(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
minus(x, y) → help(lt(y, x), x, y)
help(true, x, y) → s(minus(x, s(y)))
help(false, x, y) → 0
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:
lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
minus(x, y) → help(lt(y, x), x, y)
help(true, x, y) → s(minus(x, s(y)))
help(false, x, y) → 0
The set Q consists of the following terms:
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
minus(x0, x1)
help(true, x0, x1)
help(false, x0, 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:
LT(s(x), s(y)) → LT(x, y)
MINUS(x, y) → HELP(lt(y, x), x, y)
MINUS(x, y) → LT(y, x)
HELP(true, x, y) → MINUS(x, s(y))
The TRS R consists of the following rules:
lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
minus(x, y) → help(lt(y, x), x, y)
help(true, x, y) → s(minus(x, s(y)))
help(false, x, y) → 0
The set Q consists of the following terms:
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
minus(x0, x1)
help(true, x0, x1)
help(false, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LT(s(x), s(y)) → LT(x, y)
The TRS R consists of the following rules:
lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
minus(x, y) → help(lt(y, x), x, y)
help(true, x, y) → s(minus(x, s(y)))
help(false, x, y) → 0
The set Q consists of the following terms:
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
minus(x0, x1)
help(true, x0, x1)
help(false, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → HELP(lt(y, x), x, y)
HELP(true, x, y) → MINUS(x, s(y))
The TRS R consists of the following rules:
lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
minus(x, y) → help(lt(y, x), x, y)
help(true, x, y) → s(minus(x, s(y)))
help(false, x, y) → 0
The set Q consists of the following terms:
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
minus(x0, x1)
help(true, x0, x1)
help(false, x0, x1)
We have to consider all minimal (P,Q,R)-chains.