(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(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS 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(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
mark(fib(z0)) → active(fib(mark(z0)))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(fib1(z0, z1)) → active(fib1(mark(z0), mark(z1)))
mark(s(z0)) → active(s(mark(z0)))
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(add(z0, z1)) → active(add(mark(z0), mark(z1)))
fib(mark(z0)) → fib(z0)
fib(active(z0)) → fib(z0)
sel(mark(z0), z1) → sel(z0, z1)
sel(z0, mark(z1)) → sel(z0, z1)
sel(active(z0), z1) → sel(z0, z1)
sel(z0, active(z1)) → sel(z0, z1)
fib1(mark(z0), z1) → fib1(z0, z1)
fib1(z0, mark(z1)) → fib1(z0, z1)
fib1(active(z0), z1) → fib1(z0, z1)
fib1(z0, active(z1)) → fib1(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
add(mark(z0), z1) → add(z0, z1)
add(z0, mark(z1)) → add(z0, z1)
add(active(z0), z1) → add(z0, z1)
add(z0, active(z1)) → add(z0, z1)
Tuples:
ACTIVE(fib(z0)) → c(MARK(sel(z0, fib1(s(0), s(0)))), SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(MARK(cons(z0, fib1(z1, add(z0, z1)))), CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(0, z0)) → c2(MARK(z0))
ACTIVE(add(s(z0), z1)) → c3(MARK(s(add(z0, z1))), S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(0, cons(z0, z1))) → c4(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(MARK(sel(z0, z2)), SEL(z0, z2))
MARK(fib(z0)) → c6(ACTIVE(fib(mark(z0))), FIB(mark(z0)), MARK(z0))
MARK(sel(z0, z1)) → c7(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(fib1(z0, z1)) → c8(ACTIVE(fib1(mark(z0), mark(z1))), FIB1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(0) → c10(ACTIVE(0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(add(z0, z1)) → c12(ACTIVE(add(mark(z0), mark(z1))), ADD(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FIB(mark(z0)) → c13(FIB(z0))
FIB(active(z0)) → c14(FIB(z0))
SEL(mark(z0), z1) → c15(SEL(z0, z1))
SEL(z0, mark(z1)) → c16(SEL(z0, z1))
SEL(active(z0), z1) → c17(SEL(z0, z1))
SEL(z0, active(z1)) → c18(SEL(z0, z1))
FIB1(mark(z0), z1) → c19(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c20(FIB1(z0, z1))
FIB1(active(z0), z1) → c21(FIB1(z0, z1))
FIB1(z0, active(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(active(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(z0, mark(z1)) → c26(CONS(z0, z1))
CONS(active(z0), z1) → c27(CONS(z0, z1))
CONS(z0, active(z1)) → c28(CONS(z0, z1))
ADD(mark(z0), z1) → c29(ADD(z0, z1))
ADD(z0, mark(z1)) → c30(ADD(z0, z1))
ADD(active(z0), z1) → c31(ADD(z0, z1))
ADD(z0, active(z1)) → c32(ADD(z0, z1))
S tuples:
ACTIVE(fib(z0)) → c(MARK(sel(z0, fib1(s(0), s(0)))), SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(MARK(cons(z0, fib1(z1, add(z0, z1)))), CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(0, z0)) → c2(MARK(z0))
ACTIVE(add(s(z0), z1)) → c3(MARK(s(add(z0, z1))), S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(0, cons(z0, z1))) → c4(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(MARK(sel(z0, z2)), SEL(z0, z2))
MARK(fib(z0)) → c6(ACTIVE(fib(mark(z0))), FIB(mark(z0)), MARK(z0))
MARK(sel(z0, z1)) → c7(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(fib1(z0, z1)) → c8(ACTIVE(fib1(mark(z0), mark(z1))), FIB1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(0) → c10(ACTIVE(0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(add(z0, z1)) → c12(ACTIVE(add(mark(z0), mark(z1))), ADD(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FIB(mark(z0)) → c13(FIB(z0))
FIB(active(z0)) → c14(FIB(z0))
SEL(mark(z0), z1) → c15(SEL(z0, z1))
SEL(z0, mark(z1)) → c16(SEL(z0, z1))
SEL(active(z0), z1) → c17(SEL(z0, z1))
SEL(z0, active(z1)) → c18(SEL(z0, z1))
FIB1(mark(z0), z1) → c19(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c20(FIB1(z0, z1))
FIB1(active(z0), z1) → c21(FIB1(z0, z1))
FIB1(z0, active(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(active(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(z0, mark(z1)) → c26(CONS(z0, z1))
CONS(active(z0), z1) → c27(CONS(z0, z1))
CONS(z0, active(z1)) → c28(CONS(z0, z1))
ADD(mark(z0), z1) → c29(ADD(z0, z1))
ADD(z0, mark(z1)) → c30(ADD(z0, z1))
ADD(active(z0), z1) → c31(ADD(z0, z1))
ADD(z0, active(z1)) → c32(ADD(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, fib, sel, fib1, s, cons, add
Defined Pair Symbols:
ACTIVE, MARK, FIB, SEL, FIB1, S, CONS, ADD
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
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(fib(z0)) → c(MARK(sel(z0, fib1(s(0), s(0)))), SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)), S(0), S(0))
ACTIVE(fib1(z0, z1)) → c1(MARK(cons(z0, fib1(z1, add(z0, z1)))), CONS(z0, fib1(z1, add(z0, z1))), FIB1(z1, add(z0, z1)), ADD(z0, z1))
ACTIVE(add(0, z0)) → c2(MARK(z0))
ACTIVE(add(s(z0), z1)) → c3(MARK(s(add(z0, z1))), S(add(z0, z1)), ADD(z0, z1))
ACTIVE(sel(0, cons(z0, z1))) → c4(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(MARK(sel(z0, z2)), SEL(z0, z2))
MARK(fib(z0)) → c6(ACTIVE(fib(mark(z0))), FIB(mark(z0)), MARK(z0))
MARK(sel(z0, z1)) → c7(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(fib1(z0, z1)) → c8(ACTIVE(fib1(mark(z0), mark(z1))), FIB1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(add(z0, z1)) → c12(ACTIVE(add(mark(z0), mark(z1))), ADD(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FIB(mark(z0)) → c13(FIB(z0))
FIB(active(z0)) → c14(FIB(z0))
SEL(mark(z0), z1) → c15(SEL(z0, z1))
SEL(z0, mark(z1)) → c16(SEL(z0, z1))
SEL(active(z0), z1) → c17(SEL(z0, z1))
SEL(z0, active(z1)) → c18(SEL(z0, z1))
FIB1(mark(z0), z1) → c19(FIB1(z0, z1))
FIB1(z0, mark(z1)) → c20(FIB1(z0, z1))
FIB1(active(z0), z1) → c21(FIB1(z0, z1))
FIB1(z0, active(z1)) → c22(FIB1(z0, z1))
S(mark(z0)) → c23(S(z0))
S(active(z0)) → c24(S(z0))
CONS(mark(z0), z1) → c25(CONS(z0, z1))
CONS(z0, mark(z1)) → c26(CONS(z0, z1))
CONS(active(z0), z1) → c27(CONS(z0, z1))
CONS(z0, active(z1)) → c28(CONS(z0, z1))
ADD(mark(z0), z1) → c29(ADD(z0, z1))
ADD(z0, mark(z1)) → c30(ADD(z0, z1))
ADD(active(z0), z1) → c31(ADD(z0, z1))
ADD(z0, active(z1)) → c32(ADD(z0, z1))
(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(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
mark(fib(z0)) → active(fib(mark(z0)))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(fib1(z0, z1)) → active(fib1(mark(z0), mark(z1)))
mark(s(z0)) → active(s(mark(z0)))
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(add(z0, z1)) → active(add(mark(z0), mark(z1)))
fib(mark(z0)) → fib(z0)
fib(active(z0)) → fib(z0)
sel(mark(z0), z1) → sel(z0, z1)
sel(z0, mark(z1)) → sel(z0, z1)
sel(active(z0), z1) → sel(z0, z1)
sel(z0, active(z1)) → sel(z0, z1)
fib1(mark(z0), z1) → fib1(z0, z1)
fib1(z0, mark(z1)) → fib1(z0, z1)
fib1(active(z0), z1) → fib1(z0, z1)
fib1(z0, active(z1)) → fib1(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
add(mark(z0), z1) → add(z0, z1)
add(z0, mark(z1)) → add(z0, z1)
add(active(z0), z1) → add(z0, z1)
add(z0, active(z1)) → add(z0, z1)
Tuples:
MARK(0) → c10(ACTIVE(0))
S tuples:
MARK(0) → c10(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, fib, sel, fib1, s, cons, add
Defined Pair Symbols:
MARK
Compound Symbols:
c10
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
MARK(0) → c10(ACTIVE(0))
(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(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
mark(fib(z0)) → active(fib(mark(z0)))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(fib1(z0, z1)) → active(fib1(mark(z0), mark(z1)))
mark(s(z0)) → active(s(mark(z0)))
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(add(z0, z1)) → active(add(mark(z0), mark(z1)))
fib(mark(z0)) → fib(z0)
fib(active(z0)) → fib(z0)
sel(mark(z0), z1) → sel(z0, z1)
sel(z0, mark(z1)) → sel(z0, z1)
sel(active(z0), z1) → sel(z0, z1)
sel(z0, active(z1)) → sel(z0, z1)
fib1(mark(z0), z1) → fib1(z0, z1)
fib1(z0, mark(z1)) → fib1(z0, z1)
fib1(active(z0), z1) → fib1(z0, z1)
fib1(z0, active(z1)) → fib1(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
add(mark(z0), z1) → add(z0, z1)
add(z0, mark(z1)) → add(z0, z1)
add(active(z0), z1) → add(z0, z1)
add(z0, active(z1)) → add(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, fib, sel, fib1, s, cons, add
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))