Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

eq → true
eq → eq
eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

[eq, true] > false > [inf₁, cons, 0, nil, s]
take₂ > [inf₁, cons, 0, nil, s]
length₁ > [inf₁, cons, 0, nil, s]

Status:

eq: []
true: []
false: []
inf₁: [1]
cons: []
take₂: [1,2]
0: []
nil: []
s: []
length₁: [1]

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

eq → false
inf(X) → cons
take(0, X) → nil
take(s, cons) → cons
length(nil) → 0
length(cons) → s

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

eq → true
eq → eq

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

eq > true

Status:

eq: []
true: []

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

eq → true

(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

eq → eq

Q is empty.