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

(0) Obligation:

Clauses:

max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Query: max(a,a,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

less(0, s(X1)).
max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(s(X), s(Y)) :- less(X, Y).

Query: max(a,a,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: max(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: less(T14, T12)\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: less(T14, T12), SCOPE: 2, CLAUSE: 0 | less(T14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true | less(T14, s(T19)), SCOPE: 2, CLAUSE: 3\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(T14, T12), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nX16 is free\nless(T14, T12) !=? less(0, s(X16))", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(T14, s(T19)), SCOPE: 2, CLAUSE: 3\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T26, T25)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: less(T35, T34)\n\nT34 is ground\nX16 is free\nless(T14, s(T34)) !=? less(0, s(X16))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: less(T44, s(T43))\n\nT43 is ground", 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];
2 -> 40 [arrowhead = none ];
40 -> 4;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 41 [arrowhead = none ];
41 -> 5;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="EVAL with clause\nmax(X10, X11, X10) :- less(X11, X10).\nand substitutionT1 -> T12,\nX10 -> T12,\nT2 -> T14,\nX11 -> T14,\nT3 -> T12,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 43 [arrowhead = none ];
43 -> 12;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 44 [arrowhead = none ];
44 -> 14;

45 [label="EVAL with clause\nmax(X37, X38, X38) :- less(X37, s(X38)).\nand substitutionT1 -> T44,\nX37 -> T44,\nT2 -> T43,\nX38 -> T43,\nT3 -> T43,\nT42 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 45 [arrowhead = none ];
45 -> 38;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 46 [arrowhead = none ];
46 -> 39;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 47 [arrowhead = none ];
47 -> 20;

48 [label="EVAL with clause\nless(0, s(X16)).\nand substitutionT14 -> 0,\nX16 -> T19,\nT12 -> s(T19)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 49 [arrowhead = none ];
49 -> 23;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 50 [arrowhead = none ];
50 -> 25;

51 [label="EVAL with clause\nless(s(X29), s(X30)) :- less(X29, X30).\nand substitutionX29 -> T35,\nT14 -> s(T35),\nX30 -> T34,\nT12 -> s(T34),\nT33 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 51 [arrowhead = none ];
51 -> 31;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 52 [arrowhead = none ];
52 -> 35;

53 [label="EVAL with clause\nless(s(X21), s(X22)) :- less(X21, X22).\nand substitutionX21 -> T26,\nT14 -> s(T26),\nT19 -> T25,\nX22 -> T25,\nT24 -> T26", 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 -> 29;

55 [label="INSTANCE with matching:\nT14 -> T26\nT12 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 55 [arrowhead = none , style=dashed];
55 -> 12[style=dashed];

56 [label="INSTANCE with matching:\nT14 -> T35\nT12 -> T34", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 56 [arrowhead = none , style=dashed];
56 -> 12[style=dashed];

57 [label="INSTANCE with matching:\nT14 -> T44\nT12 -> s(T43)", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 57 [arrowhead = none , style=dashed];
57 -> 12[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f4_in(z0), z0)
U1(f4_out1(z0, z1), z2) → f2_out1(z0, z1)
U1(f4_out2(z0, z1), z2) → f2_out1(z0, z1)
f12_in(s(z0)) → f12_out1(0)
f12_in(s(z0)) → U2(f12_in(z0), s(z0))
f12_in(s(z0)) → U3(f12_in(z0), s(z0))
U2(f12_out1(z0), s(z1)) → f12_out1(s(z0))
U3(f12_out1(z0), s(z1)) → f12_out1(s(z0))
f5_in(z0) → U4(f12_in(z0), z0)
U4(f12_out1(z0), z1) → f5_out1(z1, z0)
f6_in(z0) → U5(f12_in(s(z0)), z0)
U5(f12_out1(z0), z1) → f6_out1(z0, z1)
f4_in(z0) → U6(f5_in(z0), f6_in(z0), z0)
U6(f5_out1(z0, z1), z2, z3) → f4_out1(z0, z1)
U6(z0, f6_out1(z1, z2), z3) → f4_out2(z1, z2)

Tuples:

F2_IN(z0) → c(U1'(f4_in(z0), z0), F4_IN(z0))
F12_IN(s(z0)) → c4(U2'(f12_in(z0), s(z0)), F12_IN(z0))
F12_IN(s(z0)) → c5(U3'(f12_in(z0), s(z0)), F12_IN(z0))
F5_IN(z0) → c8(U4'(f12_in(z0), z0), F12_IN(z0))
F6_IN(z0) → c10(U5'(f12_in(s(z0)), z0), F12_IN(s(z0)))
F4_IN(z0) → c12(U6'(f5_in(z0), f6_in(z0), z0), F5_IN(z0), F6_IN(z0))

S tuples:

F2_IN(z0) → c(U1'(f4_in(z0), z0), F4_IN(z0))
F12_IN(s(z0)) → c4(U2'(f12_in(z0), s(z0)), F12_IN(z0))
F12_IN(s(z0)) → c5(U3'(f12_in(z0), s(z0)), F12_IN(z0))
F5_IN(z0) → c8(U4'(f12_in(z0), z0), F12_IN(z0))
F6_IN(z0) → c10(U5'(f12_in(s(z0)), z0), F12_IN(s(z0)))
F4_IN(z0) → c12(U6'(f5_in(z0), f6_in(z0), z0), F5_IN(z0), F6_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f12_in, U2, U3, f5_in, U4, f6_in, U5, f4_in, U6

Defined Pair Symbols:

F2_IN, F12_IN, F5_IN, F6_IN, F4_IN

Compound Symbols:

c, c4, c5, c8, c10, c12

(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) → U1(f4_in(z0), z0)
U1(f4_out1(z0, z1), z2) → f2_out1(z0, z1)
U1(f4_out2(z0, z1), z2) → f2_out1(z0, z1)
f12_in(s(z0)) → f12_out1(0)
f12_in(s(z0)) → U2(f12_in(z0), s(z0))
f12_in(s(z0)) → U3(f12_in(z0), s(z0))
U2(f12_out1(z0), s(z1)) → f12_out1(s(z0))
U3(f12_out1(z0), s(z1)) → f12_out1(s(z0))
f5_in(z0) → U4(f12_in(z0), z0)
U4(f12_out1(z0), z1) → f5_out1(z1, z0)
f6_in(z0) → U5(f12_in(s(z0)), z0)
U5(f12_out1(z0), z1) → f6_out1(z0, z1)
f4_in(z0) → U6(f5_in(z0), f6_in(z0), z0)
U6(f5_out1(z0, z1), z2, z3) → f4_out1(z0, z1)
U6(z0, f6_out1(z1, z2), z3) → f4_out2(z1, z2)

Tuples:

F12_IN(s(z0)) → c4(U2'(f12_in(z0), s(z0)), F12_IN(z0))
F12_IN(s(z0)) → c5(U3'(f12_in(z0), s(z0)), F12_IN(z0))
F2_IN(z0) → c1(U1'(f4_in(z0), z0))
F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(U4'(f12_in(z0), z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(U5'(f12_in(s(z0)), z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(U6'(f5_in(z0), f6_in(z0), z0))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))

S tuples:

F12_IN(s(z0)) → c4(U2'(f12_in(z0), s(z0)), F12_IN(z0))
F12_IN(s(z0)) → c5(U3'(f12_in(z0), s(z0)), F12_IN(z0))
F2_IN(z0) → c1(U1'(f4_in(z0), z0))
F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(U4'(f12_in(z0), z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(U5'(f12_in(s(z0)), z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(U6'(f5_in(z0), f6_in(z0), z0))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f12_in, U2, U3, f5_in, U4, f6_in, U5, f4_in, U6

Defined Pair Symbols:

F12_IN, F2_IN, F5_IN, F6_IN, F4_IN

Compound Symbols:

c4, c5, c1

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

Removed 6 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f4_in(z0), z0)
U1(f4_out1(z0, z1), z2) → f2_out1(z0, z1)
U1(f4_out2(z0, z1), z2) → f2_out1(z0, z1)
f12_in(s(z0)) → f12_out1(0)
f12_in(s(z0)) → U2(f12_in(z0), s(z0))
f12_in(s(z0)) → U3(f12_in(z0), s(z0))
U2(f12_out1(z0), s(z1)) → f12_out1(s(z0))
U3(f12_out1(z0), s(z1)) → f12_out1(s(z0))
f5_in(z0) → U4(f12_in(z0), z0)
U4(f12_out1(z0), z1) → f5_out1(z1, z0)
f6_in(z0) → U5(f12_in(s(z0)), z0)
U5(f12_out1(z0), z1) → f6_out1(z0, z1)
f4_in(z0) → U6(f5_in(z0), f6_in(z0), z0)
U6(f5_out1(z0, z1), z2, z3) → f4_out1(z0, z1)
U6(z0, f6_out1(z1, z2), z3) → f4_out2(z1, z2)

Tuples:

F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1

S tuples:

F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f12_in, U2, U3, f5_in, U4, f6_in, U5, f4_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F4_IN, F12_IN

Compound Symbols:

c1, c4, c5, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0) → c1(F4_IN(z0))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F4_IN(z0) → c1
F5_IN(z0) → c1(F12_IN(z0))
F5_IN(z0) → c1
F6_IN(z0) → c1(F12_IN(s(z0)))
F6_IN(z0) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f4_in(z0), z0)
U1(f4_out1(z0, z1), z2) → f2_out1(z0, z1)
U1(f4_out2(z0, z1), z2) → f2_out1(z0, z1)
f12_in(s(z0)) → f12_out1(0)
f12_in(s(z0)) → U2(f12_in(z0), s(z0))
f12_in(s(z0)) → U3(f12_in(z0), s(z0))
U2(f12_out1(z0), s(z1)) → f12_out1(s(z0))
U3(f12_out1(z0), s(z1)) → f12_out1(s(z0))
f5_in(z0) → U4(f12_in(z0), z0)
U4(f12_out1(z0), z1) → f5_out1(z1, z0)
f6_in(z0) → U5(f12_in(s(z0)), z0)
U5(f12_out1(z0), z1) → f6_out1(z0, z1)
f4_in(z0) → U6(f5_in(z0), f6_in(z0), z0)
U6(f5_out1(z0, z1), z2, z3) → f4_out1(z0, z1)
U6(z0, f6_out1(z1, z2), z3) → f4_out2(z1, z2)

Tuples:

F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1

S tuples:

F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))

K tuples:

F2_IN(z0) → c1(F4_IN(z0))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))

Defined Rule Symbols:

f2_in, U1, f12_in, U2, U3, f5_in, U4, f6_in, U5, f4_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F4_IN, F12_IN

Compound Symbols:

c1, c4, c5, 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.

F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))

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

F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1

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

POL(F12_IN(x₁)) = x₁   
POL(F2_IN(x₁)) = [3] + [3]x₁   
POL(F4_IN(x₁)) = [3] + [3]x₁   
POL(F5_IN(x₁)) = [2] + x₁   
POL(F6_IN(x₁)) = [2] + [2]x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f4_in(z0), z0)
U1(f4_out1(z0, z1), z2) → f2_out1(z0, z1)
U1(f4_out2(z0, z1), z2) → f2_out1(z0, z1)
f12_in(s(z0)) → f12_out1(0)
f12_in(s(z0)) → U2(f12_in(z0), s(z0))
f12_in(s(z0)) → U3(f12_in(z0), s(z0))
U2(f12_out1(z0), s(z1)) → f12_out1(s(z0))
U3(f12_out1(z0), s(z1)) → f12_out1(s(z0))
f5_in(z0) → U4(f12_in(z0), z0)
U4(f12_out1(z0), z1) → f5_out1(z1, z0)
f6_in(z0) → U5(f12_in(s(z0)), z0)
U5(f12_out1(z0), z1) → f6_out1(z0, z1)
f4_in(z0) → U6(f5_in(z0), f6_in(z0), z0)
U6(f5_out1(z0, z1), z2, z3) → f4_out1(z0, z1)
U6(z0, f6_out1(z1, z2), z3) → f4_out2(z1, z2)

Tuples:

F2_IN(z0) → c1(F4_IN(z0))
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1

S tuples:none
K tuples:

F2_IN(z0) → c1(F4_IN(z0))
F4_IN(z0) → c1(F5_IN(z0))
F4_IN(z0) → c1(F6_IN(z0))
F2_IN(z0) → c1
F5_IN(z0) → c1
F6_IN(z0) → c1
F4_IN(z0) → c1
F5_IN(z0) → c1(F12_IN(z0))
F6_IN(z0) → c1(F12_IN(s(z0)))
F12_IN(s(z0)) → c4(F12_IN(z0))
F12_IN(s(z0)) → c5(F12_IN(z0))

Defined Rule Symbols:

f2_in, U1, f12_in, U2, U3, f5_in, U4, f6_in, U5, f4_in, U6

Defined Pair Symbols:

F2_IN, F5_IN, F6_IN, F4_IN, F12_IN

Compound Symbols:

c1, c4, c5, c1

(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];
1 [label="1: max(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: less(T8, T6) | max(T1, T2, T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: max(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: less(T8, T6), SCOPE: 2, CLAUSE: 0 | less(T8, T6), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, T2, T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | less(T8, s(T11)), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, T2, s(T11)), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T8, T6), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, T2, T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T8, s(T11)), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T1, T2, s(T11)), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(T8, s(T11)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ?_2 | max(T1, T2, s(T11)), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T22, T21)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: less(T22, T21), SCOPE: 3, CLAUSE: 0 | less(T22, T21), SCOPE: 3, CLAUSE: 3\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | less(T22, s(T27)), SCOPE: 3, CLAUSE: 3\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T22, T21), SCOPE: 3, CLAUSE: 3\n\nT21 is ground\nX23 is free\nless(T22, T21) !=? less(0, s(X23))", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T22, s(T27)), SCOPE: 3, CLAUSE: 3\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T34, T33)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: less(T43, T42)\n\nT42 is ground\nX23 is free\nless(T22, s(T42)) !=? less(0, s(X23))", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: max(T1, T2, s(T11)), SCOPE: 1, CLAUSE: 2\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T55, s(s(T54)))\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T8, T6), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_2 | max(T1, T2, T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(T68, T67)\n\nT67 is ground\nX8 is free\nless(T8, s(T67)) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: max(T1, T2, T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free\nless(T8, T6) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T77, s(T76))\n\nT76 is ground\nX8 is free\nless(T8, T76) !=? less(0, s(X8))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: less(T84, s(T83))\n\nT83 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T83) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: less(T84, s(T83)), SCOPE: 4, CLAUSE: 0 | less(T84, s(T83)), SCOPE: 4, CLAUSE: 3\n\nT83 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T83) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true | less(T84, s(T87)), SCOPE: 4, CLAUSE: 3\n\nT87 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T87) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T84, s(T83)), SCOPE: 4, CLAUSE: 3\n\nT83 is ground\nX4 is free\nX5 is free\nX74 is free\nmax(T1, T2, T83) !=? max(X4, X5, X4)\nless(T84, s(T83)) !=? less(0, s(X74))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(T84, s(T87)), SCOPE: 4, CLAUSE: 3\n\nT87 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T87) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(T94, T93)\n\nT93 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T93) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(T103, T102)\n\nT102 is ground\nX4 is free\nX5 is free\nX74 is free\nmax(T1, T2, T102) !=? max(X4, X5, X4)\nless(T84, s(T102)) !=? less(0, s(X74))", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 57 [arrowhead = none ];
57 -> 3;

58 [label="EVAL with clause\nmax(X4, X5, X4) :- less(X5, X4).\nand substitutionT1 -> T6,\nX4 -> T6,\nT2 -> T8,\nX5 -> T8,\nT3 -> T6,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 58 [arrowhead = none ];
58 -> 7;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 59 [arrowhead = none ];
59 -> 8;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="EVAL with clause\nmax(X70, X71, X71) :- less(X70, s(X71)).\nand substitutionT1 -> T84,\nX70 -> T84,\nT2 -> T83,\nX71 -> T83,\nT3 -> T83,\nT82 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 61 [arrowhead = none ];
61 -> 47;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 62 [arrowhead = none ];
62 -> 48;

63 [label="EVAL with clause\nless(0, s(X8)).\nand substitutionT8 -> 0,\nX8 -> T11,\nT6 -> s(T11)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 63 [arrowhead = none ];
63 -> 10;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 64 [arrowhead = none ];
64 -> 11;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 65 [arrowhead = none ];
65 -> 13;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 66 [arrowhead = none ];
66 -> 40;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 67 [arrowhead = none ];
67 -> 41;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="EVAL with clause\nless(s(X17), s(X18)) :- less(X17, X18).\nand substitutionX17 -> T22,\nT8 -> s(T22),\nT11 -> T21,\nX18 -> T21,\nT20 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 72 [arrowhead = none ];
72 -> 34;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 73 [arrowhead = none ];
73 -> 19;

74 [label="EVAL with clause\nless(0, s(X23)).\nand substitutionT22 -> 0,\nX23 -> T27,\nT21 -> s(T27)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 74 [arrowhead = none ];
74 -> 22;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 76 [arrowhead = none ];
76 -> 26;

77 [label="EVAL with clause\nless(s(X36), s(X37)) :- less(X36, X37).\nand substitutionX36 -> T43,\nT22 -> s(T43),\nX37 -> T42,\nT21 -> s(T42),\nT41 -> T43", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 77 [arrowhead = none ];
77 -> 32;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 78 [arrowhead = none ];
78 -> 33;

79 [label="EVAL with clause\nless(s(X28), s(X29)) :- less(X28, X29).\nand substitutionX28 -> T34,\nT22 -> s(T34),\nT27 -> T33,\nX29 -> T33,\nT32 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 79 [arrowhead = none ];
79 -> 28;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 80 [arrowhead = none ];
80 -> 30;

81 [label="INSTANCE with matching:\nT22 -> T34\nT21 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 81 [arrowhead = none , style=dashed];
81 -> 17[style=dashed];

82 [label="INSTANCE with matching:\nT22 -> T43\nT21 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 82 [arrowhead = none , style=dashed];
82 -> 17[style=dashed];

83 [label="EVAL with clause\nmax(X44, X45, X45) :- less(X44, s(X45)).\nand substitutionT1 -> T55,\nX44 -> T55,\nT2 -> s(T54),\nX45 -> s(T54),\nT11 -> T54,\nT53 -> s(T54),\nT52 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 83 [arrowhead = none ];
83 -> 36;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 84 [arrowhead = none ];
84 -> 37;

85 [label="INSTANCE with matching:\nT22 -> T55\nT21 -> s(s(T54))", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 85 [arrowhead = none , style=dashed];
85 -> 17[style=dashed];

86 [label="EVAL with clause\nless(s(X56), s(X57)) :- less(X56, X57).\nand substitutionX56 -> T68,\nT8 -> s(T68),\nX57 -> T67,\nT6 -> s(T67),\nT66 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 86 [arrowhead = none ];
86 -> 42;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 87 [arrowhead = none ];
87 -> 43;

88 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 88 [arrowhead = none ];
88 -> 44;

89 [label="INSTANCE with matching:\nT22 -> T68\nT21 -> T67", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 89 [arrowhead = none , style=dashed];
89 -> 17[style=dashed];

90 [label="EVAL with clause\nmax(X64, X65, X65) :- less(X64, s(X65)).\nand substitutionT1 -> T77,\nX64 -> T77,\nT2 -> T76,\nX65 -> T76,\nT6 -> T76,\nT75 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 90 [arrowhead = none ];
90 -> 45;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 91 [arrowhead = none ];
91 -> 46;

92 [label="INSTANCE with matching:\nT22 -> T77\nT21 -> s(T76)", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 92 [arrowhead = none , style=dashed];
92 -> 17[style=dashed];

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 93 [arrowhead = none ];
93 -> 49;

94 [label="EVAL with clause\nless(0, s(X74)).\nand substitutionT84 -> 0,\nT83 -> T87,\nX74 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 94 [arrowhead = none ];
94 -> 50;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 95 [arrowhead = none ];
95 -> 51;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 96 [arrowhead = none ];
96 -> 52;

97 [label="EVAL with clause\nless(s(X87), s(X88)) :- less(X87, X88).\nand substitutionX87 -> T103,\nT84 -> s(T103),\nT83 -> T102,\nX88 -> T102,\nT101 -> T103", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 97 [arrowhead = none ];
97 -> 55;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 98 [arrowhead = none ];
98 -> 56;

99 [label="EVAL with clause\nless(s(X79), s(X80)) :- less(X79, X80).\nand substitutionX79 -> T94,\nT84 -> s(T94),\nT87 -> T93,\nX80 -> T93,\nT92 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 99 [arrowhead = none ];
99 -> 53;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 100 [arrowhead = none ];
100 -> 54;

101 [label="INSTANCE with matching:\nT22 -> T94\nT21 -> T93", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 101 [arrowhead = none , style=dashed];
101 -> 17[style=dashed];

102 [label="INSTANCE with matching:\nT22 -> T103\nT21 -> T102", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 102 [arrowhead = none , style=dashed];
102 -> 17[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(s(z0), 0)
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
f1_in(z0) → U2(f11_in(z0), z0)
f1_in(z0) → f1_out1(0, z0)
f1_in(z0) → U3(f17_in(z0), z0)
f1_in(z0) → U4(f17_in(z0), z0)
U1(f13_out1(z0), s(z1)) → f1_out1(s(z1), z0)
U1(f13_out3(z0, z1), s(z2)) → f1_out1(z0, z1)
U2(f11_out1(z0), z1) → f1_out1(z1, z0)
U2(f11_out3(z0, z1), z2) → f1_out1(z0, z1)
U3(f17_out1(z0), z1) → f1_out1(s(z0), z1)
U4(f17_out1(z0), z1) → f1_out1(s(z0), z1)
f17_in(s(z0)) → f17_out1(0)
f17_in(s(z0)) → U5(f17_in(z0), s(z0))
f17_in(s(z0)) → U6(f17_in(z0), s(z0))
U5(f17_out1(z0), s(z1)) → f17_out1(s(z0))
U6(f17_out1(z0), s(z1)) → f17_out1(s(z0))
f40_in(s(z0)) → U7(f17_in(z0), s(z0))
U7(f17_out1(z0), s(z1)) → f40_out1(s(z0))
f41_in(z0) → U8(f17_in(s(z0)), z0)
U8(f17_out1(z0), z1) → f41_out2(z0, z1)
f15_in(z0) → U9(f17_in(z0), z0)
U9(f17_out1(z0), z1) → f15_out1(s(z0))
f16_in(z0) → U10(f17_in(s(s(z0))), z0)
U10(f17_out1(z0), z1) → f16_out2(z0, s(z1))
f11_in(z0) → U11(f40_in(z0), f41_in(z0), z0)
U11(f40_out1(z0), z1, z2) → f11_out1(z0)
U11(z0, f41_out2(z1, z2), z3) → f11_out3(z1, z2)
f13_in(z0) → U12(f15_in(z0), f16_in(z0), z0)
U12(f15_out1(z0), z1, z2) → f13_out1(z0)
U12(z0, f16_out2(z1, z2), z3) → f13_out3(z1, z2)

Tuples:

F1_IN(s(z0)) → c1(U1'(f13_in(z0), s(z0)), F13_IN(z0))
F1_IN(z0) → c2(U2'(f11_in(z0), z0), F11_IN(z0))
F1_IN(z0) → c4(U3'(f17_in(z0), z0), F17_IN(z0))
F1_IN(z0) → c5(U4'(f17_in(z0), z0), F17_IN(z0))
F17_IN(s(z0)) → c13(U5'(f17_in(z0), s(z0)), F17_IN(z0))
F17_IN(s(z0)) → c14(U6'(f17_in(z0), s(z0)), F17_IN(z0))
F40_IN(s(z0)) → c17(U7'(f17_in(z0), s(z0)), F17_IN(z0))
F41_IN(z0) → c19(U8'(f17_in(s(z0)), z0), F17_IN(s(z0)))
F15_IN(z0) → c21(U9'(f17_in(z0), z0), F17_IN(z0))
F16_IN(z0) → c23(U10'(f17_in(s(s(z0))), z0), F17_IN(s(s(z0))))
F11_IN(z0) → c25(U11'(f40_in(z0), f41_in(z0), z0), F40_IN(z0), F41_IN(z0))
F13_IN(z0) → c28(U12'(f15_in(z0), f16_in(z0), z0), F15_IN(z0), F16_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(U1'(f13_in(z0), s(z0)), F13_IN(z0))
F1_IN(z0) → c2(U2'(f11_in(z0), z0), F11_IN(z0))
F1_IN(z0) → c4(U3'(f17_in(z0), z0), F17_IN(z0))
F1_IN(z0) → c5(U4'(f17_in(z0), z0), F17_IN(z0))
F17_IN(s(z0)) → c13(U5'(f17_in(z0), s(z0)), F17_IN(z0))
F17_IN(s(z0)) → c14(U6'(f17_in(z0), s(z0)), F17_IN(z0))
F40_IN(s(z0)) → c17(U7'(f17_in(z0), s(z0)), F17_IN(z0))
F41_IN(z0) → c19(U8'(f17_in(s(z0)), z0), F17_IN(s(z0)))
F15_IN(z0) → c21(U9'(f17_in(z0), z0), F17_IN(z0))
F16_IN(z0) → c23(U10'(f17_in(s(s(z0))), z0), F17_IN(s(s(z0))))
F11_IN(z0) → c25(U11'(f40_in(z0), f41_in(z0), z0), F40_IN(z0), F41_IN(z0))
F13_IN(z0) → c28(U12'(f15_in(z0), f16_in(z0), z0), F15_IN(z0), F16_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f17_in, U5, U6, f40_in, U7, f41_in, U8, f15_in, U9, f16_in, U10, f11_in, U11, f13_in, U12

Defined Pair Symbols:

F1_IN, F17_IN, F40_IN, F41_IN, F15_IN, F16_IN, F11_IN, F13_IN

Compound Symbols:

c1, c2, c4, c5, c13, c14, c17, c19, c21, c23, c25, c28

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

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(s(z0), 0)
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
f1_in(z0) → U2(f11_in(z0), z0)
f1_in(z0) → f1_out1(0, z0)
f1_in(z0) → U3(f17_in(z0), z0)
f1_in(z0) → U4(f17_in(z0), z0)
U1(f13_out1(z0), s(z1)) → f1_out1(s(z1), z0)
U1(f13_out3(z0, z1), s(z2)) → f1_out1(z0, z1)
U2(f11_out1(z0), z1) → f1_out1(z1, z0)
U2(f11_out3(z0, z1), z2) → f1_out1(z0, z1)
U3(f17_out1(z0), z1) → f1_out1(s(z0), z1)
U4(f17_out1(z0), z1) → f1_out1(s(z0), z1)
f17_in(s(z0)) → f17_out1(0)
f17_in(s(z0)) → U5(f17_in(z0), s(z0))
f17_in(s(z0)) → U6(f17_in(z0), s(z0))
U5(f17_out1(z0), s(z1)) → f17_out1(s(z0))
U6(f17_out1(z0), s(z1)) → f17_out1(s(z0))
f40_in(s(z0)) → U7(f17_in(z0), s(z0))
U7(f17_out1(z0), s(z1)) → f40_out1(s(z0))
f41_in(z0) → U8(f17_in(s(z0)), z0)
U8(f17_out1(z0), z1) → f41_out2(z0, z1)
f15_in(z0) → U9(f17_in(z0), z0)
U9(f17_out1(z0), z1) → f15_out1(s(z0))
f16_in(z0) → U10(f17_in(s(s(z0))), z0)
U10(f17_out1(z0), z1) → f16_out2(z0, s(z1))
f11_in(z0) → U11(f40_in(z0), f41_in(z0), z0)
U11(f40_out1(z0), z1, z2) → f11_out1(z0)
U11(z0, f41_out2(z1, z2), z3) → f11_out3(z1, z2)
f13_in(z0) → U12(f15_in(z0), f16_in(z0), z0)
U12(f15_out1(z0), z1, z2) → f13_out1(z0)
U12(z0, f16_out2(z1, z2), z3) → f13_out3(z1, z2)

Tuples:

F17_IN(s(z0)) → c13(U5'(f17_in(z0), s(z0)), F17_IN(z0))
F17_IN(s(z0)) → c14(U6'(f17_in(z0), s(z0)), F17_IN(z0))
F1_IN(s(z0)) → c(U1'(f13_in(z0), s(z0)))
F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(U2'(f11_in(z0), z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(U3'(f17_in(z0), z0))
F1_IN(z0) → c(F17_IN(z0))
F1_IN(z0) → c(U4'(f17_in(z0), z0))
F40_IN(s(z0)) → c(U7'(f17_in(z0), s(z0)))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(U8'(f17_in(s(z0)), z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(U9'(f17_in(z0), z0))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(U10'(f17_in(s(s(z0))), z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(U11'(f40_in(z0), f41_in(z0), z0))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(U12'(f15_in(z0), f16_in(z0), z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))

S tuples:

F17_IN(s(z0)) → c13(U5'(f17_in(z0), s(z0)), F17_IN(z0))
F17_IN(s(z0)) → c14(U6'(f17_in(z0), s(z0)), F17_IN(z0))
F1_IN(s(z0)) → c(U1'(f13_in(z0), s(z0)))
F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(U2'(f11_in(z0), z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(U3'(f17_in(z0), z0))
F1_IN(z0) → c(F17_IN(z0))
F1_IN(z0) → c(U4'(f17_in(z0), z0))
F40_IN(s(z0)) → c(U7'(f17_in(z0), s(z0)))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(U8'(f17_in(s(z0)), z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(U9'(f17_in(z0), z0))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(U10'(f17_in(s(s(z0))), z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(U11'(f40_in(z0), f41_in(z0), z0))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(U12'(f15_in(z0), f16_in(z0), z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f17_in, U5, U6, f40_in, U7, f41_in, U8, f15_in, U9, f16_in, U10, f11_in, U11, f13_in, U12

Defined Pair Symbols:

F17_IN, F1_IN, F40_IN, F41_IN, F15_IN, F16_IN, F11_IN, F13_IN

Compound Symbols:

c13, c14, c

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

Removed 12 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(s(z0), 0)
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
f1_in(z0) → U2(f11_in(z0), z0)
f1_in(z0) → f1_out1(0, z0)
f1_in(z0) → U3(f17_in(z0), z0)
f1_in(z0) → U4(f17_in(z0), z0)
U1(f13_out1(z0), s(z1)) → f1_out1(s(z1), z0)
U1(f13_out3(z0, z1), s(z2)) → f1_out1(z0, z1)
U2(f11_out1(z0), z1) → f1_out1(z1, z0)
U2(f11_out3(z0, z1), z2) → f1_out1(z0, z1)
U3(f17_out1(z0), z1) → f1_out1(s(z0), z1)
U4(f17_out1(z0), z1) → f1_out1(s(z0), z1)
f17_in(s(z0)) → f17_out1(0)
f17_in(s(z0)) → U5(f17_in(z0), s(z0))
f17_in(s(z0)) → U6(f17_in(z0), s(z0))
U5(f17_out1(z0), s(z1)) → f17_out1(s(z0))
U6(f17_out1(z0), s(z1)) → f17_out1(s(z0))
f40_in(s(z0)) → U7(f17_in(z0), s(z0))
U7(f17_out1(z0), s(z1)) → f40_out1(s(z0))
f41_in(z0) → U8(f17_in(s(z0)), z0)
U8(f17_out1(z0), z1) → f41_out2(z0, z1)
f15_in(z0) → U9(f17_in(z0), z0)
U9(f17_out1(z0), z1) → f15_out1(s(z0))
f16_in(z0) → U10(f17_in(s(s(z0))), z0)
U10(f17_out1(z0), z1) → f16_out2(z0, s(z1))
f11_in(z0) → U11(f40_in(z0), f41_in(z0), z0)
U11(f40_out1(z0), z1, z2) → f11_out1(z0)
U11(z0, f41_out2(z1, z2), z3) → f11_out3(z1, z2)
f13_in(z0) → U12(f15_in(z0), f16_in(z0), z0)
U12(f15_out1(z0), z1, z2) → f13_out1(z0)
U12(z0, f16_out2(z1, z2), z3) → f13_out3(z1, z2)

Tuples:

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c

S tuples:

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f17_in, U5, U6, f40_in, U7, f41_in, U8, f15_in, U9, f16_in, U10, f11_in, U11, f13_in, U12

Defined Pair Symbols:

F1_IN, F40_IN, F41_IN, F15_IN, F16_IN, F11_IN, F13_IN, F17_IN

Compound Symbols:

c, c13, c14, c

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F1_IN(z0) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F13_IN(z0) → c
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F11_IN(z0) → c
F40_IN(s(z0)) → c(F17_IN(z0))
F40_IN(s(z0)) → c
F41_IN(z0) → c(F17_IN(s(z0)))
F41_IN(z0) → c
F15_IN(z0) → c(F17_IN(z0))
F15_IN(z0) → c
F16_IN(z0) → c(F17_IN(s(s(z0))))
F16_IN(z0) → c

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(s(z0), 0)
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
f1_in(z0) → U2(f11_in(z0), z0)
f1_in(z0) → f1_out1(0, z0)
f1_in(z0) → U3(f17_in(z0), z0)
f1_in(z0) → U4(f17_in(z0), z0)
U1(f13_out1(z0), s(z1)) → f1_out1(s(z1), z0)
U1(f13_out3(z0, z1), s(z2)) → f1_out1(z0, z1)
U2(f11_out1(z0), z1) → f1_out1(z1, z0)
U2(f11_out3(z0, z1), z2) → f1_out1(z0, z1)
U3(f17_out1(z0), z1) → f1_out1(s(z0), z1)
U4(f17_out1(z0), z1) → f1_out1(s(z0), z1)
f17_in(s(z0)) → f17_out1(0)
f17_in(s(z0)) → U5(f17_in(z0), s(z0))
f17_in(s(z0)) → U6(f17_in(z0), s(z0))
U5(f17_out1(z0), s(z1)) → f17_out1(s(z0))
U6(f17_out1(z0), s(z1)) → f17_out1(s(z0))
f40_in(s(z0)) → U7(f17_in(z0), s(z0))
U7(f17_out1(z0), s(z1)) → f40_out1(s(z0))
f41_in(z0) → U8(f17_in(s(z0)), z0)
U8(f17_out1(z0), z1) → f41_out2(z0, z1)
f15_in(z0) → U9(f17_in(z0), z0)
U9(f17_out1(z0), z1) → f15_out1(s(z0))
f16_in(z0) → U10(f17_in(s(s(z0))), z0)
U10(f17_out1(z0), z1) → f16_out2(z0, s(z1))
f11_in(z0) → U11(f40_in(z0), f41_in(z0), z0)
U11(f40_out1(z0), z1, z2) → f11_out1(z0)
U11(z0, f41_out2(z1, z2), z3) → f11_out3(z1, z2)
f13_in(z0) → U12(f15_in(z0), f16_in(z0), z0)
U12(f15_out1(z0), z1, z2) → f13_out1(z0)
U12(z0, f16_out2(z1, z2), z3) → f13_out3(z1, z2)

Tuples:

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c

S tuples:

F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))

K tuples:

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))

Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f17_in, U5, U6, f40_in, U7, f41_in, U8, f15_in, U9, f16_in, U10, f11_in, U11, f13_in, U12

Defined Pair Symbols:

F1_IN, F40_IN, F41_IN, F15_IN, F16_IN, F11_IN, F13_IN, F17_IN

Compound Symbols:

c, c13, c14, c

(23) 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.

F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))

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

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c

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

POL(F11_IN(x₁)) = [3] + [2]x₁   
POL(F13_IN(x₁)) = [3] + x₁   
POL(F15_IN(x₁)) = x₁   
POL(F16_IN(x₁)) = [3] + x₁   
POL(F17_IN(x₁)) = x₁   
POL(F1_IN(x₁)) = [3] + [2]x₁   
POL(F40_IN(x₁)) = [2] + [2]x₁   
POL(F41_IN(x₁)) = [2] + [2]x₁   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(s(z0)) → f1_out1(s(z0), 0)
f1_in(s(z0)) → U1(f13_in(z0), s(z0))
f1_in(z0) → U2(f11_in(z0), z0)
f1_in(z0) → f1_out1(0, z0)
f1_in(z0) → U3(f17_in(z0), z0)
f1_in(z0) → U4(f17_in(z0), z0)
U1(f13_out1(z0), s(z1)) → f1_out1(s(z1), z0)
U1(f13_out3(z0, z1), s(z2)) → f1_out1(z0, z1)
U2(f11_out1(z0), z1) → f1_out1(z1, z0)
U2(f11_out3(z0, z1), z2) → f1_out1(z0, z1)
U3(f17_out1(z0), z1) → f1_out1(s(z0), z1)
U4(f17_out1(z0), z1) → f1_out1(s(z0), z1)
f17_in(s(z0)) → f17_out1(0)
f17_in(s(z0)) → U5(f17_in(z0), s(z0))
f17_in(s(z0)) → U6(f17_in(z0), s(z0))
U5(f17_out1(z0), s(z1)) → f17_out1(s(z0))
U6(f17_out1(z0), s(z1)) → f17_out1(s(z0))
f40_in(s(z0)) → U7(f17_in(z0), s(z0))
U7(f17_out1(z0), s(z1)) → f40_out1(s(z0))
f41_in(z0) → U8(f17_in(s(z0)), z0)
U8(f17_out1(z0), z1) → f41_out2(z0, z1)
f15_in(z0) → U9(f17_in(z0), z0)
U9(f17_out1(z0), z1) → f15_out1(s(z0))
f16_in(z0) → U10(f17_in(s(s(z0))), z0)
U10(f17_out1(z0), z1) → f16_out2(z0, s(z1))
f11_in(z0) → U11(f40_in(z0), f41_in(z0), z0)
U11(f40_out1(z0), z1, z2) → f11_out1(z0)
U11(z0, f41_out2(z1, z2), z3) → f11_out3(z1, z2)
f13_in(z0) → U12(f15_in(z0), f16_in(z0), z0)
U12(f15_out1(z0), z1, z2) → f13_out1(z0)
U12(z0, f16_out2(z1, z2), z3) → f13_out3(z1, z2)

Tuples:

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c

S tuples:none
K tuples:

F1_IN(s(z0)) → c(F13_IN(z0))
F1_IN(z0) → c(F11_IN(z0))
F1_IN(z0) → c(F17_IN(z0))
F11_IN(z0) → c(F40_IN(z0))
F11_IN(z0) → c(F41_IN(z0))
F13_IN(z0) → c(F15_IN(z0))
F13_IN(z0) → c(F16_IN(z0))
F1_IN(s(z0)) → c
F1_IN(z0) → c
F40_IN(s(z0)) → c
F41_IN(z0) → c
F15_IN(z0) → c
F16_IN(z0) → c
F11_IN(z0) → c
F13_IN(z0) → c
F40_IN(s(z0)) → c(F17_IN(z0))
F41_IN(z0) → c(F17_IN(s(z0)))
F15_IN(z0) → c(F17_IN(z0))
F16_IN(z0) → c(F17_IN(s(s(z0))))
F17_IN(s(z0)) → c13(F17_IN(z0))
F17_IN(s(z0)) → c14(F17_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, U3, U4, f17_in, U5, U6, f40_in, U7, f41_in, U8, f15_in, U9, f16_in, U10, f11_in, U11, f13_in, U12

Defined Pair Symbols:

F1_IN, F40_IN, F41_IN, F15_IN, F16_IN, F11_IN, F13_IN, F17_IN

Compound Symbols:

c, c13, c14, c