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:

a(a(y, 0), 0) → y
c(c(y)) → y
c(a(c(c(y)), x)) → a(c(c(c(a(x, 0)))), y)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0, 0), 0) → z0
c(c(z0)) → z0
c(a(c(c(z0)), z1)) → a(c(c(c(a(z1, 0)))), z0)

Tuples:

C(a(c(c(z0)), z1)) → c3(A(c(c(c(a(z1, 0)))), z0), C(c(c(a(z1, 0)))), C(c(a(z1, 0))), C(a(z1, 0)), A(z1, 0))

S tuples:

C(a(c(c(z0)), z1)) → c3(A(c(c(c(a(z1, 0)))), z0), C(c(c(a(z1, 0)))), C(c(a(z1, 0))), C(a(z1, 0)), A(z1, 0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

C(a(c(c(z0)), z1)) → c3(A(c(c(c(a(z1, 0)))), z0), C(c(c(a(z1, 0)))), C(c(a(z1, 0))), C(a(z1, 0)), A(z1, 0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0, 0), 0) → z0
c(c(z0)) → z0
c(a(c(c(z0)), z1)) → a(c(c(c(a(z1, 0)))), z0)

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

a, c

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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