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 Narrowing (⇔)
↳20 QDP
↳21 DependencyGraphProof (⇔)
↳22 QDP
↳23 Narrowing (⇔)
↳24 QDP
↳25 Rewriting (⇔)
↳26 QDP
↳27 Rewriting (⇔)
↳28 QDP
↳29 QDPOrderProof (⇔)
↳30 QDP
↳31 DependencyGraphProof (⇔)
↳32 TRUE
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
CONCAT(cons(u, v), y) → CONCAT(v, y)
LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
LESS_LEAVES(u, v) → ISLEAF(u)
LESS_LEAVES(u, v) → ISLEAF(v)
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
IF2(false, u, v) → CONCAT(left(u), right(u))
IF2(false, u, v) → LEFT(u)
IF2(false, u, v) → RIGHT(u)
IF2(false, u, v) → CONCAT(left(v), right(v))
IF2(false, u, v) → LEFT(v)
IF2(false, u, v) → RIGHT(v)
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
CONCAT(cons(u, v), y) → CONCAT(v, y)
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
CONCAT(cons(u, v), y) → CONCAT(v, y)
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
CONCAT(cons(u, v), y) → CONCAT(v, y)
From the DPs we obtained the following set of size-change graphs:
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
IF1(b, false, u, v) → IF2(b, u, v)
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(u, v) → if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) → false
if1(b, false, u, v) → if2(b, u, v)
if2(true, u, v) → true
if2(false, u, v) → less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
IF1(b, false, u, v) → IF2(b, u, v)
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
less_leaves(x0, x1)
if1(x0, true, x1, x2)
if1(x0, false, x1, x2)
if2(true, x0, x1)
if2(false, x0, x1)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
LESS_LEAVES(u, v) → IF1(isLeaf(u), isLeaf(v), u, v)
IF1(b, false, u, v) → IF2(b, u, v)
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
LESS_LEAVES(y0, leaf) → IF1(isLeaf(y0), true, y0, leaf)
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
IF1(b, false, u, v) → IF2(b, u, v)
LESS_LEAVES(y0, leaf) → IF1(isLeaf(y0), true, y0, leaf)
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, u, v) → LESS_LEAVES(concat(left(u), right(u)), concat(left(v), right(v)))
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, right(cons(x0, x1))))
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1))
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, right(cons(x0, x1))))
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1))
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(left(cons(x0, x1)), x1))
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
IF1(b, false, u, v) → IF2(b, u, v)
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
IF2(false, y0, cons(x0, x1)) → LESS_LEAVES(concat(left(y0), right(y0)), concat(x0, x1))
POL(IF1(x1, x2, x3, x4)) = x1 + x4
POL(IF2(x1, x2, x3)) = x1 + x3
POL(LESS_LEAVES(x1, x2)) = 1 + x2
POL(concat(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 1 + x1 + x2
POL(false) = 1
POL(isLeaf(x1)) = 1
POL(leaf) = 0
POL(left(x1)) = 0
POL(right(x1)) = 0
POL(true) = 0
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
LESS_LEAVES(y0, cons(x0, x1)) → IF1(isLeaf(y0), false, y0, cons(x0, x1))
IF1(b, false, u, v) → IF2(b, u, v)
isLeaf(leaf) → true
isLeaf(cons(u, v)) → false
left(cons(u, v)) → u
right(cons(u, v)) → v
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
isLeaf(leaf)
isLeaf(cons(x0, x1))
left(cons(x0, x1))
right(cons(x0, x1))
concat(leaf, x0)
concat(cons(x0, x1), x2)