(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
terms(x1)  =  terms(x1)
cons(x1)  =  x1
recip(x1)  =  x1
sqr(x1)  =  sqr(x1)
0  =  0
s(x1)  =  s(x1)
add(x1, x2)  =  add(x1, x2)
dbl(x1)  =  dbl(x1)
first(x1, x2)  =  first(x1, x2)
nil  =  nil

Recursive path order with status [RPO].
Quasi-Precedence:

[terms₁, sqr₁] > 0 > nil > s₁
[terms₁, sqr₁] > add₂ > s₁
[terms₁, sqr₁] > dbl₁ > s₁
first₂ > nil > s₁

Status:

terms₁: [1]
sqr₁: [1]
0: multiset
s₁: multiset
add₂: [1,2]
dbl₁: multiset
first₂: [1,2]
nil: multiset

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

sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y)) → cons(Y)

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:

sqr₁ > recip₁ > terms₁ > cons₁

and weight map:

terms_1=3
cons_1=1
recip_1=1
sqr_1=1

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

terms(N) → cons(recip(sqr(N)))

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.