(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(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → 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 2.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[a__f_1|0, mark_1|0, f_1|1, g_1|1]
1→3[g_1|1]
1→5[a__f_1|1, f_1|2]
1→6[h_1|1]
1→7[g_1|2]
2→2[g_1|0, h_1|0, f_1|0]
3→4[h_1|1]
4→2[f_1|1]
5→2[mark_1|1, g_1|1]
5→5[a__f_1|1, f_1|2]
5→6[h_1|1]
5→7[g_1|2]
6→2[mark_1|1, g_1|1]
6→5[a__f_1|1, f_1|2]
6→6[h_1|1]
6→7[g_1|2]
7→8[h_1|2]
8→5[f_1|2]
(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:
a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))
Tuples:
A__F(z0) → c
A__F(z0) → c1
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(g(z0)) → c3
MARK(h(z0)) → c4(MARK(z0))
S tuples:
A__F(z0) → c
A__F(z0) → c1
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(g(z0)) → c3
MARK(h(z0)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, mark
Defined Pair Symbols:
A__F, MARK
Compound Symbols:
c, c1, c2, c3, c4
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
A__F(z0) → c
A__F(z0) → c1
MARK(g(z0)) → c3
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))
Tuples:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))
S tuples:
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, mark
Defined Pair Symbols:
MARK
Compound Symbols:
c2, c4
(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))
Tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
S tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, mark
Defined Pair Symbols:
MARK
Compound Symbols:
c4, c2
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
a__f(z0) → g(h(f(z0)))
a__f(z0) → f(z0)
mark(f(z0)) → a__f(mark(z0))
mark(g(z0)) → g(z0)
mark(h(z0)) → h(mark(z0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
S tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
MARK
Compound Symbols:
c4, c2
(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(MARK(x1)) = x1
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(f(x1)) = [1] + x1
POL(h(x1)) = [1] + x1
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
S tuples:none
K tuples:
MARK(h(z0)) → c4(MARK(z0))
MARK(f(z0)) → c2(MARK(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
MARK
Compound Symbols:
c4, c2
(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(14) BOUNDS(1, 1)