(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = 2 + 2·x1
POL(times(x1, x2)) = 2·x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
times(X, s(Y)) → plus(X, times(Y, X))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
trivial
and weight map:
plus_2=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:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES