(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0) → s(0)
f(s(0)) → s(s(0))
f(s(0)) → *(s(s(0)), f(0))
f(+(x, s(0))) → +(s(s(0)), f(x))
f(+(x, y)) → *(f(x), f(y))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(*(x1, x2)) = x1 + x2
POL(+(x1, x2)) = 1 + x1 + x2
POL(0) = 0
POL(f(x1)) = x1
POL(s(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(+(x, y)) → *(f(x), f(y))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0) → s(0)
f(s(0)) → s(s(0))
f(s(0)) → *(s(s(0)), f(0))
f(+(x, s(0))) → +(s(s(0)), f(x))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(*(x1, x2)) = x1 + x2
POL(+(x1, x2)) = x1 + x2
POL(0) = 0
POL(f(x1)) = 1 + x1
POL(s(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(0) → s(0)
f(s(0)) → s(s(0))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(0)) → *(s(s(0)), f(0))
f(+(x, s(0))) → +(s(s(0)), f(x))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(*(x1, x2)) = x1 + 2·x2
POL(+(x1, x2)) = 2 + 2·x1 + 2·x2
POL(0) = 0
POL(f(x1)) = 2·x1
POL(s(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(+(x, s(0))) → +(s(s(0)), f(x))
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(0)) → *(s(s(0)), f(0))
Q is empty.
(7) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(0)) → *(s(s(0)), f(0))
The set Q consists of the following terms:
f(s(0))
(9) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(0)) → F(0)
The TRS R consists of the following rules:
f(s(0)) → *(s(s(0)), f(0))
The set Q consists of the following terms:
f(s(0))
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(12) TRUE