(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0
Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c4
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, 0) → c6
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
ACTIVATE(z0) → c8
S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c4
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, 0) → c6
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
ACTIVATE(z0) → c8
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

U11', U12', U21', U22', PLUS, X, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

PLUS(z0, 0) → c4
X(z0, 0) → c6
ACTIVATE(z0) → c8

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0
Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

U11', U12', U21', U22', PLUS, X

Compound Symbols:

c, c1, c2, c3, c5, c7

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

(7) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
We considered the (Usable) Rules:

activate(z0) → z0
And the Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(PLUS(x1, x2)) = 0
POL(U11(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [3]x3
POL(U11'(x1, x2, x3)) = 0
POL(U12(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [3]x3
POL(U12'(x1, x2, x3)) = x1
POL(U21(x1, x2, x3)) = [5] + x1 + [2]x2
POL(U21'(x1, x2, x3)) = [2] + [4]x2
POL(U22(x1, x2, x3)) = [3] + [2]x1 + [5]x2 + [3]x3
POL(U22'(x1, x2, x3)) = [1] + [4]x2
POL(X(x1, x2)) = [4]x2
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c7(x1)) = x1
POL(plus(x1, x2)) = [3] + [3]x1 + [5]x2
POL(s(x1)) = [1] + x1
POL(tt) = 0
POL(x(x1, x2)) = [4]

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
We considered the (Usable) Rules:

activate(z0) → z0
And the Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(PLUS(x1, x2)) = [2]x2
POL(U11(x1, x2, x3)) = 0
POL(U11'(x1, x2, x3)) = [2]x1 + [2]x2
POL(U12(x1, x2, x3)) = 0
POL(U12'(x1, x2, x3)) = [2]x2
POL(U21(x1, x2, x3)) = 0
POL(U21'(x1, x2, x3)) = [2] + [3]x1 + x2 + [2]x3 + [3]x2·x3 + [3]x1·x3 + [2]x1·x2 + [2]x22
POL(U22(x1, x2, x3)) = 0
POL(U22'(x1, x2, x3)) = x1 + x2 + [2]x3 + [3]x2·x3 + [3]x1·x3 + [2]x1·x2 + [2]x22
POL(X(x1, x2)) = [3]x1 + [2]x22 + [3]x1·x2
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(c7(x1)) = x1
POL(plus(x1, x2)) = 0
POL(s(x1)) = [2] + x1
POL(tt) = [2]
POL(x(x1, x2)) = 0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

(11) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
Now S is empty