(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
g(h(x1)) → g(f(s(x1)))
f(s(s(s(x1)))) → h(f(s(h(x1))))
f(h(x1)) → h(f(s(h(x1))))
h(x1) → x1
f(f(s(s(x1)))) → s(s(s(f(f(x1)))))
b(a(x1)) → a(b(x1))
a(a(a(x1))) → b(a(a(b(x1))))
b(b(b(b(x1)))) → a(x1)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(h(z0)) → g(f(s(z0)))
f(s(s(s(z0)))) → h(f(s(h(z0))))
f(h(z0)) → h(f(s(h(z0))))
f(f(s(s(z0)))) → s(s(s(f(f(z0)))))
h(z0) → z0
b(a(z0)) → a(b(z0))
b(b(b(b(z0)))) → a(z0)
a(a(a(z0))) → b(a(a(b(z0))))
Tuples:
G(h(z0)) → c(G(f(s(z0))), F(s(z0)))
F(s(s(s(z0)))) → c1(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
F(h(z0)) → c2(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
F(f(s(s(z0)))) → c3(F(f(z0)), F(z0))
B(a(z0)) → c5(A(b(z0)), B(z0))
B(b(b(b(z0)))) → c6(A(z0))
A(a(a(z0))) → c7(B(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0))
S tuples:
G(h(z0)) → c(G(f(s(z0))), F(s(z0)))
F(s(s(s(z0)))) → c1(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
F(h(z0)) → c2(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
F(f(s(s(z0)))) → c3(F(f(z0)), F(z0))
B(a(z0)) → c5(A(b(z0)), B(z0))
B(b(b(b(z0)))) → c6(A(z0))
A(a(a(z0))) → c7(B(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:
g, f, h, b, a
Defined Pair Symbols:
G, F, B, A
Compound Symbols:
c, c1, c2, c3, c5, c6, c7
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
G(h(z0)) → c(G(f(s(z0))), F(s(z0)))
F(h(z0)) → c2(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
F(f(s(s(z0)))) → c3(F(f(z0)), F(z0))
B(a(z0)) → c5(A(b(z0)), B(z0))
B(b(b(b(z0)))) → c6(A(z0))
A(a(a(z0))) → c7(B(a(a(b(z0)))), A(a(b(z0))), A(b(z0)), B(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(h(z0)) → g(f(s(z0)))
f(s(s(s(z0)))) → h(f(s(h(z0))))
f(h(z0)) → h(f(s(h(z0))))
f(f(s(s(z0)))) → s(s(s(f(f(z0)))))
h(z0) → z0
b(a(z0)) → a(b(z0))
b(b(b(b(z0)))) → a(z0)
a(a(a(z0))) → b(a(a(b(z0))))
Tuples:
F(s(s(s(z0)))) → c1(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
S tuples:
F(s(s(s(z0)))) → c1(H(f(s(h(z0)))), F(s(h(z0))), H(z0))
K tuples:none
Defined Rule Symbols:
g, f, h, b, a
Defined Pair Symbols:
F
Compound Symbols:
c1
(5) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(h(z0)) → g(f(s(z0)))
f(s(s(s(z0)))) → h(f(s(h(z0))))
f(h(z0)) → h(f(s(h(z0))))
f(f(s(s(z0)))) → s(s(s(f(f(z0)))))
h(z0) → z0
b(a(z0)) → a(b(z0))
b(b(b(b(z0)))) → a(z0)
a(a(a(z0))) → b(a(a(b(z0))))
Tuples:
F(s(s(s(z0)))) → c1(F(s(h(z0))))
S tuples:
F(s(s(s(z0)))) → c1(F(s(h(z0))))
K tuples:none
Defined Rule Symbols:
g, f, h, b, a
Defined Pair Symbols:
F
Compound Symbols:
c1
(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F(s(s(s(z0)))) → c1(F(s(h(z0))))
We considered the (Usable) Rules:
h(z0) → z0
And the Tuples:
F(s(s(s(z0)))) → c1(F(s(h(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = [2]x1 + [2]x12
POL(c1(x1)) = x1
POL(h(x1)) = [3] + x1
POL(s(x1)) = [3] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(h(z0)) → g(f(s(z0)))
f(s(s(s(z0)))) → h(f(s(h(z0))))
f(h(z0)) → h(f(s(h(z0))))
f(f(s(s(z0)))) → s(s(s(f(f(z0)))))
h(z0) → z0
b(a(z0)) → a(b(z0))
b(b(b(b(z0)))) → a(z0)
a(a(a(z0))) → b(a(a(b(z0))))
Tuples:
F(s(s(s(z0)))) → c1(F(s(h(z0))))
S tuples:none
K tuples:
F(s(s(s(z0)))) → c1(F(s(h(z0))))
Defined Rule Symbols:
g, f, h, b, a
Defined Pair Symbols:
F
Compound Symbols:
c1
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))