(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(fst(0, Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0)
active(len(cons(X, Z))) → mark(s(len(Z)))
mark(fst(X1, X2)) → active(fst(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(s(X)) → active(s(X))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
mark(len(X)) → active(len(mark(X)))
fst(mark(X1), X2) → fst(X1, X2)
fst(X1, mark(X2)) → fst(X1, X2)
fst(active(X1), X2) → fst(X1, X2)
fst(X1, active(X2)) → fst(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)
from(mark(X)) → from(X)
from(active(X)) → from(X)
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)
len(mark(X)) → len(X)
len(active(X)) → len(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fst(0, z0)) → mark(nil)
active(fst(s(z0), cons(z1, z2))) → mark(cons(z1, fst(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(len(nil)) → mark(0)
active(len(cons(z0, z1))) → mark(s(len(z1)))
mark(fst(z0, z1)) → active(fst(mark(z0), mark(z1)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(s(z0)) → active(s(z0))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(add(z0, z1)) → active(add(mark(z0), mark(z1)))
mark(len(z0)) → active(len(mark(z0)))
fst(mark(z0), z1) → fst(z0, z1)
fst(z0, mark(z1)) → fst(z0, z1)
fst(active(z0), z1) → fst(z0, z1)
fst(z0, active(z1)) → fst(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)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
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)
len(mark(z0)) → len(z0)
len(active(z0)) → len(z0)
Tuples:
ACTIVE(fst(0, z0)) → c(MARK(nil))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(MARK(cons(z1, fst(z0, z2))), CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(add(0, z0)) → c3(MARK(z0))
ACTIVE(add(s(z0), z1)) → c4(MARK(s(add(z0, z1))), S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(nil)) → c5(MARK(0))
ACTIVE(len(cons(z0, z1))) → c6(MARK(s(len(z1))), S(len(z1)), LEN(z1))
MARK(fst(z0, z1)) → c7(ACTIVE(fst(mark(z0), mark(z1))), FST(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c8(ACTIVE(0))
MARK(nil) → c9(ACTIVE(nil))
MARK(s(z0)) → c10(ACTIVE(s(z0)), S(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c12(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) → c13(ACTIVE(add(mark(z0), mark(z1))), ADD(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(len(z0)) → c14(ACTIVE(len(mark(z0))), LEN(mark(z0)), MARK(z0))
FST(mark(z0), z1) → c15(FST(z0, z1))
FST(z0, mark(z1)) → c16(FST(z0, z1))
FST(active(z0), z1) → c17(FST(z0, z1))
FST(z0, active(z1)) → c18(FST(z0, z1))
S(mark(z0)) → c19(S(z0))
S(active(z0)) → c20(S(z0))
CONS(mark(z0), z1) → c21(CONS(z0, z1))
CONS(z0, mark(z1)) → c22(CONS(z0, z1))
CONS(active(z0), z1) → c23(CONS(z0, z1))
CONS(z0, active(z1)) → c24(CONS(z0, z1))
FROM(mark(z0)) → c25(FROM(z0))
FROM(active(z0)) → c26(FROM(z0))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(active(z0), z1) → c29(ADD(z0, z1))
ADD(z0, active(z1)) → c30(ADD(z0, z1))
LEN(mark(z0)) → c31(LEN(z0))
LEN(active(z0)) → c32(LEN(z0))
S tuples:
ACTIVE(fst(0, z0)) → c(MARK(nil))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(MARK(cons(z1, fst(z0, z2))), CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(add(0, z0)) → c3(MARK(z0))
ACTIVE(add(s(z0), z1)) → c4(MARK(s(add(z0, z1))), S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(nil)) → c5(MARK(0))
ACTIVE(len(cons(z0, z1))) → c6(MARK(s(len(z1))), S(len(z1)), LEN(z1))
MARK(fst(z0, z1)) → c7(ACTIVE(fst(mark(z0), mark(z1))), FST(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c8(ACTIVE(0))
MARK(nil) → c9(ACTIVE(nil))
MARK(s(z0)) → c10(ACTIVE(s(z0)), S(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c12(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) → c13(ACTIVE(add(mark(z0), mark(z1))), ADD(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(len(z0)) → c14(ACTIVE(len(mark(z0))), LEN(mark(z0)), MARK(z0))
FST(mark(z0), z1) → c15(FST(z0, z1))
FST(z0, mark(z1)) → c16(FST(z0, z1))
FST(active(z0), z1) → c17(FST(z0, z1))
FST(z0, active(z1)) → c18(FST(z0, z1))
S(mark(z0)) → c19(S(z0))
S(active(z0)) → c20(S(z0))
CONS(mark(z0), z1) → c21(CONS(z0, z1))
CONS(z0, mark(z1)) → c22(CONS(z0, z1))
CONS(active(z0), z1) → c23(CONS(z0, z1))
CONS(z0, active(z1)) → c24(CONS(z0, z1))
FROM(mark(z0)) → c25(FROM(z0))
FROM(active(z0)) → c26(FROM(z0))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(active(z0), z1) → c29(ADD(z0, z1))
ADD(z0, active(z1)) → c30(ADD(z0, z1))
LEN(mark(z0)) → c31(LEN(z0))
LEN(active(z0)) → c32(LEN(z0))
K tuples:none
Defined Rule Symbols:
active, mark, fst, s, cons, from, add, len
Defined Pair Symbols:
ACTIVE, MARK, FST, S, CONS, FROM, ADD, LEN
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(fst(0, z0)) → c(MARK(nil))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(MARK(cons(z1, fst(z0, z2))), CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(add(0, z0)) → c3(MARK(z0))
ACTIVE(add(s(z0), z1)) → c4(MARK(s(add(z0, z1))), S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(nil)) → c5(MARK(0))
ACTIVE(len(cons(z0, z1))) → c6(MARK(s(len(z1))), S(len(z1)), LEN(z1))
MARK(fst(z0, z1)) → c7(ACTIVE(fst(mark(z0), mark(z1))), FST(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c10(ACTIVE(s(z0)), S(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c12(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(add(z0, z1)) → c13(ACTIVE(add(mark(z0), mark(z1))), ADD(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(len(z0)) → c14(ACTIVE(len(mark(z0))), LEN(mark(z0)), MARK(z0))
FST(mark(z0), z1) → c15(FST(z0, z1))
FST(z0, mark(z1)) → c16(FST(z0, z1))
FST(active(z0), z1) → c17(FST(z0, z1))
FST(z0, active(z1)) → c18(FST(z0, z1))
S(mark(z0)) → c19(S(z0))
S(active(z0)) → c20(S(z0))
CONS(mark(z0), z1) → c21(CONS(z0, z1))
CONS(z0, mark(z1)) → c22(CONS(z0, z1))
CONS(active(z0), z1) → c23(CONS(z0, z1))
CONS(z0, active(z1)) → c24(CONS(z0, z1))
FROM(mark(z0)) → c25(FROM(z0))
FROM(active(z0)) → c26(FROM(z0))
ADD(mark(z0), z1) → c27(ADD(z0, z1))
ADD(z0, mark(z1)) → c28(ADD(z0, z1))
ADD(active(z0), z1) → c29(ADD(z0, z1))
ADD(z0, active(z1)) → c30(ADD(z0, z1))
LEN(mark(z0)) → c31(LEN(z0))
LEN(active(z0)) → c32(LEN(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fst(0, z0)) → mark(nil)
active(fst(s(z0), cons(z1, z2))) → mark(cons(z1, fst(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(len(nil)) → mark(0)
active(len(cons(z0, z1))) → mark(s(len(z1)))
mark(fst(z0, z1)) → active(fst(mark(z0), mark(z1)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(s(z0)) → active(s(z0))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(add(z0, z1)) → active(add(mark(z0), mark(z1)))
mark(len(z0)) → active(len(mark(z0)))
fst(mark(z0), z1) → fst(z0, z1)
fst(z0, mark(z1)) → fst(z0, z1)
fst(active(z0), z1) → fst(z0, z1)
fst(z0, active(z1)) → fst(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)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
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)
len(mark(z0)) → len(z0)
len(active(z0)) → len(z0)
Tuples:
MARK(0) → c8(ACTIVE(0))
MARK(nil) → c9(ACTIVE(nil))
S tuples:
MARK(0) → c8(ACTIVE(0))
MARK(nil) → c9(ACTIVE(nil))
K tuples:none
Defined Rule Symbols:
active, mark, fst, s, cons, from, add, len
Defined Pair Symbols:
MARK
Compound Symbols:
c8, c9
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(nil) → c9(ACTIVE(nil))
MARK(0) → c8(ACTIVE(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(fst(0, z0)) → mark(nil)
active(fst(s(z0), cons(z1, z2))) → mark(cons(z1, fst(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(len(nil)) → mark(0)
active(len(cons(z0, z1))) → mark(s(len(z1)))
mark(fst(z0, z1)) → active(fst(mark(z0), mark(z1)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(s(z0)) → active(s(z0))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(add(z0, z1)) → active(add(mark(z0), mark(z1)))
mark(len(z0)) → active(len(mark(z0)))
fst(mark(z0), z1) → fst(z0, z1)
fst(z0, mark(z1)) → fst(z0, z1)
fst(active(z0), z1) → fst(z0, z1)
fst(z0, active(z1)) → fst(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)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
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)
len(mark(z0)) → len(z0)
len(active(z0)) → len(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, fst, s, cons, from, add, len
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))