(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a__f(x1)) = 1 + 2·x1
POL(a__if(x1, x2, x3)) = x1 + 2·x2 + x3
POL(c) = 0
POL(f(x1)) = 1 + 2·x1
POL(false) = 1
POL(if(x1, x2, x3)) = x1 + 2·x2 + x3
POL(mark(x1)) = x1
POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__if(false, X, Y) → mark(Y)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a__f(x1)) = 1 + 2·x1
POL(a__if(x1, x2, x3)) = x1 + 2·x2 + x3
POL(c) = 0
POL(f(x1)) = 1 + 2·x1
POL(false) = 0
POL(if(x1, x2, x3)) = x1 + 2·x2 + x3
POL(mark(x1)) = 2·x1
POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(f(X)) → a__f(mark(X))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
mark1 > aif3 > if3 > af1 > true > c > false > f1
and weight map:
c=1
true=1
false=1
a__f_1=4
mark_1=0
f_1=1
a__if_3=0
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:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES