0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 UsableRulesProof (⇔)
↳16 QDP
↳17 QReductionProof (⇔)
↳18 QDP
↳19 QDPSizeChangeProof (⇔)
↳20 TRUE
↳21 QDP
↳22 UsableRulesProof (⇔)
↳23 QDP
↳24 QReductionProof (⇔)
↳25 QDP
↳26 Instantiation (⇔)
↳27 QDP
↳28 NonInfProof (⇔)
↳29 QDP
↳30 DependencyGraphProof (⇔)
↳31 TRUE
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))
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))
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)
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)
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))
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)
LT(s(x), s(y)) → LT(x, y)
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))
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)
LT(s(x), s(y)) → LT(x, y)
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)
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)
LT(s(x), s(y)) → LT(x, y)
From the DPs we obtained the following set of size-change graphs:
PLUS(s(x), y) → PLUS(x, y)
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))
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)
PLUS(s(x), y) → PLUS(x, y)
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)
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)
PLUS(s(x), y) → PLUS(x, y)
From the DPs we obtained the following set of size-change graphs:
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)
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))
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)
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)
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
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)
fib(x0)
fibiter(x0, x1, x2, x3)
if(false, x0, x1, x2, x3)
if(true, x0, x1, x2, x3)
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)
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
FIBITER(z0, s(z1), z3, y_0) → IF(lt(s(z1), z0), z0, s(z1), z3, y_0)
IF(true, b, c, x, y) → FIBITER(b, s(c), y, plus(x, y))
FIBITER(z0, s(z1), z3, y_0) → IF(lt(s(z1), z0), z0, s(z1), z3, y_0)
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
(1) (IF(lt(x5, x4), x4, x5, x6, x7)=IF(true, x8, x9, x10, x11) ⇒ IF(true, x8, x9, x10, x11)≥FIBITER(x8, s(x9), x11, plus(x10, x11)))
(2) (lt(x5, x4)=true ⇒ IF(true, x4, x5, x6, x7)≥FIBITER(x4, s(x5), x7, plus(x6, x7)))
(3) (true=true ⇒ IF(true, s(x24), 0, x6, x7)≥FIBITER(s(x24), s(0), x7, plus(x6, x7)))
(4) (lt(x27, x26)=true∧(∀x28,x29:lt(x27, x26)=true ⇒ IF(true, x26, x27, x28, x29)≥FIBITER(x26, s(x27), x29, plus(x28, x29))) ⇒ IF(true, s(x26), s(x27), x6, x7)≥FIBITER(s(x26), s(s(x27)), x7, plus(x6, x7)))
(5) (IF(true, s(x24), 0, x6, x7)≥FIBITER(s(x24), s(0), x7, plus(x6, x7)))
(6) (IF(true, x26, x27, x6, x7)≥FIBITER(x26, s(x27), x7, plus(x6, x7)) ⇒ IF(true, s(x26), s(x27), x6, x7)≥FIBITER(s(x26), s(s(x27)), x7, plus(x6, x7)))
(7) (FIBITER(x12, s(x13), x15, plus(x14, x15))=FIBITER(x16, x17, x18, x19) ⇒ FIBITER(x16, x17, x18, x19)≥IF(lt(x17, x16), x16, x17, x18, x19))
(8) (FIBITER(x12, s(x13), x15, x19)≥IF(lt(s(x13), x12), x12, s(x13), x15, x19))
POL(0) = 1
POL(FIBITER(x1, x2, x3, x4)) = x1 - x2
POL(IF(x1, x2, x3, x4, x5)) = -1 + x1 + x2 - x3
POL(c) = -1
POL(false) = 0
POL(lt(x1, x2)) = 1
POL(plus(x1, x2)) = 0
POL(s(x1)) = 1 + x1
POL(true) = 1
The following pairs are in Pbound:
IF(true, b, c, x, y) → FIBITER(b, s(c), y, plus(x, y))
The following rules are usable:
IF(true, b, c, x, y) → FIBITER(b, s(c), y, plus(x, y))
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
lt(0, s(y)) → true
FIBITER(b, c, x, y) → IF(lt(c, b), b, c, x, y)
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))