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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔, 0 ms)
↳4 CdtProblem
↳5 SIsEmptyProof (⇔, 0 ms)
↳6 BOUNDS(O(1), O(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))
if(x, if(x, y, z), z) → if(x, y, z)
if(x, y, if(x, y, z)) → if(x, y, z)

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), z3, z4) → if(z0, if(z1, z3, z4), if(z2, z3, z4))
if(z0, if(z0, z1, z2), z2) → if(z0, z1, z2)
if(z0, z1, if(z0, z1, z2)) → if(z0, z1, z2)
Tuples:

IF(if(z0, z1, z2), z3, z4) → c3(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
IF(z0, if(z0, z1, z2), z2) → c4(IF(z0, z1, z2))
IF(z0, z1, if(z0, z1, z2)) → c5(IF(z0, z1, z2))
S tuples:

IF(if(z0, z1, z2), z3, z4) → c3(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
IF(z0, if(z0, z1, z2), z2) → c4(IF(z0, z1, z2))
IF(z0, z1, if(z0, z1, z2)) → c5(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

if

Defined Pair Symbols:

IF

Compound Symbols:

c3, c4, c5

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

IF(if(z0, z1, z2), z3, z4) → c3(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
IF(z0, if(z0, z1, z2), z2) → c4(IF(z0, z1, z2))
IF(z0, z1, if(z0, z1, z2)) → c5(IF(z0, z1, z2))

(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), z3, z4) → if(z0, if(z1, z3, z4), if(z2, z3, z4))
if(z0, if(z0, z1, z2), z2) → if(z0, z1, z2)
if(z0, z1, if(z0, z1, z2)) → if(z0, z1, z2)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

if

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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