(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
mark(s(X)) → active(s(mark(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)
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)
length(mark(X)) → length(X)
length(active(X)) → length(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(nil)) → c2(MARK(0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c6(ACTIVE(0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c8(ACTIVE(tt))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c10(ACTIVE(nil))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
S tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(nil)) → c2(MARK(0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c6(ACTIVE(0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c8(ACTIVE(tt))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(nil) → c10(ACTIVE(nil))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
K tuples:none
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
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
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
MARK(0) → c6(ACTIVE(0))
ACTIVE(length(nil)) → c2(MARK(0))
MARK(tt) → c8(ACTIVE(tt))
MARK(nil) → c10(ACTIVE(nil))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
S tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
K tuples:none
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c, c1, c3, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23
(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
ACTIVE(and(tt, z0)) → c1(MARK(z0))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
And the Tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(AND(x1, x2)) = 0
POL(CONS(x1, x2)) = 0
POL(LENGTH(x1)) = 0
POL(MARK(x1)) = [2]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [3]x1 + [2]x2
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2, x3)) = x1 + x2 + x3
POL(c4(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [4]x1 + [4]x2
POL(length(x1)) = [4]x1
POL(mark(x1)) = x1
POL(nil) = [4]
POL(s(x1)) = [2]x1
POL(tt) = [3]
POL(zeros) = 0
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
S tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c, c1, c3, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23
(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
And the Tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = [2]x1
POL(AND(x1, x2)) = [2]
POL(CONS(x1, x2)) = [5]x2
POL(LENGTH(x1)) = 0
POL(MARK(x1)) = [3]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [5] + [4]x1 + [4]x2
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2, x3)) = x1 + x2 + x3
POL(c4(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [4]x1 + [5]x2
POL(length(x1)) = [3]x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = [3]x1
POL(tt) = 0
POL(zeros) = 0
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
S tuples:
ACTIVE(zeros) → c(MARK(cons(0, zeros)), CONS(0, zeros))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c, c1, c3, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
zeros) →
c(
MARK(
cons(
0,
zeros)),
CONS(
0,
zeros)) by
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
S tuples:
ACTIVE(length(cons(z0, z1))) → c3(MARK(s(length(z1))), S(length(z1)), LENGTH(z1))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c3, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
length(
cons(
z0,
z1))) →
c3(
MARK(
s(
length(
z1))),
S(
length(
z1)),
LENGTH(
z1)) by
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c5, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
MARK(
cons(
z0,
z1)) →
c5(
ACTIVE(
cons(
mark(
z0),
z1)),
CONS(
mark(
z0),
z1),
MARK(
z0)) by
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5
(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
And the Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = [2]x1
POL(AND(x1, x2)) = [4]x2
POL(CONS(x1, x2)) = x2
POL(LENGTH(x1)) = x1
POL(MARK(x1)) = [3]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [4]x1 + [4]x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [5]x1 + [5]x2
POL(length(x1)) = [4]x1
POL(mark(x1)) = x1
POL(nil) = [2]
POL(s(x1)) = [3]x1
POL(tt) = 0
POL(zeros) = 0
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5
(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
And the Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(AND(x1, x2)) = [1]
POL(CONS(x1, x2)) = [2]x2
POL(LENGTH(x1)) = 0
POL(MARK(x1)) = [2]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [2] + [2]x1 + [2]x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [4]x1 + [4]x2
POL(length(x1)) = [2]x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = [2]x1
POL(tt) = 0
POL(zeros) = 0
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5
(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
And the Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(AND(x1, x2)) = x1 + [2]x2
POL(CONS(x1, x2)) = [3]x2
POL(LENGTH(x1)) = 0
POL(MARK(x1)) = [2]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [5]x1 + [2]x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [4]x1 + [4]x2
POL(length(x1)) = [2]x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = [2]x1
POL(tt) = [2]
POL(zeros) = 0
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c7, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5
(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
MARK(
and(
z0,
z1)) →
c7(
ACTIVE(
and(
mark(
z0),
z1)),
AND(
mark(
z0),
z1),
MARK(
z0)) by
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(length(z0)) → c9(ACTIVE(length(mark(z0))), LENGTH(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c9, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c7, c7
(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
MARK(
length(
z0)) →
c9(
ACTIVE(
length(
mark(
z0))),
LENGTH(
mark(
z0)),
MARK(
z0)) by
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c7, c7, c9, c9
(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
And the Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = [2]x1
POL(AND(x1, x2)) = [4]x1 + [4]x2
POL(CONS(x1, x2)) = [2]x1 + [4]x2
POL(LENGTH(x1)) = 0
POL(MARK(x1)) = [4]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [4]x1 + [2]x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1)) = x1
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9) = 0
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [4]x1 + [4]x2
POL(length(x1)) = [4]x1
POL(mark(x1)) = x1
POL(nil) = [4]
POL(s(x1)) = [2]x1
POL(tt) = 0
POL(zeros) = 0
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c7, c7, c9, c9
(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
We considered the (Usable) Rules:
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(zeros) → active(zeros)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(mark(z0)))
mark(0) → active(0)
mark(tt) → active(tt)
mark(nil) → active(nil)
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(length(nil)) → mark(0)
s(active(z0)) → s(z0)
s(mark(z0)) → s(z0)
length(active(z0)) → length(z0)
length(mark(z0)) → length(z0)
cons(z0, mark(z1)) → cons(z0, z1)
cons(mark(z0), z1) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
And the Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = [2]x1
POL(AND(x1, x2)) = x2
POL(CONS(x1, x2)) = 0
POL(LENGTH(x1)) = 0
POL(MARK(x1)) = [4]x1
POL(S(x1)) = 0
POL(active(x1)) = x1
POL(and(x1, x2)) = [2]x1 + [4]x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(c5(x1, x2, x3)) = x1 + x2 + x3
POL(c7(x1)) = x1
POL(c7(x1, x2, x3)) = x1 + x2 + x3
POL(c9) = 0
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(cons(x1, x2)) = [2]x1 + [4]x2
POL(length(x1)) = [3]x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = [2]x1
POL(tt) = [1]
POL(zeros) = 0
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(zeros) → mark(cons(0, zeros))
active(and(tt, z0)) → mark(z0)
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
mark(zeros) → active(zeros)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(length(z0)) → active(length(mark(z0)))
mark(nil) → active(nil)
mark(s(z0)) → active(s(mark(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)
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(and(z0, z1)) → c7(ACTIVE(and(z0, z1)), AND(mark(z0), z1), MARK(z0))
MARK(and(zeros, x1)) → c7(ACTIVE(and(active(zeros), x1)), AND(mark(zeros), x1), MARK(zeros))
MARK(and(cons(z0, z1), x1)) → c7(ACTIVE(and(active(cons(mark(z0), z1)), x1)), AND(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(and(0, x1)) → c7(ACTIVE(and(active(0), x1)), AND(mark(0), x1), MARK(0))
MARK(and(and(z0, z1), x1)) → c7(ACTIVE(and(active(and(mark(z0), z1)), x1)), AND(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(and(tt, x1)) → c7(ACTIVE(and(active(tt), x1)), AND(mark(tt), x1), MARK(tt))
MARK(and(length(z0), x1)) → c7(ACTIVE(and(active(length(mark(z0))), x1)), AND(mark(length(z0)), x1), MARK(length(z0)))
MARK(and(nil, x1)) → c7(ACTIVE(and(active(nil), x1)), AND(mark(nil), x1), MARK(nil))
MARK(and(s(z0), x1)) → c7(ACTIVE(and(active(s(mark(z0))), x1)), AND(mark(s(z0)), x1), MARK(s(z0)))
MARK(and(x0, x1)) → c7(AND(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
S tuples:
MARK(zeros) → c4(ACTIVE(zeros))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
CONS(mark(z0), z1) → c12(CONS(z0, z1))
CONS(z0, mark(z1)) → c13(CONS(z0, z1))
CONS(active(z0), z1) → c14(CONS(z0, z1))
CONS(z0, active(z1)) → c15(CONS(z0, z1))
AND(mark(z0), z1) → c16(AND(z0, z1))
AND(z0, mark(z1)) → c17(AND(z0, z1))
AND(active(z0), z1) → c18(AND(z0, z1))
AND(z0, active(z1)) → c19(AND(z0, z1))
LENGTH(mark(z0)) → c20(LENGTH(z0))
LENGTH(active(z0)) → c21(LENGTH(z0))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
ACTIVE(zeros) → c(MARK(cons(0, zeros)))
ACTIVE(length(cons(x0, x1))) → c3(MARK(s(length(x1))), LENGTH(x1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(z0, z1)), CONS(mark(z0), z1), MARK(z0))
MARK(cons(zeros, x1)) → c5(ACTIVE(cons(active(zeros), x1)), CONS(mark(zeros), x1), MARK(zeros))
MARK(cons(cons(z0, z1), x1)) → c5(ACTIVE(cons(active(cons(mark(z0), z1)), x1)), CONS(mark(cons(z0, z1)), x1), MARK(cons(z0, z1)))
MARK(cons(0, x1)) → c5(ACTIVE(cons(active(0), x1)), CONS(mark(0), x1), MARK(0))
MARK(cons(length(z0), x1)) → c5(ACTIVE(cons(active(length(mark(z0))), x1)), CONS(mark(length(z0)), x1), MARK(length(z0)))
MARK(cons(s(z0), x1)) → c5(ACTIVE(cons(active(s(mark(z0))), x1)), CONS(mark(s(z0)), x1), MARK(s(z0)))
MARK(cons(x0, x1)) → c5(CONS(mark(x0), x1))
MARK(length(z0)) → c9(ACTIVE(length(z0)), LENGTH(mark(z0)), MARK(z0))
MARK(length(zeros)) → c9(ACTIVE(length(active(zeros))), LENGTH(mark(zeros)), MARK(zeros))
MARK(length(cons(z0, z1))) → c9(ACTIVE(length(active(cons(mark(z0), z1)))), LENGTH(mark(cons(z0, z1))), MARK(cons(z0, z1)))
MARK(length(0)) → c9(ACTIVE(length(active(0))), LENGTH(mark(0)), MARK(0))
MARK(length(and(z0, z1))) → c9(ACTIVE(length(active(and(mark(z0), z1)))), LENGTH(mark(and(z0, z1))), MARK(and(z0, z1)))
MARK(length(length(z0))) → c9(ACTIVE(length(active(length(mark(z0))))), LENGTH(mark(length(z0))), MARK(length(z0)))
MARK(length(s(z0))) → c9(ACTIVE(length(active(s(mark(z0))))), LENGTH(mark(s(z0))), MARK(s(z0)))
MARK(length(x0)) → c9
K tuples:
ACTIVE(and(tt, z0)) → c1(MARK(z0))
MARK(and(z0, z1)) → c7(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(cons(nil, x1)) → c5(ACTIVE(cons(active(nil), x1)), CONS(mark(nil), x1), MARK(nil))
MARK(cons(and(z0, z1), x1)) → c5(ACTIVE(cons(active(and(mark(z0), z1)), x1)), CONS(mark(and(z0, z1)), x1), MARK(and(z0, z1)))
MARK(cons(tt, x1)) → c5(ACTIVE(cons(active(tt), x1)), CONS(mark(tt), x1), MARK(tt))
MARK(length(nil)) → c9(ACTIVE(length(active(nil))), LENGTH(mark(nil)), MARK(nil))
MARK(length(tt)) → c9(ACTIVE(length(active(tt))), LENGTH(mark(tt)), MARK(tt))
Defined Rule Symbols:
active, mark, cons, and, length, s
Defined Pair Symbols:
ACTIVE, MARK, CONS, AND, LENGTH, S
Compound Symbols:
c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c3, c5, c5, c7, c7, c9, c9
(29) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 4.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6]
transitions:
zeros0() → 0
00() → 0
tt0() → 0
nil0() → 0
active0(0) → 1
mark0(0) → 2
cons0(0, 0) → 3
and0(0, 0) → 4
length0(0) → 5
s0(0) → 6
01() → 8
zeros1() → 9
cons1(8, 9) → 7
mark1(7) → 1
zeros1() → 10
active1(10) → 2
01() → 11
active1(11) → 2
tt1() → 12
active1(12) → 2
nil1() → 13
active1(13) → 2
02() → 15
zeros2() → 16
cons2(15, 16) → 14
mark2(14) → 2
mark2(8) → 18
cons2(18, 9) → 17
active2(17) → 1
mark3(15) → 20
cons3(20, 16) → 19
active3(19) → 2
02() → 21
active2(21) → 18
cons3(8, 9) → 17
cons3(21, 9) → 17
03() → 22
active3(22) → 20
cons4(15, 16) → 19
cons4(22, 16) → 19
(30) BOUNDS(O(1), O(n^1))