(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:
f(g(x), g(y)) → f(p(f(g(x), s(y))), g(s(p(x))))
p(0) → g(0)
g(s(p(x))) → p(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:
f(g(x), g(y)) → f(p(f(g(x), s(y))), g(s(p(x))))
g(s(p(x))) → p(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:
p(0) → g(0)
Rewrite Strategy: INNERMOST
(3) CpxTrsMatchBoundsProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[p_1|0]
1→3[g_1|1]
2→2[0|0, g_1|0]
3→2[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:
p(0) → g(0)
Tuples:
P(0) → c
S tuples:
P(0) → c
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:
P
Compound Symbols:
c
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
P(0) → c
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(0) → g(0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)