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 QDPSizeChangeProof (⇔)
↳34 TRUE
↳35 QDP
↳36 UsableRulesProof (⇔)
↳37 QDP
↳38 QReductionProof (⇔)
↳39 QDP
↳40 Instantiation (⇔)
↳41 QDP
↳42 NonInfProof (⇔)
↳43 AND
↳44 QDP
↳45 DependencyGraphProof (⇔)
↳46 TRUE
↳47 QDP
↳48 DependencyGraphProof (⇔)
↳49 TRUE
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
PLUS(s(x), y) → PLUS(x, y)
TIMES(s(x), y) → PLUS(y, times(x, y))
TIMES(s(x), y) → TIMES(x, y)
EXP(x, s(y)) → TIMES(x, exp(x, y))
EXP(x, s(y)) → EXP(x, y)
GE(s(x), s(y)) → GE(x, y)
TOWER(x, y) → TOWERITER(0, x, y, s(0))
TOWERITER(c, x, y, z) → HELP(ge(c, x), c, x, y, z)
TOWERITER(c, x, y, z) → GE(c, x)
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
HELP(false, c, x, y, z) → EXP(y, z)
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
GE(s(x), s(y)) → GE(x, y)
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
GE(s(x), s(y)) → GE(x, y)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
GE(s(x), s(y)) → GE(x, y)
From the DPs we obtained the following set of size-change graphs:
PLUS(s(x), y) → PLUS(x, y)
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
PLUS(s(x), y) → PLUS(x, y)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
PLUS(s(x), y) → PLUS(x, y)
From the DPs we obtained the following set of size-change graphs:
TIMES(s(x), y) → TIMES(x, y)
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
TIMES(s(x), y) → TIMES(x, y)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
TIMES(s(x), y) → TIMES(x, y)
From the DPs we obtained the following set of size-change graphs:
EXP(x, s(y)) → EXP(x, y)
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
EXP(x, s(y)) → EXP(x, y)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
EXP(x, s(y)) → EXP(x, y)
From the DPs we obtained the following set of size-change graphs:
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
TOWERITER(c, x, y, z) → HELP(ge(c, x), c, x, y, z)
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
tower(x, y) → towerIter(0, x, y, s(0))
towerIter(c, x, y, z) → help(ge(c, x), c, x, y, z)
help(true, c, x, y, z) → z
help(false, c, x, y, z) → towerIter(s(c), x, y, exp(y, z))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
TOWERITER(c, x, y, z) → HELP(ge(c, x), c, x, y, z)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
tower(x0, x1)
towerIter(x0, x1, x2, x3)
help(true, x0, x1, x2, x3)
help(false, x0, x1, x2, x3)
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
TOWERITER(c, x, y, z) → HELP(ge(c, x), c, x, y, z)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
TOWERITER(s(z0), z1, z2, y_0) → HELP(ge(s(z0), z1), s(z0), z1, z2, y_0)
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
TOWERITER(s(z0), z1, z2, y_0) → HELP(ge(s(z0), z1), s(z0), z1, z2, y_0)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
(1) (HELP(ge(x4, x5), x4, x5, x6, x7)=HELP(false, x8, x9, x10, x11) ⇒ HELP(false, x8, x9, x10, x11)≥TOWERITER(s(x8), x9, x10, exp(x10, x11)))
(2) (ge(x4, x5)=false ⇒ HELP(false, x4, x5, x6, x7)≥TOWERITER(s(x4), x5, x6, exp(x6, x7)))
(3) (false=false ⇒ HELP(false, 0, s(x25), x6, x7)≥TOWERITER(s(0), s(x25), x6, exp(x6, x7)))
(4) (ge(x27, x26)=false∧(∀x28,x29:ge(x27, x26)=false ⇒ HELP(false, x27, x26, x28, x29)≥TOWERITER(s(x27), x26, x28, exp(x28, x29))) ⇒ HELP(false, s(x27), s(x26), x6, x7)≥TOWERITER(s(s(x27)), s(x26), x6, exp(x6, x7)))
(5) (HELP(false, 0, s(x25), x6, x7)≥TOWERITER(s(0), s(x25), x6, exp(x6, x7)))
(6) (HELP(false, x27, x26, x6, x7)≥TOWERITER(s(x27), x26, x6, exp(x6, x7)) ⇒ HELP(false, s(x27), s(x26), x6, x7)≥TOWERITER(s(s(x27)), s(x26), x6, exp(x6, x7)))
(7) (TOWERITER(s(x12), x13, x14, exp(x14, x15))=TOWERITER(x16, x17, x18, x19) ⇒ TOWERITER(x16, x17, x18, x19)≥HELP(ge(x16, x17), x16, x17, x18, x19))
(8) (TOWERITER(s(x12), x13, x14, x19)≥HELP(ge(s(x12), x13), s(x12), x13, x14, x19))
POL(0) = 1
POL(HELP(x1, x2, x3, x4, x5)) = -1 - x1 - x2 + x3 + x4
POL(TOWERITER(x1, x2, x3, x4)) = -1 - x1 + x2 + x3
POL(c) = -2
POL(exp(x1, x2)) = 0
POL(false) = 1
POL(ge(x1, x2)) = 1
POL(plus(x1, x2)) = x2
POL(s(x1)) = 1 + x1
POL(times(x1, x2)) = 0
POL(true) = 1
The following pairs are in Pbound:
TOWERITER(c, x, y, z) → HELP(ge(c, x), c, x, y, z)
The following rules are usable:
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
ge(x, y) → ge(s(x), s(y))
false → ge(0, s(x))
true → ge(x, 0)
HELP(false, c, x, y, z) → TOWERITER(s(c), x, y, exp(y, z))
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
TOWERITER(c, x, y, z) → HELP(ge(c, x), c, x, y, z)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))