Left Termination proof of /tmp/tmpEssCoF/minus3.pl

Left Termination of the query pattern q_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

            ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                    ↳13 QDP

                        ↳14 QDPSizeChangeProof (⇔)

                            ↳15 TRUE

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

            ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                    ↳20 QDP

                        ↳21 QDPSizeChangeProof (⇔)

                            ↳22 TRUE

(0) Obligation:

Clauses:

q(X, Y) :- m(X, Y, Z).
m(X, 0, X).
m(0, Y, 0) :- !.
m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))).
p(0, 0).
p(s(X), X).

Queries:

q(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: m(T3, T4, X5)\n\nT3 is ground\nT4 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T3, T4, X5), SCOPE: 2, CLAUSE: 1 | m(T3, T4, X5), SCOPE: 2, CLAUSE: 2 | m(T3, T4, X5), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT3 is ground\nT4 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: m(T3, T4, X5), SCOPE: 2, CLAUSE: 1\n\nT3 is ground\nT4 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: m(T3, T4, X5), SCOPE: 2, CLAUSE: 2 | m(T3, T4, X5), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT3 is ground\nT4 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX5 is free\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: !_2 | m(0, T6, X5), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT6 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: m(T3, T4, X5), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nT4 is ground\nX5 is free; \nX10 is free; \nm(T3, T4, X5) !=? m(0, X10, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T7, X17), ','(p(T8, X18), m(X17, X18, X19)))\n\nT7 is ground\nT8 is ground\nX5 is free; \nX10 is free; \nX19 is free; \nX17 is free; \nX18 is free; \nm(T7, T8, X5) !=? m(0, X10, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T7, X17), ','(p(T8, X18), m(X17, X18, X19))), SCOPE: 3, CLAUSE: 4 | ','(p(T7, X17), ','(p(T8, X18), m(X17, X18, X19))), SCOPE: 3, CLAUSE: 5\n\nT7 is ground\nT8 is ground\nX5 is free; \nX10 is free; \nX19 is free; \nX17 is free; \nX18 is free; \nm(T7, T8, X5) !=? m(0, X10, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(T7, X17), ','(p(T8, X18), m(X17, X18, X19))), SCOPE: 3, CLAUSE: 5\n\nT7 is ground\nT8 is ground\nX5 is free; \nX10 is free; \nX19 is free; \nX17 is free; \nX18 is free; \nm(T7, T8, X5) !=? m(0, X10, 0)\np(T7, X17) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T8, X18), m(T9, X18, X19))\n\nT8 is ground\nT9 is ground\nX5 is free; \nX10 is free; \nX19 is free; \nX17 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\nX5 is free\nX10 is free; \nX17 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T8, X18), m(T9, X18, X19)), SCOPE: 4, CLAUSE: 4 | ','(p(T8, X18), m(T9, X18, X19)), SCOPE: 4, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nX19 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: m(T9, 0, X19) | ','(p(0, X18), m(T9, X18, X19)), SCOPE: 4, CLAUSE: 5\n\nT9 is ground\nX19 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T8, X18), m(T9, X18, X19)), SCOPE: 4, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nX19 is free; \nX18 is free; \np(T8, X18) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: m(T9, 0, X19)\n\nT9 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(0, X18), m(T9, X18, X19)), SCOPE: 4, CLAUSE: 5\n\nT9 is ground\nX19 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: m(T9, 0, X19), SCOPE: 5, CLAUSE: 1 | m(T9, 0, X19), SCOPE: 5, CLAUSE: 2 | m(T9, 0, X19), SCOPE: 5, CLAUSE: 3 | ?_5\n\nT9 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: m(T9, 0, X19), SCOPE: 5, CLAUSE: 1\n\nT9 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: m(T9, 0, X19), SCOPE: 5, CLAUSE: 2 | m(T9, 0, X19), SCOPE: 5, CLAUSE: 3 | ?_5\n\nT9 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: !_5 | m(0, 0, X19), SCOPE: 5, CLAUSE: 3 | ?_5\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: m(T9, 0, X19), SCOPE: 5, CLAUSE: 3\n\nT9 is ground\nX19 is free; \nX26 is free; \nm(T9, 0, X19) !=? m(0, X26, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(T11, X33), ','(p(0, X34), m(X33, X34, X35)))\n\nT11 is ground\nX19 is free; \nX26 is free; \nX35 is free; \nX33 is free; \nX34 is free; \nm(T11, 0, X19) !=? m(0, X26, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T11, X33), ','(p(0, X34), m(X33, X34, X35))), SCOPE: 6, CLAUSE: 4 | ','(p(T11, X33), ','(p(0, X34), m(X33, X34, X35))), SCOPE: 6, CLAUSE: 5\n\nT11 is ground\nX19 is free; \nX26 is free; \nX35 is free; \nX33 is free; \nX34 is free; \nm(T11, 0, X19) !=? m(0, X26, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T11, X33), ','(p(0, X34), m(X33, X34, X35))), SCOPE: 6, CLAUSE: 5\n\nT11 is ground\nX19 is free; \nX26 is free; \nX35 is free; \nX33 is free; \nX34 is free; \nm(T11, 0, X19) !=? m(0, X26, 0)\np(T11, X33) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(0, X34), m(T12, X34, X35))\n\nT12 is ground\nX19 is free; \nX26 is free; \nX35 is free; \nX33 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX19 is free\nX26 is free; \nX33 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(0, X34), m(T12, X34, X35)), SCOPE: 7, CLAUSE: 4 | ','(p(0, X34), m(T12, X34, X35)), SCOPE: 7, CLAUSE: 5\n\nT12 is ground\nX35 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: m(T12, 0, X35) | ','(p(0, X34), m(T12, X34, X35)), SCOPE: 7, CLAUSE: 5\n\nT12 is ground\nX35 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: m(T9, T13, X19)\n\nT9 is ground\nT13 is ground\nX19 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\nX18 is free\nX40 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 43 [arrowhead = none ];
43 -> 2;

44 [label="EVAL with clause\nq(X3, X4) :- m(X3, X4, X5).\nand substitution\nT1 -> T3\nX3 -> T3\nT2 -> T4\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 44 [arrowhead = none ];
44 -> 3;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 45 [arrowhead = none ];
45 -> 4;

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 46 [arrowhead = none ];
46 -> 5;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 47 [arrowhead = none ];
47 -> 6;

48 [label="EVAL with clause\nm(X8, 0, X8).\nand substitution\nT3 -> T5\nX8 -> T5\nT4 -> 0\nX5 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 48 [arrowhead = none ];
48 -> 7;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 49 [arrowhead = none ];
49 -> 8;

50 [label="EVAL with clause\nm(0, X10, 0) :- !.\nand substitution\nT3 -> 0\nT4 -> T6\nX10 -> T6\nX5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 50 [arrowhead = none ];
50 -> 10;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 51 [arrowhead = none ];
51 -> 11;

52 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 52 [arrowhead = none ];
52 -> 9;

53 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 53 [arrowhead = none ];
53 -> 12;

54 [label="EVAL with clause\nm(X14, X15, X16) :- ','(p(X14, X17), ','(p(X15, X18), m(X17, X18, X16))).\nand substitution\nT3 -> T7\nX14 -> T7\nT4 -> T8\nX15 -> T8\nX5 -> X19\nX16 -> X19", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 54 [arrowhead = none ];
54 -> 14;

55 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 55 [arrowhead = none ];
55 -> 13;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 56 [arrowhead = none ];
56 -> 15;

57 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T7, T8, X5), m(0, X10, 0))", fontsize=14, style = filled, fillcolor = white];
15 -> 57 [arrowhead = none ];
57 -> 16;

58 [label="EVAL with clause\np(s(X21), X21).\nand substitution\nX21 -> T9\nT7 -> s(T9)\nX17 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 58 [arrowhead = none ];
58 -> 17;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 59 [arrowhead = none ];
59 -> 18;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 60 [arrowhead = none ];
60 -> 19;

61 [label="EVAL with clause\np(0, 0).\nand substitution\nT8 -> 0\nX18 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 61 [arrowhead = none ];
61 -> 20;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 62 [arrowhead = none ];
62 -> 21;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 63 [arrowhead = none ];
63 -> 22;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 64 [arrowhead = none ];
64 -> 23;

65 [label="EVAL with clause\np(s(X40), X40).\nand substitution\nX40 -> T13\nT8 -> s(T13)\nX18 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 65 [arrowhead = none ];
65 -> 41;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 66 [arrowhead = none ];
66 -> 42;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 67 [arrowhead = none ];
67 -> 24;

68 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 68 [arrowhead = none ];
68 -> 40;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 70 [arrowhead = none ];
70 -> 26;

71 [label="EVAL with clause\nm(X24, 0, X24).\nand substitution\nT9 -> T10\nX24 -> T10\nX19 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 71 [arrowhead = none ];
71 -> 27;

72 [label="EVAL with clause\nm(0, X26, 0) :- !.\nand substitution\nT9 -> 0\nX26 -> 0\nX19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 72 [arrowhead = none ];
72 -> 29;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 73 [arrowhead = none ];
73 -> 30;

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 74 [arrowhead = none ];
74 -> 28;

75 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 75 [arrowhead = none ];
75 -> 31;

76 [label="EVAL with clause\nm(X30, X31, X32) :- ','(p(X30, X33), ','(p(X31, X34), m(X33, X34, X32))).\nand substitution\nT9 -> T11\nX30 -> T11\nX31 -> 0\nX19 -> X35\nX32 -> X35", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 76 [arrowhead = none ];
76 -> 33;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 77 [arrowhead = none ];
77 -> 32;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 78 [arrowhead = none ];
78 -> 34;

79 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T11, 0, X19), m(0, X26, 0))", fontsize=14, style = filled, fillcolor = white];
34 -> 79 [arrowhead = none ];
79 -> 35;

80 [label="EVAL with clause\np(s(X37), X37).\nand substitution\nX37 -> T12\nT11 -> s(T12)\nX33 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 80 [arrowhead = none ];
80 -> 36;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 81 [arrowhead = none ];
81 -> 37;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 82 [arrowhead = none ];
82 -> 38;

83 [label="EVAL with clause\np(0, 0).\nand substitution\nX34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 83 [arrowhead = none ];
83 -> 39;

84 [label="INSTANCE with matching:\nT9 -> T12\nX19 -> X35\nX18 -> X34", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 84 [arrowhead = none , style=dashed];
84 -> 20[style=dashed];

85 [label="INSTANCE with matching:\nT3 -> T9\nT4 -> T13\nX5 -> X19", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 85 [arrowhead = none , style=dashed];
85 -> 3[style=dashed];

}

(2) Obligation:

Clauses:

m20(T10, T10, X18).
m20(0, 0, X18).
m20(s(T12), X35, X18) :- m20(T12, X35, X34).
m3(T5, 0, T5).
m3(0, T6, 0).
m3(s(T10), 0, T10).
m3(s(0), 0, 0).
m3(s(s(T12)), 0, X35) :- m20(T12, X35, X34).
m3(s(T9), s(T13), X19) :- m3(T9, T13, X19).
q1(T5, 0).
q1(0, T6).
q1(s(T10), 0).
q1(s(0), 0).
q1(s(s(T12)), 0) :- m20(T12, X35, X34).
q1(s(T9), s(T13)) :- m3(T9, T13, X19).

Queries:

q1(g,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
q1_in: (b,b)
m20_in: (b,f,f)
m3_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

q1_in_gg(T5, 0) → q1_out_gg(T5, 0)
q1_in_gg(0, T6) → q1_out_gg(0, T6)
q1_in_gg(s(T10), 0) → q1_out_gg(s(T10), 0)
q1_in_gg(s(0), 0) → q1_out_gg(s(0), 0)
q1_in_gg(s(s(T12)), 0) → U4_gg(T12, m20_in_gaa(T12, X35, X34))
m20_in_gaa(T10, T10, X18) → m20_out_gaa(T10, T10, X18)
m20_in_gaa(0, 0, X18) → m20_out_gaa(0, 0, X18)
m20_in_gaa(s(T12), X35, X18) → U1_gaa(T12, X35, X18, m20_in_gaa(T12, X35, X34))
U1_gaa(T12, X35, X18, m20_out_gaa(T12, X35, X34)) → m20_out_gaa(s(T12), X35, X18)
U4_gg(T12, m20_out_gaa(T12, X35, X34)) → q1_out_gg(s(s(T12)), 0)
q1_in_gg(s(T9), s(T13)) → U5_gg(T9, T13, m3_in_gga(T9, T13, X19))
m3_in_gga(T5, 0, T5) → m3_out_gga(T5, 0, T5)
m3_in_gga(0, T6, 0) → m3_out_gga(0, T6, 0)
m3_in_gga(s(T10), 0, T10) → m3_out_gga(s(T10), 0, T10)
m3_in_gga(s(0), 0, 0) → m3_out_gga(s(0), 0, 0)
m3_in_gga(s(s(T12)), 0, X35) → U2_gga(T12, X35, m20_in_gaa(T12, X35, X34))
U2_gga(T12, X35, m20_out_gaa(T12, X35, X34)) → m3_out_gga(s(s(T12)), 0, X35)
m3_in_gga(s(T9), s(T13), X19) → U3_gga(T9, T13, X19, m3_in_gga(T9, T13, X19))
U3_gga(T9, T13, X19, m3_out_gga(T9, T13, X19)) → m3_out_gga(s(T9), s(T13), X19)
U5_gg(T9, T13, m3_out_gga(T9, T13, X19)) → q1_out_gg(s(T9), s(T13))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

q1_in_gg(T5, 0) → q1_out_gg(T5, 0)
q1_in_gg(0, T6) → q1_out_gg(0, T6)
q1_in_gg(s(T10), 0) → q1_out_gg(s(T10), 0)
q1_in_gg(s(0), 0) → q1_out_gg(s(0), 0)
q1_in_gg(s(s(T12)), 0) → U4_gg(T12, m20_in_gaa(T12, X35, X34))
m20_in_gaa(T10, T10, X18) → m20_out_gaa(T10, T10, X18)
m20_in_gaa(0, 0, X18) → m20_out_gaa(0, 0, X18)
m20_in_gaa(s(T12), X35, X18) → U1_gaa(T12, X35, X18, m20_in_gaa(T12, X35, X34))
U1_gaa(T12, X35, X18, m20_out_gaa(T12, X35, X34)) → m20_out_gaa(s(T12), X35, X18)
U4_gg(T12, m20_out_gaa(T12, X35, X34)) → q1_out_gg(s(s(T12)), 0)
q1_in_gg(s(T9), s(T13)) → U5_gg(T9, T13, m3_in_gga(T9, T13, X19))
m3_in_gga(T5, 0, T5) → m3_out_gga(T5, 0, T5)
m3_in_gga(0, T6, 0) → m3_out_gga(0, T6, 0)
m3_in_gga(s(T10), 0, T10) → m3_out_gga(s(T10), 0, T10)
m3_in_gga(s(0), 0, 0) → m3_out_gga(s(0), 0, 0)
m3_in_gga(s(s(T12)), 0, X35) → U2_gga(T12, X35, m20_in_gaa(T12, X35, X34))
U2_gga(T12, X35, m20_out_gaa(T12, X35, X34)) → m3_out_gga(s(s(T12)), 0, X35)
m3_in_gga(s(T9), s(T13), X19) → U3_gga(T9, T13, X19, m3_in_gga(T9, T13, X19))
U3_gga(T9, T13, X19, m3_out_gga(T9, T13, X19)) → m3_out_gga(s(T9), s(T13), X19)
U5_gg(T9, T13, m3_out_gga(T9, T13, X19)) → q1_out_gg(s(T9), s(T13))

(5) DependencyPairsProof (EQUIVALENT transformation)

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

Q1_IN_GG(s(s(T12)), 0) → U4_GG(T12, m20_in_gaa(T12, X35, X34))
Q1_IN_GG(s(s(T12)), 0) → M20_IN_GAA(T12, X35, X34)
M20_IN_GAA(s(T12), X35, X18) → U1_GAA(T12, X35, X18, m20_in_gaa(T12, X35, X34))
M20_IN_GAA(s(T12), X35, X18) → M20_IN_GAA(T12, X35, X34)
Q1_IN_GG(s(T9), s(T13)) → U5_GG(T9, T13, m3_in_gga(T9, T13, X19))
Q1_IN_GG(s(T9), s(T13)) → M3_IN_GGA(T9, T13, X19)
M3_IN_GGA(s(s(T12)), 0, X35) → U2_GGA(T12, X35, m20_in_gaa(T12, X35, X34))
M3_IN_GGA(s(s(T12)), 0, X35) → M20_IN_GAA(T12, X35, X34)
M3_IN_GGA(s(T9), s(T13), X19) → U3_GGA(T9, T13, X19, m3_in_gga(T9, T13, X19))
M3_IN_GGA(s(T9), s(T13), X19) → M3_IN_GGA(T9, T13, X19)

The TRS R consists of the following rules:

q1_in_gg(T5, 0) → q1_out_gg(T5, 0)
q1_in_gg(0, T6) → q1_out_gg(0, T6)
q1_in_gg(s(T10), 0) → q1_out_gg(s(T10), 0)
q1_in_gg(s(0), 0) → q1_out_gg(s(0), 0)
q1_in_gg(s(s(T12)), 0) → U4_gg(T12, m20_in_gaa(T12, X35, X34))
m20_in_gaa(T10, T10, X18) → m20_out_gaa(T10, T10, X18)
m20_in_gaa(0, 0, X18) → m20_out_gaa(0, 0, X18)
m20_in_gaa(s(T12), X35, X18) → U1_gaa(T12, X35, X18, m20_in_gaa(T12, X35, X34))
U1_gaa(T12, X35, X18, m20_out_gaa(T12, X35, X34)) → m20_out_gaa(s(T12), X35, X18)
U4_gg(T12, m20_out_gaa(T12, X35, X34)) → q1_out_gg(s(s(T12)), 0)
q1_in_gg(s(T9), s(T13)) → U5_gg(T9, T13, m3_in_gga(T9, T13, X19))
m3_in_gga(T5, 0, T5) → m3_out_gga(T5, 0, T5)
m3_in_gga(0, T6, 0) → m3_out_gga(0, T6, 0)
m3_in_gga(s(T10), 0, T10) → m3_out_gga(s(T10), 0, T10)
m3_in_gga(s(0), 0, 0) → m3_out_gga(s(0), 0, 0)
m3_in_gga(s(s(T12)), 0, X35) → U2_gga(T12, X35, m20_in_gaa(T12, X35, X34))
U2_gga(T12, X35, m20_out_gaa(T12, X35, X34)) → m3_out_gga(s(s(T12)), 0, X35)
m3_in_gga(s(T9), s(T13), X19) → U3_gga(T9, T13, X19, m3_in_gga(T9, T13, X19))
U3_gga(T9, T13, X19, m3_out_gga(T9, T13, X19)) → m3_out_gga(s(T9), s(T13), X19)
U5_gg(T9, T13, m3_out_gga(T9, T13, X19)) → q1_out_gg(s(T9), s(T13))

The argument filtering Pi contains the following mapping:
q1_in_gg(x1, x2) = q1_in_gg(x1, x2)
0 = 0
q1_out_gg(x1, x2) = q1_out_gg
s(x1) = s(x1)
U4_gg(x1, x2) = U4_gg(x2)
m20_in_gaa(x1, x2, x3) = m20_in_gaa(x1)
m20_out_gaa(x1, x2, x3) = m20_out_gaa(x2)
U1_gaa(x1, x2, x3, x4) = U1_gaa(x4)
U5_gg(x1, x2, x3) = U5_gg(x3)
m3_in_gga(x1, x2, x3) = m3_in_gga(x1, x2)
m3_out_gga(x1, x2, x3) = m3_out_gga(x3)
U2_gga(x1, x2, x3) = U2_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
Q1_IN_GG(x1, x2) = Q1_IN_GG(x1, x2)
U4_GG(x1, x2) = U4_GG(x2)
M20_IN_GAA(x1, x2, x3) = M20_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4) = U1_GAA(x4)
U5_GG(x1, x2, x3) = U5_GG(x3)
M3_IN_GGA(x1, x2, x3) = M3_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)

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

(6) Obligation:

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

Q1_IN_GG(s(s(T12)), 0) → U4_GG(T12, m20_in_gaa(T12, X35, X34))
Q1_IN_GG(s(s(T12)), 0) → M20_IN_GAA(T12, X35, X34)
M20_IN_GAA(s(T12), X35, X18) → U1_GAA(T12, X35, X18, m20_in_gaa(T12, X35, X34))
M20_IN_GAA(s(T12), X35, X18) → M20_IN_GAA(T12, X35, X34)
Q1_IN_GG(s(T9), s(T13)) → U5_GG(T9, T13, m3_in_gga(T9, T13, X19))
Q1_IN_GG(s(T9), s(T13)) → M3_IN_GGA(T9, T13, X19)
M3_IN_GGA(s(s(T12)), 0, X35) → U2_GGA(T12, X35, m20_in_gaa(T12, X35, X34))
M3_IN_GGA(s(s(T12)), 0, X35) → M20_IN_GAA(T12, X35, X34)
M3_IN_GGA(s(T9), s(T13), X19) → U3_GGA(T9, T13, X19, m3_in_gga(T9, T13, X19))
M3_IN_GGA(s(T9), s(T13), X19) → M3_IN_GGA(T9, T13, X19)

The TRS R consists of the following rules:

q1_in_gg(T5, 0) → q1_out_gg(T5, 0)
q1_in_gg(0, T6) → q1_out_gg(0, T6)
q1_in_gg(s(T10), 0) → q1_out_gg(s(T10), 0)
q1_in_gg(s(0), 0) → q1_out_gg(s(0), 0)
q1_in_gg(s(s(T12)), 0) → U4_gg(T12, m20_in_gaa(T12, X35, X34))
m20_in_gaa(T10, T10, X18) → m20_out_gaa(T10, T10, X18)
m20_in_gaa(0, 0, X18) → m20_out_gaa(0, 0, X18)
m20_in_gaa(s(T12), X35, X18) → U1_gaa(T12, X35, X18, m20_in_gaa(T12, X35, X34))
U1_gaa(T12, X35, X18, m20_out_gaa(T12, X35, X34)) → m20_out_gaa(s(T12), X35, X18)
U4_gg(T12, m20_out_gaa(T12, X35, X34)) → q1_out_gg(s(s(T12)), 0)
q1_in_gg(s(T9), s(T13)) → U5_gg(T9, T13, m3_in_gga(T9, T13, X19))
m3_in_gga(T5, 0, T5) → m3_out_gga(T5, 0, T5)
m3_in_gga(0, T6, 0) → m3_out_gga(0, T6, 0)
m3_in_gga(s(T10), 0, T10) → m3_out_gga(s(T10), 0, T10)
m3_in_gga(s(0), 0, 0) → m3_out_gga(s(0), 0, 0)
m3_in_gga(s(s(T12)), 0, X35) → U2_gga(T12, X35, m20_in_gaa(T12, X35, X34))
U2_gga(T12, X35, m20_out_gaa(T12, X35, X34)) → m3_out_gga(s(s(T12)), 0, X35)
m3_in_gga(s(T9), s(T13), X19) → U3_gga(T9, T13, X19, m3_in_gga(T9, T13, X19))
U3_gga(T9, T13, X19, m3_out_gga(T9, T13, X19)) → m3_out_gga(s(T9), s(T13), X19)
U5_gg(T9, T13, m3_out_gga(T9, T13, X19)) → q1_out_gg(s(T9), s(T13))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

M20_IN_GAA(s(T12), X35, X18) → M20_IN_GAA(T12, X35, X34)

The TRS R consists of the following rules:

q1_in_gg(T5, 0) → q1_out_gg(T5, 0)
q1_in_gg(0, T6) → q1_out_gg(0, T6)
q1_in_gg(s(T10), 0) → q1_out_gg(s(T10), 0)
q1_in_gg(s(0), 0) → q1_out_gg(s(0), 0)
q1_in_gg(s(s(T12)), 0) → U4_gg(T12, m20_in_gaa(T12, X35, X34))
m20_in_gaa(T10, T10, X18) → m20_out_gaa(T10, T10, X18)
m20_in_gaa(0, 0, X18) → m20_out_gaa(0, 0, X18)
m20_in_gaa(s(T12), X35, X18) → U1_gaa(T12, X35, X18, m20_in_gaa(T12, X35, X34))
U1_gaa(T12, X35, X18, m20_out_gaa(T12, X35, X34)) → m20_out_gaa(s(T12), X35, X18)
U4_gg(T12, m20_out_gaa(T12, X35, X34)) → q1_out_gg(s(s(T12)), 0)
q1_in_gg(s(T9), s(T13)) → U5_gg(T9, T13, m3_in_gga(T9, T13, X19))
m3_in_gga(T5, 0, T5) → m3_out_gga(T5, 0, T5)
m3_in_gga(0, T6, 0) → m3_out_gga(0, T6, 0)
m3_in_gga(s(T10), 0, T10) → m3_out_gga(s(T10), 0, T10)
m3_in_gga(s(0), 0, 0) → m3_out_gga(s(0), 0, 0)
m3_in_gga(s(s(T12)), 0, X35) → U2_gga(T12, X35, m20_in_gaa(T12, X35, X34))
U2_gga(T12, X35, m20_out_gaa(T12, X35, X34)) → m3_out_gga(s(s(T12)), 0, X35)
m3_in_gga(s(T9), s(T13), X19) → U3_gga(T9, T13, X19, m3_in_gga(T9, T13, X19))
U3_gga(T9, T13, X19, m3_out_gga(T9, T13, X19)) → m3_out_gga(s(T9), s(T13), X19)
U5_gg(T9, T13, m3_out_gga(T9, T13, X19)) → q1_out_gg(s(T9), s(T13))

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

M20_IN_GAA(s(T12), X35, X18) → M20_IN_GAA(T12, X35, X34)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
M20_IN_GAA(x1, x2, x3) = M20_IN_GAA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

M20_IN_GAA(s(T12)) → M20_IN_GAA(T12)

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

(14) 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:

M20_IN_GAA(s(T12)) → M20_IN_GAA(T12)
The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

M3_IN_GGA(s(T9), s(T13), X19) → M3_IN_GGA(T9, T13, X19)

The TRS R consists of the following rules:

q1_in_gg(T5, 0) → q1_out_gg(T5, 0)
q1_in_gg(0, T6) → q1_out_gg(0, T6)
q1_in_gg(s(T10), 0) → q1_out_gg(s(T10), 0)
q1_in_gg(s(0), 0) → q1_out_gg(s(0), 0)
q1_in_gg(s(s(T12)), 0) → U4_gg(T12, m20_in_gaa(T12, X35, X34))
m20_in_gaa(T10, T10, X18) → m20_out_gaa(T10, T10, X18)
m20_in_gaa(0, 0, X18) → m20_out_gaa(0, 0, X18)
m20_in_gaa(s(T12), X35, X18) → U1_gaa(T12, X35, X18, m20_in_gaa(T12, X35, X34))
U1_gaa(T12, X35, X18, m20_out_gaa(T12, X35, X34)) → m20_out_gaa(s(T12), X35, X18)
U4_gg(T12, m20_out_gaa(T12, X35, X34)) → q1_out_gg(s(s(T12)), 0)
q1_in_gg(s(T9), s(T13)) → U5_gg(T9, T13, m3_in_gga(T9, T13, X19))
m3_in_gga(T5, 0, T5) → m3_out_gga(T5, 0, T5)
m3_in_gga(0, T6, 0) → m3_out_gga(0, T6, 0)
m3_in_gga(s(T10), 0, T10) → m3_out_gga(s(T10), 0, T10)
m3_in_gga(s(0), 0, 0) → m3_out_gga(s(0), 0, 0)
m3_in_gga(s(s(T12)), 0, X35) → U2_gga(T12, X35, m20_in_gaa(T12, X35, X34))
U2_gga(T12, X35, m20_out_gaa(T12, X35, X34)) → m3_out_gga(s(s(T12)), 0, X35)
m3_in_gga(s(T9), s(T13), X19) → U3_gga(T9, T13, X19, m3_in_gga(T9, T13, X19))
U3_gga(T9, T13, X19, m3_out_gga(T9, T13, X19)) → m3_out_gga(s(T9), s(T13), X19)
U5_gg(T9, T13, m3_out_gga(T9, T13, X19)) → q1_out_gg(s(T9), s(T13))

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

M3_IN_GGA(s(T9), s(T13), X19) → M3_IN_GGA(T9, T13, X19)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
M3_IN_GGA(x1, x2, x3) = M3_IN_GGA(x1, x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

M3_IN_GGA(s(T9), s(T13)) → M3_IN_GGA(T9, T13)

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

(21) 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:

M3_IN_GGA(s(T9), s(T13)) → M3_IN_GGA(T9, T13)
The graph contains the following edges 1 > 1, 2 > 2

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE