Term Rewriting System R:
[a, s, t, u]
circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(s, id) -> s
circ(id, s) -> s
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
subst(a, id) -> a
msubst(a, id) -> a
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

CIRC(cons(a, s), t) -> MSUBST(a, t)
CIRC(cons(a, s), t) -> CIRC(s, t)
CIRC(cons(lift, s), cons(a, t)) -> CIRC(s, t)
CIRC(cons(lift, s), cons(lift, t)) -> CIRC(s, t)
CIRC(circ(s, t), u) -> CIRC(s, circ(t, u))
CIRC(circ(s, t), u) -> CIRC(t, u)
CIRC(cons(lift, s), circ(cons(lift, t), u)) -> CIRC(cons(lift, circ(s, t)), u)
CIRC(cons(lift, s), circ(cons(lift, t), u)) -> CIRC(s, t)
MSUBST(msubst(a, s), t) -> MSUBST(a, circ(s, t))
MSUBST(msubst(a, s), t) -> CIRC(s, t)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

CIRC(cons(lift, s), circ(cons(lift, t), u)) -> CIRC(s, t)
CIRC(cons(lift, s), circ(cons(lift, t), u)) -> CIRC(cons(lift, circ(s, t)), u)
CIRC(circ(s, t), u) -> CIRC(t, u)
CIRC(circ(s, t), u) -> CIRC(s, circ(t, u))
CIRC(cons(lift, s), cons(lift, t)) -> CIRC(s, t)
CIRC(cons(lift, s), cons(a, t)) -> CIRC(s, t)
CIRC(cons(a, s), t) -> CIRC(s, t)
MSUBST(msubst(a, s), t) -> CIRC(s, t)
MSUBST(msubst(a, s), t) -> MSUBST(a, circ(s, t))
CIRC(cons(a, s), t) -> MSUBST(a, t)

Rules:

circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(s, id) -> s
circ(id, s) -> s
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
subst(a, id) -> a
msubst(a, id) -> a
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

Termination of R could not be shown.
Duration:
0:01 minutes