(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
mark(__(z0, z1)) → active(__(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(isNePal(z0)) → active(isNePal(mark(z0)))
__(mark(z0), z1) → __(z0, z1)
__(z0, mark(z1)) → __(z0, z1)
__(active(z0), z1) → __(z0, z1)
__(z0, active(z1)) → __(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
isNePal(mark(z0)) → isNePal(z0)
isNePal(active(z0)) → isNePal(z0)
Tuples:
ACTIVE(__(__(z0, z1), z2)) → c(MARK(__(z0, __(z1, z2))), __'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, nil)) → c1(MARK(z0))
ACTIVE(__(nil, z0)) → c2(MARK(z0))
ACTIVE(and(tt, z0)) → c3(MARK(z0))
ACTIVE(isNePal(__(z0, __(z1, z0)))) → c4(MARK(tt))
MARK(__(z0, z1)) → c5(ACTIVE(__(mark(z0), mark(z1))), __'(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c6(ACTIVE(nil))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c8(ACTIVE(tt))
MARK(isNePal(z0)) → c9(ACTIVE(isNePal(mark(z0))), ISNEPAL(mark(z0)), MARK(z0))
__'(mark(z0), z1) → c10(__'(z0, z1))
__'(z0, mark(z1)) → c11(__'(z0, z1))
__'(active(z0), z1) → c12(__'(z0, z1))
__'(z0, active(z1)) → c13(__'(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
AND(z0, mark(z1)) → c15(AND(z0, z1))
AND(active(z0), z1) → c16(AND(z0, z1))
AND(z0, active(z1)) → c17(AND(z0, z1))
ISNEPAL(mark(z0)) → c18(ISNEPAL(z0))
ISNEPAL(active(z0)) → c19(ISNEPAL(z0))
S tuples:
ACTIVE(__(__(z0, z1), z2)) → c(MARK(__(z0, __(z1, z2))), __'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, nil)) → c1(MARK(z0))
ACTIVE(__(nil, z0)) → c2(MARK(z0))
ACTIVE(and(tt, z0)) → c3(MARK(z0))
ACTIVE(isNePal(__(z0, __(z1, z0)))) → c4(MARK(tt))
MARK(__(z0, z1)) → c5(ACTIVE(__(mark(z0), mark(z1))), __'(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c6(ACTIVE(nil))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c8(ACTIVE(tt))
MARK(isNePal(z0)) → c9(ACTIVE(isNePal(mark(z0))), ISNEPAL(mark(z0)), MARK(z0))
__'(mark(z0), z1) → c10(__'(z0, z1))
__'(z0, mark(z1)) → c11(__'(z0, z1))
__'(active(z0), z1) → c12(__'(z0, z1))
__'(z0, active(z1)) → c13(__'(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
AND(z0, mark(z1)) → c15(AND(z0, z1))
AND(active(z0), z1) → c16(AND(z0, z1))
AND(z0, active(z1)) → c17(AND(z0, z1))
ISNEPAL(mark(z0)) → c18(ISNEPAL(z0))
ISNEPAL(active(z0)) → c19(ISNEPAL(z0))
K tuples:none
Defined Rule Symbols:
active, mark, __, and, isNePal
Defined Pair Symbols:
ACTIVE, MARK, __', AND, ISNEPAL
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(__(__(z0, z1), z2)) → c(MARK(__(z0, __(z1, z2))), __'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, nil)) → c1(MARK(z0))
ACTIVE(__(nil, z0)) → c2(MARK(z0))
ACTIVE(and(tt, z0)) → c3(MARK(z0))
ACTIVE(isNePal(__(z0, __(z1, z0)))) → c4(MARK(tt))
MARK(__(z0, z1)) → c5(ACTIVE(__(mark(z0), mark(z1))), __'(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNePal(z0)) → c9(ACTIVE(isNePal(mark(z0))), ISNEPAL(mark(z0)), MARK(z0))
__'(mark(z0), z1) → c10(__'(z0, z1))
__'(z0, mark(z1)) → c11(__'(z0, z1))
__'(active(z0), z1) → c12(__'(z0, z1))
__'(z0, active(z1)) → c13(__'(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
AND(z0, mark(z1)) → c15(AND(z0, z1))
AND(active(z0), z1) → c16(AND(z0, z1))
AND(z0, active(z1)) → c17(AND(z0, z1))
ISNEPAL(mark(z0)) → c18(ISNEPAL(z0))
ISNEPAL(active(z0)) → c19(ISNEPAL(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
mark(__(z0, z1)) → active(__(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(isNePal(z0)) → active(isNePal(mark(z0)))
__(mark(z0), z1) → __(z0, z1)
__(z0, mark(z1)) → __(z0, z1)
__(active(z0), z1) → __(z0, z1)
__(z0, active(z1)) → __(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
isNePal(mark(z0)) → isNePal(z0)
isNePal(active(z0)) → isNePal(z0)
Tuples:
MARK(nil) → c6(ACTIVE(nil))
MARK(tt) → c8(ACTIVE(tt))
S tuples:
MARK(nil) → c6(ACTIVE(nil))
MARK(tt) → c8(ACTIVE(tt))
K tuples:none
Defined Rule Symbols:
active, mark, __, and, isNePal
Defined Pair Symbols:
MARK
Compound Symbols:
c6, c8
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(nil) → c6(ACTIVE(nil))
MARK(tt) → c8(ACTIVE(tt))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
mark(__(z0, z1)) → active(__(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(isNePal(z0)) → active(isNePal(mark(z0)))
__(mark(z0), z1) → __(z0, z1)
__(z0, mark(z1)) → __(z0, z1)
__(active(z0), z1) → __(z0, z1)
__(z0, active(z1)) → __(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
isNePal(mark(z0)) → isNePal(z0)
isNePal(active(z0)) → isNePal(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, __, and, isNePal
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))