(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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) 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
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing nodes:
MARK(g(z0)) → c7
MARK(f(z0)) → c6(A__F(z0))
MARK(c) → c5(A__C)
A__C → c2
A__F(z0) → c4
A__C → c1(A__F(g(c)))
A__F(g(z0)) → c3
(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:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a__c, a__f, mark
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)