(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:
p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(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 1.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[p_1|0, q_1|0]
1→3[q_1|1]
1→5[q_1|1]
1→7[p_1|1]
1→9[p_1|1]
2→2[f_1|0, g_1|0]
3→4[f_1|1]
4→2[g_1|1]
5→6[g_1|1]
6→2[f_1|1]
7→8[f_1|1]
8→2[g_1|1]
9→10[g_1|1]
10→2[f_1|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:
p(f(f(z0))) → q(f(g(z0)))
p(g(g(z0))) → q(g(f(z0)))
q(f(f(z0))) → p(f(g(z0)))
q(g(g(z0))) → p(g(f(z0)))
Tuples:
P(f(f(z0))) → c(Q(f(g(z0))))
P(g(g(z0))) → c1(Q(g(f(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))
Q(g(g(z0))) → c3(P(g(f(z0))))
S tuples:
P(f(f(z0))) → c(Q(f(g(z0))))
P(g(g(z0))) → c1(Q(g(f(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))
Q(g(g(z0))) → c3(P(g(f(z0))))
K tuples:none
Defined Rule Symbols:
p, q
Defined Pair Symbols:
P, Q
Compound Symbols:
c, c1, c2, c3
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
P(f(f(z0))) → c(Q(f(g(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))
P(g(g(z0))) → c1(Q(g(f(z0))))
Q(g(g(z0))) → c3(P(g(f(z0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(f(f(z0))) → q(f(g(z0)))
p(g(g(z0))) → q(g(f(z0)))
q(f(f(z0))) → p(f(g(z0)))
q(g(g(z0))) → p(g(f(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
p, q
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)