### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(fib(X)) → fib(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(fib1(X1, X2)) → fib1(active(X1), X2)
active(fib1(X1, X2)) → fib1(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
fib(mark(X)) → mark(fib(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
proper(fib(X)) → fib(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) → fib1(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

### (1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(sel(0, cons(z0, z1))) → c4
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(0) → c34
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(sel(0, cons(z0, z1))) → c4
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(0) → c34
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, fib, sel, fib1, s, cons, add, proper, top

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38

### (3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

PROPER(0) → c34
ACTIVE(sel(0, cons(z0, z1))) → c4

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, fib, sel, fib1, s, cons, add, proper, top

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c, c1, c3, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38

### (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
K tuples:none
Defined Rule Symbols:

active, fib, sel, fib1, s, cons, add, proper, top

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3

### (9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(fib(z0)) → c6(FIB(active(z0)), ACTIVE(z0)) by

ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6

### (11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c11(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11

### (13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(cons(z0, z1)) → c12(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12

### (15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(fib(z0)) → c30(FIB(proper(z0)), PROPER(z0)) by

PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(0)) → c30(FIB(ok(0)), PROPER(0))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(0)) → c30(FIB(ok(0)), PROPER(0))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(0)) → c30(FIB(ok(0)), PROPER(0))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30

### (17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c31, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30

### (19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(sel(z0, z1)) → c31(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31

### (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c32, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31

### (23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(fib1(z0, z1)) → c32(FIB1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, fib, fib1, s, cons, add, proper

Defined Pair Symbols:

ACTIVE, FIB, SEL, FIB1, S, CONS, ADD, PROPER, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c33, c35, c36, c37, c38, c, c1, c3, c6, c11, c12, c30, c30, c31, c31, c32

### (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fib(z0)) → mark(sel(z0, fib1(s(0), s(0))))
active(fib1(z0, z1)) → mark(cons(z0, fib1(z1, add(z0, z1))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(fib(z0)) → fib(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
active(fib1(z0, z1)) → fib1(active(z0), z1)
active(fib1(z0, z1)) → fib1(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
fib(mark(z0)) → mark(fib(z0))
fib(ok(z0)) → ok(fib(z0))
fib1(mark(z0), z1) → mark(fib1(z0, z1))
fib1(ok(z0), ok(z1)) → ok(fib1(z0, z1))
fib1(z0, mark(z1)) → mark(fib1(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(fib(z0)) → fib(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
proper(fib1(z0, z1)) → fib1(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(mark(sel(z0, fib1(s(0), s(0)))), x1), ACTIVE(fib(z0)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(mark(cons(z0, fib1(z1, add(z0, z1)))), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c12(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c12(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(fib(z0), x1)) → c12(CONS(fib(active(z0)), x1), ACTIVE(fib(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c12(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(active(z0), z1), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(fib1(z0, z1), x1)) → c12(CONS(fib1(z0, active(z1)), x1), ACTIVE(fib1(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c12(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c12(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
PROPER(fib(fib(z0))) → c30(FIB(fib(proper(z0))), PROPER(fib(z0)))
PROPER(fib(sel(z0, z1))) → c30(FIB(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(fib(fib1(z0, z1))) → c30(FIB(fib1(proper(z0), proper(z1))), PROPER(fib1(z0, z1)))
PROPER(fib(s(z0))) → c30(FIB(s(proper(z0))), PROPER(s(z0)))
PROPER(fib(cons(z0, z1))) → c30(FIB(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(fib(0)) → c30(FIB(ok(0)))
PROPER(sel(x0, fib(z0))) → c31(SEL(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(sel(x0, sel(z0, z1))) → c31(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(x0, fib1(z0, z1))) → c31(SEL(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(sel(x0, s(z0))) → c31(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, cons(z0, z1))) → c31(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(fib(z0), x1)) → c31(SEL(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c31(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(fib1(z0, z1), x1)) → c31(SEL(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c31(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c31(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c31(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c31(SEL(ok(0), proper(x1)), PROPER(x1))
PROPER(fib1(x0, fib(z0))) → c32(FIB1(proper(x0), fib(proper(z0))), PROPER(x0), PROPER(fib(z0)))
PROPER(fib1(x0, sel(z0, z1))) → c32(FIB1(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(fib1(x0, fib1(z0, z1))) → c32(FIB1(proper(x0), fib1(proper(z0), proper(z1))), PROPER(x0), PROPER(fib1(z0, z1)))
PROPER(fib1(x0, s(z0))) → c32(FIB1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(fib1(x0, cons(z0, z1))) → c32(FIB1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(fib1(fib(z0), x1)) → c32(FIB1(fib(proper(z0)), proper(x1)), PROPER(fib(z0)), PROPER(x1))
PROPER(fib1(sel(z0, z1), x1)) → c32(FIB1(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(fib1(fib1(z0, z1), x1)) → c32(FIB1(fib1(proper(z0), proper(z1)), proper(x1)), PROPER(fib1(z0, z1)), PROPER(x1))
PROPER(fib1(s(z0), x1)) → c32(FIB1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(fib1(cons(z0, z1), x1)) → c32(FIB1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(fib1(x0, 0)) → c32(FIB1(proper(x0), ok(0)), PROPER(x0))
PROPER(fib1(0, x1)) → c32(FIB1(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c5(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c7(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c8(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(fib1(z0, z1)) → c9(FIB1(active(z0), z1), ACTIVE(z0))
ACTIVE(fib1(z0, z1)) → c10(FIB1(z0, active(z1)), ACTIVE(z1))
FIB(mark(z0)) → c15(FIB(z0))
FIB(ok(z0)) → c16(FIB(z0))
SEL(mark(z0), z1) → c17(SEL(z0, z1))
SEL(z0, mark(z1)) → c18(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
FIB1(mark(z0), z1) → c20(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c21(FIB1(z0, z1))
FIB1(ok(z0), ok(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(ok(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c26(CONS(z0, z1))
PROPER(s(z0)) → c33(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fib(z0)) → c(SEL(z0, fib1(s(0), s(0))))
ACTIVE(fib(fib(z0))) → c6(FIB(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(sel(0, cons(z0, z1)))) → c6(FIB(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(fib(sel(s(z0), cons(z1, z2)))) → c6(FIB(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(fib(fib(z0))) → c6(FIB(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(fib(sel(z0, z1))) → c6(FIB(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(fib1(z0, z1))) → c6(FIB(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(fib(s(z0))) → c6(FIB(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(fib(cons(z0, z1))) → c6(FIB(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(fib(z0))) → c11(S(mark(sel(z0, fib1(s(0), s(0))))), ACTIVE(fib(z0)))
ACTIVE(s(fib1(z0, z1))) → c11(S(mark(cons(z0, fib1(z1, add(z0, z1))))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(sel(0, cons(z0, z1)))) → c11(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c11(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(fib(z0))) → c11(S(fib(active(z0))), ACTIVE(fib(z0)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c11(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(active(z0), z1)), ACTIVE(fib1(z0, z1)))
ACTIVE(s(fib1(z0, z1))) → c11(S(fib1(z0, active(z1))), ACTIVE(fib1(z0, z1)))
ACTIVE(s(s(z0))) → c11(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c11(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))