(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(app(nil, z0)) → mark(z0)
active(app(cons(z0, z1), z2)) → mark(cons(z0, app(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(zWadr(nil, z0)) → mark(nil)
active(zWadr(z0, nil)) → mark(nil)
active(zWadr(cons(z0, z1), cons(z2, z3))) → mark(cons(app(z2, cons(z0, nil)), zWadr(z1, z3)))
active(prefix(z0)) → mark(cons(nil, zWadr(z0, prefix(z0))))
mark(app(z0, z1)) → active(app(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(s(z0)) → active(s(mark(z0)))
mark(zWadr(z0, z1)) → active(zWadr(mark(z0), mark(z1)))
mark(prefix(z0)) → active(prefix(mark(z0)))
app(mark(z0), z1) → app(z0, z1)
app(z0, mark(z1)) → app(z0, z1)
app(active(z0), z1) → app(z0, z1)
app(z0, active(z1)) → app(z0, z1)
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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
zWadr(mark(z0), z1) → zWadr(z0, z1)
zWadr(z0, mark(z1)) → zWadr(z0, z1)
zWadr(active(z0), z1) → zWadr(z0, z1)
zWadr(z0, active(z1)) → zWadr(z0, z1)
prefix(mark(z0)) → prefix(z0)
prefix(active(z0)) → prefix(z0)
Tuples:
ACTIVE(app(nil, z0)) → c(MARK(z0))
ACTIVE(app(cons(z0, z1), z2)) → c1(MARK(cons(z0, app(z1, z2))), CONS(z0, app(z1, z2)), APP(z1, z2))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(zWadr(nil, z0)) → c3(MARK(nil))
ACTIVE(zWadr(z0, nil)) → c4(MARK(nil))
ACTIVE(zWadr(cons(z0, z1), cons(z2, z3))) → c5(MARK(cons(app(z2, cons(z0, nil)), zWadr(z1, z3))), CONS(app(z2, cons(z0, nil)), zWadr(z1, z3)), APP(z2, cons(z0, nil)), CONS(z0, nil), ZWADR(z1, z3))
ACTIVE(prefix(z0)) → c6(MARK(cons(nil, zWadr(z0, prefix(z0)))), CONS(nil, zWadr(z0, prefix(z0))), ZWADR(z0, prefix(z0)), PREFIX(z0))
MARK(app(z0, z1)) → c7(ACTIVE(app(mark(z0), mark(z1))), APP(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c8(ACTIVE(nil))
MARK(cons(z0, z1)) → c9(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c10(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) → c12(ACTIVE(zWadr(mark(z0), mark(z1))), ZWADR(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(prefix(z0)) → c13(ACTIVE(prefix(mark(z0))), PREFIX(mark(z0)), MARK(z0))
APP(mark(z0), z1) → c14(APP(z0, z1))
APP(z0, mark(z1)) → c15(APP(z0, z1))
APP(active(z0), z1) → c16(APP(z0, z1))
APP(z0, active(z1)) → c17(APP(z0, z1))
CONS(mark(z0), z1) → c18(CONS(z0, z1))
CONS(z0, mark(z1)) → c19(CONS(z0, z1))
CONS(active(z0), z1) → c20(CONS(z0, z1))
CONS(z0, active(z1)) → c21(CONS(z0, z1))
FROM(mark(z0)) → c22(FROM(z0))
FROM(active(z0)) → c23(FROM(z0))
S(mark(z0)) → c24(S(z0))
S(active(z0)) → c25(S(z0))
ZWADR(mark(z0), z1) → c26(ZWADR(z0, z1))
ZWADR(z0, mark(z1)) → c27(ZWADR(z0, z1))
ZWADR(active(z0), z1) → c28(ZWADR(z0, z1))
ZWADR(z0, active(z1)) → c29(ZWADR(z0, z1))
PREFIX(mark(z0)) → c30(PREFIX(z0))
PREFIX(active(z0)) → c31(PREFIX(z0))
S tuples:
ACTIVE(app(nil, z0)) → c(MARK(z0))
ACTIVE(app(cons(z0, z1), z2)) → c1(MARK(cons(z0, app(z1, z2))), CONS(z0, app(z1, z2)), APP(z1, z2))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(zWadr(nil, z0)) → c3(MARK(nil))
ACTIVE(zWadr(z0, nil)) → c4(MARK(nil))
ACTIVE(zWadr(cons(z0, z1), cons(z2, z3))) → c5(MARK(cons(app(z2, cons(z0, nil)), zWadr(z1, z3))), CONS(app(z2, cons(z0, nil)), zWadr(z1, z3)), APP(z2, cons(z0, nil)), CONS(z0, nil), ZWADR(z1, z3))
ACTIVE(prefix(z0)) → c6(MARK(cons(nil, zWadr(z0, prefix(z0)))), CONS(nil, zWadr(z0, prefix(z0))), ZWADR(z0, prefix(z0)), PREFIX(z0))
MARK(app(z0, z1)) → c7(ACTIVE(app(mark(z0), mark(z1))), APP(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c8(ACTIVE(nil))
MARK(cons(z0, z1)) → c9(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c10(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) → c12(ACTIVE(zWadr(mark(z0), mark(z1))), ZWADR(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(prefix(z0)) → c13(ACTIVE(prefix(mark(z0))), PREFIX(mark(z0)), MARK(z0))
APP(mark(z0), z1) → c14(APP(z0, z1))
APP(z0, mark(z1)) → c15(APP(z0, z1))
APP(active(z0), z1) → c16(APP(z0, z1))
APP(z0, active(z1)) → c17(APP(z0, z1))
CONS(mark(z0), z1) → c18(CONS(z0, z1))
CONS(z0, mark(z1)) → c19(CONS(z0, z1))
CONS(active(z0), z1) → c20(CONS(z0, z1))
CONS(z0, active(z1)) → c21(CONS(z0, z1))
FROM(mark(z0)) → c22(FROM(z0))
FROM(active(z0)) → c23(FROM(z0))
S(mark(z0)) → c24(S(z0))
S(active(z0)) → c25(S(z0))
ZWADR(mark(z0), z1) → c26(ZWADR(z0, z1))
ZWADR(z0, mark(z1)) → c27(ZWADR(z0, z1))
ZWADR(active(z0), z1) → c28(ZWADR(z0, z1))
ZWADR(z0, active(z1)) → c29(ZWADR(z0, z1))
PREFIX(mark(z0)) → c30(PREFIX(z0))
PREFIX(active(z0)) → c31(PREFIX(z0))
K tuples:none
Defined Rule Symbols:
active, mark, app, cons, from, s, zWadr, prefix
Defined Pair Symbols:
ACTIVE, MARK, APP, CONS, FROM, S, ZWADR, PREFIX
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
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(app(nil, z0)) → c(MARK(z0))
ACTIVE(app(cons(z0, z1), z2)) → c1(MARK(cons(z0, app(z1, z2))), CONS(z0, app(z1, z2)), APP(z1, z2))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(zWadr(nil, z0)) → c3(MARK(nil))
ACTIVE(zWadr(z0, nil)) → c4(MARK(nil))
ACTIVE(zWadr(cons(z0, z1), cons(z2, z3))) → c5(MARK(cons(app(z2, cons(z0, nil)), zWadr(z1, z3))), CONS(app(z2, cons(z0, nil)), zWadr(z1, z3)), APP(z2, cons(z0, nil)), CONS(z0, nil), ZWADR(z1, z3))
ACTIVE(prefix(z0)) → c6(MARK(cons(nil, zWadr(z0, prefix(z0)))), CONS(nil, zWadr(z0, prefix(z0))), ZWADR(z0, prefix(z0)), PREFIX(z0))
MARK(app(z0, z1)) → c7(ACTIVE(app(mark(z0), mark(z1))), APP(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(cons(z0, z1)) → c9(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c10(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(zWadr(z0, z1)) → c12(ACTIVE(zWadr(mark(z0), mark(z1))), ZWADR(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(prefix(z0)) → c13(ACTIVE(prefix(mark(z0))), PREFIX(mark(z0)), MARK(z0))
APP(mark(z0), z1) → c14(APP(z0, z1))
APP(z0, mark(z1)) → c15(APP(z0, z1))
APP(active(z0), z1) → c16(APP(z0, z1))
APP(z0, active(z1)) → c17(APP(z0, z1))
CONS(mark(z0), z1) → c18(CONS(z0, z1))
CONS(z0, mark(z1)) → c19(CONS(z0, z1))
CONS(active(z0), z1) → c20(CONS(z0, z1))
CONS(z0, active(z1)) → c21(CONS(z0, z1))
FROM(mark(z0)) → c22(FROM(z0))
FROM(active(z0)) → c23(FROM(z0))
S(mark(z0)) → c24(S(z0))
S(active(z0)) → c25(S(z0))
ZWADR(mark(z0), z1) → c26(ZWADR(z0, z1))
ZWADR(z0, mark(z1)) → c27(ZWADR(z0, z1))
ZWADR(active(z0), z1) → c28(ZWADR(z0, z1))
ZWADR(z0, active(z1)) → c29(ZWADR(z0, z1))
PREFIX(mark(z0)) → c30(PREFIX(z0))
PREFIX(active(z0)) → c31(PREFIX(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(app(nil, z0)) → mark(z0)
active(app(cons(z0, z1), z2)) → mark(cons(z0, app(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(zWadr(nil, z0)) → mark(nil)
active(zWadr(z0, nil)) → mark(nil)
active(zWadr(cons(z0, z1), cons(z2, z3))) → mark(cons(app(z2, cons(z0, nil)), zWadr(z1, z3)))
active(prefix(z0)) → mark(cons(nil, zWadr(z0, prefix(z0))))
mark(app(z0, z1)) → active(app(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(s(z0)) → active(s(mark(z0)))
mark(zWadr(z0, z1)) → active(zWadr(mark(z0), mark(z1)))
mark(prefix(z0)) → active(prefix(mark(z0)))
app(mark(z0), z1) → app(z0, z1)
app(z0, mark(z1)) → app(z0, z1)
app(active(z0), z1) → app(z0, z1)
app(z0, active(z1)) → app(z0, z1)
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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
zWadr(mark(z0), z1) → zWadr(z0, z1)
zWadr(z0, mark(z1)) → zWadr(z0, z1)
zWadr(active(z0), z1) → zWadr(z0, z1)
zWadr(z0, active(z1)) → zWadr(z0, z1)
prefix(mark(z0)) → prefix(z0)
prefix(active(z0)) → prefix(z0)
Tuples:
MARK(nil) → c8(ACTIVE(nil))
S tuples:
MARK(nil) → c8(ACTIVE(nil))
K tuples:none
Defined Rule Symbols:
active, mark, app, cons, from, s, zWadr, prefix
Defined Pair Symbols:
MARK
Compound Symbols:
c8
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
MARK(nil) → c8(ACTIVE(nil))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(app(nil, z0)) → mark(z0)
active(app(cons(z0, z1), z2)) → mark(cons(z0, app(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(zWadr(nil, z0)) → mark(nil)
active(zWadr(z0, nil)) → mark(nil)
active(zWadr(cons(z0, z1), cons(z2, z3))) → mark(cons(app(z2, cons(z0, nil)), zWadr(z1, z3)))
active(prefix(z0)) → mark(cons(nil, zWadr(z0, prefix(z0))))
mark(app(z0, z1)) → active(app(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(s(z0)) → active(s(mark(z0)))
mark(zWadr(z0, z1)) → active(zWadr(mark(z0), mark(z1)))
mark(prefix(z0)) → active(prefix(mark(z0)))
app(mark(z0), z1) → app(z0, z1)
app(z0, mark(z1)) → app(z0, z1)
app(active(z0), z1) → app(z0, z1)
app(z0, active(z1)) → app(z0, z1)
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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
zWadr(mark(z0), z1) → zWadr(z0, z1)
zWadr(z0, mark(z1)) → zWadr(z0, z1)
zWadr(active(z0), z1) → zWadr(z0, z1)
zWadr(z0, active(z1)) → zWadr(z0, z1)
prefix(mark(z0)) → prefix(z0)
prefix(active(z0)) → prefix(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, app, cons, from, s, zWadr, prefix
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))