Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(0, cons(X, Z)) -> mark(X)
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_first}(0, Z) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_sel1}(s(X), cons(Y, Z)) -> a_{_sel1}(mark(X), mark(Z))
a_{_sel1}(0, cons(X, Z)) -> a_{_quote}(X)
a_{_sel1}(X1, X2) -> sel1(X1, X2)
a_{_first1}(0, Z) -> nil1
a_{_first1}(s(X), cons(Y, Z)) -> cons1(a_{_quote}(Y), a_{_first1}(mark(X), mark(Z)))
a_{_first1}(X1, X2) -> first1(X1, X2)
a_{_quote}(0) -> 01
a_{_quote}(s(X)) -> s1(a_{_quote}(X))
a_{_quote}(sel(X, Z)) -> a_{_sel1}(mark(X), mark(Z))
a_{_quote}(X) -> quote(X)
a_{_quote1}(cons(X, Z)) -> cons1(a_{_quote}(X), a_{_quote1}(Z))
a_{_quote1}(nil) -> nil1
a_{_quote1}(first(X, Z)) -> a_{_first1}(mark(X), mark(Z))
a_{_quote1}(X) -> quote1(X)
a_{_unquote}(01) -> 0
a_{_unquote}(s1(X)) -> s(a_{_unquote}(mark(X)))
a_{_unquote}(X) -> unquote(X)
a_{_unquote1}(nil1) -> nil
a_{_unquote1}(cons1(X, Z)) -> a_{_fcons}(a_{_unquote}(mark(X)), a_{_unquote1}(mark(Z)))
a_{_unquote1}(X) -> unquote1(X)
a_{_fcons}(X, Z) -> cons(mark(X), Z)
a_{_fcons}(X1, X2) -> fcons(X1, X2)
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(sel1(X1, X2)) -> a_{_sel1}(mark(X1), mark(X2))
mark(quote(X)) -> a_{_quote}(X)
mark(first1(X1, X2)) -> a_{_first1}(mark(X1), mark(X2))
mark(quote1(X)) -> a_{_quote1}(X)
mark(unquote(X)) -> a_{_unquote}(mark(X))
mark(unquote1(X)) -> a_{_unquote1}(mark(X))
mark(fcons(X1, X2)) -> a_{_fcons}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(nil) -> nil
mark(nil1) -> nil1
mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
mark(01) -> 01
mark(s1(X)) -> s1(mark(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_SEL}(0, cons(X, Z)) -> MARK(X)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
A_{_FROM}(X) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> A_{_SEL1}(mark(X), mark(Z))
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_SEL1}(0, cons(X, Z)) -> A_{_QUOTE}(X)
A_{_FIRST1}(s(X), cons(Y, Z)) -> A_{_QUOTE}(Y)
A_{_FIRST1}(s(X), cons(Y, Z)) -> A_{_FIRST1}(mark(X), mark(Z))
A_{_FIRST1}(s(X), cons(Y, Z)) -> MARK(X)
A_{_FIRST1}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_QUOTE}(s(X)) -> A_{_QUOTE}(X)
A_{_QUOTE}(sel(X, Z)) -> A_{_SEL1}(mark(X), mark(Z))
A_{_QUOTE}(sel(X, Z)) -> MARK(X)
A_{_QUOTE}(sel(X, Z)) -> MARK(Z)
A_{_QUOTE1}(cons(X, Z)) -> A_{_QUOTE}(X)
A_{_QUOTE1}(cons(X, Z)) -> A_{_QUOTE1}(Z)
A_{_QUOTE1}(first(X, Z)) -> A_{_FIRST1}(mark(X), mark(Z))
A_{_QUOTE1}(first(X, Z)) -> MARK(X)
A_{_QUOTE1}(first(X, Z)) -> MARK(Z)
A_{_UNQUOTE}(s1(X)) -> A_{_UNQUOTE}(mark(X))
A_{_UNQUOTE}(s1(X)) -> MARK(X)
A_{_UNQUOTE1}(cons1(X, Z)) -> A_{_FCONS}(a_{_unquote}(mark(X)), a_{_unquote1}(mark(Z)))
A_{_UNQUOTE1}(cons1(X, Z)) -> A_{_UNQUOTE}(mark(X))
A_{_UNQUOTE1}(cons1(X, Z)) -> MARK(X)
A_{_UNQUOTE1}(cons1(X, Z)) -> A_{_UNQUOTE1}(mark(Z))
A_{_UNQUOTE1}(cons1(X, Z)) -> MARK(Z)
A_{_FCONS}(X, Z) -> 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(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(sel1(X1, X2)) -> A_{_SEL1}(mark(X1), mark(X2))
MARK(sel1(X1, X2)) -> MARK(X1)
MARK(sel1(X1, X2)) -> MARK(X2)
MARK(quote(X)) -> A_{_QUOTE}(X)
MARK(first1(X1, X2)) -> A_{_FIRST1}(mark(X1), mark(X2))
MARK(first1(X1, X2)) -> MARK(X1)
MARK(first1(X1, X2)) -> MARK(X2)
MARK(quote1(X)) -> A_{_QUOTE1}(X)
MARK(unquote(X)) -> A_{_UNQUOTE}(mark(X))
MARK(unquote(X)) -> MARK(X)
MARK(unquote1(X)) -> A_{_UNQUOTE1}(mark(X))
MARK(unquote1(X)) -> MARK(X)
MARK(fcons(X1, X2)) -> A_{_FCONS}(mark(X1), mark(X2))
MARK(fcons(X1, X2)) -> MARK(X1)
MARK(fcons(X1, X2)) -> MARK(X2)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(cons1(X1, X2)) -> MARK(X1)
MARK(cons1(X1, X2)) -> MARK(X2)
MARK(s1(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

A_{_SEL}(0, cons(X, Z)) -> MARK(X)
A_{_SEL1}(0, cons(X, Z)) -> A_{_QUOTE}(X)
A_{_QUOTE1}(first(X, Z)) -> MARK(Z)
A_{_QUOTE1}(first(X, Z)) -> MARK(X)
A_{_FIRST1}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_FIRST1}(s(X), cons(Y, Z)) -> MARK(X)
A_{_FIRST1}(s(X), cons(Y, Z)) -> A_{_FIRST1}(mark(X), mark(Z))
A_{_QUOTE1}(first(X, Z)) -> A_{_FIRST1}(mark(X), mark(Z))
A_{_QUOTE1}(cons(X, Z)) -> A_{_QUOTE1}(Z)
A_{_UNQUOTE1}(cons1(X, Z)) -> MARK(Z)
A_{_UNQUOTE1}(cons1(X, Z)) -> A_{_UNQUOTE1}(mark(Z))
A_{_UNQUOTE1}(cons1(X, Z)) -> MARK(X)
A_{_UNQUOTE1}(cons1(X, Z)) -> A_{_UNQUOTE}(mark(X))
MARK(s1(X)) -> MARK(X)
MARK(cons1(X1, X2)) -> MARK(X2)
MARK(cons1(X1, X2)) -> MARK(X1)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(fcons(X1, X2)) -> MARK(X2)
MARK(fcons(X1, X2)) -> MARK(X1)
MARK(fcons(X1, X2)) -> A_{_FCONS}(mark(X1), mark(X2))
MARK(unquote1(X)) -> MARK(X)
A_{_FCONS}(X, Z) -> MARK(X)
A_{_UNQUOTE1}(cons1(X, Z)) -> A_{_FCONS}(a_{_unquote}(mark(X)), a_{_unquote1}(mark(Z)))
MARK(unquote1(X)) -> A_{_UNQUOTE1}(mark(X))
MARK(unquote(X)) -> MARK(X)
A_{_UNQUOTE}(s1(X)) -> MARK(X)
A_{_UNQUOTE}(s1(X)) -> A_{_UNQUOTE}(mark(X))
MARK(unquote(X)) -> A_{_UNQUOTE}(mark(X))
A_{_QUOTE}(sel(X, Z)) -> MARK(Z)
A_{_QUOTE1}(cons(X, Z)) -> A_{_QUOTE}(X)
MARK(quote1(X)) -> A_{_QUOTE1}(X)
MARK(first1(X1, X2)) -> MARK(X2)
MARK(first1(X1, X2)) -> MARK(X1)
A_{_QUOTE}(sel(X, Z)) -> MARK(X)
A_{_FIRST1}(s(X), cons(Y, Z)) -> A_{_QUOTE}(Y)
MARK(first1(X1, X2)) -> A_{_FIRST1}(mark(X1), mark(X2))
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_QUOTE}(sel(X, Z)) -> A_{_SEL1}(mark(X), mark(Z))
A_{_QUOTE}(s(X)) -> A_{_QUOTE}(X)
MARK(quote(X)) -> A_{_QUOTE}(X)
MARK(sel1(X1, X2)) -> MARK(X2)
MARK(sel1(X1, X2)) -> MARK(X1)
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> A_{_SEL1}(mark(X), mark(Z))
MARK(sel1(X1, X2)) -> A_{_SEL1}(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))

Rules:

a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(0, cons(X, Z)) -> mark(X)
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_first}(0, Z) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_sel1}(s(X), cons(Y, Z)) -> a_{_sel1}(mark(X), mark(Z))
a_{_sel1}(0, cons(X, Z)) -> a_{_quote}(X)
a_{_sel1}(X1, X2) -> sel1(X1, X2)
a_{_first1}(0, Z) -> nil1
a_{_first1}(s(X), cons(Y, Z)) -> cons1(a_{_quote}(Y), a_{_first1}(mark(X), mark(Z)))
a_{_first1}(X1, X2) -> first1(X1, X2)
a_{_quote}(0) -> 01
a_{_quote}(s(X)) -> s1(a_{_quote}(X))
a_{_quote}(sel(X, Z)) -> a_{_sel1}(mark(X), mark(Z))
a_{_quote}(X) -> quote(X)
a_{_quote1}(cons(X, Z)) -> cons1(a_{_quote}(X), a_{_quote1}(Z))
a_{_quote1}(nil) -> nil1
a_{_quote1}(first(X, Z)) -> a_{_first1}(mark(X), mark(Z))
a_{_quote1}(X) -> quote1(X)
a_{_unquote}(01) -> 0
a_{_unquote}(s1(X)) -> s(a_{_unquote}(mark(X)))
a_{_unquote}(X) -> unquote(X)
a_{_unquote1}(nil1) -> nil
a_{_unquote1}(cons1(X, Z)) -> a_{_fcons}(a_{_unquote}(mark(X)), a_{_unquote1}(mark(Z)))
a_{_unquote1}(X) -> unquote1(X)
a_{_fcons}(X, Z) -> cons(mark(X), Z)
a_{_fcons}(X1, X2) -> fcons(X1, X2)
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(sel1(X1, X2)) -> a_{_sel1}(mark(X1), mark(X2))
mark(quote(X)) -> a_{_quote}(X)
mark(first1(X1, X2)) -> a_{_first1}(mark(X1), mark(X2))
mark(quote1(X)) -> a_{_quote1}(X)
mark(unquote(X)) -> a_{_unquote}(mark(X))
mark(unquote1(X)) -> a_{_unquote1}(mark(X))
mark(fcons(X1, X2)) -> a_{_fcons}(mark(X1), mark(X2))
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(nil) -> nil
mark(nil1) -> nil1
mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
mark(01) -> 01
mark(s1(X)) -> s1(mark(X))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:07 minutes