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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 8 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtUsableRulesProof (⇔, 0 ms)
↳22 CdtProblem
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳24 CdtProblem
↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳28 BOUNDS(1, 1)

(0) Obligation:

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


The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
H(0) → c1
H(g(z0, z1)) → c2
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
H(0) → c1
H(g(z0, z1)) → c2
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F, H

Compound Symbols:

c, c1, c2

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

Removed 2 trailing nodes:

H(g(z0, z1)) → c2
H(0) → c1

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(0, 1, z0) → f(h(z0), h(z0), z0)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(0, 1, z0) → c(F(h(z0), h(z0), z0)) by

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, 0) → c(F(h(0), 0, 0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1))) by

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
S tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(0, 1, 0) → c(F(0, h(0), 0)) by

F(0, 1, 0) → c(F(0, 0, 0))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, 0) → c(F(0, 0, 0))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

(21) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

h(0) → 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1))) by

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:none

Compound Symbols:none

(27) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(28) BOUNDS(1, 1)