(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(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:
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(x, y))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fib(x0)
fibiter(x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
if(true, x0, x1, x2, x3)
(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:
PLUS(s(x), y) → PLUS(x, y)
LT(s(x), s(y)) → LT(x, y)
FIB(x) → FIBITER(x, 0, 0, s(0))
FIBITER(b, c, x, y) → IF(lt(c, b), b, c, x, y)
FIBITER(b, c, x, y) → LT(c, b)
IF(true, b, c, x, y) → FIBITER(b, s(c), y, plus(x, y))
IF(true, b, c, x, y) → PLUS(x, y)
The TRS R consists of the following rules:
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(x, y))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fib(x0)
fibiter(x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
if(true, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes.
(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:
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(x, y))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fib(x0)
fibiter(x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
if(true, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(x), y) → PLUS(x, y)
The TRS R consists of the following rules:
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(x, y))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fib(x0)
fibiter(x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
if(true, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
IF(true, b, c, x, y) → FIBITER(b, s(c), y, plus(x, y))
FIBITER(b, c, x, y) → IF(lt(c, b), b, c, x, y)
The TRS R consists of the following rules:
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fib(x) → fibiter(x, 0, 0, s(0))
fibiter(b, c, x, y) → if(lt(c, b), b, c, x, y)
if(false, b, c, x, y) → x
if(true, b, c, x, y) → fibiter(b, s(c), y, plus(x, y))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fib(x0)
fibiter(x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
if(true, x0, x1, x2, x3)
We have to consider all minimal (P,Q,R)-chains.