Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(cons(te(msubst(te(a), sortSu(t))), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(cons(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(t), sortSu(u)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t))))), sortSu(u)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(t), sortSu(u)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(t)))

Rules:

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

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))

six new Dependency Pairs are created:

SORTSU(circ(sortSu(circ(sortSu(s), sortSu(cons(te(a'), sortSu(s''))))), sortSu(u'))) -> SORTSU(circ(sortSu(s), sortSu(cons(te(msubst(te(a'), sortSu(u'))), sortSu(circ(sortSu(s''), sortSu(u')))))))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(cons(sop(lift), sortSu(s''))))), sortSu(cons(te(a'), sortSu(t''))))) -> SORTSU(circ(sortSu(s), sortSu(cons(te(a'), sortSu(circ(sortSu(s''), sortSu(t'')))))))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(cons(sop(lift), sortSu(s''))))), sortSu(cons(sop(lift), sortSu(t''))))) -> SORTSU(circ(sortSu(s), sortSu(cons(sop(lift), sortSu(circ(sortSu(s''), sortSu(t'')))))))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(circ(sortSu(s''), sortSu(t''))))), sortSu(u''))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(s''), sortSu(circ(sortSu(t''), sortSu(u'')))))))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t'))), sortSu(id))) -> SORTSU(circ(sortSu(s), sortSu(t')))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(cons(sop(lift), sortSu(s''))))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t''))), sortSu(u''))))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(cons(sop(lift), sortSu(circ(sortSu(s''), sortSu(t''))))), sortSu(u'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polynomial Ordering

Dependency Pairs:

SORTSU(circ(sortSu(circ(sortSu(s), sortSu(circ(sortSu(s''), sortSu(t''))))), sortSu(u''))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(s''), sortSu(circ(sortSu(t''), sortSu(u'')))))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(t)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(t), sortSu(u)))

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

SORTSU(circ(sortSu(circ(sortSu(s), sortSu(circ(sortSu(s''), sortSu(t''))))), sortSu(u''))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(s''), sortSu(circ(sortSu(t''), sortSu(u'')))))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(t)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(t), sortSu(u)))

Additionally, the following usable rules w.r.t. the implicit AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(circ(x₁, x₂)) = x₁ + x₁·x₂

POL(TE(x₁)) = x₁

POL(lift) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(subst(x₁, x₂)) = 1 + x₁ + x₂

POL(msubst(x₁, x₂)) = x₁ + x₁·x₂ + x₂

POL(SORTSU(x₁)) = x₁

POL(te(x₁)) = x₁

POL(id) = 0

POL(sortSu(x₁)) = 1 + x₁

POL(sop(x₁)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Polo

...

               →DP Problem 3

                 ↳Dependency Graph

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:51 minutes

POL(circ(x₁, x₂))	= x₁ + x₁·x₂
POL(TE(x₁))	= x₁
POL(lift)	= 0
POL(cons(x₁, x₂))	= x₁ + x₂
POL(subst(x₁, x₂))	= 1 + x₁ + x₂
POL(msubst(x₁, x₂))	= x₁ + x₁·x₂ + x₂
POL(SORTSU(x₁))	= x₁
POL(te(x₁))	= x₁
POL(id)	= 0
POL(sortSu(x₁))	= 1 + x₁
POL(sop(x₁))	= 0