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:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

(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:

+(*(x, y), *(x, z)) → *(x, +(y, z))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))

Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

+'

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.

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

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

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

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

POL(*(x₁, x₂)) = [1] + x₂   
POL(+'(x₁, x₂)) = x₂   
POL(c(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

S tuples:none
K tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

+'

Compound Symbols:

c

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

The set S is empty

(10) BOUNDS(1, 1)