Runtime Complexity of the query pattern
star(g,g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

Clauses:

star(U, []) :- !.
star([], W) :- ','(!, =(W, [])).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
=(X, X).

Query: star(g,g)

(1) BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

star(U, []) :- !.
star([], W) :- ','(!, user_defined_=(W, [])).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
user_defined_=(X, X).

Query: star(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];
1 [label="1: star(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: star(T1, T2), SCOPE: 1, CLAUSE: 0 | star(T1, T2), SCOPE: 1, CLAUSE: 1 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: !_1 | star(T4, []), SCOPE: 1, CLAUSE: 1 | star(T4, []), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: star(T1, T2), SCOPE: 1, CLAUSE: 1 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX2 is free\nstar(T1, T2) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_1, user_defined_=(T6, [])) | star([], T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX2 is free\nstar([], T6) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX2 is free\nX4 is free\nstar(T1, T2) !=? star(X2, [])\nstar(T1, T2) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: user_defined_=(T6, [])\n\nT6 is ground\nX2 is free\nstar([], T6) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: user_defined_=(T6, []), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX2 is free\nstar([], T6) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(app(T10, X10, T11), star(T10, X10))\n\nT10 is ground\nT11 is ground\nX2 is free\nX4 is free\nX10 is free\nstar(T10, T11) !=? star(X2, [])\nstar(T10, T11) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(app(T10, X10, T11), star(T10, X10)), SCOPE: 3, CLAUSE: 3 | ','(app(T10, X10, T11), star(T10, X10)), SCOPE: 3, CLAUSE: 4\n\nT10 is ground\nT11 is ground\nX2 is free\nX4 is free\nX10 is free\nstar(T10, T11) !=? star(X2, [])\nstar(T10, T11) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(app(T10, X10, T11), star(T10, X10)), SCOPE: 3, CLAUSE: 4\n\nT10 is ground\nT11 is ground\nX2 is free\nX4 is free\nX10 is free\nstar(T10, T11) !=? star(X2, [])\nstar(T10, T11) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(app(T20, X27, T21), star(.(T19, T20), X27))\n\nT19 is ground\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T20, X27, T21)\n\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: star(.(T19, T20), T24)\n\nT19 is ground\nT20 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T20, X27, T21), SCOPE: 4, CLAUSE: 3 | app(T20, X27, T21), SCOPE: 4, CLAUSE: 4\n\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true | app([], X27, T29), SCOPE: 4, CLAUSE: 4\n\nT29 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app(T20, X27, T21), SCOPE: 4, CLAUSE: 4\n\nT20 is ground\nT21 is ground\nX27 is free\nX40 is free\napp(T20, X27, T21) !=? app([], X40, X40)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app([], X27, T29), SCOPE: 4, CLAUSE: 4\n\nT29 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app(T37, X60, T38)\n\nT37 is ground\nT38 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 40 [arrowhead = none ];
40 -> 3;

41 [label="EVAL with clause\nstar(X2, []) :- !_1.\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 41 [arrowhead = none ];
41 -> 5;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 42 [arrowhead = none ];
42 -> 7;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 43 [arrowhead = none ];
43 -> 9;

44 [label="EVAL with clause\nstar([], X4) :- ','(!_1, user_defined_=(X4, [])).\nand substitutionT1 -> [],\nT2 -> T6,\nX4 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 44 [arrowhead = none ];
44 -> 13;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 45 [arrowhead = none ];
45 -> 15;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 46 [arrowhead = none ];
46 -> 11;

47 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 47 [arrowhead = none ];
47 -> 17;

48 [label="ONLY EVAL with clause\nstar(X8, X9) :- ','(app(X8, X10, X9), star(X8, X10)).\nand substitutionT1 -> T10,\nX8 -> T10,\nT2 -> T11,\nX9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 48 [arrowhead = none ];
48 -> 23;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 49 [arrowhead = none ];
49 -> 19;

50 [label="BACKTRACK\nfor clause: user_defined_=(X, X)\nwith clash: (star([], T6), star(X2, []))", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 50 [arrowhead = none ];
50 -> 21;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="BACKTRACK\nfor clause: app([], L, L)\nwith clash: (star(T10, T11), star([], X4))", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="EVAL with clause\napp(.(X23, X24), X25, .(X23, X26)) :- app(X24, X25, X26).\nand substitutionX23 -> T19,\nX24 -> T20,\nT10 -> .(T19, T20),\nX10 -> X27,\nX25 -> X27,\nX26 -> T21,\nT11 -> .(T19, T21)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 53 [arrowhead = none ];
53 -> 27;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 54 [arrowhead = none ];
54 -> 28;

55 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 55 [arrowhead = none ];
55 -> 29;

56 [label="SPLIT 2\nnew knowledge:\nT20 is ground\nT24 is ground\nT21 is ground\nreplacements:X27 -> T24", fontsize=14, style = filled, fillcolor = violet];
27 -> 56 [arrowhead = none ];
56 -> 30;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 57 [arrowhead = none ];
57 -> 31;

58 [label="INSTANCE with matching:\nT1 -> .(T19, T20)\nT2 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

59 [label="EVAL with clause\napp([], X40, X40).\nand substitutionT20 -> [],\nX27 -> T29,\nX40 -> T29,\nT21 -> T29,\nX41 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 59 [arrowhead = none ];
59 -> 34;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 60 [arrowhead = none ];
60 -> 35;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 61 [arrowhead = none ];
61 -> 36;

62 [label="EVAL with clause\napp(.(X56, X57), X58, .(X56, X59)) :- app(X57, X58, X59).\nand substitutionX56 -> T36,\nX57 -> T37,\nT20 -> .(T36, T37),\nX27 -> X60,\nX58 -> X60,\nX59 -> T38,\nT21 -> .(T36, T38)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 62 [arrowhead = none ];
62 -> 38;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 63 [arrowhead = none ];
63 -> 39;

64 [label="BACKTRACK\nfor clause: app(.(X, L), M, .(X, N)) :- app(L, M, N)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 64 [arrowhead = none ];
64 -> 37;

65 [label="INSTANCE with matching:\nT20 -> T37\nX27 -> X60\nT21 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 65 [arrowhead = none , style=dashed];
65 -> 29[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f27_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)
f27_in(z0, z1, z2) → U3(f29_in(z0, z1), z0, z1, z2)
U3(f29_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z0), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f27_out1(z3)

Tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(U1'(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2)), F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f29_in(z1, z2), .(z0, z1), .(z0, z2)), F29_IN(z1, z2))
F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
U3'(f29_out1(z0), z1, z2, z3) → c7(U4'(f1_in(.(z3, z1), z0), z1, z2, z3, z0), F1_IN(.(z3, z1), z0))

S tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(U1'(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2)), F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f29_in(z1, z2), .(z0, z1), .(z0, z2)), F29_IN(z1, z2))
F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
U3'(f29_out1(z0), z1, z2, z3) → c7(U4'(f1_in(.(z3, z1), z0), z1, z2, z3, z0), F1_IN(.(z3, z1), z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f29_in, U2, f27_in, U3, U4

Defined Pair Symbols:

F1_IN, F29_IN, F27_IN, U3'

Compound Symbols:

c1, c4, c6, c7

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f27_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)
f27_in(z0, z1, z2) → U3(f29_in(z0, z1), z0, z1, z2)
U3(f29_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z0), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f27_out1(z3)

Tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

S tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f29_in, U2, f27_in, U3, U4

Defined Pair Symbols:

F27_IN, F1_IN, F29_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(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.

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))

We considered the (Usable) Rules:

f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)

And the Tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F1_IN(x₁, x₂)) = [2]x₂   
POL(F27_IN(x₁, x₂, x₃)) = [2] + [2]x₂ + x₃   
POL(F29_IN(x₁, x₂)) = [1]   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3'(x₁, x₂, x₃, x₄)) = [2]x₁ + x₄   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f29_in(x₁, x₂)) = x₂   
POL(f29_out1(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f27_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)
f27_in(z0, z1, z2) → U3(f29_in(z0, z1), z0, z1, z2)
U3(f29_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z0), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f27_out1(z3)

Tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

S tuples:

F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

K tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))

Defined Rule Symbols:

f1_in, U1, f29_in, U2, f27_in, U3, U4

Defined Pair Symbols:

F27_IN, F1_IN, F29_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f27_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)
f27_in(z0, z1, z2) → U3(f29_in(z0, z1), z0, z1, z2)
U3(f29_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z0), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f27_out1(z3)

Tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

S tuples:

F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))

K tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

Defined Rule Symbols:

f1_in, U1, f29_in, U2, f27_in, U3, U4

Defined Pair Symbols:

F27_IN, F1_IN, F29_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))

We considered the (Usable) Rules:

f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)

And the Tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

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

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F1_IN(x₁, x₂)) = x₂²   
POL(F27_IN(x₁, x₂, x₃)) = [1] + x₂ + x₃ + x₃² + x₂·x₃ + x₂²   
POL(F29_IN(x₁, x₂)) = x₂   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3'(x₁, x₂, x₃, x₄)) = x₁·x₄ + x₁²   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f29_in(x₁, x₂)) = x₂   
POL(f29_out1(x₁)) = x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f27_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f27_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f29_in([], z0) → f29_out1(z0)
f29_in(.(z0, z1), .(z0, z2)) → U2(f29_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f29_out1(z0), .(z1, z2), .(z1, z3)) → f29_out1(z0)
f27_in(z0, z1, z2) → U3(f29_in(z0, z1), z0, z1, z2)
U3(f29_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z0), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f27_out1(z3)

Tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))

S tuples:none
K tuples:

F27_IN(z0, z1, z2) → c6(U3'(f29_in(z0, z1), z0, z1, z2), F29_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F27_IN(z1, z2, z0))
U3'(f29_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z0))
F29_IN(.(z0, z1), .(z0, z2)) → c4(F29_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f29_in, U2, f27_in, U3, U4

Defined Pair Symbols:

F27_IN, F1_IN, F29_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))

(15) 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: star(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: star(T1, T2), SCOPE: 1, CLAUSE: 0 | star(T1, T2), SCOPE: 1, CLAUSE: 1 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: !_1 | star(T5, []), SCOPE: 1, CLAUSE: 1 | star(T5, []), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: star(T1, T2), SCOPE: 1, CLAUSE: 1 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nstar(T1, T2) !=? star(X3, [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(!_1, user_defined_=(T8, [])) | star([], T8), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nX3 is free\nstar([], T8) !=? star(X3, [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX6 is free\nstar(T1, T2) !=? star(X3, [])\nstar(T1, T2) !=? star([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: user_defined_=(T8, [])\n\nT8 is ground\nX3 is free\nstar([], T8) !=? star(X3, [])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: user_defined_=(T8, []), SCOPE: 2, CLAUSE: 5\n\nT8 is ground\nX3 is free\nstar([], T8) !=? star(X3, [])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(app(T19, X24, T20), star(T19, X24))\n\nT19 is ground\nT20 is ground\nX3 is free\nX6 is free\nX24 is free\nstar(T19, T20) !=? star(X3, [])\nstar(T19, T20) !=? star([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T19, X24, T20)\n\nT19 is ground\nT20 is ground\nX3 is free\nX6 is free\nX24 is free\nstar(T19, T20) !=? star(X3, [])\nstar(T19, T20) !=? star([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: star(T19, T26)\n\nT19 is ground\nT20 is ground\nT26 is ground\nX3 is free\nX6 is free\nstar(T19, T20) !=? star(X3, [])\nstar(T19, T20) !=? star([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app(T19, X24, T20), SCOPE: 3, CLAUSE: 3 | app(T19, X24, T20), SCOPE: 3, CLAUSE: 4\n\nT19 is ground\nT20 is ground\nX3 is free\nX6 is free\nX24 is free\nstar(T19, T20) !=? star(X3, [])\nstar(T19, T20) !=? star([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T19, X24, T20), SCOPE: 3, CLAUSE: 4\n\nT19 is ground\nT20 is ground\nX3 is free\nX6 is free\nX24 is free\nstar(T19, T20) !=? star(X3, [])\nstar(T19, T20) !=? star([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app(T40, X59, T41)\n\nT40 is ground\nT41 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app(T40, X59, T41), SCOPE: 4, CLAUSE: 3 | app(T40, X59, T41), SCOPE: 4, CLAUSE: 4\n\nT40 is ground\nT41 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true | app([], X59, T46), SCOPE: 4, CLAUSE: 4\n\nT46 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: app(T40, X59, T41), SCOPE: 4, CLAUSE: 4\n\nT40 is ground\nT41 is ground\nX59 is free\nX68 is free\napp(T40, X59, T41) !=? app([], X68, X68)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app([], X59, T46), SCOPE: 4, CLAUSE: 4\n\nT46 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: app(T54, X88, T55)\n\nT54 is ground\nT55 is ground\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 51 [arrowhead = none ];
51 -> 4;

52 [label="EVAL with clause\nstar(X3, []) :- !_1.\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 52 [arrowhead = none ];
52 -> 6;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 53 [arrowhead = none ];
53 -> 8;

54 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 54 [arrowhead = none ];
54 -> 10;

55 [label="EVAL with clause\nstar([], X6) :- ','(!_1, user_defined_=(X6, [])).\nand substitutionT1 -> [],\nT2 -> T8,\nX6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 55 [arrowhead = none ];
55 -> 14;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 56 [arrowhead = none ];
56 -> 16;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 57 [arrowhead = none ];
57 -> 12;

58 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 58 [arrowhead = none ];
58 -> 18;

59 [label="ONLY EVAL with clause\nstar(X22, X23) :- ','(app(X22, X24, X23), star(X22, X24)).\nand substitutionT1 -> T19,\nX22 -> T19,\nT2 -> T20,\nX23 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 59 [arrowhead = none ];
59 -> 26;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 60 [arrowhead = none ];
60 -> 20;

61 [label="BACKTRACK\nfor clause: user_defined_=(X, X)\nwith clash: (star([], T8), star(X3, []))", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 61 [arrowhead = none ];
61 -> 22;

62 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 62 [arrowhead = none ];
62 -> 32;

63 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nT26 is ground\nT20 is ground\nreplacements:X24 -> T26", fontsize=14, style = filled, fillcolor = violet];
26 -> 63 [arrowhead = none ];
63 -> 33;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 64 [arrowhead = none ];
64 -> 40;

65 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 65 [arrowhead = none , style=dashed];
65 -> 2[style=dashed];

66 [label="BACKTRACK\nfor clause: app([], L, L)\nwith clash: (star(T19, T20), star([], X6))", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 66 [arrowhead = none ];
66 -> 41;

67 [label="EVAL with clause\napp(.(X55, X56), X57, .(X55, X58)) :- app(X56, X57, X58).\nand substitutionX55 -> T39,\nX56 -> T40,\nT19 -> .(T39, T40),\nX24 -> X59,\nX57 -> X59,\nX58 -> T41,\nT20 -> .(T39, T41)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 67 [arrowhead = none ];
67 -> 42;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 68 [arrowhead = none ];
68 -> 43;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 69 [arrowhead = none ];
69 -> 44;

70 [label="EVAL with clause\napp([], X68, X68).\nand substitutionT40 -> [],\nX59 -> T46,\nX68 -> T46,\nT41 -> T46,\nX69 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 70 [arrowhead = none ];
70 -> 45;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 71 [arrowhead = none ];
71 -> 46;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 72 [arrowhead = none ];
72 -> 47;

73 [label="EVAL with clause\napp(.(X84, X85), X86, .(X84, X87)) :- app(X85, X86, X87).\nand substitutionX84 -> T53,\nX85 -> T54,\nT40 -> .(T53, T54),\nX59 -> X88,\nX86 -> X88,\nX87 -> T55,\nT41 -> .(T53, T55)", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 73 [arrowhead = none ];
73 -> 49;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 74 [arrowhead = none ];
74 -> 50;

75 [label="BACKTRACK\nfor clause: app(.(X, L), M, .(X, N)) :- app(L, M, N)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 75 [arrowhead = none ];
75 -> 48;

76 [label="INSTANCE with matching:\nT40 -> T54\nX59 -> X88\nT41 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 76 [arrowhead = none , style=dashed];
76 -> 42[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(z0, z1) → U1(f26_in(z0, z1), z0, z1)
U1(f26_out1(z0), z1, z2) → f2_out1
f42_in([], z0) → f42_out1(z0)
f42_in(.(z0, z1), .(z0, z2)) → U2(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f42_out1(z0), .(z1, z2), .(z1, z3)) → f42_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f42_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f26_in(z0, z1) → U4(f32_in(z0, z1), z0, z1)
U4(f32_out1(z0), z1, z2) → U5(f2_in(z1, z0), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f26_out1(z2)

Tuples:

F2_IN(z0, z1) → c1(U1'(f26_in(z0, z1), z0, z1), F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f42_in(z1, z2), .(z0, z1), .(z0, z2)), F42_IN(z1, z2))
F32_IN(.(z0, z1), .(z0, z2)) → c6(U3'(f42_in(z1, z2), .(z0, z1), .(z0, z2)), F42_IN(z1, z2))
F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2) → c9(U5'(f2_in(z1, z0), z1, z2, z0), F2_IN(z1, z0))

S tuples:

F2_IN(z0, z1) → c1(U1'(f26_in(z0, z1), z0, z1), F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f42_in(z1, z2), .(z0, z1), .(z0, z2)), F42_IN(z1, z2))
F32_IN(.(z0, z1), .(z0, z2)) → c6(U3'(f42_in(z1, z2), .(z0, z1), .(z0, z2)), F42_IN(z1, z2))
F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2) → c9(U5'(f2_in(z1, z0), z1, z2, z0), F2_IN(z1, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f42_in, U2, f32_in, U3, f26_in, U4, U5

Defined Pair Symbols:

F2_IN, F42_IN, F32_IN, F26_IN, U4'

Compound Symbols:

c1, c4, c6, c8, c9

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

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(z0, z1) → U1(f26_in(z0, z1), z0, z1)
U1(f26_out1(z0), z1, z2) → f2_out1
f42_in([], z0) → f42_out1(z0)
f42_in(.(z0, z1), .(z0, z2)) → U2(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f42_out1(z0), .(z1, z2), .(z1, z3)) → f42_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f42_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f26_in(z0, z1) → U4(f32_in(z0, z1), z0, z1)
U4(f32_out1(z0), z1, z2) → U5(f2_in(z1, z0), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f26_out1(z2)

Tuples:

F2_IN(z0, z1) → c1(U1'(f26_in(z0, z1), z0, z1), F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f42_in(z1, z2), .(z0, z1), .(z0, z2)), F42_IN(z1, z2))
F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2) → c9(U5'(f2_in(z1, z0), z1, z2, z0), F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c(U3'(f42_in(z1, z2), .(z0, z1), .(z0, z2)))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))

S tuples:

F2_IN(z0, z1) → c1(U1'(f26_in(z0, z1), z0, z1), F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f42_in(z1, z2), .(z0, z1), .(z0, z2)), F42_IN(z1, z2))
F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2) → c9(U5'(f2_in(z1, z0), z1, z2, z0), F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c(U3'(f42_in(z1, z2), .(z0, z1), .(z0, z2)))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f42_in, U2, f32_in, U3, f26_in, U4, U5

Defined Pair Symbols:

F2_IN, F42_IN, F26_IN, U4', F32_IN

Compound Symbols:

c1, c4, c8, c9, c

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

Removed 4 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(z0, z1) → U1(f26_in(z0, z1), z0, z1)
U1(f26_out1(z0), z1, z2) → f2_out1
f42_in([], z0) → f42_out1(z0)
f42_in(.(z0, z1), .(z0, z2)) → U2(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f42_out1(z0), .(z1, z2), .(z1, z3)) → f42_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f42_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f26_in(z0, z1) → U4(f32_in(z0, z1), z0, z1)
U4(f32_out1(z0), z1, z2) → U5(f2_in(z1, z0), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f26_out1(z2)

Tuples:

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
F2_IN(z0, z1) → c1(F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

S tuples:

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
F2_IN(z0, z1) → c1(F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f42_in, U2, f32_in, U3, f26_in, U4, U5

Defined Pair Symbols:

F26_IN, F32_IN, F2_IN, F42_IN, U4'

Compound Symbols:

c8, c, c1, c4, c9, c

(21) 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) → c1(F26_IN(z0, z1))

We considered the (Usable) Rules:

f32_in(.(z0, z1), .(z0, z2)) → U3(f42_in(z1, z2), .(z0, z1), .(z0, z2))
f42_in([], z0) → f42_out1(z0)
f42_in(.(z0, z1), .(z0, z2)) → U2(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f42_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
U2(f42_out1(z0), .(z1, z2), .(z1, z3)) → f42_out1(z0)

And the Tuples:

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
F2_IN(z0, z1) → c1(F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F26_IN(x₁, x₂)) = x₂   
POL(F2_IN(x₁, x₂)) = [1] + x₂   
POL(F32_IN(x₁, x₂)) = 0   
POL(F42_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = [1] + x₁   
POL(U3(x₁, x₂, x₃)) = [1] + x₁   
POL(U4'(x₁, x₂, x₃)) = x₁   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f32_in(x₁, x₂)) = x₂   
POL(f32_out1(x₁)) = [1] + x₁   
POL(f42_in(x₁, x₂)) = x₂   
POL(f42_out1(x₁)) = x₁   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(z0, z1) → U1(f26_in(z0, z1), z0, z1)
U1(f26_out1(z0), z1, z2) → f2_out1
f42_in([], z0) → f42_out1(z0)
f42_in(.(z0, z1), .(z0, z2)) → U2(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f42_out1(z0), .(z1, z2), .(z1, z3)) → f42_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f42_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f26_in(z0, z1) → U4(f32_in(z0, z1), z0, z1)
U4(f32_out1(z0), z1, z2) → U5(f2_in(z1, z0), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f26_out1(z2)

Tuples:

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
F2_IN(z0, z1) → c1(F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

S tuples:

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

K tuples:

F2_IN(z0, z1) → c1(F26_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f42_in, U2, f32_in, U3, f26_in, U4, U5

Defined Pair Symbols:

F26_IN, F32_IN, F2_IN, F42_IN, U4'

Compound Symbols:

c8, c, c1, c4, c9, c

(23) CdtKnowledgeProof (EQUIVALENT transformation)

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

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c
F2_IN(z0, z1) → c1(F26_IN(z0, z1))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(z0, z1) → U1(f26_in(z0, z1), z0, z1)
U1(f26_out1(z0), z1, z2) → f2_out1
f42_in([], z0) → f42_out1(z0)
f42_in(.(z0, z1), .(z0, z2)) → U2(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f42_out1(z0), .(z1, z2), .(z1, z3)) → f42_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f42_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f42_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f26_in(z0, z1) → U4(f32_in(z0, z1), z0, z1)
U4(f32_out1(z0), z1, z2) → U5(f2_in(z1, z0), z1, z2, z0)
U5(f2_out1, z0, z1, z2) → f26_out1(z2)

Tuples:

F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
F2_IN(z0, z1) → c1(F26_IN(z0, z1))
F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

S tuples:

F42_IN(.(z0, z1), .(z0, z2)) → c4(F42_IN(z1, z2))

K tuples:

F2_IN(z0, z1) → c1(F26_IN(z0, z1))
F26_IN(z0, z1) → c8(U4'(f32_in(z0, z1), z0, z1), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c(F42_IN(z1, z2))
U4'(f32_out1(z0), z1, z2) → c9(F2_IN(z1, z0))
F32_IN(.(z0, z1), .(z0, z2)) → c

Defined Rule Symbols:

f2_in, U1, f42_in, U2, f32_in, U3, f26_in, U4, U5

Defined Pair Symbols:

F26_IN, F32_IN, F2_IN, F42_IN, U4'

Compound Symbols:

c8, c, c1, c4, c9, c