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

(0) Obligation:

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


The TRS R consists of the following rules:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 1th argument of plus: times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))

(2) Obligation:

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


The TRS R consists of the following rules:

times(X, s(Y)) → plus(X, times(Y, X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, s(z1)) → plus(z0, times(z1, z0))

Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

S tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

K tuples:none
Defined Rule Symbols:

times

Defined Pair Symbols:

TIMES

Compound Symbols:

c

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

times(z0, s(z1)) → plus(z0, times(z1, z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

S tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

TIMES

Compound Symbols:

c

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

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

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

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

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

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

POL(TIMES(x₁, x₂)) = x₁ + x₂   
POL(c(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

S tuples:none
K tuples:

TIMES(z0, s(z1)) → c(TIMES(z1, z0))

Defined Rule Symbols:none

Defined Pair Symbols:

TIMES

Compound Symbols:

c

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

The set S is empty

(10) BOUNDS(1, 1)