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
  ↳7 RisEmptyProof (⇔)
    ↳8 TRUE

(0) Obligation:

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

app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) → app(app(f, x), y)
add → app(curry, plus)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(add) = 2   
POL(app(x1, x2)) = x1 + x2   
POL(curry) = 1   
POL(plus) = 0   
POL(s) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

app(app(plus, 0), y) → y
app(app(app(curry, f), x), y) → app(app(f, x), y)
add → app(curry, plus)


(2) Obligation:

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

app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(app(x1, x2)) = 1 + 2·x1 + x2   
POL(plus) = 2   
POL(s) = 2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))


(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

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE