The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).

(0) Obligation:

The Runtime Complexity (full) 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: FULL

(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

(2) 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

(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(c) → c5(A__C)
A__C → c2
A__C → c1(A__F(g(c)))
MARK(g(z0)) → c7
A__F(g(z0)) → c3
MARK(f(z0)) → c6(A__F(z0))
A__F(z0) → c4

(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)