(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, N)) → mark(N)
active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) → mark(isNat(V1))
active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U21(z0, z1, z2)) → active(U21(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(0) → active(0)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U21(mark(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, mark(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, mark(z2)) → U21(z0, z1, z2)
U21(active(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, active(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, active(z2)) → U21(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(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)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
Tuples:
ACTIVE(U11(tt, z0)) → c(MARK(z0))
ACTIVE(U21(tt, z0, z1)) → c1(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c2(MARK(z0))
ACTIVE(isNat(0)) → c3(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c4(MARK(and(isNat(z0), isNat(z1))), AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(MARK(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(MARK(U11(isNat(z0), z0)), U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(MARK(U21(and(isNat(z1), isNat(z0)), z1, z0)), U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(tt) → c9(ACTIVE(tt))
MARK(U21(z0, z1, z2)) → c10(ACTIVE(U21(mark(z0), z1, z2)), U21'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c12(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c13(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c14(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(0) → c15(ACTIVE(0))
U11'(mark(z0), z1) → c16(U11'(z0, z1))
U11'(z0, mark(z1)) → c17(U11'(z0, z1))
U11'(active(z0), z1) → c18(U11'(z0, z1))
U11'(z0, active(z1)) → c19(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c20(U21'(z0, z1, z2))
U21'(z0, mark(z1), z2) → c21(U21'(z0, z1, z2))
U21'(z0, z1, mark(z2)) → c22(U21'(z0, z1, z2))
U21'(active(z0), z1, z2) → c23(U21'(z0, z1, z2))
U21'(z0, active(z1), z2) → c24(U21'(z0, z1, z2))
U21'(z0, z1, active(z2)) → c25(U21'(z0, z1, z2))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
PLUS(mark(z0), z1) → c28(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c29(PLUS(z0, z1))
PLUS(active(z0), z1) → c30(PLUS(z0, z1))
PLUS(z0, active(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c32(AND(z0, z1))
AND(z0, mark(z1)) → c33(AND(z0, z1))
AND(active(z0), z1) → c34(AND(z0, z1))
AND(z0, active(z1)) → c35(AND(z0, z1))
ISNAT(mark(z0)) → c36(ISNAT(z0))
ISNAT(active(z0)) → c37(ISNAT(z0))
S tuples:
ACTIVE(U11(tt, z0)) → c(MARK(z0))
ACTIVE(U21(tt, z0, z1)) → c1(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c2(MARK(z0))
ACTIVE(isNat(0)) → c3(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c4(MARK(and(isNat(z0), isNat(z1))), AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(MARK(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(MARK(U11(isNat(z0), z0)), U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(MARK(U21(and(isNat(z1), isNat(z0)), z1, z0)), U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(tt) → c9(ACTIVE(tt))
MARK(U21(z0, z1, z2)) → c10(ACTIVE(U21(mark(z0), z1, z2)), U21'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c12(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c13(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c14(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(0) → c15(ACTIVE(0))
U11'(mark(z0), z1) → c16(U11'(z0, z1))
U11'(z0, mark(z1)) → c17(U11'(z0, z1))
U11'(active(z0), z1) → c18(U11'(z0, z1))
U11'(z0, active(z1)) → c19(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c20(U21'(z0, z1, z2))
U21'(z0, mark(z1), z2) → c21(U21'(z0, z1, z2))
U21'(z0, z1, mark(z2)) → c22(U21'(z0, z1, z2))
U21'(active(z0), z1, z2) → c23(U21'(z0, z1, z2))
U21'(z0, active(z1), z2) → c24(U21'(z0, z1, z2))
U21'(z0, z1, active(z2)) → c25(U21'(z0, z1, z2))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
PLUS(mark(z0), z1) → c28(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c29(PLUS(z0, z1))
PLUS(active(z0), z1) → c30(PLUS(z0, z1))
PLUS(z0, active(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c32(AND(z0, z1))
AND(z0, mark(z1)) → c33(AND(z0, z1))
AND(active(z0), z1) → c34(AND(z0, z1))
AND(z0, active(z1)) → c35(AND(z0, z1))
ISNAT(mark(z0)) → c36(ISNAT(z0))
ISNAT(active(z0)) → c37(ISNAT(z0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U21, s, plus, and, isNat
Defined Pair Symbols:
ACTIVE, MARK, U11', U21', S, PLUS, AND, ISNAT
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(U11(tt, z0)) → c(MARK(z0))
ACTIVE(U21(tt, z0, z1)) → c1(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c2(MARK(z0))
ACTIVE(isNat(0)) → c3(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c4(MARK(and(isNat(z0), isNat(z1))), AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(MARK(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(MARK(U11(isNat(z0), z0)), U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(MARK(U21(and(isNat(z1), isNat(z0)), z1, z0)), U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
MARK(U11(z0, z1)) → c8(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U21(z0, z1, z2)) → c10(ACTIVE(U21(mark(z0), z1, z2)), U21'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c12(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c13(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c14(ACTIVE(isNat(z0)), ISNAT(z0))
U11'(mark(z0), z1) → c16(U11'(z0, z1))
U11'(z0, mark(z1)) → c17(U11'(z0, z1))
U11'(active(z0), z1) → c18(U11'(z0, z1))
U11'(z0, active(z1)) → c19(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c20(U21'(z0, z1, z2))
U21'(z0, mark(z1), z2) → c21(U21'(z0, z1, z2))
U21'(z0, z1, mark(z2)) → c22(U21'(z0, z1, z2))
U21'(active(z0), z1, z2) → c23(U21'(z0, z1, z2))
U21'(z0, active(z1), z2) → c24(U21'(z0, z1, z2))
U21'(z0, z1, active(z2)) → c25(U21'(z0, z1, z2))
S(mark(z0)) → c26(S(z0))
S(active(z0)) → c27(S(z0))
PLUS(mark(z0), z1) → c28(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c29(PLUS(z0, z1))
PLUS(active(z0), z1) → c30(PLUS(z0, z1))
PLUS(z0, active(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c32(AND(z0, z1))
AND(z0, mark(z1)) → c33(AND(z0, z1))
AND(active(z0), z1) → c34(AND(z0, z1))
AND(z0, active(z1)) → c35(AND(z0, z1))
ISNAT(mark(z0)) → c36(ISNAT(z0))
ISNAT(active(z0)) → c37(ISNAT(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U21(z0, z1, z2)) → active(U21(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(0) → active(0)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U21(mark(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, mark(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, mark(z2)) → U21(z0, z1, z2)
U21(active(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, active(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, active(z2)) → U21(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(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)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
Tuples:
MARK(tt) → c9(ACTIVE(tt))
MARK(0) → c15(ACTIVE(0))
S tuples:
MARK(tt) → c9(ACTIVE(tt))
MARK(0) → c15(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U21, s, plus, and, isNat
Defined Pair Symbols:
MARK
Compound Symbols:
c9, c15
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c15(ACTIVE(0))
MARK(tt) → c9(ACTIVE(tt))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U21(z0, z1, z2)) → active(U21(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(0) → active(0)
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U21(mark(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, mark(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, mark(z2)) → U21(z0, z1, z2)
U21(active(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, active(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, active(z2)) → U21(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(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)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, U11, U21, s, plus, and, isNat
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))