(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(activate(x1)) = x1
POL(c) = 0
POL(f(x1)) = 2 + 2·x1
POL(false) = 2
POL(if(x1, x2, x3)) = 2·x1 + x2 + x3
POL(n__f(x1)) = 2 + 2·x1
POL(n__true) = 0
POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(false, X, Y) → activate(Y)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(activate(x1)) = 2 + 2·x1
POL(c) = 0
POL(f(x1)) = 2 + 2·x1
POL(if(x1, x2, x3)) = x1 + x2 + x3
POL(n__f(x1)) = 2 + 2·x1
POL(n__true) = 0
POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(true, X, Y) → X
true → n__true
activate(n__true) → true
activate(X) → X
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
activate(n__f(X)) → f(activate(X))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(activate(x1)) = 2·x1
POL(c) = 0
POL(f(x1)) = 2 + 2·x1
POL(if(x1, x2, x3)) = x1 + x2 + x3
POL(n__f(x1)) = 2 + 2·x1
POL(n__true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
activate(n__f(X)) → f(activate(X))
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
ntrue > nf1 > c > f1 > if3
and weight map:
c=1
n__true=1
f_1=3
n__f_1=1
if_3=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:
f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
(8) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) YES