Left Termination of the query pattern q_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

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

Queries:

q(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | q(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: q(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: q(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: m(T4, T5, T7)\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: m(T4, T5, T7), SCOPE: 2, CLAUSE: 0 | m(T4, T5, T7), SCOPE: 2, CLAUSE: 1 | m(T4, T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, =(T10, T8)) | m(T8, 0, T7), SCOPE: 2, CLAUSE: 1 | m(T8, 0, T7), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: m(T4, T5, T7), SCOPE: 2, CLAUSE: 1 | m(T4, T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT4 is ground\nT5 is ground\nX12 is free; \nX13 is free; \nm(T4, T5, T7) !=? m(X12, 0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: =(T10, T8)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: =(T10, T8), SCOPE: 3, CLAUSE: 7\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(!_2, =(T14, 0)) | m(0, T12, T7), SCOPE: 2, CLAUSE: 2 | ?_2\n\nT12 is ground\nX12 is free; \nX13 is free; \nm(0, T12, T7) !=? m(X12, 0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: m(T4, T5, T7), SCOPE: 2, CLAUSE: 2\n\nT4 is ground\nT5 is ground\nX12 is free; \nX13 is free; \nX18 is free; \nX19 is free; \nm(T4, T5, T7) !=? m(X12, 0, X13)\nm(T4, T5, T7) !=? m(0, X18, X19)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: =(T14, 0)\n\nX12 is free\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: =(T14, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T16, X28), ','(p(T17, X29), m(X28, X29, T19)))\n\nT16 is ground\nT17 is ground\nX12 is free; \nX13 is free; \nX18 is free; \nX19 is free; \nX28 is free; \nX29 is free; \nm(T16, T17, T7) !=? m(X12, 0, X13)\nm(T16, T17, T7) !=? m(0, X18, X19)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T16, X28), ','(p(T17, X29), m(X28, X29, T19))), SCOPE: 5, CLAUSE: 3 | ','(p(T16, X28), ','(p(T17, X29), m(X28, X29, T19))), SCOPE: 5, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX12 is free; \nX13 is free; \nX18 is free; \nX19 is free; \nX28 is free; \nX29 is free; \nm(T16, T17, T7) !=? m(X12, 0, X13)\nm(T16, T17, T7) !=? m(0, X18, X19)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T16, X28), ','(p(T17, X29), m(X28, X29, T19))), SCOPE: 5, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX12 is free; \nX13 is free; \nX18 is free; \nX19 is free; \nX28 is free; \nX29 is free; \nm(T16, T17, T7) !=? m(X12, 0, X13)\nm(T16, T17, T7) !=? m(0, X18, X19)\np(T16, X28) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T17, X29), m(T20, X29, T19))\n\nT17 is ground\nT20 is ground\nX12 is free; \nX13 is free; \nX18 is free; \nX19 is free; \nX28 is free; \nX29 is free; \nm(s(T20), T17, T7) !=? m(X12, 0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\nX12 is free\nX13 is free; \nX18 is free; \nX19 is free; \nX28 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T17, X29), m(T20, X29, T19)), SCOPE: 6, CLAUSE: 3 | ','(p(T17, X29), m(T20, X29, T19)), SCOPE: 6, CLAUSE: 4\n\nT17 is ground\nT20 is ground\nX12 is free; \nX13 is free; \nX29 is free; \nm(s(T20), T17, T7) !=? m(X12, 0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(p(T17, X29), m(T20, X29, T19)), SCOPE: 6, CLAUSE: 4\n\nT17 is ground\nT20 is ground\nX12 is free; \nX13 is free; \nX29 is free; \nm(s(T20), T17, T7) !=? m(X12, 0, X13)\np(T17, X29) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: m(T20, T21, T19)\n\nT20 is ground\nT21 is ground\nX12 is free; \nX13 is free; \nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\nX12 is free\nX13 is free; \nX29 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 32 [arrowhead = none ];
32 -> 2;

33 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 33 [arrowhead = none ];
33 -> 3;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 34 [arrowhead = none ];
34 -> 4;

35 [label="EVAL with clause\nq(X7, X8, X9) :- m(X7, X8, X9).\nand substitution\nT1 -> T4\nX7 -> T4\nT2 -> T5\nX8 -> T5\nT3 -> T7\nX9 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="EVAL with clause\nq(X37, X38, X39).\nand substitution\nT1 -> T22\nX37 -> T22\nT2 -> T23\nX38 -> T23\nT3 -> T24\nX39 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 36 [arrowhead = none ];
36 -> 30;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 37 [arrowhead = none ];
37 -> 6;

38 [label="EVAL with clause\nm(X12, 0, X13) :- ','(!, =(X13, X12)).\nand substitution\nT4 -> T8\nX12 -> T8\nT5 -> 0\nT7 -> T10\nX13 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 38 [arrowhead = none ];
38 -> 7;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 39 [arrowhead = none ];
39 -> 8;

40 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 40 [arrowhead = none ];
40 -> 9;

41 [label="EVAL with clause\nm(0, X18, X19) :- ','(!, =(X19, 0)).\nand substitution\nT4 -> 0\nT5 -> T12\nX18 -> T12\nT7 -> T14\nX19 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 41 [arrowhead = none ];
41 -> 14;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 42 [arrowhead = none ];
42 -> 15;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 43 [arrowhead = none ];
43 -> 10;

44 [label="EVAL with clause\n=(X15, X15).\nand substitution\nT10 -> T11\nX15 -> T11\nT8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 44 [arrowhead = none ];
44 -> 11;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 45 [arrowhead = none ];
45 -> 12;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 46 [arrowhead = none ];
46 -> 13;

47 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 47 [arrowhead = none ];
47 -> 16;

48 [label="EVAL with clause\nm(X25, X26, X27) :- ','(p(X25, X28), ','(p(X26, X29), m(X28, X29, X27))).\nand substitution\nT4 -> T16\nX25 -> T16\nT5 -> T17\nX26 -> T17\nT7 -> T19\nX27 -> T19\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 49 [arrowhead = none ];
49 -> 17;

50 [label="EVAL with clause\n=(X21, X21).\nand substitution\nT14 -> 0\nX21 -> 0\nT15 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 50 [arrowhead = none ];
50 -> 18;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 51 [arrowhead = none ];
51 -> 19;

52 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 52 [arrowhead = none ];
52 -> 20;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 53 [arrowhead = none ];
53 -> 22;

54 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T16, T17, T7), m(0, X18, X19))", fontsize=14, style = filled, fillcolor = white];
22 -> 54 [arrowhead = none ];
54 -> 23;

55 [label="EVAL with clause\np(s(X31), X31).\nand substitution\nX31 -> T20\nT16 -> s(T20)\nX28 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 55 [arrowhead = none ];
55 -> 24;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 56 [arrowhead = none ];
56 -> 25;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 57 [arrowhead = none ];
57 -> 26;

58 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(T20), T17, T7), m(X12, 0, X13))", fontsize=14, style = filled, fillcolor = white];
26 -> 58 [arrowhead = none ];
58 -> 27;

59 [label="EVAL with clause\np(s(X33), X33).\nand substitution\nX33 -> T21\nT17 -> s(T21)\nX29 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 59 [arrowhead = none ];
59 -> 28;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 60 [arrowhead = none ];
60 -> 29;

61 [label="INSTANCE with matching:\nT4 -> T20\nT5 -> T21\nT7 -> T19", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 61 [arrowhead = none , style=dashed];
61 -> 5[style=dashed];

62 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 62 [arrowhead = none ];
62 -> 31;

}

(2) Obligation:

Clauses:

m5(T11, 0, T11).
m5(0, T12, 0).
m5(s(T20), s(T21), T19) :- m5(T20, T21, T19).
q1(T11, 0, T11).
q1(0, T12, 0).
q1(s(T20), s(T21), T19) :- m5(T20, T21, T19).
q1(T22, T23, T24).

Queries:

q1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
q1_in: (b,b,f)
m5_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_gga(T11, 0, T11) → q1_out_gga(T11, 0, T11)
q1_in_gga(0, T12, 0) → q1_out_gga(0, T12, 0)
q1_in_gga(s(T20), s(T21), T19) → U2_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
m5_in_gga(T11, 0, T11) → m5_out_gga(T11, 0, T11)
m5_in_gga(0, T12, 0) → m5_out_gga(0, T12, 0)
m5_in_gga(s(T20), s(T21), T19) → U1_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
U1_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → m5_out_gga(s(T20), s(T21), T19)
U2_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → q1_out_gga(s(T20), s(T21), T19)
q1_in_gga(T22, T23, T24) → q1_out_gga(T22, T23, T24)

The argument filtering Pi contains the following mapping:
q1_in_gga(x1, x2, x3)  =  q1_in_gga(x1, x2)
0  =  0
q1_out_gga(x1, x2, x3)  =  q1_out_gga
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
m5_in_gga(x1, x2, x3)  =  m5_in_gga(x1, x2)
m5_out_gga(x1, x2, x3)  =  m5_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)

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_gga(T11, 0, T11) → q1_out_gga(T11, 0, T11)
q1_in_gga(0, T12, 0) → q1_out_gga(0, T12, 0)
q1_in_gga(s(T20), s(T21), T19) → U2_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
m5_in_gga(T11, 0, T11) → m5_out_gga(T11, 0, T11)
m5_in_gga(0, T12, 0) → m5_out_gga(0, T12, 0)
m5_in_gga(s(T20), s(T21), T19) → U1_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
U1_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → m5_out_gga(s(T20), s(T21), T19)
U2_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → q1_out_gga(s(T20), s(T21), T19)
q1_in_gga(T22, T23, T24) → q1_out_gga(T22, T23, T24)

The argument filtering Pi contains the following mapping:
q1_in_gga(x1, x2, x3)  =  q1_in_gga(x1, x2)
0  =  0
q1_out_gga(x1, x2, x3)  =  q1_out_gga
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
m5_in_gga(x1, x2, x3)  =  m5_in_gga(x1, x2)
m5_out_gga(x1, x2, x3)  =  m5_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)

(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_GGA(s(T20), s(T21), T19) → U2_GGA(T20, T21, T19, m5_in_gga(T20, T21, T19))
Q1_IN_GGA(s(T20), s(T21), T19) → M5_IN_GGA(T20, T21, T19)
M5_IN_GGA(s(T20), s(T21), T19) → U1_GGA(T20, T21, T19, m5_in_gga(T20, T21, T19))
M5_IN_GGA(s(T20), s(T21), T19) → M5_IN_GGA(T20, T21, T19)

The TRS R consists of the following rules:

q1_in_gga(T11, 0, T11) → q1_out_gga(T11, 0, T11)
q1_in_gga(0, T12, 0) → q1_out_gga(0, T12, 0)
q1_in_gga(s(T20), s(T21), T19) → U2_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
m5_in_gga(T11, 0, T11) → m5_out_gga(T11, 0, T11)
m5_in_gga(0, T12, 0) → m5_out_gga(0, T12, 0)
m5_in_gga(s(T20), s(T21), T19) → U1_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
U1_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → m5_out_gga(s(T20), s(T21), T19)
U2_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → q1_out_gga(s(T20), s(T21), T19)
q1_in_gga(T22, T23, T24) → q1_out_gga(T22, T23, T24)

The argument filtering Pi contains the following mapping:
q1_in_gga(x1, x2, x3)  =  q1_in_gga(x1, x2)
0  =  0
q1_out_gga(x1, x2, x3)  =  q1_out_gga
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
m5_in_gga(x1, x2, x3)  =  m5_in_gga(x1, x2)
m5_out_gga(x1, x2, x3)  =  m5_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
Q1_IN_GGA(x1, x2, x3)  =  Q1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
M5_IN_GGA(x1, x2, x3)  =  M5_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_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_GGA(s(T20), s(T21), T19) → U2_GGA(T20, T21, T19, m5_in_gga(T20, T21, T19))
Q1_IN_GGA(s(T20), s(T21), T19) → M5_IN_GGA(T20, T21, T19)
M5_IN_GGA(s(T20), s(T21), T19) → U1_GGA(T20, T21, T19, m5_in_gga(T20, T21, T19))
M5_IN_GGA(s(T20), s(T21), T19) → M5_IN_GGA(T20, T21, T19)

The TRS R consists of the following rules:

q1_in_gga(T11, 0, T11) → q1_out_gga(T11, 0, T11)
q1_in_gga(0, T12, 0) → q1_out_gga(0, T12, 0)
q1_in_gga(s(T20), s(T21), T19) → U2_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
m5_in_gga(T11, 0, T11) → m5_out_gga(T11, 0, T11)
m5_in_gga(0, T12, 0) → m5_out_gga(0, T12, 0)
m5_in_gga(s(T20), s(T21), T19) → U1_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
U1_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → m5_out_gga(s(T20), s(T21), T19)
U2_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → q1_out_gga(s(T20), s(T21), T19)
q1_in_gga(T22, T23, T24) → q1_out_gga(T22, T23, T24)

The argument filtering Pi contains the following mapping:
q1_in_gga(x1, x2, x3)  =  q1_in_gga(x1, x2)
0  =  0
q1_out_gga(x1, x2, x3)  =  q1_out_gga
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
m5_in_gga(x1, x2, x3)  =  m5_in_gga(x1, x2)
m5_out_gga(x1, x2, x3)  =  m5_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
Q1_IN_GGA(x1, x2, x3)  =  Q1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
M5_IN_GGA(x1, x2, x3)  =  M5_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

M5_IN_GGA(s(T20), s(T21), T19) → M5_IN_GGA(T20, T21, T19)

The TRS R consists of the following rules:

q1_in_gga(T11, 0, T11) → q1_out_gga(T11, 0, T11)
q1_in_gga(0, T12, 0) → q1_out_gga(0, T12, 0)
q1_in_gga(s(T20), s(T21), T19) → U2_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
m5_in_gga(T11, 0, T11) → m5_out_gga(T11, 0, T11)
m5_in_gga(0, T12, 0) → m5_out_gga(0, T12, 0)
m5_in_gga(s(T20), s(T21), T19) → U1_gga(T20, T21, T19, m5_in_gga(T20, T21, T19))
U1_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → m5_out_gga(s(T20), s(T21), T19)
U2_gga(T20, T21, T19, m5_out_gga(T20, T21, T19)) → q1_out_gga(s(T20), s(T21), T19)
q1_in_gga(T22, T23, T24) → q1_out_gga(T22, T23, T24)

The argument filtering Pi contains the following mapping:
q1_in_gga(x1, x2, x3)  =  q1_in_gga(x1, x2)
0  =  0
q1_out_gga(x1, x2, x3)  =  q1_out_gga
s(x1)  =  s(x1)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
m5_in_gga(x1, x2, x3)  =  m5_in_gga(x1, x2)
m5_out_gga(x1, x2, x3)  =  m5_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
M5_IN_GGA(x1, x2, x3)  =  M5_IN_GGA(x1, x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

M5_IN_GGA(s(T20), s(T21), T19) → M5_IN_GGA(T20, T21, T19)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
M5_IN_GGA(x1, x2, x3)  =  M5_IN_GGA(x1, x2)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

M5_IN_GGA(s(T20), s(T21)) → M5_IN_GGA(T20, T21)

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

(13) 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:

• M5_IN_GGA(s(T20), s(T21)) → M5_IN_GGA(T20, T21)
  The graph contains the following edges 1 > 1, 2 > 2

(14) TRUE