(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
BIN(s(x), s(y)) → BIN(x, s(y))
BIN(s(x), s(y)) → BIN(x, y)
The TRS R consists of the following rules:
bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(5) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
BIN(s(x), s(y)) → BIN(x, s(y))
BIN(s(x), s(y)) → BIN(x, y)
R is empty.
The set Q consists of the following terms:
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(7) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
bin(x0, 0)
bin(0, s(x0))
bin(s(x0), s(x1))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
BIN(s(x), s(y)) → BIN(x, s(y))
BIN(s(x), s(y)) → BIN(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- BIN(s(x), s(y)) → BIN(x, s(y))
The graph contains the following edges 1 > 1, 2 >= 2
- BIN(s(x), s(y)) → BIN(x, y)
The graph contains the following edges 1 > 1, 2 > 2
(10) TRUE