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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

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];
4 [label="4: 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];
10 [label="10: true | samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: 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];
16 [label="16: samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [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];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(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];
26 [label="26: ','(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];
28 [label="28: ','(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];
29 [label="29: ','(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];
32 [label="D: ','(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];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [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];
40 [label="40: samefringe(T15, T17)\n\nT15 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: 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];
48 [label="48: 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];
52 [label="52: 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];
53 [label="53: 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];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: 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];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(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];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 75 [arrowhead = none ];
75 -> 4;

76 [label="EVAL with clause\nsamefringe(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 77 [arrowhead = none ];
77 -> 13;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 78 [arrowhead = none ];
78 -> 16;

79 [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];
13 -> 79 [arrowhead = none ];
79 -> 20;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 80 [arrowhead = none ];
80 -> 22;

81 [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];
16 -> 81 [arrowhead = none ];
81 -> 18;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 82 [arrowhead = none ];
82 -> 24;

83 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 83 [arrowhead = none ];
83 -> 26;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 84 [arrowhead = none ];
84 -> 28;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 85 [arrowhead = none ];
85 -> 29;

86 [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];
28 -> 86 [arrowhead = none ];
86 -> 32;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 87 [arrowhead = none ];
87 -> 33;

88 [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];
29 -> 88 [arrowhead = none ];
88 -> 73;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 89 [arrowhead = none ];
89 -> 74;

90 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
32 -> 90 [arrowhead = none ];
90 -> 39;

91 [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];
32 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 92 [arrowhead = none ];
92 -> 45;

93 [label="INSTANCE with matching:\nT1 -> T15\nT2 -> T17", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 93 [arrowhead = none , style=dashed];
93 -> 1[style=dashed];

94 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 94 [arrowhead = none ];
94 -> 48;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 95 [arrowhead = none ];
95 -> 52;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 96 [arrowhead = none ];
96 -> 53;

97 [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];
52 -> 97 [arrowhead = none ];
97 -> 57;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 98 [arrowhead = none ];
98 -> 60;

99 [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];
53 -> 99 [arrowhead = none ];
99 -> 67;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 100 [arrowhead = none ];
100 -> 68;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 101 [arrowhead = none ];
101 -> 61;

102 [label="INSTANCE with matching:\nT9 -> T29\nT10 -> cons(T30, T31)\nX16 -> X44\nX17 -> X45", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 102 [arrowhead = none , style=dashed];
102 -> 39[style=dashed];

103 [label="INSTANCE with matching:\nT7 -> T38\nT8 -> cons(T39, T40)\nX14 -> X62\nX15 -> X63", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 103 [arrowhead = none , style=dashed];
103 -> 20[style=dashed];

}

(2) Obligation:

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

samefringeA_in_gg(nil, nil) → samefringeA_out_gg(nil, nil)
samefringeA_in_gg(cons(T7, T8), cons(T9, T10)) → U1_gg(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17) → U3_ggaaggaa(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
pD_in_ggaag(T9, T10, T16, T17, T15) → U5_ggaag(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
gopherC_in_ggaa(nil, T22, nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31, X44, X45) → U2_ggaa(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
U2_ggaa(T29, T30, T31, X44, X45, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)
U5_ggaag(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_ggaag(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U6_ggaag(T9, T10, T16, T17, T15, samefringeA_out_gg(T15, T17)) → pD_out_ggaag(T9, T10, T16, T17, T15)
U3_ggaaggaa(T15, T9, T10, X16, X17, pD_out_ggaag(T9, T10, X16, X17, T15)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_out_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)) → pB_out_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17)
U1_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeA_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
nil  =  nil
samefringeA_out_gg(x1, x2)  =  samefringeA_out_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
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)
pD_in_ggaag(x1, x2, x3, x4, x5)  =  pD_in_ggaag(x1, x2, x5)
U5_ggaag(x1, x2, x3, x4, x5, x6)  =  U5_ggaag(x1, x2, x5, x6)
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gopherC_out_ggaa(x1, x2, x3, x4)  =  gopherC_out_ggaa(x1, x2, x3, x4)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
U6_ggaag(x1, x2, x3, x4, x5, x6)  =  U6_ggaag(x1, x2, x3, x4, x5, x6)
pD_out_ggaag(x1, x2, x3, x4, x5)  =  pD_out_ggaag(x1, x2, x3, x4, x5)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_ggaaggaa(x1, x2, x3, x6, x7, x10)

(3) 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:

SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → U1_GG(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, X14, X15, T9, T10, X16, X17)
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → U3_GGAAGGAA(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → PD_IN_GGAAG(T9, T10, X16, X17, T15)
PD_IN_GGAAG(T9, T10, T16, T17, T15) → U5_GGAAG(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
PD_IN_GGAAG(T9, T10, T16, T17, T15) → GOPHERC_IN_GGAA(T9, T10, T16, T17)
GOPHERC_IN_GGAA(cons(T29, T30), T31, X44, X45) → U2_GGAA(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
GOPHERC_IN_GGAA(cons(T29, T30), T31, X44, X45) → GOPHERC_IN_GGAA(T29, cons(T30, T31), X44, X45)
U5_GGAAG(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_GGAAG(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U5_GGAAG(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEA_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_GGAAGGAA(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → PB_IN_GGAAGGAA(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)

The TRS R consists of the following rules:

samefringeA_in_gg(nil, nil) → samefringeA_out_gg(nil, nil)
samefringeA_in_gg(cons(T7, T8), cons(T9, T10)) → U1_gg(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17) → U3_ggaaggaa(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
pD_in_ggaag(T9, T10, T16, T17, T15) → U5_ggaag(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
gopherC_in_ggaa(nil, T22, nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31, X44, X45) → U2_ggaa(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
U2_ggaa(T29, T30, T31, X44, X45, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)
U5_ggaag(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_ggaag(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U6_ggaag(T9, T10, T16, T17, T15, samefringeA_out_gg(T15, T17)) → pD_out_ggaag(T9, T10, T16, T17, T15)
U3_ggaaggaa(T15, T9, T10, X16, X17, pD_out_ggaag(T9, T10, X16, X17, T15)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_out_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)) → pB_out_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17)
U1_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeA_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
nil  =  nil
samefringeA_out_gg(x1, x2)  =  samefringeA_out_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
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)
pD_in_ggaag(x1, x2, x3, x4, x5)  =  pD_in_ggaag(x1, x2, x5)
U5_ggaag(x1, x2, x3, x4, x5, x6)  =  U5_ggaag(x1, x2, x5, x6)
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gopherC_out_ggaa(x1, x2, x3, x4)  =  gopherC_out_ggaa(x1, x2, x3, x4)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
U6_ggaag(x1, x2, x3, x4, x5, x6)  =  U6_ggaag(x1, x2, x3, x4, x5, x6)
pD_out_ggaag(x1, x2, x3, x4, x5)  =  pD_out_ggaag(x1, x2, x3, x4, x5)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_ggaaggaa(x1, x2, x3, x6, x7, x10)
SAMEFRINGEA_IN_GG(x1, x2)  =  SAMEFRINGEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
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)
PD_IN_GGAAG(x1, x2, x3, x4, x5)  =  PD_IN_GGAAG(x1, x2, x5)
U5_GGAAG(x1, x2, x3, x4, x5, x6)  =  U5_GGAAG(x1, x2, x5, x6)
GOPHERC_IN_GGAA(x1, x2, x3, x4)  =  GOPHERC_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
U6_GGAAG(x1, x2, x3, x4, x5, x6)  =  U6_GGAAG(x1, x2, x3, x4, x5, x6)
U4_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_GGAAGGAA(x1, x2, x3, x6, x7, x10)

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

(4) Obligation:

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

SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → U1_GG(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, X14, X15, T9, T10, X16, X17)
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → U3_GGAAGGAA(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → PD_IN_GGAAG(T9, T10, X16, X17, T15)
PD_IN_GGAAG(T9, T10, T16, T17, T15) → U5_GGAAG(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
PD_IN_GGAAG(T9, T10, T16, T17, T15) → GOPHERC_IN_GGAA(T9, T10, T16, T17)
GOPHERC_IN_GGAA(cons(T29, T30), T31, X44, X45) → U2_GGAA(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
GOPHERC_IN_GGAA(cons(T29, T30), T31, X44, X45) → GOPHERC_IN_GGAA(T29, cons(T30, T31), X44, X45)
U5_GGAAG(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_GGAAG(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U5_GGAAG(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEA_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_GGAAGGAA(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → PB_IN_GGAAGGAA(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)

The TRS R consists of the following rules:

samefringeA_in_gg(nil, nil) → samefringeA_out_gg(nil, nil)
samefringeA_in_gg(cons(T7, T8), cons(T9, T10)) → U1_gg(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17) → U3_ggaaggaa(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
pD_in_ggaag(T9, T10, T16, T17, T15) → U5_ggaag(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
gopherC_in_ggaa(nil, T22, nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31, X44, X45) → U2_ggaa(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
U2_ggaa(T29, T30, T31, X44, X45, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)
U5_ggaag(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_ggaag(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U6_ggaag(T9, T10, T16, T17, T15, samefringeA_out_gg(T15, T17)) → pD_out_ggaag(T9, T10, T16, T17, T15)
U3_ggaaggaa(T15, T9, T10, X16, X17, pD_out_ggaag(T9, T10, X16, X17, T15)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_out_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)) → pB_out_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17)
U1_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeA_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
nil  =  nil
samefringeA_out_gg(x1, x2)  =  samefringeA_out_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
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)
pD_in_ggaag(x1, x2, x3, x4, x5)  =  pD_in_ggaag(x1, x2, x5)
U5_ggaag(x1, x2, x3, x4, x5, x6)  =  U5_ggaag(x1, x2, x5, x6)
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gopherC_out_ggaa(x1, x2, x3, x4)  =  gopherC_out_ggaa(x1, x2, x3, x4)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
U6_ggaag(x1, x2, x3, x4, x5, x6)  =  U6_ggaag(x1, x2, x3, x4, x5, x6)
pD_out_ggaag(x1, x2, x3, x4, x5)  =  pD_out_ggaag(x1, x2, x3, x4, x5)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_ggaaggaa(x1, x2, x3, x6, x7, x10)
SAMEFRINGEA_IN_GG(x1, x2)  =  SAMEFRINGEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
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)
PD_IN_GGAAG(x1, x2, x3, x4, x5)  =  PD_IN_GGAAG(x1, x2, x5)
U5_GGAAG(x1, x2, x3, x4, x5, x6)  =  U5_GGAAG(x1, x2, x5, x6)
GOPHERC_IN_GGAA(x1, x2, x3, x4)  =  GOPHERC_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)
U6_GGAAG(x1, x2, x3, x4, x5, x6)  =  U6_GGAAG(x1, x2, x3, x4, x5, x6)
U4_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_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(T29, T30), T31, X44, X45) → GOPHERC_IN_GGAA(T29, cons(T30, T31), X44, X45)

The TRS R consists of the following rules:

samefringeA_in_gg(nil, nil) → samefringeA_out_gg(nil, nil)
samefringeA_in_gg(cons(T7, T8), cons(T9, T10)) → U1_gg(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17) → U3_ggaaggaa(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
pD_in_ggaag(T9, T10, T16, T17, T15) → U5_ggaag(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
gopherC_in_ggaa(nil, T22, nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31, X44, X45) → U2_ggaa(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
U2_ggaa(T29, T30, T31, X44, X45, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)
U5_ggaag(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_ggaag(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U6_ggaag(T9, T10, T16, T17, T15, samefringeA_out_gg(T15, T17)) → pD_out_ggaag(T9, T10, T16, T17, T15)
U3_ggaaggaa(T15, T9, T10, X16, X17, pD_out_ggaag(T9, T10, X16, X17, T15)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_out_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)) → pB_out_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17)
U1_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeA_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
nil  =  nil
samefringeA_out_gg(x1, x2)  =  samefringeA_out_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
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)
pD_in_ggaag(x1, x2, x3, x4, x5)  =  pD_in_ggaag(x1, x2, x5)
U5_ggaag(x1, x2, x3, x4, x5, x6)  =  U5_ggaag(x1, x2, x5, x6)
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gopherC_out_ggaa(x1, x2, x3, x4)  =  gopherC_out_ggaa(x1, x2, x3, x4)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
U6_ggaag(x1, x2, x3, x4, x5, x6)  =  U6_ggaag(x1, x2, x3, x4, x5, x6)
pD_out_ggaag(x1, x2, x3, x4, x5)  =  pD_out_ggaag(x1, x2, x3, x4, x5)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_ggaaggaa(x1, x2, x3, x6, x7, x10)
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(T29, T30), T31, X44, X45) → GOPHERC_IN_GGAA(T29, cons(T30, T31), X44, X45)

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(T29, T30), T31) → GOPHERC_IN_GGAA(T29, cons(T30, T31))

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(T29, T30), T31) → GOPHERC_IN_GGAA(T29, cons(T30, T31))
  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(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, X14, X15, T9, T10, X16, X17)
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → PD_IN_GGAAG(T9, T10, X16, X17, T15)
PD_IN_GGAAG(T9, T10, T16, T17, T15) → U5_GGAAG(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
U5_GGAAG(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEA_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → PB_IN_GGAAGGAA(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)

The TRS R consists of the following rules:

samefringeA_in_gg(nil, nil) → samefringeA_out_gg(nil, nil)
samefringeA_in_gg(cons(T7, T8), cons(T9, T10)) → U1_gg(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17) → U3_ggaaggaa(T15, T9, T10, X16, X17, pD_in_ggaag(T9, T10, X16, X17, T15))
pD_in_ggaag(T9, T10, T16, T17, T15) → U5_ggaag(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
gopherC_in_ggaa(nil, T22, nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31, X44, X45) → U2_ggaa(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
U2_ggaa(T29, T30, T31, X44, X45, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)
U5_ggaag(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → U6_ggaag(T9, T10, T16, T17, T15, samefringeA_in_gg(T15, T17))
U6_ggaag(T9, T10, T16, T17, T15, samefringeA_out_gg(T15, T17)) → pD_out_ggaag(T9, T10, T16, T17, T15)
U3_ggaaggaa(T15, T9, T10, X16, X17, pD_out_ggaag(T9, T10, X16, X17, T15)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U4_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_out_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)) → pB_out_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17)
U1_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeA_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeA_in_gg(x1, x2)  =  samefringeA_in_gg(x1, x2)
nil  =  nil
samefringeA_out_gg(x1, x2)  =  samefringeA_out_gg(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
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)
pD_in_ggaag(x1, x2, x3, x4, x5)  =  pD_in_ggaag(x1, x2, x5)
U5_ggaag(x1, x2, x3, x4, x5, x6)  =  U5_ggaag(x1, x2, x5, x6)
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gopherC_out_ggaa(x1, x2, x3, x4)  =  gopherC_out_ggaa(x1, x2, x3, x4)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_ggaa(x1, x2, x3, x6)
U6_ggaag(x1, x2, x3, x4, x5, x6)  =  U6_ggaag(x1, x2, x3, x4, x5, x6)
pD_out_ggaag(x1, x2, x3, x4, x5)  =  pD_out_ggaag(x1, x2, x3, x4, x5)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U4_ggaaggaa(x1, x2, x3, x6, x7, x10)
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)
PD_IN_GGAAG(x1, x2, x3, x4, x5)  =  PD_IN_GGAAG(x1, x2, x5)
U5_GGAAG(x1, x2, x3, x4, x5, x6)  =  U5_GGAAG(x1, x2, x5, x6)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, X14, X15, T9, T10, X16, X17)
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → PD_IN_GGAAG(T9, T10, X16, X17, T15)
PD_IN_GGAAG(T9, T10, T16, T17, T15) → U5_GGAAG(T9, T10, T16, T17, T15, gopherC_in_ggaa(T9, T10, T16, T17))
U5_GGAAG(T9, T10, T16, T17, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEA_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → PB_IN_GGAAGGAA(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17)

The TRS R consists of the following rules:

gopherC_in_ggaa(nil, T22, nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31, X44, X45) → U2_ggaa(T29, T30, T31, X44, X45, gopherC_in_ggaa(T29, cons(T30, T31), X44, X45))
U2_ggaa(T29, T30, T31, X44, X45, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
gopherC_in_ggaa(x1, x2, x3, x4)  =  gopherC_in_ggaa(x1, x2)
gopherC_out_ggaa(x1, x2, x3, x4)  =  gopherC_out_ggaa(x1, x2, x3, x4)
U2_ggaa(x1, x2, x3, x4, x5, x6)  =  U2_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)
PD_IN_GGAAG(x1, x2, x3, x4, x5)  =  PD_IN_GGAAG(x1, x2, x5)
U5_GGAAG(x1, x2, x3, x4, x5, x6)  =  U5_GGAAG(x1, x2, x5, x6)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, T9, T10)
PB_IN_GGAAGGAA(nil, T15, T9, T10) → PD_IN_GGAAG(T9, T10, T15)
PD_IN_GGAAG(T9, T10, T15) → U5_GGAAG(T9, T10, T15, gopherC_in_ggaa(T9, T10))
U5_GGAAG(T9, T10, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEA_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, T9, T10) → PB_IN_GGAAGGAA(T38, cons(T39, T40), T9, T10)

The TRS R consists of the following rules:

gopherC_in_ggaa(nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31) → U2_ggaa(T29, T30, T31, gopherC_in_ggaa(T29, cons(T30, T31)))
U2_ggaa(T29, T30, T31, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)

The set Q consists of the following terms:

gopherC_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2, x3)

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

(19) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

SAMEFRINGEA_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, T9, T10)

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

POL(PB_IN_GGAAGGAA(x₁, x₂, x₃, x₄)) = x₁ + x₂   
POL(PD_IN_GGAAG(x₁, x₂, x₃)) = x₃   
POL(SAMEFRINGEA_IN_GG(x₁, x₂)) = x₁   
POL(U2_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U5_GGAAG(x₁, x₂, x₃, x₄)) = x₃   
POL(cons(x₁, x₂)) = 1 + x₁ + x₂   
POL(gopherC_in_ggaa(x₁, x₂)) = 0   
POL(gopherC_out_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(nil) = 0   

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

(20) Obligation:

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

PB_IN_GGAAGGAA(nil, T15, T9, T10) → PD_IN_GGAAG(T9, T10, T15)
PD_IN_GGAAG(T9, T10, T15) → U5_GGAAG(T9, T10, T15, gopherC_in_ggaa(T9, T10))
U5_GGAAG(T9, T10, T15, gopherC_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEA_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, T9, T10) → PB_IN_GGAAGGAA(T38, cons(T39, T40), T9, T10)

The TRS R consists of the following rules:

gopherC_in_ggaa(nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31) → U2_ggaa(T29, T30, T31, gopherC_in_ggaa(T29, cons(T30, T31)))
U2_ggaa(T29, T30, T31, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)

The set Q consists of the following terms:

gopherC_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2, x3)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(22) Obligation:

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

PB_IN_GGAAGGAA(cons(T38, T39), T40, T9, T10) → PB_IN_GGAAGGAA(T38, cons(T39, T40), T9, T10)

The TRS R consists of the following rules:

gopherC_in_ggaa(nil, T22) → gopherC_out_ggaa(nil, T22, nil, T22)
gopherC_in_ggaa(cons(T29, T30), T31) → U2_ggaa(T29, T30, T31, gopherC_in_ggaa(T29, cons(T30, T31)))
U2_ggaa(T29, T30, T31, gopherC_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherC_out_ggaa(cons(T29, T30), T31, X44, X45)

The set Q consists of the following terms:

gopherC_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2, x3)

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

(23) 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.

(24) Obligation:

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

PB_IN_GGAAGGAA(cons(T38, T39), T40, T9, T10) → PB_IN_GGAAGGAA(T38, cons(T39, T40), T9, T10)

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

gopherC_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2, x3)

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

(25) 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].

gopherC_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2, x3)

(26) Obligation:

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

PB_IN_GGAAGGAA(cons(T38, T39), T40, T9, T10) → PB_IN_GGAAGGAA(T38, cons(T39, T40), T9, T10)

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

(27) 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(T38, T39), T40, T9, T10) → PB_IN_GGAAGGAA(T38, cons(T39, T40), T9, T10)
  The graph contains the following edges 1 > 1, 3 >= 3, 4 >= 4

(28) YES