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

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

plus(s(X), plus(Y, Z)) → plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) → plus(X1, plus(X3, plus(X2, X4)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

plus(s(z0), plus(z1, z2)) → plus(z0, plus(s(s(z1)), z2))
plus(s(z0), plus(z1, plus(z2, z3))) → plus(z0, plus(z2, plus(z1, z3)))

Tuples:

PLUS(s(z0), plus(z1, z2)) → c(PLUS(z0, plus(s(s(z1)), z2)), PLUS(s(s(z1)), z2))
PLUS(s(z0), plus(z1, plus(z2, z3))) → c1(PLUS(z0, plus(z2, plus(z1, z3))), PLUS(z2, plus(z1, z3)), PLUS(z1, z3))

S tuples:

PLUS(s(z0), plus(z1, z2)) → c(PLUS(z0, plus(s(s(z1)), z2)), PLUS(s(s(z1)), z2))
PLUS(s(z0), plus(z1, plus(z2, z3))) → c1(PLUS(z0, plus(z2, plus(z1, z3))), PLUS(z2, plus(z1, z3)), PLUS(z1, z3))

K tuples:none
Defined Rule Symbols:

plus

Defined Pair Symbols:

PLUS

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

PLUS(s(z0), plus(z1, z2)) → c(PLUS(z0, plus(s(s(z1)), z2)), PLUS(s(s(z1)), z2))
PLUS(s(z0), plus(z1, plus(z2, z3))) → c1(PLUS(z0, plus(z2, plus(z1, z3))), PLUS(z2, plus(z1, z3)), PLUS(z1, z3))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

plus(s(z0), plus(z1, z2)) → plus(z0, plus(s(s(z1)), z2))
plus(s(z0), plus(z1, plus(z2, z3))) → plus(z0, plus(z2, plus(z1, z3)))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

plus

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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