(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(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) 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 2.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[f_1|0, a|0, g_1|0, activate_1|0, n__f_1|1, n__a|1, n__g_1|1, f_1|1, a|1, g_1|1, n__f_1|2, n__a|2, n__g_1|2]
1→3[f_1|1, n__f_1|2]
2→2[n__f_1|0, n__a|0, n__g_1|0]
3→4[n__g_1|1]
4→5[f_1|1, n__f_1|2]
5→2[n__a|1]
(2) BOUNDS(1, n^1)
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(z0) → n__f(z0)
a → n__a
g(z0) → n__g(z0)
activate(n__f(z0)) → f(z0)
activate(n__a) → a
activate(n__g(z0)) → g(z0)
activate(z0) → z0
Tuples:
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
F(z0) → c1
A → c2
G(z0) → c3
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
ACTIVATE(z0) → c7
S tuples:
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
F(z0) → c1
A → c2
G(z0) → c3
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
ACTIVATE(z0) → c7
K tuples:none
Defined Rule Symbols:
f, a, g, activate
Defined Pair Symbols:
F, A, G, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 8 trailing nodes:
ACTIVATE(z0) → c7
F(z0) → c1
ACTIVATE(n__g(z0)) → c6(G(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__f(z0)) → c4(F(z0))
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
G(z0) → c3
A → c2
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(z0) → n__f(z0)
a → n__a
g(z0) → n__g(z0)
activate(n__f(z0)) → f(z0)
activate(n__a) → a
activate(n__g(z0)) → g(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, a, g, activate
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)