(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, 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(X1, X2)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Lexicographic path order with status [LPO].
Precedence:
fib1 > sel2 > activate1 > fib12 > cons2 > nfib12
fib1 > sel2 > activate1 > fib12 > add2 > s1 > nfib12
fib1 > 0 > nfib12
Status:
sel2: [1,2]
cons2: [1,2]
add2: [2,1]
fib12: [1,2]
nfib12: [1,2]
fib1: [1]
s1: [1]
activate1: [1]
0: []
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, 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(X1, X2)
activate(X) → X
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) TRUE