YES
proof of Transformed_CSR_04_Ex8_BLR02_FR.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 65 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   add(X1, X2) -> n__add(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[fib_1, 0] > sel_2 > activate_1 > fib1_2 > n__fib1_2 > [s_1, cons_2]
[fib_1, 0] > sel_2 > activate_1 > fib1_2 > [n__add_2, add_2] > [s_1, cons_2]


Status:
fib_1: multiset status
sel_2: [1,2]
fib1_2: multiset status
s_1: multiset status
0: multiset status
cons_2: multiset status
n__fib1_2: multiset status
n__add_2: [1,2]
add_2: [1,2]
activate_1: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(X) -> X




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   add(X1, X2) -> n__add(X1, X2)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:add_2 > n__add_2

and weight map:

   add_2=0
   n__add_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   add(X1, X2) -> n__add(X1, X2)




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES