(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))
The set Q consists of the following terms:
app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(add, app(s, x)), y) → APP(add, x)
APP(app(mult, app(s, x)), y) → APP(app(add, app(app(mult, x), y)), y)
APP(app(mult, app(s, x)), y) → APP(add, app(app(mult, x), y))
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
APP(app(mult, app(s, x)), y) → APP(mult, x)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
FACT → APP(app(rec, mult), app(s, 0))
FACT → APP(rec, mult)
FACT → APP(s, 0)
The TRS R consists of the following rules:
app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))
The set Q consists of the following terms:
app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 8 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(add, app(s, x)), y) → APP(app(add, x), y)
The TRS R consists of the following rules:
app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))
The set Q consists of the following terms:
app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
The TRS R consists of the following rules:
app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))
The set Q consists of the following terms:
app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
The TRS R consists of the following rules:
app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))
The set Q consists of the following terms:
app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact
We have to consider all minimal (P,Q,R)-chains.