Runtime Complexity of the query pattern
app2(a,g,g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).

Query: app2(a,g,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

app1([], Ys, Ys).
app2([], Ys, Ys).
app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).

Query: app2(a,g,g)

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: app2(T1, T2, T3)\n\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: app2(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | app2(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | app2(T1, T6, T6), SCOPE: 1, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: app2(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nT3 is ground\nX3 is free\napp2(T1, T2, T3) !=? app2([], X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app2(T1, T6, T6), SCOPE: 1, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app2(T19, .(T15, T18), T18)\n\nT15 is ground\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app2(T36, T34, T35)\n\nT34 is ground\nT32 is ground\nT35 is ground\nX3 is free\napp2(T1, T34, .(T32, T35)) !=? app2([], X3, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 27 [arrowhead = none ];
27 -> 4;

28 [label="EVAL with clause\napp2([], X3, X3).\nand substitutionT1 -> [],\nT2 -> T6,\nX3 -> T6,\nT3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 28 [arrowhead = none ];
28 -> 6;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 29 [arrowhead = none ];
29 -> 8;

30 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 30 [arrowhead = none ];
30 -> 10;

31 [label="EVAL with clause\napp2(.(X26, X27), X28, .(X26, X29)) :- app2(X27, X28, X29).\nand substitutionX26 -> T32,\nX27 -> T36,\nT1 -> .(T32, T36),\nT2 -> T34,\nX28 -> T34,\nX29 -> T35,\nT3 -> .(T32, T35),\nT33 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 31 [arrowhead = none ];
31 -> 25;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 32 [arrowhead = none ];
32 -> 26;

33 [label="EVAL with clause\napp2(.(X12, X13), X14, .(X12, X15)) :- app2(X13, X14, X15).\nand substitutionX12 -> T15,\nX13 -> T19,\nT1 -> .(T15, T19),\nT6 -> .(T15, T18),\nX14 -> .(T15, T18),\nX15 -> T18,\nT17 -> .(T15, T18),\nT16 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 33 [arrowhead = none ];
33 -> 15;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 34 [arrowhead = none ];
34 -> 16;

35 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> .(T15, T18)\nT3 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 35 [arrowhead = none , style=dashed];
35 -> 2[style=dashed];

36 [label="INSTANCE with matching:\nT1 -> T36\nT2 -> T34\nT3 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 36 [arrowhead = none , style=dashed];
36 -> 2[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z0) → f2_out1([])
f2_in(.(z0, z1), .(z0, z1)) → U1(f2_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f2_in(z0, .(z1, z2)) → U2(f2_in(z0, z2), z0, .(z1, z2))
U1(f2_out1(z0), .(z1, z2), .(z1, z2)) → f2_out1(.(z1, z0))
U2(f2_out1(z0), z1, .(z2, z3)) → f2_out1(.(z2, z0))

Tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(U1'(f2_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(U2'(f2_in(z0, z2), z0, .(z1, z2)), F2_IN(z0, z2))

S tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(U1'(f2_in(.(z0, z1), z1), .(z0, z1), .(z0, z1)), F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(U2'(f2_in(z0, z2), z0, .(z1, z2)), F2_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2

(5) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z0) → f2_out1([])
f2_in(.(z0, z1), .(z0, z1)) → U1(f2_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f2_in(z0, .(z1, z2)) → U2(f2_in(z0, z2), z0, .(z1, z2))
U1(f2_out1(z0), .(z1, z2), .(z1, z2)) → f2_out1(.(z1, z0))
U2(f2_out1(z0), z1, .(z2, z3)) → f2_out1(.(z2, z0))

Tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(F2_IN(z0, z2))

S tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(F2_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F2_IN(.(z0, z1), .(z0, z1)) → c1(F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(F2_IN(z0, z2))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(F2_IN(z0, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₂   
POL(F2_IN(x₁, x₂)) = x₂   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z0) → f2_out1([])
f2_in(.(z0, z1), .(z0, z1)) → U1(f2_in(.(z0, z1), z1), .(z0, z1), .(z0, z1))
f2_in(z0, .(z1, z2)) → U2(f2_in(z0, z2), z0, .(z1, z2))
U1(f2_out1(z0), .(z1, z2), .(z1, z2)) → f2_out1(.(z1, z0))
U2(f2_out1(z0), z1, .(z2, z3)) → f2_out1(.(z2, z0))

Tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(F2_IN(z0, z2))

S tuples:none
K tuples:

F2_IN(.(z0, z1), .(z0, z1)) → c1(F2_IN(.(z0, z1), z1))
F2_IN(z0, .(z1, z2)) → c2(F2_IN(z0, z2))

Defined Rule Symbols:

f2_in, U1, U2

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))

(11) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: app2(T1, T2, T3)\n\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: app2(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | app2(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | app2(T1, T5, T5), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: app2(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nT3 is ground\nX2 is free\napp2(T1, T2, T3) !=? app2([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: app2(T1, T5, T5), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app2(T14, .(T10, T13), T13)\n\nT10 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app2(T14, .(T10, T13), T13), SCOPE: 2, CLAUSE: 1 | app2(T14, .(T10, T13), T13), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app2(T14, .(T10, T13), T13), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app2(T32, .(T29, .(T27, T31)), T31)\n\nT29 is ground\nT27 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app2(T47, T45, T46)\n\nT45 is ground\nT43 is ground\nT46 is ground\nX2 is free\napp2(T1, T45, .(T43, T46)) !=? app2([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app2(T47, T45, T46), SCOPE: 3, CLAUSE: 1 | app2(T47, T45, T46), SCOPE: 3, CLAUSE: 3\n\nT45 is ground\nT43 is ground\nT46 is ground\nX2 is free\napp2(T1, T45, .(T43, T46)) !=? app2([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | app2(T47, T50, T50), SCOPE: 3, CLAUSE: 3\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app2(T47, T45, T46), SCOPE: 3, CLAUSE: 3\n\nT45 is ground\nT43 is ground\nT46 is ground\nX2 is free\nX36 is free\napp2(T1, T45, .(T43, T46)) !=? app2([], X2, X2)\napp2(T47, T45, T46) !=? app2([], X36, X36)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app2(T47, T50, T50), SCOPE: 3, CLAUSE: 3\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app2(T63, .(T59, T62), T62)\n\nT59 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app2(T80, T78, T79)\n\nT43 is ground\nT78 is ground\nT76 is ground\nT79 is ground\nX2 is free\nX36 is free\napp2(T1, T78, .(T43, .(T76, T79))) !=? app2([], X2, X2)\napp2(T47, T78, .(T76, T79)) !=? app2([], X36, X36)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 3;

32 [label="EVAL with clause\napp2([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="EVAL with clause\napp2(.(X30, X31), X32, .(X30, X33)) :- app2(X31, X32, X33).\nand substitutionX30 -> T43,\nX31 -> T47,\nT1 -> .(T43, T47),\nT2 -> T45,\nX32 -> T45,\nX33 -> T46,\nT3 -> .(T43, T46),\nT44 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 35 [arrowhead = none ];
35 -> 19;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 36 [arrowhead = none ];
36 -> 20;

37 [label="EVAL with clause\napp2(.(X7, X8), X9, .(X7, X10)) :- app2(X8, X9, X10).\nand substitutionX7 -> T10,\nX8 -> T14,\nT1 -> .(T10, T14),\nT5 -> .(T10, T13),\nX9 -> .(T10, T13),\nX10 -> T13,\nT12 -> .(T10, T13),\nT11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="BACKTRACK\nfor clause: app2([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 40 [arrowhead = none ];
40 -> 14;

41 [label="EVAL with clause\napp2(.(X20, X21), X22, .(X20, X23)) :- app2(X21, X22, X23).\nand substitutionX20 -> T27,\nX21 -> T32,\nT14 -> .(T27, T32),\nT10 -> T29,\nT13 -> .(T27, T31),\nX22 -> .(T29, .(T27, T31)),\nX23 -> T31,\nT30 -> .(T27, T31),\nT28 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 41 [arrowhead = none ];
41 -> 17;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 42 [arrowhead = none ];
42 -> 18;

43 [label="INSTANCE with matching:\nT1 -> T32\nT2 -> .(T29, .(T27, T31))\nT3 -> T31", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 43 [arrowhead = none , style=dashed];
43 -> 1[style=dashed];

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 44 [arrowhead = none ];
44 -> 21;

45 [label="EVAL with clause\napp2([], X36, X36).\nand substitutionT47 -> [],\nT45 -> T50,\nX36 -> T50,\nT46 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 45 [arrowhead = none ];
45 -> 22;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 46 [arrowhead = none ];
46 -> 23;

47 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 47 [arrowhead = none ];
47 -> 24;

48 [label="EVAL with clause\napp2(.(X59, X60), X61, .(X59, X62)) :- app2(X60, X61, X62).\nand substitutionX59 -> T76,\nX60 -> T80,\nT47 -> .(T76, T80),\nT45 -> T78,\nX61 -> T78,\nX62 -> T79,\nT46 -> .(T76, T79),\nT77 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 48 [arrowhead = none ];
48 -> 29;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 49 [arrowhead = none ];
49 -> 30;

50 [label="EVAL with clause\napp2(.(X45, X46), X47, .(X45, X48)) :- app2(X46, X47, X48).\nand substitutionX45 -> T59,\nX46 -> T63,\nT47 -> .(T59, T63),\nT50 -> .(T59, T62),\nX47 -> .(T59, T62),\nX48 -> T62,\nT61 -> .(T59, T62),\nT60 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 50 [arrowhead = none ];
50 -> 27;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 51 [arrowhead = none ];
51 -> 28;

52 [label="INSTANCE with matching:\nT1 -> T63\nT2 -> .(T59, T62)\nT3 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 52 [arrowhead = none , style=dashed];
52 -> 1[style=dashed];

53 [label="INSTANCE with matching:\nT1 -> T80\nT2 -> T78\nT3 -> T79", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 53 [arrowhead = none , style=dashed];
53 -> 1[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z0) → f1_out1([])
f1_in(.(z0, .(z1, z2)), .(z0, .(z1, z2))) → U1(f1_in(.(z0, .(z1, z2)), z2), .(z0, .(z1, z2)), .(z0, .(z1, z2)))
f1_in(z0, .(z1, z0)) → f1_out1(.(z1, []))
f1_in(.(z0, z1), .(z2, .(z0, z1))) → U2(f1_in(.(z0, z1), z1), .(z0, z1), .(z2, .(z0, z1)))
f1_in(z0, .(z1, .(z2, z3))) → U3(f1_in(z0, z3), z0, .(z1, .(z2, z3)))
U1(f1_out1(z0), .(z1, .(z2, z3)), .(z1, .(z2, z3))) → f1_out1(.(z1, .(z2, z0)))
U2(f1_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f1_out1(.(z3, .(z1, z0)))
U3(f1_out1(z0), z1, .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)))

Tuples:

F1_IN(.(z0, .(z1, z2)), .(z0, .(z1, z2))) → c1(U1'(f1_in(.(z0, .(z1, z2)), z2), .(z0, .(z1, z2)), .(z0, .(z1, z2))), F1_IN(.(z0, .(z1, z2)), z2))
F1_IN(.(z0, z1), .(z2, .(z0, z1))) → c3(U2'(f1_in(.(z0, z1), z1), .(z0, z1), .(z2, .(z0, z1))), F1_IN(.(z0, z1), z1))
F1_IN(z0, .(z1, .(z2, z3))) → c4(U3'(f1_in(z0, z3), z0, .(z1, .(z2, z3))), F1_IN(z0, z3))

S tuples:

F1_IN(.(z0, .(z1, z2)), .(z0, .(z1, z2))) → c1(U1'(f1_in(.(z0, .(z1, z2)), z2), .(z0, .(z1, z2)), .(z0, .(z1, z2))), F1_IN(.(z0, .(z1, z2)), z2))
F1_IN(.(z0, z1), .(z2, .(z0, z1))) → c3(U2'(f1_in(.(z0, z1), z1), .(z0, z1), .(z2, .(z0, z1))), F1_IN(.(z0, z1), z1))
F1_IN(z0, .(z1, .(z2, z3))) → c4(U3'(f1_in(z0, z3), z0, .(z1, .(z2, z3))), F1_IN(z0, z3))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c3, c4

(13) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z0) → f1_out1([])
f1_in(.(z0, .(z1, z2)), .(z0, .(z1, z2))) → U1(f1_in(.(z0, .(z1, z2)), z2), .(z0, .(z1, z2)), .(z0, .(z1, z2)))
f1_in(z0, .(z1, z0)) → f1_out1(.(z1, []))
f1_in(.(z0, z1), .(z2, .(z0, z1))) → U2(f1_in(.(z0, z1), z1), .(z0, z1), .(z2, .(z0, z1)))
f1_in(z0, .(z1, .(z2, z3))) → U3(f1_in(z0, z3), z0, .(z1, .(z2, z3)))
U1(f1_out1(z0), .(z1, .(z2, z3)), .(z1, .(z2, z3))) → f1_out1(.(z1, .(z2, z0)))
U2(f1_out1(z0), .(z1, z2), .(z3, .(z1, z2))) → f1_out1(.(z3, .(z1, z0)))
U3(f1_out1(z0), z1, .(z2, .(z3, z4))) → f1_out1(.(z2, .(z3, z0)))

Tuples:

F1_IN(.(z0, .(z1, z2)), .(z0, .(z1, z2))) → c1(F1_IN(.(z0, .(z1, z2)), z2))
F1_IN(.(z0, z1), .(z2, .(z0, z1))) → c3(F1_IN(.(z0, z1), z1))
F1_IN(z0, .(z1, .(z2, z3))) → c4(F1_IN(z0, z3))

S tuples:

F1_IN(.(z0, .(z1, z2)), .(z0, .(z1, z2))) → c1(F1_IN(.(z0, .(z1, z2)), z2))
F1_IN(.(z0, z1), .(z2, .(z0, z1))) → c3(F1_IN(.(z0, z1), z1))
F1_IN(z0, .(z1, .(z2, z3))) → c4(F1_IN(z0, z3))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c3, c4