(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), 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(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))
The set Q consists of the following terms:
app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod
(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(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) → APP(plus, x)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
APP(app(times, app(s, x)), y) → APP(times, x)
SUM → APP(app(fold, add), 0)
SUM → APP(fold, add)
PROD → APP(app(fold, mul), app(s, 0))
PROD → APP(fold, mul)
PROD → APP(s, 0)
The TRS R consists of the following rules:
app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))
The set Q consists of the following terms:
app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod
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 10 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
The TRS R consists of the following rules:
app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))
The set Q consists of the following terms:
app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod
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(times, app(s, x)), y) → APP(app(times, x), y)
The TRS R consists of the following rules:
app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))
The set Q consists of the following terms:
app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod
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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
The TRS R consists of the following rules:
app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))
The set Q consists of the following terms:
app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod
We have to consider all minimal (P,Q,R)-chains.