Runtime Complexity (innermost) proof of /tmp/tmpAMOmj3/Ex6_GM04

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtKnowledgeProof (⇔, 0 ms)

↳8 CdtProblem

↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 220 ms)

↳10 CdtProblem

↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 87 ms)

↳12 CdtProblem

↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 103 ms)

↳14 CdtProblem

↳15 SIsEmptyProof (⇔, 0 ms)

↳16 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
mark(c) → active(c)
mark(f(X)) → active(f(X))
mark(g(X)) → active(g(X))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(c) → c1(MARK(f(g(c))), F(g(c)), G(c))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(c) → c3(ACTIVE(c))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c5(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

S tuples:

ACTIVE(c) → c1(MARK(f(g(c))), F(g(c)), G(c))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(c) → c3(ACTIVE(c))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c5(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

MARK(c) → c3(ACTIVE(c))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(c) → c1(MARK(f(g(c))), F(g(c)), G(c))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c5(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

S tuples:

ACTIVE(c) → c1(MARK(f(g(c))), F(g(c)), G(c))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
MARK(g(z0)) → c5(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c1, c2, c4, c5, c6, c7, c8, c9

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

S tuples:

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

K tuples:none
Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c2, c4, c6, c7, c8, c9, c1, c5

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

ACTIVE(c) → c1(MARK(f(g(c))))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(g(z0)) → c5(G(z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))

K tuples:

ACTIVE(c) → c1(MARK(f(g(c))))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(g(z0)) → c5(G(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c2, c4, c6, c7, c8, c9, c1, c5

(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(active(z0)) → c9(G(z0))

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

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

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

POL(ACTIVE(x₁)) = [2]x₁
POL(F(x₁)) = 0
POL(G(x₁)) = [4]x₁
POL(MARK(x₁)) = [2]x₁
POL(active(x₁)) = [4] + [4]x₁
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(f(x₁)) = [2]x₁
POL(g(x₁)) = [2]x₁
POL(mark(x₁)) = [4]x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

S tuples:

F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))

K tuples:

ACTIVE(c) → c1(MARK(f(g(c))))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(g(z0)) → c5(G(z0))
G(active(z0)) → c9(G(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c2, c4, c6, c7, c8, c9, c1, c5

(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.

F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))

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

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

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

POL(ACTIVE(x₁)) = 0
POL(F(x₁)) = [4]x₁
POL(G(x₁)) = 0
POL(MARK(x₁)) = [4]x₁
POL(active(x₁)) = [4] + [4]x₁
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(f(x₁)) = [2]x₁
POL(g(x₁)) = 0
POL(mark(x₁)) = [4] + [4]x₁

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

S tuples:

G(mark(z0)) → c8(G(z0))

K tuples:

ACTIVE(c) → c1(MARK(f(g(c))))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(g(z0)) → c5(G(z0))
G(active(z0)) → c9(G(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c2, c4, c6, c7, c8, c9, c1, c5

(13) 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))

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

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

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

POL(ACTIVE(x₁)) = [4] + [2]x₁
POL(F(x₁)) = [5]x₁
POL(G(x₁)) = [5]x₁
POL(MARK(x₁)) = [4] + [4]x₁
POL(active(x₁)) = [2]x₁
POL(c) = 0
POL(c1(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(f(x₁)) = [3]x₁
POL(g(x₁)) = [4]x₁
POL(mark(x₁)) = [1] + [2]x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(c) → mark(f(g(c)))
active(f(g(z0))) → mark(g(z0))
mark(c) → active(c)
mark(f(z0)) → active(f(z0))
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)

Tuples:

ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
ACTIVE(c) → c1(MARK(f(g(c))))
MARK(g(z0)) → c5(G(z0))

S tuples:none
K tuples:

ACTIVE(c) → c1(MARK(f(g(c))))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(z0))
ACTIVE(f(g(z0))) → c2(MARK(g(z0)), G(z0))
MARK(g(z0)) → c5(G(z0))
G(active(z0)) → c9(G(z0))
F(mark(z0)) → c6(F(z0))
F(active(z0)) → c7(F(z0))
G(mark(z0)) → c8(G(z0))

Defined Rule Symbols:

active, mark, f, g

Defined Pair Symbols:

ACTIVE, MARK, F, G

Compound Symbols:

c2, c4, c6, c7, c8, c9, c1, c5

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

(7) CdtKnowledgeProof (EQUIVALENT transformation)

(8) Obligation:

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(10) Obligation:

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(12) Obligation:

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(14) Obligation:

(15) SIsEmptyProof (EQUIVALENT transformation)

(16) BOUNDS(O(1), O(1))