Left Termination proof of /tmp/tmpArbNuc/samefringe.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 84 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 23 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 MRRProof (⇔, 84 ms)

        ↳22 QDP

        ↳23 DependencyGraphProof (⇔, 2 ms)

        ↳24 QDP

        ↳25 UsableRulesProof (⇔, 0 ms)

        ↳26 QDP

        ↳27 QReductionProof (⇔, 0 ms)

        ↳28 QDP

        ↳29 QDPSizeChangeProof (⇔, 0 ms)

        ↳30 YES

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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="7: ','(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="15: 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:

Clauses:

gopherA(nil, T22, nil, T22).
gopherA(cons(T29, T30), T31, X44, X45) :- gopherA(T29, cons(T30, T31), X44, X45).
pB(nil, T15, nil, T15, T9, T10, X16, X17) :- gopherA(T9, T10, X16, X17).
pB(nil, T15, nil, T15, T9, T10, T16, T17) :- ','(gopherA(T9, T10, T16, T17), samefringeC(T15, T17)).
pB(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) :- pB(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17).
samefringeC(nil, nil).
samefringeC(cons(T7, T8), cons(T9, T10)) :- pB(T7, T8, X14, X15, T9, T10, X16, X17).

Query: samefringeC(g,g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
samefringeC_in: (b,b)
pB_in: (b,b,f,f,b,b,f,f)
gopherA_in: (b,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

samefringeC_in_gg(nil, nil) → samefringeC_out_gg(nil, nil)
samefringeC_in_gg(cons(T7, T8), cons(T9, T10)) → U6_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) → U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)
U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_out_ggaa(T9, T10, X16, X17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17) → U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_out_gg(T15, T17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U5_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)
U6_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeC_out_gg(cons(T7, T8), cons(T9, T10))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

samefringeC_in_gg(nil, nil) → samefringeC_out_gg(nil, nil)
samefringeC_in_gg(cons(T7, T8), cons(T9, T10)) → U6_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) → U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)
U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_out_ggaa(T9, T10, X16, X17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17) → U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_out_gg(T15, T17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U5_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)
U6_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeC_out_gg(cons(T7, T8), cons(T9, T10))

(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:

SAMEFRINGEC_IN_GG(cons(T7, T8), cons(T9, T10)) → U6_GG(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
SAMEFRINGEC_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) → U2_GGAAGGAA(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → GOPHERA_IN_GGAA(T9, T10, X16, X17)
GOPHERA_IN_GGAA(cons(T29, T30), T31, X44, X45) → U1_GGAA(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
GOPHERA_IN_GGAA(cons(T29, T30), T31, X44, X45) → GOPHERA_IN_GGAA(T29, cons(T30, T31), X44, X45)
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, T16, T17) → U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_GGAAGGAA(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEC_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_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:

samefringeC_in_gg(nil, nil) → samefringeC_out_gg(nil, nil)
samefringeC_in_gg(cons(T7, T8), cons(T9, T10)) → U6_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) → U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)
U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_out_ggaa(T9, T10, X16, X17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17) → U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_out_gg(T15, T17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U5_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)
U6_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeC_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeC_in_gg(x1, x2) = samefringeC_in_gg(x1, x2)
nil = nil
samefringeC_out_gg(x1, x2) = samefringeC_out_gg
cons(x1, x2) = cons(x1, x2)
U6_gg(x1, x2, x3, x4, x5) = U6_gg(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, x6)
gopherA_in_ggaa(x1, x2, x3, x4) = gopherA_in_ggaa(x1, x2)
gopherA_out_ggaa(x1, x2, x3, x4) = gopherA_out_ggaa(x3, x4)
U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x6)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = pB_out_ggaaggaa(x3, x4, x7, x8)
U3_ggaaggaa(x1, x2, x3, x4, x5, x6) = U3_ggaaggaa(x1, x6)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6) = U4_ggaaggaa(x1, x4, x5, x6)
U5_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U5_ggaaggaa(x10)
SAMEFRINGEC_IN_GG(x1, x2) = SAMEFRINGEC_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5) = U6_GG(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, x6)
GOPHERA_IN_GGAA(x1, x2, x3, x4) = GOPHERA_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6) = U1_GGAA(x6)
U3_GGAAGGAA(x1, x2, x3, x4, x5, x6) = U3_GGAAGGAA(x1, x6)
U4_GGAAGGAA(x1, x2, x3, x4, x5, x6) = U4_GGAAGGAA(x1, x4, x5, x6)
U5_GGAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U5_GGAAGGAA(x10)

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

(6) Obligation:

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

SAMEFRINGEC_IN_GG(cons(T7, T8), cons(T9, T10)) → U6_GG(T7, T8, T9, T10, pB_in_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17))
SAMEFRINGEC_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) → U2_GGAAGGAA(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, X16, X17) → GOPHERA_IN_GGAA(T9, T10, X16, X17)
GOPHERA_IN_GGAA(cons(T29, T30), T31, X44, X45) → U1_GGAA(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
GOPHERA_IN_GGAA(cons(T29, T30), T31, X44, X45) → GOPHERA_IN_GGAA(T29, cons(T30, T31), X44, X45)
PB_IN_GGAAGGAA(nil, T15, nil, T15, T9, T10, T16, T17) → U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_GGAAGGAA(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEC_IN_GG(T15, T17)
PB_IN_GGAAGGAA(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_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:

samefringeC_in_gg(nil, nil) → samefringeC_out_gg(nil, nil)
samefringeC_in_gg(cons(T7, T8), cons(T9, T10)) → U6_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) → U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)
U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_out_ggaa(T9, T10, X16, X17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17) → U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_out_gg(T15, T17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U5_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)
U6_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeC_out_gg(cons(T7, T8), cons(T9, T10))

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

GOPHERA_IN_GGAA(cons(T29, T30), T31, X44, X45) → GOPHERA_IN_GGAA(T29, cons(T30, T31), X44, X45)

The TRS R consists of the following rules:

samefringeC_in_gg(nil, nil) → samefringeC_out_gg(nil, nil)
samefringeC_in_gg(cons(T7, T8), cons(T9, T10)) → U6_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) → U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)
U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_out_ggaa(T9, T10, X16, X17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17) → U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_out_gg(T15, T17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U5_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)
U6_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeC_out_gg(cons(T7, T8), cons(T9, T10))

(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:

GOPHERA_IN_GGAA(cons(T29, T30), T31, X44, X45) → GOPHERA_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)
GOPHERA_IN_GGAA(x1, x2, x3, x4) = GOPHERA_IN_GGAA(x1, x2)

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:

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

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:

GOPHERA_IN_GGAA(cons(T29, T30), T31) → GOPHERA_IN_GGAA(T29, cons(T30, T31))
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

SAMEFRINGEC_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, T16, T17) → U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEC_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:

samefringeC_in_gg(nil, nil) → samefringeC_out_gg(nil, nil)
samefringeC_in_gg(cons(T7, T8), cons(T9, T10)) → U6_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) → U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_in_ggaa(T9, T10, X16, X17))
gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)
U2_ggaaggaa(T15, T9, T10, X16, X17, gopherA_out_ggaa(T9, T10, X16, X17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, X16, X17)
pB_in_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17) → U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_ggaaggaa(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_in_gg(T15, T17))
U4_ggaaggaa(T15, T9, T10, T16, T17, samefringeC_out_gg(T15, T17)) → pB_out_ggaaggaa(nil, T15, nil, T15, T9, T10, T16, T17)
pB_in_ggaaggaa(cons(T38, T39), T40, X62, X63, T9, T10, X16, X17) → U5_ggaaggaa(T38, T39, T40, X62, X63, T9, T10, X16, X17, pB_in_ggaaggaa(T38, cons(T39, T40), X62, X63, T9, T10, X16, X17))
U5_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)
U6_gg(T7, T8, T9, T10, pB_out_ggaaggaa(T7, T8, X14, X15, T9, T10, X16, X17)) → samefringeC_out_gg(cons(T7, T8), cons(T9, T10))

The argument filtering Pi contains the following mapping:
samefringeC_in_gg(x1, x2) = samefringeC_in_gg(x1, x2)
nil = nil
samefringeC_out_gg(x1, x2) = samefringeC_out_gg
cons(x1, x2) = cons(x1, x2)
U6_gg(x1, x2, x3, x4, x5) = U6_gg(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, x6)
gopherA_in_ggaa(x1, x2, x3, x4) = gopherA_in_ggaa(x1, x2)
gopherA_out_ggaa(x1, x2, x3, x4) = gopherA_out_ggaa(x3, x4)
U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x6)
pB_out_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = pB_out_ggaaggaa(x3, x4, x7, x8)
U3_ggaaggaa(x1, x2, x3, x4, x5, x6) = U3_ggaaggaa(x1, x6)
U4_ggaaggaa(x1, x2, x3, x4, x5, x6) = U4_ggaaggaa(x1, x4, x5, x6)
U5_ggaaggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U5_ggaaggaa(x10)
SAMEFRINGEC_IN_GG(x1, x2) = SAMEFRINGEC_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, x6)

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

(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:

SAMEFRINGEC_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, T16, T17) → U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_in_ggaa(T9, T10, T16, T17))
U3_GGAAGGAA(T15, T9, T10, T16, T17, gopherA_out_ggaa(T9, T10, T16, T17)) → SAMEFRINGEC_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:

gopherA_in_ggaa(nil, T22, nil, T22) → gopherA_out_ggaa(nil, T22, nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31, X44, X45) → U1_ggaa(T29, T30, T31, X44, X45, gopherA_in_ggaa(T29, cons(T30, T31), X44, X45))
U1_ggaa(T29, T30, T31, X44, X45, gopherA_out_ggaa(T29, cons(T30, T31), X44, X45)) → gopherA_out_ggaa(cons(T29, T30), T31, X44, X45)

The argument filtering Pi contains the following mapping:
nil = nil
cons(x1, x2) = cons(x1, x2)
gopherA_in_ggaa(x1, x2, x3, x4) = gopherA_in_ggaa(x1, x2)
gopherA_out_ggaa(x1, x2, x3, x4) = gopherA_out_ggaa(x3, x4)
U1_ggaa(x1, x2, x3, x4, x5, x6) = U1_ggaa(x6)
SAMEFRINGEC_IN_GG(x1, x2) = SAMEFRINGEC_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, x6)

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:

SAMEFRINGEC_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, T9, T10)
PB_IN_GGAAGGAA(nil, T15, T9, T10) → U3_GGAAGGAA(T15, gopherA_in_ggaa(T9, T10))
U3_GGAAGGAA(T15, gopherA_out_ggaa(T16, T17)) → SAMEFRINGEC_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:

gopherA_in_ggaa(nil, T22) → gopherA_out_ggaa(nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31) → U1_ggaa(gopherA_in_ggaa(T29, cons(T30, T31)))
U1_ggaa(gopherA_out_ggaa(X44, X45)) → gopherA_out_ggaa(X44, X45)

The set Q consists of the following terms:

gopherA_in_ggaa(x0, x1)
U1_ggaa(x0)

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

(21) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

PB_IN_GGAAGGAA(nil, T15, T9, T10) → U3_GGAAGGAA(T15, gopherA_in_ggaa(T9, T10))

Used ordering: Polynomial interpretation [POLO]:

POL(PB_IN_GGAAGGAA(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(SAMEFRINGEC_IN_GG(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U1_ggaa(x₁)) = x₁
POL(U3_GGAAGGAA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(gopherA_in_ggaa(x₁, x₂)) = x₁ + x₂
POL(gopherA_out_ggaa(x₁, x₂)) = x₁ + x₂
POL(nil) = 2

(22) Obligation:

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

SAMEFRINGEC_IN_GG(cons(T7, T8), cons(T9, T10)) → PB_IN_GGAAGGAA(T7, T8, T9, T10)
U3_GGAAGGAA(T15, gopherA_out_ggaa(T16, T17)) → SAMEFRINGEC_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:

gopherA_in_ggaa(nil, T22) → gopherA_out_ggaa(nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31) → U1_ggaa(gopherA_in_ggaa(T29, cons(T30, T31)))
U1_ggaa(gopherA_out_ggaa(X44, X45)) → gopherA_out_ggaa(X44, X45)

The set Q consists of the following terms:

gopherA_in_ggaa(x0, x1)
U1_ggaa(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(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)

The TRS R consists of the following rules:

gopherA_in_ggaa(nil, T22) → gopherA_out_ggaa(nil, T22)
gopherA_in_ggaa(cons(T29, T30), T31) → U1_ggaa(gopherA_in_ggaa(T29, cons(T30, T31)))
U1_ggaa(gopherA_out_ggaa(X44, X45)) → gopherA_out_ggaa(X44, X45)

The set Q consists of the following terms:

gopherA_in_ggaa(x0, x1)
U1_ggaa(x0)

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

(25) 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.

(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.
The set Q consists of the following terms:

gopherA_in_ggaa(x0, x1)
U1_ggaa(x0)

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

(27) 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].

gopherA_in_ggaa(x0, x1)
U1_ggaa(x0)

(28) 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.

(29) 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

(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) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) MRRProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Obligation:

(25) UsableRulesProof (EQUIVALENT transformation)

(26) Obligation:

(27) QReductionProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPSizeChangeProof (EQUIVALENT transformation)

(30) YES