(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:
a__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(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 3.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[a__f_1|0, a__c_1|0, a__h_1|0, mark_1|0, f_1|1, d_1|1, c_1|1, h_1|1, a__c_1|1, g_1|1, d_1|2, c_1|2]
1→3[a__c_1|1, d_1|2, c_1|2]
1→6[a__c_1|1, d_1|2, c_1|2]
1→7[a__f_1|1, f_1|2]
1→8[a__h_1|1, h_1|2]
1→9[a__c_1|2, d_1|3, c_1|3]
1→10[a__c_1|2, d_1|3, c_1|3]
2→2[f_1|0, g_1|0, d_1|0, c_1|0, h_1|0]
3→4[f_1|1]
4→5[g_1|1]
5→2[f_1|1]
6→2[d_1|1]
7→2[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]
7→7[a__f_1|1, f_1|2]
7→8[a__h_1|1, h_1|2]
7→9[a__c_1|2, d_1|3, c_1|3]
7→10[a__c_1|2, d_1|3, c_1|3]
8→2[mark_1|1, a__c_1|1, g_1|1, d_1|1, d_1|2, c_1|2]
8→7[a__f_1|1, f_1|2]
8→8[a__h_1|1, h_1|2]
8→9[a__c_1|2, d_1|3, c_1|3]
8→10[a__c_1|2, d_1|3, c_1|3]
9→8[d_1|2]
10→11[f_1|2]
11→12[g_1|2]
12→7[f_1|2]