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

(0) Obligation:

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

[c₂, g₁] > s₁ > f₁

Status:

f₁: [1]
c₂: [2,1]
s₁: [1]
g₁: [1]

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

(2) Obligation:

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

(3) RisEmptyProof (EQUIVALENT transformation)

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

(4) TRUE