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