(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
f(c(s(x), y)) → f(c(x, s(y)))
g(c(x, s(y))) → g(c(s(x), y))
g(s(f(x))) → g(f(x))
Rewrite Strategy: INNERMOST
(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)
The following rules are not reachable from basic terms in the dependency graph and can be removed:
g(s(f(x))) → g(f(x))
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
Rewrite Strategy: INNERMOST
(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
c0(0, 0) → 0
s0(0) → 0
g0(0) → 1
f0(0) → 2
s1(0) → 4
c1(4, 0) → 3
g1(3) → 1
s1(0) → 6
c1(0, 6) → 5
f1(5) → 2
s1(4) → 4
s1(6) → 6
(4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(c(z0, s(z1))) → g(c(s(z0), z1))
f(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
K tuples:none
Defined Rule Symbols:
g, f
Defined Pair Symbols:
G, F
Compound Symbols:
c1, c2
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
g(c(z0, s(z1))) → g(c(s(z0), z1))
f(c(s(z0), z1)) → f(c(z0, s(z1)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
G, F
Compound Symbols:
c1, c2
(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
We considered the (Usable) Rules:none
And the Tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = x1
POL(G(x1)) = 0
POL(c(x1, x2)) = x1
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
K tuples:
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
Defined Rule Symbols:none
Defined Pair Symbols:
G, F
Compound Symbols:
c1, c2
(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
We considered the (Usable) Rules:none
And the Tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = 0
POL(G(x1)) = [3]x1
POL(c(x1, x2)) = x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [3] + x1
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:none
K tuples:
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
Defined Rule Symbols:none
Defined Pair Symbols:
G, F
Compound Symbols:
c1, c2
(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(14) BOUNDS(1, 1)