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))
↳ QTRS
↳ DependencyPairsProof
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))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(times, app(s, x)), y) → APP(times, x)
SUM → APP(fold, add)
PROD → APP(fold, mul)
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
APP(app(plus, app(s, x)), y) → APP(plus, x)
PROD → APP(s, 0)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
SUM → APP(app(fold, add), 0)
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))
PROD → APP(app(fold, mul), app(s, 0))
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), 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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(times, app(s, x)), y) → APP(times, x)
SUM → APP(fold, add)
PROD → APP(fold, mul)
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
APP(app(plus, app(s, x)), y) → APP(plus, x)
PROD → APP(s, 0)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
SUM → APP(app(fold, add), 0)
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))
PROD → APP(app(fold, mul), app(s, 0))
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), 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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
The value of delta used in the strict ordering is 2.
POL(APP(x1, x2)) = (2)x_1
POL(plus) = 0
POL(app(x1, x2)) = 1 + x_2
POL(s) = 3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
The value of delta used in the strict ordering is 2.
POL(APP(x1, x2)) = (2)x_1
POL(times) = 0
POL(app(x1, x2)) = 1 + x_2
POL(s) = 3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
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))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
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(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), 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))