(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(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(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, p_1|0, 0|1, cons_1|1, s_1|1]
1→3[cons_1|1]
1→4[f_1|1]
1→7[cons_1|2]
2→2[0|0, cons_1|0, s_1|0]
3→2[0|1]
4→5[p_1|1]
4→2[0|2]
5→6[s_1|1]
6→2[0|1]
7→2[0|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:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:
F(0) → c
F(s(0)) → c1(F(p(s(0))), P(s(0)))
P(s(z0)) → c2
S tuples:
F(0) → c
F(s(0)) → c1(F(p(s(0))), P(s(0)))
P(s(z0)) → c2
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F, P
Compound Symbols:
c, c1, c2
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(0) → c
P(s(z0)) → c2
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
S tuples:
F(s(0)) → c1(F(p(s(0))), P(s(0)))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))))
S tuples:
F(s(0)) → c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:
f, p
Defined Pair Symbols:
F
Compound Symbols:
c1
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
f(0) → cons(0)
f(s(0)) → f(p(s(0)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(p(s(0))))
S tuples:
F(s(0)) → c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:
F
Compound Symbols:
c1
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
s(
0)) →
c1(
F(
p(
s(
0)))) by
F(s(0)) → c1(F(0))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
Tuples:
F(s(0)) → c1(F(0))
S tuples:
F(s(0)) → c1(F(0))
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:
F
Compound Symbols:
c1
(13) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(s(0)) → c1(F(0))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
p
Defined Pair Symbols:none
Compound Symbols:none
(15) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(16) BOUNDS(1, 1)