50, T49)
U6_GGA(T49, minus19_out_gga(T50)) → DIV59_IN_GGA(T50, T49)

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

POL(0) = 0
POL(DIV59_IN_GGA(x₁, x₂)) = x₁
POL(U1_gga(x₁)) = 1 + x₁
POL(U2_gga(x₁)) = 1 + x₁
POL(U4_GGA(x₁, x₂)) = x₂
POL(U6_GGA(x₁, x₂)) = x₂
POL(minus19_in_gga(x₁, x₂)) = x₁
POL(minus19_out_gga(x₁)) = 1 + x₁
POL(minus33_in_gga(x₁, x₂)) = 1 + x₁
POL(minus33_out_gga(x₁)) = x₁
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented:

minus19_in_gga(s(0), s(T30)) → minus19_out_gga(0)
minus19_in_gga(s(T35), s(0)) → minus19_out_gga(T35)
minus19_in_gga(s(s(T39)), s(s(T41))) → U2_gga(minus33_in_gga(T39, T41))
minus33_in_gga(0, T30) → minus33_out_gga(0)
minus33_in_gga(T35, 0) → minus33_out_gga(T35)
minus33_in_gga(s(T39), s(T41)) → U1_gga(minus33_in_gga(T39, T41))
U2_gga(minus33_out_gga(X90)) → minus19_out_gga(X90)
U1_gga(minus33_out_gga(X90)) → minus33_out_gga(X90)

(24) Obligation:

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

DIV59_IN_GGA(T48, T49) → U4_GGA(T49, minus19_in_gga(T48, T49))
DIV59_IN_GGA(T48, T49) → U6_GGA(T49, minus19_in_gga(T48, T49))

The TRS R consists of the following rules:

minus19_in_gga(s(0), s(T30)) → minus19_out_gga(0)
minus19_in_gga(s(T35), s(0)) → minus19_out_gga(T35)
minus19_in_gga(s(s(T39)), s(s(T41))) → U2_gga(minus33_in_gga(T39, T41))
U2_gga(minus33_out_gga(X90)) → minus19_out_gga(X90)
minus33_in_gga(0, T30) → minus33_out_gga(0)
minus33_in_gga(T35, 0) → minus33_out_gga(T35)
minus33_in_gga(s(T39), s(T41)) → U1_gga(minus33_in_gga(T39, T41))
U1_gga(minus33_out_gga(X90)) → minus33_out_gga(X90)

The set Q consists of the following terms:

minus19_in_gga(x0, x1)
U2_gga(x0)
minus33_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) PredefinedPredicateTransformerProof (SOUND 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) TRUE

(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