Runtime Complexity of the query pattern
app3_a(g,g,g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app([], Ys, Ys).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: app3_a(g,g,g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

app([], Ys, Ys).
app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: app3_a(g,g,g,a)

(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: app3_a(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: app3_a(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app(T16, T17, X16), app(X16, T18, T20))\n\nT16 is ground\nT17 is ground\nT18 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app(T16, T17, X16)\n\nT16 is ground\nT17 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app(T23, T18, T20)\n\nT18 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(T16, T17, X16), SCOPE: 2, CLAUSE: 0 | app(T16, T17, X16), SCOPE: 2, CLAUSE: 3\n\nT16 is ground\nT17 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true | app([], T28, X16), SCOPE: 2, CLAUSE: 3\n\nT28 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T16, T17, X16), SCOPE: 2, CLAUSE: 3\n\nT16 is ground\nT17 is ground\nX16 is free\nX23 is free\napp(T16, T17, X16) !=? app([], X23, X23)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app([], T28, X16), SCOPE: 2, CLAUSE: 3\n\nT28 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app(T36, T37, X42)\n\nT36 is ground\nT37 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T23, T18, T20), SCOPE: 3, CLAUSE: 0 | app(T23, T18, T20), SCOPE: 3, CLAUSE: 3\n\nT18 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true | app([], T44, T20), SCOPE: 3, CLAUSE: 3\n\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T23, T18, T20), SCOPE: 3, CLAUSE: 3\n\nT18 is ground\nT23 is ground\nX49 is free\napp(T23, T18, T20) !=? app([], X49, X49)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app([], T44, T20), SCOPE: 3, CLAUSE: 3\n\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: app(T54, T55, T57)\n\nT54 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 41 [arrowhead = none ];
41 -> 4;

42 [label="ONLY EVAL with clause\napp3_a(X12, X13, X14, X15) :- ','(app(X12, X13, X16), app(X16, X14, X15)).\nand substitutionT1 -> T16,\nX12 -> T16,\nT2 -> T17,\nX13 -> T17,\nT3 -> T18,\nX14 -> T18,\nT4 -> T20,\nX15 -> T20,\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 42 [arrowhead = none ];
42 -> 7;

43 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 43 [arrowhead = none ];
43 -> 13;

44 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nT17 is ground\nT23 is ground\nreplacements:X16 -> T23", fontsize=14, style = filled, fillcolor = violet];
7 -> 44 [arrowhead = none ];
44 -> 14;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 45 [arrowhead = none ];
45 -> 18;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 46 [arrowhead = none ];
46 -> 29;

47 [label="EVAL with clause\napp([], X23, X23).\nand substitutionT16 -> [],\nT17 -> T28,\nX23 -> T28,\nX16 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 47 [arrowhead = none ];
47 -> 20;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="EVAL with clause\napp(.(X38, X39), X40, .(X38, X41)) :- app(X39, X40, X41).\nand substitutionX38 -> T35,\nX39 -> T36,\nT16 -> .(T35, T36),\nT17 -> T37,\nX40 -> T37,\nX41 -> X42,\nX16 -> .(T35, X42)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 50 [arrowhead = none ];
50 -> 26;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 51 [arrowhead = none ];
51 -> 27;

52 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 52 [arrowhead = none ];
52 -> 23;

53 [label="INSTANCE with matching:\nT16 -> T36\nT17 -> T37\nX16 -> X42", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 53 [arrowhead = none , style=dashed];
53 -> 13[style=dashed];

54 [label="EVAL with clause\napp([], X49, X49).\nand substitutionT23 -> [],\nT18 -> T44,\nX49 -> T44,\nT20 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 54 [arrowhead = none ];
54 -> 30;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 55 [arrowhead = none ];
55 -> 31;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 56 [arrowhead = none ];
56 -> 32;

57 [label="EVAL with clause\napp(.(X62, X63), X64, .(X62, X65)) :- app(X63, X64, X65).\nand substitutionX62 -> T53,\nX63 -> T54,\nT23 -> .(T53, T54),\nT18 -> T55,\nX64 -> T55,\nX65 -> T57,\nT20 -> .(T53, T57),\nT56 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 57 [arrowhead = none ];
57 -> 39;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 58 [arrowhead = none ];
58 -> 40;

59 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 59 [arrowhead = none ];
59 -> 33;

60 [label="INSTANCE with matching:\nT23 -> T54\nT18 -> T55\nT20 -> T57", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 60 [arrowhead = none , style=dashed];
60 -> 14[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1, z2) → U1(f7_in(z0, z1, z2), z0, z1, z2)
U1(f7_out1(z0, z1), z2, z3, z4) → f2_out1(z1)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
f14_in([], z0) → f14_out1(z0)
f14_in(.(z0, z1), z2) → U3(f14_in(z1, z2), .(z0, z1), z2)
U3(f14_out1(z0), .(z1, z2), z3) → f14_out1(.(z1, z0))
f7_in(z0, z1, z2) → U4(f13_in(z0, z1), z0, z1, z2)
U4(f13_out1(z0), z1, z2, z3) → U5(f14_in(z0, z3), z1, z2, z3, z0)
U5(f14_out1(z0), z1, z2, z3, z4) → f7_out1(z4, z0)

Tuples:

F2_IN(z0, z1, z2) → c(U1'(f7_in(z0, z1, z2), z0, z1, z2), F7_IN(z0, z1, z2))
F13_IN(.(z0, z1), z2) → c3(U2'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(U3'(f14_in(z1, z2), .(z0, z1), z2), F14_IN(z1, z2))
F7_IN(z0, z1, z2) → c8(U4'(f13_in(z0, z1), z0, z1, z2), F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c9(U5'(f14_in(z0, z3), z1, z2, z3, z0), F14_IN(z0, z3))

S tuples:

F2_IN(z0, z1, z2) → c(U1'(f7_in(z0, z1, z2), z0, z1, z2), F7_IN(z0, z1, z2))
F13_IN(.(z0, z1), z2) → c3(U2'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(U3'(f14_in(z1, z2), .(z0, z1), z2), F14_IN(z1, z2))
F7_IN(z0, z1, z2) → c8(U4'(f13_in(z0, z1), z0, z1, z2), F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c9(U5'(f14_in(z0, z3), z1, z2, z3, z0), F14_IN(z0, z3))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f13_in, U2, f14_in, U3, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F13_IN, F14_IN, F7_IN, U4'

Compound Symbols:

c, c3, c6, c8, c9

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1, z2) → U1(f7_in(z0, z1, z2), z0, z1, z2)
U1(f7_out1(z0, z1), z2, z3, z4) → f2_out1(z1)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
f14_in([], z0) → f14_out1(z0)
f14_in(.(z0, z1), z2) → U3(f14_in(z1, z2), .(z0, z1), z2)
U3(f14_out1(z0), .(z1, z2), z3) → f14_out1(.(z1, z0))
f7_in(z0, z1, z2) → U4(f13_in(z0, z1), z0, z1, z2)
U4(f13_out1(z0), z1, z2, z3) → U5(f14_in(z0, z3), z1, z2, z3, z0)
U5(f14_out1(z0), z1, z2, z3, z4) → f7_out1(z4, z0)

Tuples:

F13_IN(.(z0, z1), z2) → c3(U2'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(U3'(f14_in(z1, z2), .(z0, z1), z2), F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1(U1'(f7_in(z0, z1, z2), z0, z1, z2))
F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(U5'(f14_in(z0, z3), z1, z2, z3, z0))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))

S tuples:

F13_IN(.(z0, z1), z2) → c3(U2'(f13_in(z1, z2), .(z0, z1), z2), F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(U3'(f14_in(z1, z2), .(z0, z1), z2), F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1(U1'(f7_in(z0, z1, z2), z0, z1, z2))
F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(U5'(f14_in(z0, z3), z1, z2, z3, z0))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f13_in, U2, f14_in, U3, f7_in, U4, U5

Defined Pair Symbols:

F13_IN, F14_IN, F2_IN, F7_IN, U4'

Compound Symbols:

c3, c6, c1

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1, z2) → U1(f7_in(z0, z1, z2), z0, z1, z2)
U1(f7_out1(z0, z1), z2, z3, z4) → f2_out1(z1)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
f14_in([], z0) → f14_out1(z0)
f14_in(.(z0, z1), z2) → U3(f14_in(z1, z2), .(z0, z1), z2)
U3(f14_out1(z0), .(z1, z2), z3) → f14_out1(.(z1, z0))
f7_in(z0, z1, z2) → U4(f13_in(z0, z1), z0, z1, z2)
U4(f13_out1(z0), z1, z2, z3) → U5(f14_in(z0, z3), z1, z2, z3, z0)
U5(f14_out1(z0), z1, z2, z3, z4) → f7_out1(z4, z0)

Tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

S tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f13_in, U2, f14_in, U3, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F7_IN, U4', F13_IN, F14_IN

Compound Symbols:

c1, c3, c6, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
U4'(f13_out1(z0), z1, z2, z3) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1, z2) → U1(f7_in(z0, z1, z2), z0, z1, z2)
U1(f7_out1(z0, z1), z2, z3, z4) → f2_out1(z1)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
f14_in([], z0) → f14_out1(z0)
f14_in(.(z0, z1), z2) → U3(f14_in(z1, z2), .(z0, z1), z2)
U3(f14_out1(z0), .(z1, z2), z3) → f14_out1(.(z1, z0))
f7_in(z0, z1, z2) → U4(f13_in(z0, z1), z0, z1, z2)
U4(f13_out1(z0), z1, z2, z3) → U5(f14_in(z0, z3), z1, z2, z3, z0)
U5(f14_out1(z0), z1, z2, z3, z4) → f7_out1(z4, z0)

Tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

S tuples:

F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))

K tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

Defined Rule Symbols:

f2_in, U1, f13_in, U2, f14_in, U3, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F7_IN, U4', F13_IN, F14_IN

Compound Symbols:

c1, c3, c6, c1

(11) 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.

F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))

We considered the (Usable) Rules:

f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))

And the Tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F13_IN(x₁, x₂)) = [2] + [2]x₁ + [2]x₂   
POL(F14_IN(x₁, x₂)) = 0   
POL(F2_IN(x₁, x₂, x₃)) = [2] + [2]x₁ + [2]x₂ + [2]x₃   
POL(F7_IN(x₁, x₂, x₃)) = [2] + [2]x₁ + [2]x₂ + x₃   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄)) = [2]x₂ + [2]x₃ + x₄   
POL([]) = 0   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(f13_in(x₁, x₂)) = 0   
POL(f13_out1(x₁)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1, z2) → U1(f7_in(z0, z1, z2), z0, z1, z2)
U1(f7_out1(z0, z1), z2, z3, z4) → f2_out1(z1)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
f14_in([], z0) → f14_out1(z0)
f14_in(.(z0, z1), z2) → U3(f14_in(z1, z2), .(z0, z1), z2)
U3(f14_out1(z0), .(z1, z2), z3) → f14_out1(.(z1, z0))
f7_in(z0, z1, z2) → U4(f13_in(z0, z1), z0, z1, z2)
U4(f13_out1(z0), z1, z2, z3) → U5(f14_in(z0, z3), z1, z2, z3, z0)
U5(f14_out1(z0), z1, z2, z3, z4) → f7_out1(z4, z0)

Tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

S tuples:

F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))

K tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))

Defined Rule Symbols:

f2_in, U1, f13_in, U2, f14_in, U3, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F7_IN, U4', F13_IN, F14_IN

Compound Symbols:

c1, c3, c6, c1

(13) 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.

F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))

We considered the (Usable) Rules:

f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))

And the Tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F13_IN(x₁, x₂)) = [1] + x₂   
POL(F14_IN(x₁, x₂)) = [1] + x₁   
POL(F2_IN(x₁, x₂, x₃)) = [3] + [3]x₁ + [2]x₂ + x₃   
POL(F7_IN(x₁, x₂, x₃)) = [2] + [3]x₁ + [2]x₂   
POL(U2(x₁, x₂, x₃)) = [2] + x₁   
POL(U4'(x₁, x₂, x₃, x₄)) = [1] + x₁   
POL([]) = [1]   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(f13_in(x₁, x₂)) = [1] + [2]x₁ + x₂   
POL(f13_out1(x₁)) = x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1, z2) → U1(f7_in(z0, z1, z2), z0, z1, z2)
U1(f7_out1(z0, z1), z2, z3, z4) → f2_out1(z1)
f13_in([], z0) → f13_out1(z0)
f13_in(.(z0, z1), z2) → U2(f13_in(z1, z2), .(z0, z1), z2)
U2(f13_out1(z0), .(z1, z2), z3) → f13_out1(.(z1, z0))
f14_in([], z0) → f14_out1(z0)
f14_in(.(z0, z1), z2) → U3(f14_in(z1, z2), .(z0, z1), z2)
U3(f14_out1(z0), .(z1, z2), z3) → f14_out1(.(z1, z0))
f7_in(z0, z1, z2) → U4(f13_in(z0, z1), z0, z1, z2)
U4(f13_out1(z0), z1, z2, z3) → U5(f14_in(z0, z3), z1, z2, z3, z0)
U5(f14_out1(z0), z1, z2, z3, z4) → f7_out1(z4, z0)

Tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1

S tuples:none
K tuples:

F2_IN(z0, z1, z2) → c1(F7_IN(z0, z1, z2))
F7_IN(z0, z1, z2) → c1(U4'(f13_in(z0, z1), z0, z1, z2))
F7_IN(z0, z1, z2) → c1(F13_IN(z0, z1))
U4'(f13_out1(z0), z1, z2, z3) → c1(F14_IN(z0, z3))
F2_IN(z0, z1, z2) → c1
U4'(f13_out1(z0), z1, z2, z3) → c1
F13_IN(.(z0, z1), z2) → c3(F13_IN(z1, z2))
F14_IN(.(z0, z1), z2) → c6(F14_IN(z1, z2))

Defined Rule Symbols:

f2_in, U1, f13_in, U2, f14_in, U3, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F7_IN, U4', F13_IN, F14_IN

Compound Symbols:

c1, c3, c6, c1

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))

(17) 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: app3_a(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: app3_a(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app(T9, T10, X9), app(X9, T11, T13))\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(T9, T10, X9), app(X9, T11, T13)), SCOPE: 2, CLAUSE: 0 | ','(app(T9, T10, X9), app(X9, T11, T13)), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: app(T16, T11, T13) | ','(app([], T16, X9), app(X9, T11, T13)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT16 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(app(T9, T10, X9), app(X9, T11, T13)), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nT11 is ground\nX9 is free\nX12 is free\napp(T9, T10, X9) !=? app([], X12, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app(T16, T11, T13)\n\nT11 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app([], T16, X9), app(X9, T11, T13)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT16 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app(T16, T11, T13), SCOPE: 3, CLAUSE: 0 | app(T16, T11, T13), SCOPE: 3, CLAUSE: 3\n\nT11 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | app([], T23, T13), SCOPE: 3, CLAUSE: 3\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app(T16, T11, T13), SCOPE: 3, CLAUSE: 3\n\nT11 is ground\nT16 is ground\nX19 is free\napp(T16, T11, T13) !=? app([], X19, X19)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app([], T23, T13), SCOPE: 3, CLAUSE: 3\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(T33, T34, T36)\n\nT33 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(app(T46, T47, X56), app(.(T45, X56), T11, T13))\n\nT11 is ground\nT45 is ground\nT46 is ground\nT47 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T46, T47, X56)\n\nT46 is ground\nT47 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app(.(T45, T50), T11, T13)\n\nT11 is ground\nT45 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app(T46, T47, X56), SCOPE: 4, CLAUSE: 0 | app(T46, T47, X56), SCOPE: 4, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true | app([], T55, X56), SCOPE: 4, CLAUSE: 3\n\nT55 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app(T46, T47, X56), SCOPE: 4, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nX56 is free\nX63 is free\napp(T46, T47, X56) !=? app([], X63, X63)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app([], T55, X56), SCOPE: 4, CLAUSE: 3\n\nT55 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app(T63, T64, X82)\n\nT63 is ground\nT64 is ground\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 47 [arrowhead = none ];
47 -> 3;

48 [label="ONLY EVAL with clause\napp3_a(X5, X6, X7, X8) :- ','(app(X5, X6, X9), app(X9, X7, X8)).\nand substitutionT1 -> T9,\nX5 -> T9,\nT2 -> T10,\nX6 -> T10,\nT3 -> T11,\nX7 -> T11,\nT4 -> T13,\nX8 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 48 [arrowhead = none ];
48 -> 5;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 49 [arrowhead = none ];
49 -> 6;

50 [label="EVAL with clause\napp([], X12, X12).\nand substitutionT9 -> [],\nT10 -> T16,\nX12 -> T16,\nX9 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 50 [arrowhead = none ];
50 -> 8;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 51 [arrowhead = none ];
51 -> 9;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 52 [arrowhead = none ];
52 -> 10;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 53 [arrowhead = none ];
53 -> 11;

54 [label="EVAL with clause\napp(.(X52, X53), X54, .(X52, X55)) :- app(X53, X54, X55).\nand substitutionX52 -> T45,\nX53 -> T46,\nT9 -> .(T45, T46),\nT10 -> T47,\nX54 -> T47,\nX55 -> X56,\nX9 -> .(T45, X56)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 54 [arrowhead = none ];
54 -> 34;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 55 [arrowhead = none ];
55 -> 35;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 56 [arrowhead = none ];
56 -> 12;

57 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 57 [arrowhead = none ];
57 -> 28;

58 [label="EVAL with clause\napp([], X19, X19).\nand substitutionT16 -> [],\nT11 -> T23,\nX19 -> T23,\nT13 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 58 [arrowhead = none ];
58 -> 15;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 59 [arrowhead = none ];
59 -> 16;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 60 [arrowhead = none ];
60 -> 17;

61 [label="EVAL with clause\napp(.(X32, X33), X34, .(X32, X35)) :- app(X33, X34, X35).\nand substitutionX32 -> T32,\nX33 -> T33,\nT16 -> .(T32, T33),\nT11 -> T34,\nX34 -> T34,\nX35 -> T36,\nT13 -> .(T32, T36),\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 62 [arrowhead = none ];
62 -> 25;

63 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 63 [arrowhead = none ];
63 -> 19;

64 [label="INSTANCE with matching:\nT16 -> T33\nT11 -> T34\nT13 -> T36", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 64 [arrowhead = none , style=dashed];
64 -> 10[style=dashed];

65 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 65 [arrowhead = none ];
65 -> 36;

66 [label="SPLIT 2\nnew knowledge:\nT46 is ground\nT47 is ground\nT50 is ground\nreplacements:X56 -> T50", fontsize=14, style = filled, fillcolor = violet];
34 -> 66 [arrowhead = none ];
66 -> 37;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 67 [arrowhead = none ];
67 -> 38;

68 [label="INSTANCE with matching:\nT16 -> .(T45, T50)", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 68 [arrowhead = none , style=dashed];
68 -> 10[style=dashed];

69 [label="EVAL with clause\napp([], X63, X63).\nand substitutionT46 -> [],\nT47 -> T55,\nX63 -> T55,\nX56 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 69 [arrowhead = none ];
69 -> 41;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 70 [arrowhead = none ];
70 -> 42;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 71 [arrowhead = none ];
71 -> 43;

72 [label="EVAL with clause\napp(.(X78, X79), X80, .(X78, X81)) :- app(X79, X80, X81).\nand substitutionX78 -> T62,\nX79 -> T63,\nT46 -> .(T62, T63),\nT47 -> T64,\nX80 -> T64,\nX81 -> X82,\nX56 -> .(T62, X82)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 72 [arrowhead = none ];
72 -> 45;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 73 [arrowhead = none ];
73 -> 46;

74 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 74 [arrowhead = none ];
74 -> 44;

75 [label="INSTANCE with matching:\nT46 -> T63\nT47 -> T64\nX56 -> X82", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 75 [arrowhead = none , style=dashed];
75 -> 36[style=dashed];

}

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], z0, z1) → U1(f8_in(z0, z1), [], z0, z1)
f1_in(.(z0, z1), z2, z3) → U2(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3)
U1(f8_out1(z0), [], z1, z2) → f1_out1(z0)
U1(f8_out2(z0, z1), [], z2, z3) → f1_out1(z1)
U2(f34_out1(z0, z1), .(z2, z3), z4, z5) → f1_out1(z1)
f10_in([], z0) → f10_out1(z0)
f10_in(.(z0, z1), z2) → U3(f10_in(z1, z2), .(z0, z1), z2)
U3(f10_out1(z0), .(z1, z2), z3) → f10_out1(.(z1, z0))
f36_in([], z0) → f36_out1(z0)
f36_in(.(z0, z1), z2) → U4(f36_in(z1, z2), .(z0, z1), z2)
U4(f36_out1(z0), .(z1, z2), z3) → f36_out1(.(z1, z0))
f34_in(z0, z1, z2, z3) → U5(f36_in(z0, z1), z0, z1, z2, z3)
U5(f36_out1(z0), z1, z2, z3, z4) → U6(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0)
U6(f10_out1(z0), z1, z2, z3, z4, z5) → f34_out1(z5, z0)
f8_in(z0, z1) → U7(f10_in(z0, z1), f11_in(z0, z1), z0, z1)
U7(f10_out1(z0), z1, z2, z3) → f8_out1(z0)
U7(z0, f11_out1(z1, z2), z3, z4) → f8_out2(z1, z2)

Tuples:

F1_IN([], z0, z1) → c(U1'(f8_in(z0, z1), [], z0, z1), F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c1(U2'(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z0, z3))
F10_IN(.(z0, z1), z2) → c6(U3'(f10_in(z1, z2), .(z0, z1), z2), F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(U4'(f36_in(z1, z2), .(z0, z1), z2), F36_IN(z1, z2))
F34_IN(z0, z1, z2, z3) → c11(U5'(f36_in(z0, z1), z0, z1, z2, z3), F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c12(U6'(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0), F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c14(U7'(f10_in(z0, z1), f11_in(z0, z1), z0, z1), F10_IN(z0, z1))

S tuples:

F1_IN([], z0, z1) → c(U1'(f8_in(z0, z1), [], z0, z1), F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c1(U2'(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3), F34_IN(z1, z2, z0, z3))
F10_IN(.(z0, z1), z2) → c6(U3'(f10_in(z1, z2), .(z0, z1), z2), F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(U4'(f36_in(z1, z2), .(z0, z1), z2), F36_IN(z1, z2))
F34_IN(z0, z1, z2, z3) → c11(U5'(f36_in(z0, z1), z0, z1, z2, z3), F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c12(U6'(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0), F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c14(U7'(f10_in(z0, z1), f11_in(z0, z1), z0, z1), F10_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f10_in, U3, f36_in, U4, f34_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F1_IN, F10_IN, F36_IN, F34_IN, U5', F8_IN

Compound Symbols:

c, c1, c6, c9, c11, c12, c14

(19) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], z0, z1) → U1(f8_in(z0, z1), [], z0, z1)
f1_in(.(z0, z1), z2, z3) → U2(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3)
U1(f8_out1(z0), [], z1, z2) → f1_out1(z0)
U1(f8_out2(z0, z1), [], z2, z3) → f1_out1(z1)
U2(f34_out1(z0, z1), .(z2, z3), z4, z5) → f1_out1(z1)
f10_in([], z0) → f10_out1(z0)
f10_in(.(z0, z1), z2) → U3(f10_in(z1, z2), .(z0, z1), z2)
U3(f10_out1(z0), .(z1, z2), z3) → f10_out1(.(z1, z0))
f36_in([], z0) → f36_out1(z0)
f36_in(.(z0, z1), z2) → U4(f36_in(z1, z2), .(z0, z1), z2)
U4(f36_out1(z0), .(z1, z2), z3) → f36_out1(.(z1, z0))
f34_in(z0, z1, z2, z3) → U5(f36_in(z0, z1), z0, z1, z2, z3)
U5(f36_out1(z0), z1, z2, z3, z4) → U6(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0)
U6(f10_out1(z0), z1, z2, z3, z4, z5) → f34_out1(z5, z0)
f8_in(z0, z1) → U7(f10_in(z0, z1), f11_in(z0, z1), z0, z1)
U7(f10_out1(z0), z1, z2, z3) → f8_out1(z0)
U7(z0, f11_out1(z1, z2), z3, z4) → f8_out2(z1, z2)

Tuples:

F10_IN(.(z0, z1), z2) → c6(U3'(f10_in(z1, z2), .(z0, z1), z2), F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(U4'(f36_in(z1, z2), .(z0, z1), z2), F36_IN(z1, z2))
F1_IN([], z0, z1) → c2(U1'(f8_in(z0, z1), [], z0, z1))
F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(U2'(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(U6'(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(U7'(f10_in(z0, z1), f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))

S tuples:

F10_IN(.(z0, z1), z2) → c6(U3'(f10_in(z1, z2), .(z0, z1), z2), F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(U4'(f36_in(z1, z2), .(z0, z1), z2), F36_IN(z1, z2))
F1_IN([], z0, z1) → c2(U1'(f8_in(z0, z1), [], z0, z1))
F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(U2'(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(U6'(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(U7'(f10_in(z0, z1), f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f10_in, U3, f36_in, U4, f34_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F10_IN, F36_IN, F1_IN, F34_IN, U5', F8_IN

Compound Symbols:

c6, c9, c2

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

Removed 6 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], z0, z1) → U1(f8_in(z0, z1), [], z0, z1)
f1_in(.(z0, z1), z2, z3) → U2(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3)
U1(f8_out1(z0), [], z1, z2) → f1_out1(z0)
U1(f8_out2(z0, z1), [], z2, z3) → f1_out1(z1)
U2(f34_out1(z0, z1), .(z2, z3), z4, z5) → f1_out1(z1)
f10_in([], z0) → f10_out1(z0)
f10_in(.(z0, z1), z2) → U3(f10_in(z1, z2), .(z0, z1), z2)
U3(f10_out1(z0), .(z1, z2), z3) → f10_out1(.(z1, z0))
f36_in([], z0) → f36_out1(z0)
f36_in(.(z0, z1), z2) → U4(f36_in(z1, z2), .(z0, z1), z2)
U4(f36_out1(z0), .(z1, z2), z3) → f36_out1(.(z1, z0))
f34_in(z0, z1, z2, z3) → U5(f36_in(z0, z1), z0, z1, z2, z3)
U5(f36_out1(z0), z1, z2, z3, z4) → U6(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0)
U6(f10_out1(z0), z1, z2, z3, z4, z5) → f34_out1(z5, z0)
f8_in(z0, z1) → U7(f10_in(z0, z1), f11_in(z0, z1), z0, z1)
U7(f10_out1(z0), z1, z2, z3) → f8_out1(z0)
U7(z0, f11_out1(z1, z2), z3, z4) → f8_out2(z1, z2)

Tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2

S tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f10_in, U3, f36_in, U4, f34_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F1_IN, F34_IN, U5', F8_IN, F10_IN, F36_IN

Compound Symbols:

c2, c6, c9, c2

(23) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F8_IN(z0, z1) → c2
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], z0, z1) → U1(f8_in(z0, z1), [], z0, z1)
f1_in(.(z0, z1), z2, z3) → U2(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3)
U1(f8_out1(z0), [], z1, z2) → f1_out1(z0)
U1(f8_out2(z0, z1), [], z2, z3) → f1_out1(z1)
U2(f34_out1(z0, z1), .(z2, z3), z4, z5) → f1_out1(z1)
f10_in([], z0) → f10_out1(z0)
f10_in(.(z0, z1), z2) → U3(f10_in(z1, z2), .(z0, z1), z2)
U3(f10_out1(z0), .(z1, z2), z3) → f10_out1(.(z1, z0))
f36_in([], z0) → f36_out1(z0)
f36_in(.(z0, z1), z2) → U4(f36_in(z1, z2), .(z0, z1), z2)
U4(f36_out1(z0), .(z1, z2), z3) → f36_out1(.(z1, z0))
f34_in(z0, z1, z2, z3) → U5(f36_in(z0, z1), z0, z1, z2, z3)
U5(f36_out1(z0), z1, z2, z3, z4) → U6(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0)
U6(f10_out1(z0), z1, z2, z3, z4, z5) → f34_out1(z5, z0)
f8_in(z0, z1) → U7(f10_in(z0, z1), f11_in(z0, z1), z0, z1)
U7(f10_out1(z0), z1, z2, z3) → f8_out1(z0)
U7(z0, f11_out1(z1, z2), z3, z4) → f8_out2(z1, z2)

Tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2

S tuples:

F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))

K tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2

Defined Rule Symbols:

f1_in, U1, U2, f10_in, U3, f36_in, U4, f34_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F1_IN, F34_IN, U5', F8_IN, F10_IN, F36_IN

Compound Symbols:

c2, c6, c9, c2

(25) 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.

F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))

We considered the (Usable) Rules:

f36_in([], z0) → f36_out1(z0)
f36_in(.(z0, z1), z2) → U4(f36_in(z1, z2), .(z0, z1), z2)
U4(f36_out1(z0), .(z1, z2), z3) → f36_out1(.(z1, z0))

And the Tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2

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

POL(.(x₁, x₂)) = [2] + x₂   
POL(F10_IN(x₁, x₂)) = [1] + x₂   
POL(F1_IN(x₁, x₂, x₃)) = [2]x₁ + [3]x₂ + [3]x₃   
POL(F34_IN(x₁, x₂, x₃, x₄)) = [2] + [2]x₁ + [3]x₂ + [2]x₄   
POL(F36_IN(x₁, x₂)) = [1] + x₁ + [2]x₂   
POL(F8_IN(x₁, x₂)) = [2] + x₁ + [2]x₂   
POL(U4(x₁, x₂, x₃)) = [2] + x₃   
POL(U5'(x₁, x₂, x₃, x₄, x₅)) = [1] + [2]x₂ + x₅   
POL([]) = [2]   
POL(c2) = 0   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f36_in(x₁, x₂)) = [3] + x₁ + [2]x₂   
POL(f36_out1(x₁)) = [2]   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], z0, z1) → U1(f8_in(z0, z1), [], z0, z1)
f1_in(.(z0, z1), z2, z3) → U2(f34_in(z1, z2, z0, z3), .(z0, z1), z2, z3)
U1(f8_out1(z0), [], z1, z2) → f1_out1(z0)
U1(f8_out2(z0, z1), [], z2, z3) → f1_out1(z1)
U2(f34_out1(z0, z1), .(z2, z3), z4, z5) → f1_out1(z1)
f10_in([], z0) → f10_out1(z0)
f10_in(.(z0, z1), z2) → U3(f10_in(z1, z2), .(z0, z1), z2)
U3(f10_out1(z0), .(z1, z2), z3) → f10_out1(.(z1, z0))
f36_in([], z0) → f36_out1(z0)
f36_in(.(z0, z1), z2) → U4(f36_in(z1, z2), .(z0, z1), z2)
U4(f36_out1(z0), .(z1, z2), z3) → f36_out1(.(z1, z0))
f34_in(z0, z1, z2, z3) → U5(f36_in(z0, z1), z0, z1, z2, z3)
U5(f36_out1(z0), z1, z2, z3, z4) → U6(f10_in(.(z3, z0), z4), z1, z2, z3, z4, z0)
U6(f10_out1(z0), z1, z2, z3, z4, z5) → f34_out1(z5, z0)
f8_in(z0, z1) → U7(f10_in(z0, z1), f11_in(z0, z1), z0, z1)
U7(f10_out1(z0), z1, z2, z3) → f8_out1(z0)
U7(z0, f11_out1(z1, z2), z3, z4) → f8_out2(z1, z2)

Tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2

S tuples:

F10_IN(.(z0, z1), z2) → c6(F10_IN(z1, z2))

K tuples:

F1_IN([], z0, z1) → c2(F8_IN(z0, z1))
F1_IN(.(z0, z1), z2, z3) → c2(F34_IN(z1, z2, z0, z3))
F34_IN(z0, z1, z2, z3) → c2(U5'(f36_in(z0, z1), z0, z1, z2, z3))
F34_IN(z0, z1, z2, z3) → c2(F36_IN(z0, z1))
U5'(f36_out1(z0), z1, z2, z3, z4) → c2(F10_IN(.(z3, z0), z4))
F8_IN(z0, z1) → c2(F10_IN(z0, z1))
F1_IN([], z0, z1) → c2
F1_IN(.(z0, z1), z2, z3) → c2
U5'(f36_out1(z0), z1, z2, z3, z4) → c2
F8_IN(z0, z1) → c2
F36_IN(.(z0, z1), z2) → c9(F36_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, U2, f10_in, U3, f36_in, U4, f34_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F1_IN, F34_IN, U5', F8_IN, F10_IN, F36_IN

Compound Symbols:

c2, c6, c9, c2