(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)
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(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)
The set Q consists of the following terms:
app(app(const, x0), x1)
app(app(app(subst, x0), x1), x2)
app(app(fix, x0), x1)
(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(subst, f), g), x) → APP(app(f, x), app(g, x))
APP(app(app(subst, f), g), x) → APP(f, x)
APP(app(app(subst, f), g), x) → APP(g, x)
APP(app(fix, f), x) → APP(app(f, app(fix, f)), x)
APP(app(fix, f), x) → APP(f, app(fix, f))
The TRS R consists of the following rules:
app(app(const, x), y) → x
app(app(app(subst, f), g), x) → app(app(f, x), app(g, x))
app(app(fix, f), x) → app(app(f, app(fix, f)), x)
The set Q consists of the following terms:
app(app(const, x0), x1)
app(app(app(subst, x0), x1), x2)
app(app(fix, x0), x1)
We have to consider all minimal (P,Q,R)-chains.