Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, X1, X2, Y, YS]
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_minus}(X, 0) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(mark(X), mark(Y))
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_quot}(0, s(Y)) -> 0
a_{_quot}(s(X), s(Y)) -> s(a_{_quot}(a_{_minus}(mark(X), mark(Y)), s(mark(Y))))
a_{_quot}(X1, X2) -> quot(X1, X2)
a_{_zWquot}(XS, nil) -> nil
a_{_zWquot}(nil, XS) -> nil
a_{_zWquot}(cons(X, XS), cons(Y, YS)) -> cons(a_{_quot}(mark(X), mark(Y)), zWquot(XS, YS))
a_{_zWquot}(X1, X2) -> zWquot(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(minus(X1, X2)) -> a_{_minus}(mark(X1), mark(X2))
mark(quot(X1, X2)) -> a_{_quot}(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> a_{_zWquot}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_FROM}(X) -> MARK(X)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_MINUS}(s(X), s(Y)) -> A_{_MINUS}(mark(X), mark(Y))
A_{_MINUS}(s(X), s(Y)) -> MARK(X)
A_{_MINUS}(s(X), s(Y)) -> MARK(Y)
A_{_QUOT}(s(X), s(Y)) -> A_{_QUOT}(a_{_minus}(mark(X), mark(Y)), s(mark(Y)))
A_{_QUOT}(s(X), s(Y)) -> A_{_MINUS}(mark(X), mark(Y))
A_{_QUOT}(s(X), s(Y)) -> MARK(X)
A_{_QUOT}(s(X), s(Y)) -> MARK(Y)
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> A_{_QUOT}(mark(X), mark(Y))
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> MARK(X)
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> MARK(Y)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(minus(X1, X2)) -> A_{_MINUS}(mark(X1), mark(X2))
MARK(minus(X1, X2)) -> MARK(X1)
MARK(minus(X1, X2)) -> MARK(X2)
MARK(quot(X1, X2)) -> A_{_QUOT}(mark(X1), mark(X2))
MARK(quot(X1, X2)) -> MARK(X1)
MARK(quot(X1, X2)) -> MARK(X2)
MARK(zWquot(X1, X2)) -> A_{_ZWQUOT}(mark(X1), mark(X2))
MARK(zWquot(X1, X2)) -> MARK(X1)
MARK(zWquot(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

Dependency Pairs:

A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> MARK(Y)
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> MARK(X)
A_{_QUOT}(s(X), s(Y)) -> MARK(Y)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(zWquot(X1, X2)) -> MARK(X2)
MARK(zWquot(X1, X2)) -> MARK(X1)
A_{_QUOT}(s(X), s(Y)) -> MARK(X)
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> A_{_QUOT}(mark(X), mark(Y))
MARK(zWquot(X1, X2)) -> A_{_ZWQUOT}(mark(X1), mark(X2))
MARK(quot(X1, X2)) -> MARK(X2)
MARK(quot(X1, X2)) -> MARK(X1)
A_{_MINUS}(s(X), s(Y)) -> MARK(Y)
A_{_QUOT}(s(X), s(Y)) -> A_{_MINUS}(mark(X), mark(Y))
A_{_QUOT}(s(X), s(Y)) -> A_{_QUOT}(a_{_minus}(mark(X), mark(Y)), s(mark(Y)))
MARK(quot(X1, X2)) -> A_{_QUOT}(mark(X1), mark(X2))
MARK(minus(X1, X2)) -> MARK(X2)
MARK(minus(X1, X2)) -> MARK(X1)
A_{_MINUS}(s(X), s(Y)) -> MARK(X)
A_{_MINUS}(s(X), s(Y)) -> A_{_MINUS}(mark(X), mark(Y))
MARK(minus(X1, X2)) -> A_{_MINUS}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_FROM}(X) -> MARK(X)

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_minus}(X, 0) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(mark(X), mark(Y))
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_quot}(0, s(Y)) -> 0
a_{_quot}(s(X), s(Y)) -> s(a_{_quot}(a_{_minus}(mark(X), mark(Y)), s(mark(Y))))
a_{_quot}(X1, X2) -> quot(X1, X2)
a_{_zWquot}(XS, nil) -> nil
a_{_zWquot}(nil, XS) -> nil
a_{_zWquot}(cons(X, XS), cons(Y, YS)) -> cons(a_{_quot}(mark(X), mark(Y)), zWquot(XS, YS))
a_{_zWquot}(X1, X2) -> zWquot(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(minus(X1, X2)) -> a_{_minus}(mark(X1), mark(X2))
mark(quot(X1, X2)) -> a_{_quot}(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> a_{_zWquot}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

A_{_QUOT}(s(X), s(Y)) -> A_{_QUOT}(a_{_minus}(mark(X), mark(Y)), s(mark(Y)))

Additionally, the following usable rules for innermost can be oriented:

mark(from(X)) -> a_{_from}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(minus(X1, X2)) -> a_{_minus}(mark(X1), mark(X2))
mark(quot(X1, X2)) -> a_{_quot}(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> a_{_zWquot}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil
a_{_zWquot}(XS, nil) -> nil
a_{_zWquot}(nil, XS) -> nil
a_{_zWquot}(cons(X, XS), cons(Y, YS)) -> cons(a_{_quot}(mark(X), mark(Y)), zWquot(XS, YS))
a_{_zWquot}(X1, X2) -> zWquot(X1, X2)
a_{_quot}(0, s(Y)) -> 0
a_{_quot}(s(X), s(Y)) -> s(a_{_quot}(a_{_minus}(mark(X), mark(Y)), s(mark(Y))))
a_{_quot}(X1, X2) -> quot(X1, X2)
a_{_minus}(X, 0) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(mark(X), mark(Y))
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 0

POL(MARK(x₁)) = 1

POL(a__sel(x₁, x₂)) = 1

POL(a__zWquot(x₁, x₂)) = 1

POL(minus(x₁, x₂)) = 0

POL(A__FROM(x₁)) = 1

POL(sel(x₁, x₂)) = 0

POL(mark(x₁)) = 1

POL(a__from(x₁)) = 1

POL(a__quot(x₁, x₂)) = x₂

POL(A__ZWQUOT(x₁, x₂)) = x₁

POL(A__QUOT(x₁, x₂)) = x₁

POL(0) = 0

POL(cons(x₁, x₂)) = 1

POL(zWquot(x₁, x₂)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(quot(x₁, x₂)) = 0

POL(nil) = 0

POL(s(x₁)) = 1

POL(A__MINUS(x₁, x₂)) = 1

POL(A__SEL(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 2

             ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> MARK(Y)
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> MARK(X)
A_{_QUOT}(s(X), s(Y)) -> MARK(Y)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(zWquot(X1, X2)) -> MARK(X2)
MARK(zWquot(X1, X2)) -> MARK(X1)
A_{_QUOT}(s(X), s(Y)) -> MARK(X)
A_{_ZWQUOT}(cons(X, XS), cons(Y, YS)) -> A_{_QUOT}(mark(X), mark(Y))
MARK(zWquot(X1, X2)) -> A_{_ZWQUOT}(mark(X1), mark(X2))
MARK(quot(X1, X2)) -> MARK(X2)
MARK(quot(X1, X2)) -> MARK(X1)
A_{_MINUS}(s(X), s(Y)) -> MARK(Y)
A_{_QUOT}(s(X), s(Y)) -> A_{_MINUS}(mark(X), mark(Y))
MARK(quot(X1, X2)) -> A_{_QUOT}(mark(X1), mark(X2))
MARK(minus(X1, X2)) -> MARK(X2)
MARK(minus(X1, X2)) -> MARK(X1)
A_{_MINUS}(s(X), s(Y)) -> MARK(X)
A_{_MINUS}(s(X), s(Y)) -> A_{_MINUS}(mark(X), mark(Y))
MARK(minus(X1, X2)) -> A_{_MINUS}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_FROM}(X) -> MARK(X)

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_minus}(X, 0) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(mark(X), mark(Y))
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_quot}(0, s(Y)) -> 0
a_{_quot}(s(X), s(Y)) -> s(a_{_quot}(a_{_minus}(mark(X), mark(Y)), s(mark(Y))))
a_{_quot}(X1, X2) -> quot(X1, X2)
a_{_zWquot}(XS, nil) -> nil
a_{_zWquot}(nil, XS) -> nil
a_{_zWquot}(cons(X, XS), cons(Y, YS)) -> cons(a_{_quot}(mark(X), mark(Y)), zWquot(XS, YS))
a_{_zWquot}(X1, X2) -> zWquot(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(minus(X1, X2)) -> a_{_minus}(mark(X1), mark(X2))
mark(quot(X1, X2)) -> a_{_quot}(mark(X1), mark(X2))
mark(zWquot(X1, X2)) -> a_{_zWquot}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:02 minutes

POL(from(x₁))	= 0
POL(MARK(x₁))	= 1
POL(a__sel(x₁, x₂))	= 1
POL(a__zWquot(x₁, x₂))	= 1
POL(minus(x₁, x₂))	= 0
POL(A__FROM(x₁))	= 1
POL(sel(x₁, x₂))	= 0
POL(mark(x₁))	= 1
POL(a__from(x₁))	= 1
POL(a__quot(x₁, x₂))	= x₂
POL(A__ZWQUOT(x₁, x₂))	= x₁
POL(A__QUOT(x₁, x₂))	= x₁
POL(0)	= 0
POL(cons(x₁, x₂))	= 1
POL(zWquot(x₁, x₂))	= 0
POL(a__minus(x₁, x₂))	= 0
POL(quot(x₁, x₂))	= 0
POL(nil)	= 0
POL(s(x₁))	= 1
POL(A__MINUS(x₁, x₂))	= 1
POL(A__SEL(x₁, x₂))	= x₂