The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

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

active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(c) → c2(MARK(d))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

S tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(c) → c2(MARK(d))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

K tuples:none
Defined Rule Symbols:

active, mark, g, h

Defined Pair Symbols:

ACTIVE, MARK, G, H

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11

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

Removed 3 trailing nodes:

MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
ACTIVE(c) → c2(MARK(d))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

S tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

K tuples:none
Defined Rule Symbols:

active, mark, g, h

Defined Pair Symbols:

ACTIVE, MARK, G, H

Compound Symbols:

c1, c3, c4, c5, c8, c9, c10, c11

(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(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
H(mark(z0)) → c10(H(z0))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = [4] + [3]x₁   
POL(G(x₁)) = 0   
POL(H(x₁)) = x₁   
POL(MARK(x₁)) = [4] + [4]x₁   
POL(active(x₁)) = [2]x₁   
POL(c) = 0   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d) = [2]   
POL(g(x₁)) = [4] + [5]x₁   
POL(h(x₁)) = [3]x₁   
POL(mark(x₁)) = [2] + [2]x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

S tuples:

MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(active(z0)) → c11(H(z0))

K tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
H(mark(z0)) → c10(H(z0))

Defined Rule Symbols:

active, mark, g, h

Defined Pair Symbols:

ACTIVE, MARK, G, H

Compound Symbols:

c1, c3, c4, c5, c8, c9, c10, c11

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
H(mark(z0)) → c10(H(z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

S tuples:

G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(active(z0)) → c11(H(z0))

K tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
H(mark(z0)) → c10(H(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))

Defined Rule Symbols:

active, mark, g, h

Defined Pair Symbols:

ACTIVE, MARK, G, H

Compound Symbols:

c1, c3, c4, c5, c8, c9, c10, c11

(9) 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.

G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = 0   
POL(G(x₁)) = x₁   
POL(H(x₁)) = 0   
POL(MARK(x₁)) = [2]x₁   
POL(active(x₁)) = [1] + [2]x₁   
POL(c) = 0   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d) = [5]   
POL(g(x₁)) = [4]x₁   
POL(h(x₁)) = 0   
POL(mark(x₁)) = [4] + [2]x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

S tuples:

H(active(z0)) → c11(H(z0))

K tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
H(mark(z0)) → c10(H(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

Defined Rule Symbols:

active, mark, g, h

Defined Pair Symbols:

ACTIVE, MARK, G, H

Compound Symbols:

c1, c3, c4, c5, c8, c9, c10, c11

(11) 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.

H(active(z0)) → c11(H(z0))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = x₁   
POL(G(x₁)) = x₁   
POL(H(x₁)) = x₁   
POL(MARK(x₁)) = [4]x₁   
POL(active(x₁)) = [1] + [2]x₁   
POL(c) = 0   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(d) = [4]   
POL(g(x₁)) = [5]x₁   
POL(h(x₁)) = x₁   
POL(mark(x₁)) = [2]x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)

Tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))

S tuples:none
K tuples:

ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
H(mark(z0)) → c10(H(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(active(z0)) → c11(H(z0))

Defined Rule Symbols:

active, mark, g, h

Defined Pair Symbols:

ACTIVE, MARK, G, H

Compound Symbols:

c1, c3, c4, c5, c8, c9, c10, c11

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))