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

(0) Obligation:

Clauses:

overlap(Xs, Ys) :- ','(member2(X, Xs), member1(X, Ys)).
has_a_or_b(Xs) :- overlap(Xs, .(a, .(b, []))).
member1(X, .(Y, Xs)) :- member1(X, Xs).
member1(X, .(X, Xs)).
member2(X, .(Y, Xs)) :- member2(X, Xs).
member2(X, .(X, Xs)).

Query: overlap(g,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

member1(X, .(X, Xs)).
member2(X, .(X, Xs)).
overlap(Xs, Ys) :- ','(member2(X, Xs), member1(X, Ys)).
has_a_or_b(Xs) :- overlap(Xs, .(a, .(b, []))).
member1(X, .(Y, Xs)) :- member1(X, Xs).
member2(X, .(Y, Xs)) :- member2(X, Xs).

Query: overlap(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: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: overlap(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(member2(X5, T5), member1(X5, T6))\n\nT5 is ground\nT6 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(member2(X5, T5), member1(X5, T6)), SCOPE: 2, CLAUSE: 1 | ','(member2(X5, T5), member1(X5, T6)), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nT6 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: member1(T11, T6) | ','(member2(X5, .(T11, T12)), member1(X5, T6)), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nT11 is ground\nT12 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(member2(X5, T5), member1(X5, T6)), SCOPE: 2, CLAUSE: 5\n\nT5 is ground\nT6 is ground\nX5 is free\nX12 is free\nX13 is free\nmember2(X5, T5) !=? member2(X12, .(X12, X13))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: member1(T11, T6)\n\nT6 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(member2(X5, .(T11, T12)), member1(X5, T6)), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nT11 is ground\nT12 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: member1(T11, T6), SCOPE: 3, CLAUSE: 0 | member1(T11, T6), SCOPE: 3, CLAUSE: 4\n\nT6 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | member1(T25, .(T25, T26)), SCOPE: 3, CLAUSE: 4\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: member1(T11, T6), SCOPE: 3, CLAUSE: 4\n\nT6 is ground\nT11 is ground\nX27 is free\nX28 is free\nmember1(T11, T6) !=? member1(X27, .(X27, X28))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: member1(T25, .(T25, T26)), SCOPE: 3, CLAUSE: 4\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: member1(T35, T36)\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: member1(T47, T49)\n\nT47 is ground\nT48 is ground\nT49 is ground\nX27 is free\nX28 is free\nmember1(T47, .(T48, T49)) !=? member1(X27, .(X27, X28))", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(member2(X76, T63), member1(X76, T6))\n\nT6 is ground\nT63 is ground\nX76 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 34 [arrowhead = none ];
34 -> 3;

35 [label="ONLY EVAL with clause\noverlap(X3, X4) :- ','(member2(X5, X3), member1(X5, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 36 [arrowhead = none ];
36 -> 6;

37 [label="EVAL with clause\nmember2(X12, .(X12, X13)).\nand substitutionX5 -> T11,\nX12 -> T11,\nX14 -> T11,\nX13 -> T12,\nT5 -> .(T11, T12)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 37 [arrowhead = none ];
37 -> 8;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 38 [arrowhead = none ];
38 -> 9;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 39 [arrowhead = none ];
39 -> 10;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="BACKTRACK\nfor clause: member2(X, .(Y, Xs)) :- member2(X, Xs)\nwith clash: (member2(X5, T5), member2(X12, .(X12, X13)))", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 41 [arrowhead = none ];
41 -> 33;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 42 [arrowhead = none ];
42 -> 14;

43 [label="ONLY EVAL with clause\nmember2(X73, .(X74, X75)) :- member2(X73, X75).\nand substitutionX5 -> X76,\nX73 -> X76,\nT11 -> T62,\nX74 -> T62,\nT12 -> T63,\nX75 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 43 [arrowhead = none ];
43 -> 31;

44 [label="EVAL with clause\nmember1(X27, .(X27, X28)).\nand substitutionT11 -> T25,\nX27 -> T25,\nX28 -> T26,\nT6 -> .(T25, T26)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 45 [arrowhead = none ];
45 -> 16;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 46 [arrowhead = none ];
46 -> 18;

47 [label="EVAL with clause\nmember1(X52, .(X53, X54)) :- member1(X52, X54).\nand substitutionT11 -> T47,\nX52 -> T47,\nX53 -> T48,\nX54 -> T49,\nT6 -> .(T48, T49)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 47 [arrowhead = none ];
47 -> 25;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 48 [arrowhead = none ];
48 -> 26;

49 [label="ONLY EVAL with clause\nmember1(X39, .(X40, X41)) :- member1(X39, X41).\nand substitutionT25 -> T35,\nX39 -> T35,\nX40 -> T35,\nT26 -> T36,\nX41 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="INSTANCE with matching:\nT11 -> T35\nT6 -> T36", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 50 [arrowhead = none , style=dashed];
50 -> 10[style=dashed];

51 [label="INSTANCE with matching:\nT11 -> T47\nT6 -> T49", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 51 [arrowhead = none , style=dashed];
51 -> 10[style=dashed];

52 [label="INSTANCE with matching:\nX5 -> X76\nT5 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 52 [arrowhead = none , style=dashed];
52 -> 5[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F1_IN(z0, z1) → c(U1'(f5_in(z0, z1), z0, z1), F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(U2'(f8_in(z0, z2, z1), .(z0, z1), z2), F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(U3'(f10_in(z0, z1), z0, .(z0, z1)), F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(U4'(f10_in(z0, z2), z0, .(z1, z2)), F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(U6'(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2), F10_IN(z0, z1), F11_IN(z0, z2, z1))

S tuples:

F1_IN(z0, z1) → c(U1'(f5_in(z0, z1), z0, z1), F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(U2'(f8_in(z0, z2, z1), .(z0, z1), z2), F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(U3'(f10_in(z0, z1), z0, .(z0, z1)), F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(U4'(f10_in(z0, z2), z0, .(z1, z2)), F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(U6'(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2), F10_IN(z0, z1), F11_IN(z0, z2, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F10_IN, F11_IN, F8_IN

Compound Symbols:

c, c2, c6, c7, 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:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F5_IN(.(z0, z1), z2) → c2(U2'(f8_in(z0, z2, z1), .(z0, z1), z2), F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(U3'(f10_in(z0, z1), z0, .(z0, z1)), F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(U4'(f10_in(z0, z2), z0, .(z1, z2)), F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(U6'(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2), F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1(U1'(f5_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c1(F5_IN(z0, z1))

S tuples:

F5_IN(.(z0, z1), z2) → c2(U2'(f8_in(z0, z2, z1), .(z0, z1), z2), F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(U3'(f10_in(z0, z1), z0, .(z0, z1)), F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(U4'(f10_in(z0, z2), z0, .(z1, z2)), F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(U5'(f5_in(z1, z2), z0, z1, z2), F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(U6'(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2), F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1(U1'(f5_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c1(F5_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F5_IN, F10_IN, F11_IN, F8_IN, F1_IN

Compound Symbols:

c2, c6, c7, c10, c12, c1

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

Removed 6 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F10_IN, F11_IN, F8_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))

K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F10_IN, F11_IN, F8_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, 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.

F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))

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

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F10_IN(x₁, x₂)) = 0   
POL(F11_IN(x₁, x₂, x₃)) = x₁ + [2]x₂   
POL(F1_IN(x₁, x₂)) = [3]x₁ + [3]x₂   
POL(F5_IN(x₁, x₂)) = [2]x₁   
POL(F8_IN(x₁, x₂, x₃)) = x₁ + [2]x₃   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))

K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F10_IN, F11_IN, F8_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:

F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))

K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F10_IN, F11_IN, F8_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(15) 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.

F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))

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

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F10_IN(x₁, x₂)) = x₂   
POL(F11_IN(x₁, x₂, x₃)) = x₂·x₃   
POL(F1_IN(x₁, x₂)) = x₁·x₂   
POL(F5_IN(x₁, x₂)) = x₁·x₂   
POL(F8_IN(x₁, x₂, x₃)) = x₂ + x₂·x₃   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1(z0), z1, z2) → f1_out1
f5_in(.(z0, z1), z2) → U2(f8_in(z0, z2, z1), .(z0, z1), z2)
U2(f8_out1, .(z0, z1), z2) → f5_out1(z0)
U2(f8_out2(z0), .(z1, z2), z3) → f5_out1(z0)
f10_in(z0, .(z0, z1)) → f10_out1
f10_in(z0, .(z0, z1)) → U3(f10_in(z0, z1), z0, .(z0, z1))
f10_in(z0, .(z1, z2)) → U4(f10_in(z0, z2), z0, .(z1, z2))
U3(f10_out1, z0, .(z0, z1)) → f10_out1
U4(f10_out1, z0, .(z1, z2)) → f10_out1
f11_in(z0, z1, z2) → U5(f5_in(z1, z2), z0, z1, z2)
U5(f5_out1(z0), z1, z2, z3) → f11_out1(z0)
f8_in(z0, z1, z2) → U6(f10_in(z0, z1), f11_in(z0, z2, z1), z0, z1, z2)
U6(f10_out1, z0, z1, z2, z3) → f8_out1
U6(z0, f11_out1(z1), z2, z3, z4) → f8_out2(z1)

Tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F1_IN(z0, z1) → c1

S tuples:none
K tuples:

F1_IN(z0, z1) → c1(F5_IN(z0, z1))
F1_IN(z0, z1) → c1
F5_IN(.(z0, z1), z2) → c2(F8_IN(z0, z2, z1))
F8_IN(z0, z1, z2) → c12(F10_IN(z0, z1), F11_IN(z0, z2, z1))
F11_IN(z0, z1, z2) → c10(F5_IN(z1, z2))
F10_IN(z0, .(z0, z1)) → c6(F10_IN(z0, z1))
F10_IN(z0, .(z1, z2)) → c7(F10_IN(z0, z2))

Defined Rule Symbols:

f1_in, U1, f5_in, U2, f10_in, U3, U4, f11_in, U5, f8_in, U6

Defined Pair Symbols:

F1_IN, F5_IN, F10_IN, F11_IN, F8_IN

Compound Symbols:

c1, c2, c6, c7, c10, c12, c1

(17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(18) BOUNDS(O(1), O(1))

(19) 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: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: overlap(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(member2(X14, T11), member1(X14, T12))\n\nT11 is ground\nT12 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: member2(X14, T11)\n\nT11 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: member1(T17, T12)\n\nT12 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: member2(X14, T11), SCOPE: 2, CLAUSE: 1 | member2(X14, T11), SCOPE: 2, CLAUSE: 5\n\nT11 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | member2(X14, .(T26, T27)), SCOPE: 2, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: member2(X14, T11), SCOPE: 2, CLAUSE: 5\n\nT11 is ground\nX14 is free\nX33 is free\nX34 is free\nmember2(X14, T11) !=? member2(X33, .(X33, X34))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: member2(X14, .(T26, T27)), SCOPE: 2, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: member2(X53, T37)\n\nT37 is ground\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: member1(T17, T12), SCOPE: 3, CLAUSE: 0 | member1(T17, T12), SCOPE: 3, CLAUSE: 4\n\nT12 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | member1(T52, .(T52, T53)), SCOPE: 3, CLAUSE: 4\n\nT52 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: member1(T17, T12), SCOPE: 3, CLAUSE: 4\n\nT12 is ground\nT17 is ground\nX72 is free\nX73 is free\nmember1(T17, T12) !=? member1(X72, .(X72, X73))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: member1(T52, .(T52, T53)), SCOPE: 3, CLAUSE: 4\n\nT52 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: member1(T62, T63)\n\nT62 is ground\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: member1(T74, T76)\n\nT74 is ground\nT75 is ground\nT76 is ground\nX72 is free\nX73 is free\nmember1(T74, .(T75, T76)) !=? member1(X72, .(X72, X73))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 36 [arrowhead = none ];
36 -> 4;

37 [label="ONLY EVAL with clause\noverlap(X12, X13) :- ','(member2(X14, X12), member1(X14, X13)).\nand substitutionT1 -> T11,\nX12 -> T11,\nT2 -> T12,\nX13 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="SPLIT 2\nnew knowledge:\nT17 is ground\nT11 is ground\nreplacements:X14 -> T17", fontsize=14, style = filled, fillcolor = violet];
7 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 40 [arrowhead = none ];
40 -> 17;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 41 [arrowhead = none ];
41 -> 27;

42 [label="EVAL with clause\nmember2(X33, .(X33, X34)).\nand substitutionX14 -> T26,\nX33 -> T26,\nX35 -> T26,\nX34 -> T27,\nT11 -> .(T26, T27)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 42 [arrowhead = none ];
42 -> 19;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 43 [arrowhead = none ];
43 -> 20;

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

45 [label="BACKTRACK\nfor clause: member2(X, .(Y, Xs)) :- member2(X, Xs)\nwith clash: (member2(X14, T11), member2(X33, .(X33, X34)))", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 45 [arrowhead = none ];
45 -> 24;

46 [label="ONLY EVAL with clause\nmember2(X50, .(X51, X52)) :- member2(X50, X52).\nand substitutionX14 -> X53,\nX50 -> X53,\nT26 -> T36,\nX51 -> T36,\nT27 -> T37,\nX52 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 46 [arrowhead = none ];
46 -> 23;

47 [label="INSTANCE with matching:\nX14 -> X53\nT11 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 47 [arrowhead = none , style=dashed];
47 -> 12[style=dashed];

48 [label="EVAL with clause\nmember1(X72, .(X72, X73)).\nand substitutionT17 -> T52,\nX72 -> T52,\nX73 -> T53,\nT12 -> .(T52, T53)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 48 [arrowhead = none ];
48 -> 28;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 49 [arrowhead = none ];
49 -> 29;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 50 [arrowhead = none ];
50 -> 30;

51 [label="EVAL with clause\nmember1(X97, .(X98, X99)) :- member1(X97, X99).\nand substitutionT17 -> T74,\nX97 -> T74,\nX98 -> T75,\nX99 -> T76,\nT12 -> .(T75, T76)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 51 [arrowhead = none ];
51 -> 34;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 52 [arrowhead = none ];
52 -> 35;

53 [label="ONLY EVAL with clause\nmember1(X84, .(X85, X86)) :- member1(X84, X86).\nand substitutionT52 -> T62,\nX84 -> T62,\nX85 -> T62,\nT53 -> T63,\nX86 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 53 [arrowhead = none ];
53 -> 32;

54 [label="INSTANCE with matching:\nT17 -> T62\nT12 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 54 [arrowhead = none , style=dashed];
54 -> 13[style=dashed];

55 [label="INSTANCE with matching:\nT17 -> T74\nT12 -> T76", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 55 [arrowhead = none , style=dashed];
55 -> 13[style=dashed];

}

(20) Obligation:

Complex Complexity Dependency Tuples Problem
MAX

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Took the maximum complexity of the problems.

(22) Complex Obligation (MAX)

(23) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(25) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(U10'(f13_in(z0, z2), z1, z2, z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))

S tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(27) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

S tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
U2'(f12_out1(z0), z1, z2) → c3
F2_IN(z0, z1) → c7

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(28) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F2_IN(z0, z1) → c7
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U2'(f12_out1(z0), z1, z2) → c3
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U9'(f12_out1(z0), z1, z2) → c7
U5'(f12_out1(z0), z1, z2) → c4

Now S is empty

(29) BOUNDS(O(1), O(1))

(30) Obligation:

Complex Complexity Dependency Tuples Problem
MULTIPLY

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

(31) MultiplicationProof (BOTH BOUNDS(ID, ID) transformation)

Multiplied the complexity of the problems.

(32) Complex Obligation (MULT)

(33) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(35) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(U10'(f13_in(z0, z2), z1, z2, z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))

S tuples:

F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(37) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

S tuples:

F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(38) 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.

F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4

We considered the (Usable) Rules:

f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

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

POL(.(x₁, x₂)) = x₂   
POL(F12_IN(x₁)) = 0   
POL(F13_IN(x₁, x₂)) = [1]   
POL(F2_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [2] + [3]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = x₂ + [2]x₃   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [1] + x₃   
POL(U9'(x₁, x₂, x₃)) = [2] + [2]x₃   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = x₁   
POL(f12_out1(x₁)) = 0   

(39) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

S tuples:

F12_IN(.(z0, z1)) → c1(F12_IN(z1))

K tuples:

F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(40) 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(.(z0, z1)) → c1(F12_IN(z1))

We considered the (Usable) Rules:

f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)

And the Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

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

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F12_IN(x₁)) = [1] + [2]x₁   
POL(F13_IN(x₁, x₂)) = 0   
POL(F2_IN(x₁, x₂)) = [2] + [3]x₁ + [3]x₂   
POL(F7_IN(x₁, x₂)) = [1] + [3]x₁ + [2]x₂   
POL(U2'(x₁, x₂, x₃)) = x₂ + [2]x₃   
POL(U4(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [1] + x₃   
POL(U9'(x₁, x₂, x₃)) = [1] + x₃   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3) = 0   
POL(c4) = 0   
POL(c5(x₁)) = x₁   
POL(c7) = 0   
POL(c7(x₁)) = x₁   
POL(f12_in(x₁)) = x₁   
POL(f12_out1(x₁)) = 0   

(41) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

S tuples:none
K tuples:

F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4
F12_IN(.(z0, z1)) → c1(F12_IN(z1))

Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7

(42) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(43) BOUNDS(O(1), O(1))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F2_IN(z0, z1) → c(U1'(f7_in(z0, z1), z0, z1), F7_IN(z0, z1))
F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
F7_IN(z0, z1) → c3(U5'(f12_in(z0), z0, z1), F12_IN(z0))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

S tuples:

F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c6(U10'(f13_in(z0, z2), z1, z2, z0), F13_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F2_IN, F7_IN, U2', F12_IN, U5', F13_IN, U9'

Compound Symbols:

c, c2, c3, c1, c3, c4, c2, c5, c6

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

Split RHS of tuples not part of any SCC

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
U2'(f12_out1(z0), z1, z2) → c3(U3'(f13_in(z0, z2), z1, z2, z0))
F12_IN(.(z0, z1)) → c1(U4'(f12_in(z1), .(z0, z1)), F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4(U6'(f13_in(z0, z2), z1, z2, z0))
F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(U1'(f7_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(U10'(f13_in(z0, z2), z1, z2, z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))

S tuples:

F13_IN(z0, .(z0, z1)) → c1(U7'(f13_in(z0, z1), z0, .(z0, z1)), F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(U8'(f13_in(z0, z2), z0, .(z1, z2)), F13_IN(z0, z2))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c7(U10'(f13_in(z0, z2), z1, z2, z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, U2', F12_IN, U5', F13_IN, F2_IN, U9'

Compound Symbols:

c2, c3, c1, c4, c2, c5, c7

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

Removed 7 trailing tuple parts

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f7_in(z0, z1), z0, z1)
U1(f7_out1(z0), z1, z2) → f2_out1
f7_in(z0, z1) → U2(f12_in(z0), z0, z1)
f7_in(z0, z1) → U5(f12_in(z0), z0, z1)
f7_in(z0, z1) → U9(f12_in(z0), z0, z1)
U2(f12_out1(z0), z1, z2) → U3(f13_in(z0, z2), z1, z2, z0)
U3(f13_out1, z0, z1, z2) → f7_out1(z2)
f12_in(.(z0, z1)) → f12_out1(z0)
f12_in(.(z0, z1)) → U4(f12_in(z1), .(z0, z1))
U4(f12_out1(z0), .(z1, z2)) → f12_out1(z0)
U5(f12_out1(z0), z1, z2) → U6(f13_in(z0, z2), z1, z2, z0)
U6(f13_out1, z0, z1, z2) → f7_out1(z2)
f13_in(z0, .(z0, z1)) → f13_out1
f13_in(z0, .(z0, z1)) → U7(f13_in(z0, z1), z0, .(z0, z1))
f13_in(z0, .(z1, z2)) → U8(f13_in(z0, z2), z0, .(z1, z2))
U7(f13_out1, z0, .(z0, z1)) → f13_out1
U8(f13_out1, z0, .(z1, z2)) → f13_out1
U9(f12_out1(z0), z1, z2) → U10(f13_in(z0, z2), z1, z2, z0)
U10(f13_out1, z0, z1, z2) → f7_out1(z2)

Tuples:

F7_IN(z0, z1) → c2(U2'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
F2_IN(z0, z1) → c7(F7_IN(z0, z1))
F7_IN(z0, z1) → c7(U5'(f12_in(z0), z0, z1))
F7_IN(z0, z1) → c7(F12_IN(z0))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
U2'(f12_out1(z0), z1, z2) → c3
F12_IN(.(z0, z1)) → c1(F12_IN(z1))
U5'(f12_out1(z0), z1, z2) → c4
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
F2_IN(z0, z1) → c7
U9'(f12_out1(z0), z1, z2) → c7

S tuples:

F7_IN(z0, z1) → c5(U9'(f12_in(z0), z0, z1))
U9'(f12_out1(z0), z1, z2) → c7(F13_IN(z0, z2))
F13_IN(z0, .(z0, z1)) → c1(F13_IN(z0, z1))
F13_IN(z0, .(z1, z2)) → c2(F13_IN(z0, z2))
U9'(f12_out1(z0), z1, z2) → c7

K tuples:none
Defined Rule Symbols:

f2_in, U1, f7_in, U2, U3, f12_in, U4, U5, U6, f13_in, U7, U8, U9, U10

Defined Pair Symbols:

F7_IN, F2_IN, U9', U2', F12_IN, U5', F13_IN

Compound Symbols:

c2, c5, c7, c3, c1, c4, c7