(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
an__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
[f1, nf1, activate1] > a > c1 > [ng1, g1]
[f1, nf1, activate1] > a > na > [ng1, g1]

Status:
f1: [1]
a: []
c1: [1]
nf1: [1]
ng1: [1]
na: []
g1: [1]
activate1: [1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f(f(a)) → c(n__f(n__g(n__f(n__a))))
an__a
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
activate(X) → X


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
g1 > ng1
activate1 > f1 > nf1 > ng1

Status:
f1: [1]
nf1: [1]
g1: [1]
ng1: [1]
activate1: [1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))


(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE