(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w))))))
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w)))
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
t(t(x)) → t(c(t(x), x))
t(x) → x
t(x) → c(0, c(0, c(0, c(0, c(0, 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:
h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w))))))
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w)))
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
t(t(x)) → t(c(t(x), x))
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
t(x) → c(0, c(0, c(0, c(0, c(0, x)))))
t(x) → x
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]
transitions:
c0(0, 0) → 0
00() → 0
t0(0) → 1
01() → 2
c1(2, 0) → 4
c1(2, 4) → 3
c1(2, 3) → 3
c1(2, 3) → 1
0 → 1
(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:
t(z0) → c(0, c(0, c(0, c(0, c(0, z0)))))
t(z0) → z0
Tuples:
T(z0) → c1
T(z0) → c2
S tuples:
T(z0) → c1
T(z0) → c2
K tuples:none
Defined Rule Symbols:
t
Defined Pair Symbols:
T
Compound Symbols:
c1, c2
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
T(z0) → c1
T(z0) → c2
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
t(z0) → c(0, c(0, c(0, c(0, c(0, z0)))))
t(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
t
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)