Termination w.r.t. Q proof of /tmp/tmprHQM4b/Ex23_Luc06

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 RisEmptyProof (⇔)

↳6 TRUE

(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)
a → n__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, n_{_f₁}, activate₁] > a > c₁ > [n_{_g₁}, g₁]
[f₁, n_{_f₁}, activate₁] > a > n_{_a} > [n_{_g₁}, g₁]

Status:

f₁: [1]
a: []
c₁: [1]
n_{_f₁}: [1]
n_{_g₁}: [1]
n_{_a}: []
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))))
a → n__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 > n_{_g₁}
activate₁ > f₁ > n_{_f₁} > n_{_g₁}

Status:

f₁: [1]
n_{_f₁}: [1]
g₁: [1]
n_{_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(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)