(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(z0, s(z1))) → f(c(s(z0), z1))
g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c1, c2
(3) CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 of 2 dangling nodes:
G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(z0, s(z1))) → f(c(s(z0), z1))
g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c1
(5) 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.
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
We considered the (Usable) Rules:none
And the Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = [4]x1
POL(c(x1, x2)) = x2
POL(c1(x1)) = x1
POL(s(x1)) = [4] + x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(z0, s(z1))) → f(c(s(z0), z1))
g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:none
K tuples:
F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c1
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))