(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
b(b(0, y), x) → y
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(b(0, z0), z1) → z0
c(c(c(z0))) → c(c(a(a(c(b(0, z0)), 0), 0)))
a(z0, 0) → b(z0, 0)
Tuples:
C(c(c(z0))) → c2(C(c(a(a(c(b(0, z0)), 0), 0))), C(a(a(c(b(0, z0)), 0), 0)), A(a(c(b(0, z0)), 0), 0), A(c(b(0, z0)), 0), C(b(0, z0)), B(0, z0))
A(z0, 0) → c3(B(z0, 0))
S tuples:
C(c(c(z0))) → c2(C(c(a(a(c(b(0, z0)), 0), 0))), C(a(a(c(b(0, z0)), 0), 0)), A(a(c(b(0, z0)), 0), 0), A(c(b(0, z0)), 0), C(b(0, z0)), B(0, z0))
A(z0, 0) → c3(B(z0, 0))
K tuples:none
Defined Rule Symbols:
b, c, a
Defined Pair Symbols:
C, A
Compound Symbols:
c2, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
C(c(c(z0))) → c2(C(c(a(a(c(b(0, z0)), 0), 0))), C(a(a(c(b(0, z0)), 0), 0)), A(a(c(b(0, z0)), 0), 0), A(c(b(0, z0)), 0), C(b(0, z0)), B(0, z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(b(0, z0), z1) → z0
c(c(c(z0))) → c(c(a(a(c(b(0, z0)), 0), 0)))
a(z0, 0) → b(z0, 0)
Tuples:
A(z0, 0) → c3(B(z0, 0))
S tuples:
A(z0, 0) → c3(B(z0, 0))
K tuples:none
Defined Rule Symbols:
b, c, a
Defined Pair Symbols:
A
Compound Symbols:
c3
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
A(z0, 0) → c3(B(z0, 0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(b(0, z0), z1) → z0
c(c(c(z0))) → c(c(a(a(c(b(0, z0)), 0), 0)))
a(z0, 0) → b(z0, 0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
b, c, a
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))