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

(0) Obligation:

Clauses:

gopher(nil, nil).
gopher(cons(nil, Y), cons(nil, Y)).
gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).
samefringe(nil, nil).
samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1))).

Query: samefringe(g,g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: samefringe(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: samefringe(T1, T2), SCOPE: 1, CLAUSE: 3 | samefringe(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: samefringe(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nsamefringe(T1, T2) !=? samefringe(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17)))\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 0 | ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 1 | ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 1 | ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(T15, X17))\n\nT9 is ground\nT10 is ground\nT15 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="C: gopher(cons(T9, T10), cons(X16, X17))\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: samefringe(T15, T17)\n\nT15 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 0 | gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 1 | gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 1 | gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: gopher(cons(T29, cons(T30, T31)), cons(X44, X45))\n\nT29 is ground\nT30 is ground\nT31 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(gopher(cons(T38, cons(T39, T40)), cons(X62, X63)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X63, X17)))\n\nT9 is ground\nT10 is ground\nT38 is ground\nT39 is ground\nT40 is ground\nX16 is free\nX17 is free\nX62 is free\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 28 [arrowhead = none ];
28 -> 2;

29 [label="EVAL with clause\nsamefringe(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 31 [arrowhead = none ];
31 -> 5;

32 [label="EVAL with clause\nsamefringe(cons(X10, X11), cons(X12, X13)) :- ','(gopher(cons(X10, X11), cons(X14, X15)), ','(gopher(cons(X12, X13), cons(X16, X17)), samefringe(X15, X17))).\nand substitutionX10 -> T7,\nX11 -> T8,\nT1 -> cons(T7, T8),\nX12 -> T9,\nX13 -> T10,\nT2 -> cons(T9, T10)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 7;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="BACKTRACK\nfor clause: samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 6;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="EVAL with clause\ngopher(cons(nil, X22), cons(nil, X22)).\nand substitutionT7 -> nil,\nT8 -> T15,\nX22 -> T15,\nX14 -> nil,\nX15 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 40 [arrowhead = none ];
40 -> 14;

41 [label="EVAL with clause\ngopher(cons(cons(X58, X59), X60), X61) :- gopher(cons(X58, cons(X59, X60)), X61).\nand substitutionX58 -> T38,\nX59 -> T39,\nT7 -> cons(T38, T39),\nT8 -> T40,\nX60 -> T40,\nX14 -> X62,\nX15 -> X63,\nX61 -> cons(X62, X63)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 41 [arrowhead = none ];
41 -> 26;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 42 [arrowhead = none ];
42 -> 27;

43 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 43 [arrowhead = none ];
43 -> 15;

44 [label="SPLIT 2\nnew knowledge:\nT9 is ground\nT10 is ground\nT16 is ground\nT17 is ground\nreplacements:X16 -> T16,\nX17 -> T17", fontsize=14, style = filled, fillcolor = violet];
13 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 45 [arrowhead = none ];
45 -> 17;

46 [label="INSTANCE with matching:\nT1 -> T15\nT2 -> T17", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 46 [arrowhead = none , style=dashed];
46 -> 1[style=dashed];

47 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 47 [arrowhead = none ];
47 -> 18;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 48 [arrowhead = none ];
48 -> 19;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 49 [arrowhead = none ];
49 -> 20;

50 [label="EVAL with clause\ngopher(cons(nil, X27), cons(nil, X27)).\nand substitutionT9 -> nil,\nT10 -> T22,\nX27 -> T22,\nX16 -> nil,\nX17 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 50 [arrowhead = none ];
50 -> 21;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 51 [arrowhead = none ];
51 -> 22;

52 [label="EVAL with clause\ngopher(cons(cons(X40, X41), X42), X43) :- gopher(cons(X40, cons(X41, X42)), X43).\nand substitutionX40 -> T29,\nX41 -> T30,\nT9 -> cons(T29, T30),\nT10 -> T31,\nX42 -> T31,\nX16 -> X44,\nX17 -> X45,\nX43 -> cons(X44, X45)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 52 [arrowhead = none ];
52 -> 24;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 53 [arrowhead = none ];
53 -> 25;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 54 [arrowhead = none ];
54 -> 23;

55 [label="INSTANCE with matching:\nT9 -> T29\nT10 -> cons(T30, T31)\nX16 -> X44\nX17 -> X45", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 55 [arrowhead = none , style=dashed];
55 -> 15[style=dashed];

56 [label="INSTANCE with matching:\nT7 -> T38\nT8 -> cons(T39, T40)\nX14 -> X62\nX15 -> X63", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 56 [arrowhead = none , style=dashed];
56 -> 7[style=dashed];

}

(2) Obligation:

Triples:

gopherC(cons(X1, X2), X3, X4, X5) :- gopherC(X1, cons(X2, X3), X4, X5).
pB(nil, X1, nil, X1, X2, X3, X4, X5) :- gopherC(X2, X3, X4, X5).
pB(nil, X1, nil, X1, X2, X3, X4, X5) :- ','(gophercC(X2, X3, X4, X5), samefringeA(X1, X5)).
pB(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) :- pB(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9).
samefringeA(cons(X1, X2), cons(X3, X4)) :- pB(X1, X2, X5, X6, X3, X4, X7, X8).

Clauses:

samefringecA(nil, nil).
samefringecA(cons(X1, X2), cons(X3, X4)) :- qcB(X1, X2, X5, X6, X3, X4, X7, X8).
gophercC(nil, X1, nil, X1).
gophercC(cons(X1, X2), X3, X4, X5) :- gophercC(X1, cons(X2, X3), X4, X5).
qcB(nil, X1, nil, X1, X2, X3, X4, X5) :- ','(gophercC(X2, X3, X4, X5), samefringecA(X1, X5)).
qcB(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) :- qcB(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9).

Afs:

samefringeA(x1, x2)  =  samefringeA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
samefringeA_in: (b,b)
pB_in: (b,b,f,f,b,b,f,f)
gopherC_in: (b,b,f,f)
gophercC_in: (b,b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → U6_GG(X1, X2, X3, X4, pB_in_ggaaggaa(X1, X2, X5, X6, X3, X4, X7, X8))
SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → PB_IN_GGAAGGAA(X1, X2, X5, X6, X3, X4, X7, X8)
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → U2_GGAAGGAA(X1, X2, X3, X4, X5, gopherC_in_ggaa(X2, X3, X4, X5))
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → GOPHERC_IN_GGAA(X2, X3, X4, X5)
GOPHERC_IN_GGAA(cons(X1, X2), X3, X4, X5) → U1_GGAA(X1, X2, X3, X4, X5, gopherC_in_ggaa(X1, cons(X2, X3), X4, X5))
GOPHERC_IN_GGAA(cons(X1, X2), X3, X4, X5) → GOPHERC_IN_GGAA(X1, cons(X2, X3), X4, X5)
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_in_ggaa(X2, X3, X4, X5))
U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_out_ggaa(X2, X3, X4, X5)) → U4_GGAAGGAA(X1, X2, X3, X4, X5, samefringeA_in_gg(X1, X5))
U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_out_ggaa(X2, X3, X4, X5)) → SAMEFRINGEA_IN_GG(X1, X5)
PB_IN_GGAAGGAA(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) → U5_GGAAGGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, pB_in_ggaaggaa(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9))
PB_IN_GGAAGGAA(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1, nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3, X4, X5) → U9_ggaa(X1, X2, X3, X4, X5, gophercC_in_ggaa(X1, cons(X2, X3), X4, X5))
U9_ggaa(X1, X2, X3, X4, X5, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
pB_in_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaaggaa(x1, x2, x5, x6)
nil  =  nil
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gophercC_in_ggaa(x1, x2, x3, x4)  =  gophercC_in_ggaa(x1, x2)
gophercC_out_ggaa(x1, x2, x3, x4)  =  gophercC_out_ggaa(x1, x2, x3, x4)
U9_ggaa(x1, x2, x3, x4, x5, x6)  =  U9_ggaa(x1, x2, x3, x6)
SAMEFRINGEA_IN_GG(x1, x2)  =  SAMEFRINGEA_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5)  =  U6_GG(x1, x2, x3, x4, x5)
PB_IN_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGGAA(x1, x2, x5, x6)
U2_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAAGGAA(x1, x2, x3, x6)
GOPHERC_IN_GGAA(x1, x2, x3, x4)  =  GOPHERC_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6)  =  U1_GGAA(x1, x2, x3, x6)
U3_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGAA(x1, x2, x3, x6)
U4_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAGGAA(x1, x2, x3, x4, x5, x6)
U5_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U5_GGAAGGAA(x1, x2, x3, x6, x7, x10)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → U6_GG(X1, X2, X3, X4, pB_in_ggaaggaa(X1, X2, X5, X6, X3, X4, X7, X8))
SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → PB_IN_GGAAGGAA(X1, X2, X5, X6, X3, X4, X7, X8)
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → U2_GGAAGGAA(X1, X2, X3, X4, X5, gopherC_in_ggaa(X2, X3, X4, X5))
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → GOPHERC_IN_GGAA(X2, X3, X4, X5)
GOPHERC_IN_GGAA(cons(X1, X2), X3, X4, X5) → U1_GGAA(X1, X2, X3, X4, X5, gopherC_in_ggaa(X1, cons(X2, X3), X4, X5))
GOPHERC_IN_GGAA(cons(X1, X2), X3, X4, X5) → GOPHERC_IN_GGAA(X1, cons(X2, X3), X4, X5)
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_in_ggaa(X2, X3, X4, X5))
U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_out_ggaa(X2, X3, X4, X5)) → U4_GGAAGGAA(X1, X2, X3, X4, X5, samefringeA_in_gg(X1, X5))
U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_out_ggaa(X2, X3, X4, X5)) → SAMEFRINGEA_IN_GG(X1, X5)
PB_IN_GGAAGGAA(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) → U5_GGAAGGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, pB_in_ggaaggaa(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9))
PB_IN_GGAAGGAA(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1, nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3, X4, X5) → U9_ggaa(X1, X2, X3, X4, X5, gophercC_in_ggaa(X1, cons(X2, X3), X4, X5))
U9_ggaa(X1, X2, X3, X4, X5, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
pB_in_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaaggaa(x1, x2, x5, x6)
nil  =  nil
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gophercC_in_ggaa(x1, x2, x3, x4)  =  gophercC_in_ggaa(x1, x2)
gophercC_out_ggaa(x1, x2, x3, x4)  =  gophercC_out_ggaa(x1, x2, x3, x4)
U9_ggaa(x1, x2, x3, x4, x5, x6)  =  U9_ggaa(x1, x2, x3, x6)
SAMEFRINGEA_IN_GG(x1, x2)  =  SAMEFRINGEA_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5)  =  U6_GG(x1, x2, x3, x4, x5)
PB_IN_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGGAA(x1, x2, x5, x6)
U2_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAAGGAA(x1, x2, x3, x6)
GOPHERC_IN_GGAA(x1, x2, x3, x4)  =  GOPHERC_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6)  =  U1_GGAA(x1, x2, x3, x6)
U3_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGAA(x1, x2, x3, x6)
U4_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAGGAA(x1, x2, x3, x4, x5, x6)
U5_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U5_GGAAGGAA(x1, x2, x3, x6, x7, x10)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GOPHERC_IN_GGAA(cons(X1, X2), X3, X4, X5) → GOPHERC_IN_GGAA(X1, cons(X2, X3), X4, X5)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1, nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3, X4, X5) → U9_ggaa(X1, X2, X3, X4, X5, gophercC_in_ggaa(X1, cons(X2, X3), X4, X5))
U9_ggaa(X1, X2, X3, X4, X5, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
gophercC_in_ggaa(x1, x2, x3, x4)  =  gophercC_in_ggaa(x1, x2)
gophercC_out_ggaa(x1, x2, x3, x4)  =  gophercC_out_ggaa(x1, x2, x3, x4)
U9_ggaa(x1, x2, x3, x4, x5, x6)  =  U9_ggaa(x1, x2, x3, x6)
GOPHERC_IN_GGAA(x1, x2, x3, x4)  =  GOPHERC_IN_GGAA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

GOPHERC_IN_GGAA(cons(X1, X2), X3, X4, X5) → GOPHERC_IN_GGAA(X1, cons(X2, X3), X4, X5)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

GOPHERC_IN_GGAA(cons(X1, X2), X3) → GOPHERC_IN_GGAA(X1, cons(X2, X3))

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

(12) 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:

• GOPHERC_IN_GGAA(cons(X1, X2), X3) → GOPHERC_IN_GGAA(X1, cons(X2, X3))
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → PB_IN_GGAAGGAA(X1, X2, X5, X6, X3, X4, X7, X8)
PB_IN_GGAAGGAA(nil, X1, nil, X1, X2, X3, X4, X5) → U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_in_ggaa(X2, X3, X4, X5))
U3_GGAAGGAA(X1, X2, X3, X4, X5, gophercC_out_ggaa(X2, X3, X4, X5)) → SAMEFRINGEA_IN_GG(X1, X5)
PB_IN_GGAAGGAA(cons(X1, X2), X3, X4, X5, X6, X7, X8, X9) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X4, X5, X6, X7, X8, X9)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1, nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3, X4, X5) → U9_ggaa(X1, X2, X3, X4, X5, gophercC_in_ggaa(X1, cons(X2, X3), X4, X5))
U9_ggaa(X1, X2, X3, X4, X5, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
gophercC_in_ggaa(x1, x2, x3, x4)  =  gophercC_in_ggaa(x1, x2)
gophercC_out_ggaa(x1, x2, x3, x4)  =  gophercC_out_ggaa(x1, x2, x3, x4)
U9_ggaa(x1, x2, x3, x4, x5, x6)  =  U9_ggaa(x1, x2, x3, x6)
SAMEFRINGEA_IN_GG(x1, x2)  =  SAMEFRINGEA_IN_GG(x1, x2)
PB_IN_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGGAA(x1, x2, x5, x6)
U3_GGAAGGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGAA(x1, x2, x3, x6)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → PB_IN_GGAAGGAA(X1, X2, X3, X4)
PB_IN_GGAAGGAA(nil, X1, X2, X3) → U3_GGAAGGAA(X1, X2, X3, gophercC_in_ggaa(X2, X3))
U3_GGAAGGAA(X1, X2, X3, gophercC_out_ggaa(X2, X3, X4, X5)) → SAMEFRINGEA_IN_GG(X1, X5)
PB_IN_GGAAGGAA(cons(X1, X2), X3, X6, X7) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X6, X7)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3) → U9_ggaa(X1, X2, X3, gophercC_in_ggaa(X1, cons(X2, X3)))
U9_ggaa(X1, X2, X3, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The set Q consists of the following terms:

gophercC_in_ggaa(x0, x1)
U9_ggaa(x0, x1, x2, x3)

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

(17) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PB_IN_GGAAGGAA(nil, X1, X2, X3) → U3_GGAAGGAA(X1, X2, X3, gophercC_in_ggaa(X2, X3))
U3_GGAAGGAA(X1, X2, X3, gophercC_out_ggaa(X2, X3, X4, X5)) → SAMEFRINGEA_IN_GG(X1, X5)

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

POL(PB_IN_GGAAGGAA(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂   
POL(SAMEFRINGEA_IN_GG(x₁, x₂)) = x₁   
POL(U3_GGAAGGAA(x₁, x₂, x₃, x₄)) = 1 + x₁   
POL(U9_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(cons(x₁, x₂)) = 1 + x₁ + x₂   
POL(gophercC_in_ggaa(x₁, x₂)) = 0   
POL(gophercC_out_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(nil) = 1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(18) Obligation:

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

SAMEFRINGEA_IN_GG(cons(X1, X2), cons(X3, X4)) → PB_IN_GGAAGGAA(X1, X2, X3, X4)
PB_IN_GGAAGGAA(cons(X1, X2), X3, X6, X7) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X6, X7)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3) → U9_ggaa(X1, X2, X3, gophercC_in_ggaa(X1, cons(X2, X3)))
U9_ggaa(X1, X2, X3, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The set Q consists of the following terms:

gophercC_in_ggaa(x0, x1)
U9_ggaa(x0, x1, x2, x3)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(20) Obligation:

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

PB_IN_GGAAGGAA(cons(X1, X2), X3, X6, X7) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X6, X7)

The TRS R consists of the following rules:

gophercC_in_ggaa(nil, X1) → gophercC_out_ggaa(nil, X1, nil, X1)
gophercC_in_ggaa(cons(X1, X2), X3) → U9_ggaa(X1, X2, X3, gophercC_in_ggaa(X1, cons(X2, X3)))
U9_ggaa(X1, X2, X3, gophercC_out_ggaa(X1, cons(X2, X3), X4, X5)) → gophercC_out_ggaa(cons(X1, X2), X3, X4, X5)

The set Q consists of the following terms:

gophercC_in_ggaa(x0, x1)
U9_ggaa(x0, x1, x2, x3)

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

(21) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(22) Obligation:

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

PB_IN_GGAAGGAA(cons(X1, X2), X3, X6, X7) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X6, X7)

R is empty.
The set Q consists of the following terms:

gophercC_in_ggaa(x0, x1)
U9_ggaa(x0, x1, x2, x3)

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

(23) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

gophercC_in_ggaa(x0, x1)
U9_ggaa(x0, x1, x2, x3)

(24) Obligation:

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

PB_IN_GGAAGGAA(cons(X1, X2), X3, X6, X7) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X6, X7)

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

(25) 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:

• PB_IN_GGAAGGAA(cons(X1, X2), X3, X6, X7) → PB_IN_GGAAGGAA(X1, cons(X2, X3), X6, X7)
  The graph contains the following edges 1 > 1, 3 >= 3, 4 >= 4

(26) YES