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))

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)


   R
     ↳RRRI
       →TRS2
         ↳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)
MSUBST(msubst(a, s), t) -> MSUBST(a, circ(s, t))
MSUBST(msubst(a, s), t) -> CIRC(s, t)

Furthermore, R contains one SCC. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


Dependency Pairs:

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





We number the DPs as follows:
  1. CIRC(circ(s, t), u) -> CIRC(t, u)
  2. CIRC(circ(s, t), u) -> CIRC(s, circ(t, u))
  3. CIRC(cons(lift, s), cons(lift, t)) -> CIRC(s, t)
  4. CIRC(cons(lift, s), cons(a, t)) -> CIRC(s, t)
  5. CIRC(cons(a, s), t) -> CIRC(s, t)
  6. MSUBST(msubst(a, s), t) -> CIRC(s, t)
  7. MSUBST(msubst(a, s), t) -> MSUBST(a, circ(s, t))
  8. CIRC(cons(a, s), t) -> MSUBST(a, t)
and get the following Size-Change Graph(s):
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1
2=2
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1
2>2
{6} , {6}
1>1
2=2
{7} , {7}
1>1
{8} , {8}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1
2=2
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1
2>2
{7} , {7}
1>1
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1
{6} , {8}
1>1
2=2
{8} , {6}
1>1
2=2
{5, 4, 3, 2, 1} , {6}
1>1
2>2
{5, 4, 3, 2, 1} , {6}
1>1
2=2
{7} , {8}
1>1
{6} , {8}
1>1
2>2
{8} , {5, 4, 3, 2, 1}
1>1
2>2
{6} , {7}
1>1
{8} , {5, 4, 3, 2, 1}
1>1
2=2
{8} , {6}
1>1
{8} , {5, 4, 3, 2, 1}
1>1
{6} , {8}
1>1
{5, 4, 3, 2, 1} , {6}
1>1
{8} , {6}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
msubst(x1, x2) -> msubst(x1, x2)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes