(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.

(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.

(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