(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(c(x1))) → b(c(b(a(x1))))
b(x1) → d(a(x1))
b(a(c(d(x1)))) → a(a(a(x1)))
c(x1) → x1
b(x1) → c(d(x1))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(c(z0))) → b(c(b(a(z0))))
b(z0) → d(a(z0))
b(a(c(d(z0)))) → a(a(a(z0)))
b(z0) → c(d(z0))
c(z0) → z0
Tuples:
A(a(c(z0))) → c1(B(c(b(a(z0)))), C(b(a(z0))), B(a(z0)), A(z0))
B(z0) → c2(A(z0))
B(a(c(d(z0)))) → c3(A(a(a(z0))), A(a(z0)), A(z0))
B(z0) → c4(C(d(z0)))
S tuples:
A(a(c(z0))) → c1(B(c(b(a(z0)))), C(b(a(z0))), B(a(z0)), A(z0))
B(z0) → c2(A(z0))
B(a(c(d(z0)))) → c3(A(a(a(z0))), A(a(z0)), A(z0))
B(z0) → c4(C(d(z0)))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2, c3, c4
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(a(c(z0))) → c1(B(c(b(a(z0)))), C(b(a(z0))), B(a(z0)), A(z0))
B(a(c(d(z0)))) → c3(A(a(a(z0))), A(a(z0)), A(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(c(z0))) → b(c(b(a(z0))))
b(z0) → d(a(z0))
b(a(c(d(z0)))) → a(a(a(z0)))
b(z0) → c(d(z0))
c(z0) → z0
Tuples:
B(z0) → c2(A(z0))
B(z0) → c4(C(d(z0)))
S tuples:
B(z0) → c2(A(z0))
B(z0) → c4(C(d(z0)))
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:
B
Compound Symbols:
c2, c4
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 2 of 2 dangling nodes:
B(z0) → c2(A(z0))
B(z0) → c4(C(d(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(c(z0))) → b(c(b(a(z0))))
b(z0) → d(a(z0))
b(a(c(d(z0)))) → a(a(a(z0)))
b(z0) → c(d(z0))
c(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b, c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))