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 QDPSizeChangeProof (⇔)
↳27 TRUE
↳28 QDP
↳29 UsableRulesProof (⇔)
↳30 QDP
↳31 QReductionProof (⇔)
↳32 QDP
↳33 Instantiation (⇔)
↳34 QDP
↳35 NonInfProof (⇔)
↳36 QDP
↳37 DependencyGraphProof (⇔)
↳38 TRUE
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
GT(s(x), s(y)) → GT(x, y)
PLUS(s(x), y) → PLUS(x, y)
DOUBLE(s(x)) → DOUBLE(x)
AVERAGE(x, y) → AVER(plus(x, y), 0)
AVERAGE(x, y) → PLUS(x, y)
AVER(sum, z) → IF(gt(sum, double(z)), sum, z)
AVER(sum, z) → GT(sum, double(z))
AVER(sum, z) → DOUBLE(z)
IF(true, sum, z) → AVER(sum, s(z))
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
DOUBLE(s(x)) → DOUBLE(x)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
DOUBLE(s(x)) → DOUBLE(x)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
DOUBLE(s(x)) → DOUBLE(x)
From the DPs we obtained the following set of size-change graphs:
PLUS(s(x), y) → PLUS(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
PLUS(s(x), y) → PLUS(x, y)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
PLUS(s(x), y) → PLUS(x, y)
From the DPs we obtained the following set of size-change graphs:
GT(s(x), s(y)) → GT(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
GT(s(x), s(y)) → GT(x, y)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
GT(s(x), s(y)) → GT(x, y)
From the DPs we obtained the following set of size-change graphs:
AVER(sum, z) → IF(gt(sum, double(z)), sum, z)
IF(true, sum, z) → AVER(sum, s(z))
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
double(0) → 0
double(s(x)) → s(s(double(x)))
average(x, y) → aver(plus(x, y), 0)
aver(sum, z) → if(gt(sum, double(z)), sum, z)
if(true, sum, z) → aver(sum, s(z))
if(false, sum, z) → z
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
AVER(sum, z) → IF(gt(sum, double(z)), sum, z)
IF(true, sum, z) → AVER(sum, s(z))
double(0) → 0
double(s(x)) → s(s(double(x)))
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
double(0)
double(s(x0))
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
plus(0, x0)
plus(s(x0), x1)
average(x0, x1)
aver(x0, x1)
if(true, x0, x1)
if(false, x0, x1)
AVER(sum, z) → IF(gt(sum, double(z)), sum, z)
IF(true, sum, z) → AVER(sum, s(z))
double(0) → 0
double(s(x)) → s(s(double(x)))
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
double(0)
double(s(x0))
AVER(z0, s(z1)) → IF(gt(z0, double(s(z1))), z0, s(z1))
IF(true, sum, z) → AVER(sum, s(z))
AVER(z0, s(z1)) → IF(gt(z0, double(s(z1))), z0, s(z1))
double(0) → 0
double(s(x)) → s(s(double(x)))
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
double(0)
double(s(x0))
(1) (IF(gt(x2, double(x3)), x2, x3)=IF(true, x4, x5) ⇒ AVER(x2, x3)≥IF(gt(x2, double(x3)), x2, x3))
(2) (double(x3)=x12∧gt(x2, x12)=true ⇒ AVER(x2, x3)≥IF(gt(x2, double(x3)), x2, x3))
(3) (true=true∧double(x3)=0 ⇒ AVER(s(x14), x3)≥IF(gt(s(x14), double(x3)), s(x14), x3))
(4) (gt(x16, x15)=true∧double(x3)=s(x15)∧(∀x17:gt(x16, x15)=true∧double(x17)=x15 ⇒ AVER(x16, x17)≥IF(gt(x16, double(x17)), x16, x17)) ⇒ AVER(s(x16), x3)≥IF(gt(s(x16), double(x3)), s(x16), x3))
(5) (double(x3)=0 ⇒ AVER(s(x14), x3)≥IF(gt(s(x14), double(x3)), s(x14), x3))
(6) (s(s(double(x19)))=s(x15)∧gt(x16, x15)=true∧(∀x17:gt(x16, x15)=true∧double(x17)=x15 ⇒ AVER(x16, x17)≥IF(gt(x16, double(x17)), x16, x17))∧(∀x20,x21,x22:double(x19)=s(x20)∧gt(x21, x20)=true∧(∀x22:gt(x21, x20)=true∧double(x22)=x20 ⇒ AVER(x21, x22)≥IF(gt(x21, double(x22)), x21, x22)) ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19)) ⇒ AVER(s(x16), s(x19))≥IF(gt(s(x16), double(s(x19))), s(x16), s(x19)))
(7) (0=0 ⇒ AVER(s(x14), 0)≥IF(gt(s(x14), double(0)), s(x14), 0))
(8) (AVER(s(x14), 0)≥IF(gt(s(x14), double(0)), s(x14), 0))
(9) (s(double(x19))=x15∧gt(x16, x15)=true∧(∀x20,x21:double(x19)=s(x20)∧gt(x21, x20)=true ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19)) ⇒ AVER(s(x16), s(x19))≥IF(gt(s(x16), double(s(x19))), s(x16), s(x19)))
(10) (true=true∧s(double(x19))=0∧(∀x20,x21:double(x19)=s(x20)∧gt(x21, x20)=true ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19)) ⇒ AVER(s(s(x24)), s(x19))≥IF(gt(s(s(x24)), double(s(x19))), s(s(x24)), s(x19)))
(11) (gt(x26, x25)=true∧s(double(x19))=s(x25)∧(∀x20,x21:double(x19)=s(x20)∧gt(x21, x20)=true ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19))∧(∀x27,x28,x29:gt(x26, x25)=true∧s(double(x27))=x25∧(∀x28,x29:double(x27)=s(x28)∧gt(x29, x28)=true ⇒ AVER(s(x29), x27)≥IF(gt(s(x29), double(x27)), s(x29), x27)) ⇒ AVER(s(x26), s(x27))≥IF(gt(s(x26), double(s(x27))), s(x26), s(x27))) ⇒ AVER(s(s(x26)), s(x19))≥IF(gt(s(s(x26)), double(s(x19))), s(s(x26)), s(x19)))
(12) (gt(x26, x25)=true∧double(x19)=x25∧(∀x20,x21:double(x19)=s(x20)∧gt(x21, x20)=true ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19)) ⇒ AVER(s(s(x26)), s(x19))≥IF(gt(s(s(x26)), double(s(x19))), s(s(x26)), s(x19)))
(13) (true=true∧double(x19)=0∧(∀x20,x21:double(x19)=s(x20)∧gt(x21, x20)=true ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19)) ⇒ AVER(s(s(s(x31))), s(x19))≥IF(gt(s(s(s(x31))), double(s(x19))), s(s(s(x31))), s(x19)))
(14) (gt(x33, x32)=true∧double(x19)=s(x32)∧(∀x20,x21:double(x19)=s(x20)∧gt(x21, x20)=true ⇒ AVER(s(x21), x19)≥IF(gt(s(x21), double(x19)), s(x21), x19))∧(∀x34,x35,x36:gt(x33, x32)=true∧double(x34)=x32∧(∀x35,x36:double(x34)=s(x35)∧gt(x36, x35)=true ⇒ AVER(s(x36), x34)≥IF(gt(s(x36), double(x34)), s(x36), x34)) ⇒ AVER(s(s(x33)), s(x34))≥IF(gt(s(s(x33)), double(s(x34))), s(s(x33)), s(x34))) ⇒ AVER(s(s(s(x33))), s(x19))≥IF(gt(s(s(s(x33))), double(s(x19))), s(s(s(x33))), s(x19)))
(15) (double(x19)=0 ⇒ AVER(s(s(s(x31))), s(x19))≥IF(gt(s(s(s(x31))), double(s(x19))), s(s(s(x31))), s(x19)))
(16) (AVER(s(x33), x19)≥IF(gt(s(x33), double(x19)), s(x33), x19)∧(∀x34,x35,x36:gt(x33, x32)=true∧double(x34)=x32∧(∀x35,x36:double(x34)=s(x35)∧gt(x36, x35)=true ⇒ AVER(s(x36), x34)≥IF(gt(s(x36), double(x34)), s(x36), x34)) ⇒ AVER(s(s(x33)), s(x34))≥IF(gt(s(s(x33)), double(s(x34))), s(s(x33)), s(x34))) ⇒ AVER(s(s(s(x33))), s(x19))≥IF(gt(s(s(s(x33))), double(s(x19))), s(s(s(x33))), s(x19)))
(17) (0=0 ⇒ AVER(s(s(s(x31))), s(0))≥IF(gt(s(s(s(x31))), double(s(0))), s(s(s(x31))), s(0)))
(18) (AVER(s(s(s(x31))), s(0))≥IF(gt(s(s(s(x31))), double(s(0))), s(s(s(x31))), s(0)))
(19) (AVER(s(x33), x19)≥IF(gt(s(x33), double(x19)), s(x33), x19) ⇒ AVER(s(s(s(x33))), s(x19))≥IF(gt(s(s(s(x33))), double(s(x19))), s(s(s(x33))), s(x19)))
(20) (AVER(x6, s(x7))=AVER(x8, x9) ⇒ IF(true, x6, x7)≥AVER(x6, s(x7)))
(21) (IF(true, x6, x7)≥AVER(x6, s(x7)))
POL(0) = 0
POL(AVER(x1, x2)) = x1 - x2
POL(IF(x1, x2, x3)) = -1 + x2 - x3
POL(c) = -1
POL(double(x1)) = 2
POL(false) = 1
POL(gt(x1, x2)) = x1 + x2
POL(s(x1)) = 2 + x1
POL(true) = 0
The following pairs are in Pbound:
AVER(sum, z) → IF(gt(sum, double(z)), sum, z)
IF(true, sum, z) → AVER(sum, s(z))
There are no usable rules
AVER(sum, z) → IF(gt(sum, double(z)), sum, z)
IF(true, sum, z) → AVER(sum, s(z))
double(0) → 0
double(s(x)) → s(s(double(x)))
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
double(0)
double(s(x0))