(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:
a__c → a__f(g(c))
a__f(g(X)) → g(X)
mark(c) → a__c
mark(f(X)) → a__f(X)
mark(g(X)) → g(X)
a__c → c
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 3.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[a__c|0, a__f_1|0, mark_1|0, c|1, f_1|1, g_1|1, a__c|1, a__f_1|1, c|2, f_1|2]
1→3[a__f_1|1, f_1|2]
1→5[a__f_1|2, f_1|3]
1→4[g_1|2]
1→6[g_1|3]
2→2[g_1|0, c|0, f_1|0]
3→4[g_1|1]
4→2[c|1]
5→6[g_1|2]
6→2[c|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__c → a__f(g(c))
a__c → c
a__f(g(z0)) → g(z0)
a__f(z0) → f(z0)
mark(c) → a__c
mark(f(z0)) → a__f(z0)
mark(g(z0)) → g(z0)
Tuples:
A__C → c1(A__F(g(c)))
A__C → c2
A__F(g(z0)) → c3
A__F(z0) → c4
MARK(c) → c5(A__C)
MARK(f(z0)) → c6(A__F(z0))
MARK(g(z0)) → c7
S tuples:
A__C → c1(A__F(g(c)))
A__C → c2
A__F(g(z0)) → c3
A__F(z0) → c4
MARK(c) → c5(A__C)
MARK(f(z0)) → c6(A__F(z0))
MARK(g(z0)) → c7
K tuples:none
Defined Rule Symbols:
a__c, a__f, mark
Defined Pair Symbols:
A__C, A__F, MARK
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing nodes:
MARK(f(z0)) → c6(A__F(z0))
A__C → c2
A__C → c1(A__F(g(c)))
A__F(z0) → c4
MARK(g(z0)) → c7
A__F(g(z0)) → c3
MARK(c) → c5(A__C)
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__c → a__f(g(c))
a__c → c
a__f(g(z0)) → g(z0)
a__f(z0) → f(z0)
mark(c) → a__c
mark(f(z0)) → a__f(z0)
mark(g(z0)) → g(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a__c, a__f, mark
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)