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:

if(true, x, y) → x
if(false, x, y) → y
if(x, y, y) → y
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

if(true, z0, z1) → z0
if(false, z0, z1) → z1
if(z0, z1, z1) → z1
if(if(z0, z1, z2), u, v) → if(z0, if(z1, u, v), if(z2, u, v))

Tuples:

IF(if(z0, z1, z2), u, v) → c3(IF(z0, if(z1, u, v), if(z2, u, v)), IF(z1, u, v), IF(z2, u, v))

S tuples:

IF(if(z0, z1, z2), u, v) → c3(IF(z0, if(z1, u, v), if(z2, u, v)), IF(z1, u, v), IF(z2, u, v))

K tuples:none
Defined Rule Symbols:

if

Defined Pair Symbols:

IF

Compound Symbols:

c3

(3) 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.

IF(if(z0, z1, z2), u, v) → c3(IF(z0, if(z1, u, v), if(z2, u, v)), IF(z1, u, v), IF(z2, u, v))

We considered the (Usable) Rules:

if(true, z0, z1) → z0
if(false, z0, z1) → z1
if(if(z0, z1, z2), u, v) → if(z0, if(z1, u, v), if(z2, u, v))
if(z0, z1, z1) → z1

And the Tuples:

IF(if(z0, z1, z2), u, v) → c3(IF(z0, if(z1, u, v), if(z2, u, v)), IF(z1, u, v), IF(z2, u, v))

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

POL(IF(x₁, x₂, x₃)) = x₁   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(false) = 0   
POL(if(x₁, x₂, x₃)) = [1] + [4]x₁ + x₂ + [2]x₃   
POL(true) = 0   
POL(u) = 0   
POL(v) = 0   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

if(true, z0, z1) → z0
if(false, z0, z1) → z1
if(z0, z1, z1) → z1
if(if(z0, z1, z2), u, v) → if(z0, if(z1, u, v), if(z2, u, v))

Tuples:

IF(if(z0, z1, z2), u, v) → c3(IF(z0, if(z1, u, v), if(z2, u, v)), IF(z1, u, v), IF(z2, u, v))

S tuples:none
K tuples:

IF(if(z0, z1, z2), u, v) → c3(IF(z0, if(z1, u, v), if(z2, u, v)), IF(z1, u, v), IF(z2, u, v))

Defined Rule Symbols:

if

Defined Pair Symbols:

IF

Compound Symbols:

c3

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))