R
↳Dependency Pair Analysis
ACTIVE(leq(s(X), s(Y))) -> LEQ(X, Y)
ACTIVE(diff(X, Y)) -> IF(leq(X, Y), 0, s(diff(p(X), Y)))
ACTIVE(diff(X, Y)) -> LEQ(X, Y)
ACTIVE(diff(X, Y)) -> S(diff(p(X), Y))
ACTIVE(diff(X, Y)) -> DIFF(p(X), Y)
ACTIVE(diff(X, Y)) -> P(X)
ACTIVE(p(X)) -> P(active(X))
ACTIVE(p(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(leq(X1, X2)) -> LEQ(active(X1), X2)
ACTIVE(leq(X1, X2)) -> ACTIVE(X1)
ACTIVE(leq(X1, X2)) -> LEQ(X1, active(X2))
ACTIVE(leq(X1, X2)) -> ACTIVE(X2)
ACTIVE(if(X1, X2, X3)) -> IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(diff(X1, X2)) -> DIFF(active(X1), X2)
ACTIVE(diff(X1, X2)) -> ACTIVE(X1)
ACTIVE(diff(X1, X2)) -> DIFF(X1, active(X2))
ACTIVE(diff(X1, X2)) -> ACTIVE(X2)
P(mark(X)) -> P(X)
P(ok(X)) -> P(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
LEQ(mark(X1), X2) -> LEQ(X1, X2)
LEQ(X1, mark(X2)) -> LEQ(X1, X2)
LEQ(ok(X1), ok(X2)) -> LEQ(X1, X2)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
DIFF(mark(X1), X2) -> DIFF(X1, X2)
DIFF(X1, mark(X2)) -> DIFF(X1, X2)
DIFF(ok(X1), ok(X2)) -> DIFF(X1, X2)
PROPER(p(X)) -> P(proper(X))
PROPER(p(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(leq(X1, X2)) -> LEQ(proper(X1), proper(X2))
PROPER(leq(X1, X2)) -> PROPER(X1)
PROPER(leq(X1, X2)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> IF(proper(X1), proper(X2), proper(X3))
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(diff(X1, X2)) -> DIFF(proper(X1), proper(X2))
PROPER(diff(X1, X2)) -> PROPER(X1)
PROPER(diff(X1, X2)) -> PROPER(X2)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
LEQ(ok(X1), ok(X2)) -> LEQ(X1, X2)
LEQ(X1, mark(X2)) -> LEQ(X1, X2)
LEQ(mark(X1), X2) -> LEQ(X1, X2)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
LEQ(ok(X1), ok(X2)) -> LEQ(X1, X2)
POL(LEQ(x1, x2)) = x1 POL(mark(x1)) = x1 POL(ok(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 9
↳Polynomial Ordering
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
LEQ(X1, mark(X2)) -> LEQ(X1, X2)
LEQ(mark(X1), X2) -> LEQ(X1, X2)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
LEQ(X1, mark(X2)) -> LEQ(X1, X2)
POL(LEQ(x1, x2)) = x2 POL(mark(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 9
↳Polo
...
→DP Problem 10
↳Polynomial Ordering
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
LEQ(mark(X1), X2) -> LEQ(X1, X2)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
LEQ(mark(X1), X2) -> LEQ(X1, X2)
POL(LEQ(x1, x2)) = x1 POL(mark(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 9
↳Polo
...
→DP Problem 11
↳Dependency Graph
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polynomial Ordering
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
POL(mark(x1)) = 0 POL(ok(x1)) = 1 + x1 POL(IF(x1, x2, x3)) = x2
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 12
↳Polynomial Ordering
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
POL(mark(x1)) = 1 + x1 POL(IF(x1, x2, x3)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 12
↳Polo
...
→DP Problem 13
↳Dependency Graph
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polynomial Ordering
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
S(ok(X)) -> S(X)
POL(S(x1)) = x1 POL(mark(x1)) = x1 POL(ok(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 14
↳Polynomial Ordering
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
S(mark(X)) -> S(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
S(mark(X)) -> S(X)
POL(S(x1)) = x1 POL(mark(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 14
↳Polo
...
→DP Problem 15
↳Dependency Graph
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polynomial Ordering
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
DIFF(ok(X1), ok(X2)) -> DIFF(X1, X2)
DIFF(X1, mark(X2)) -> DIFF(X1, X2)
DIFF(mark(X1), X2) -> DIFF(X1, X2)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
DIFF(ok(X1), ok(X2)) -> DIFF(X1, X2)
POL(DIFF(x1, x2)) = x1 POL(mark(x1)) = x1 POL(ok(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 16
↳Polynomial Ordering
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
DIFF(X1, mark(X2)) -> DIFF(X1, X2)
DIFF(mark(X1), X2) -> DIFF(X1, X2)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
DIFF(X1, mark(X2)) -> DIFF(X1, X2)
POL(DIFF(x1, x2)) = x2 POL(mark(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 16
↳Polo
...
→DP Problem 17
↳Polynomial Ordering
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
DIFF(mark(X1), X2) -> DIFF(X1, X2)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
DIFF(mark(X1), X2) -> DIFF(X1, X2)
POL(DIFF(x1, x2)) = x1 POL(mark(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 16
↳Polo
...
→DP Problem 18
↳Dependency Graph
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polynomial Ordering
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
P(ok(X)) -> P(X)
P(mark(X)) -> P(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
P(ok(X)) -> P(X)
POL(P(x1)) = x1 POL(mark(x1)) = x1 POL(ok(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 19
↳Polynomial Ordering
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
P(mark(X)) -> P(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
P(mark(X)) -> P(X)
POL(P(x1)) = x1 POL(mark(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 19
↳Polo
...
→DP Problem 20
↳Dependency Graph
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polynomial Ordering
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
ACTIVE(diff(X1, X2)) -> ACTIVE(X2)
ACTIVE(diff(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(leq(X1, X2)) -> ACTIVE(X2)
ACTIVE(leq(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(p(X)) -> ACTIVE(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
ACTIVE(diff(X1, X2)) -> ACTIVE(X2)
ACTIVE(diff(X1, X2)) -> ACTIVE(X1)
POL(ACTIVE(x1)) = x1 POL(if(x1, x2, x3)) = x1 POL(leq(x1, x2)) = x1 + x2 POL(s(x1)) = x1 POL(diff(x1, x2)) = 1 + x1 + x2 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 21
↳Polynomial Ordering
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(leq(X1, X2)) -> ACTIVE(X2)
ACTIVE(leq(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(p(X)) -> ACTIVE(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
POL(ACTIVE(x1)) = x1 POL(if(x1, x2, x3)) = 1 + x1 POL(leq(x1, x2)) = x1 + x2 POL(s(x1)) = x1 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 21
↳Polo
...
→DP Problem 22
↳Polynomial Ordering
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
ACTIVE(leq(X1, X2)) -> ACTIVE(X2)
ACTIVE(leq(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(p(X)) -> ACTIVE(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
ACTIVE(leq(X1, X2)) -> ACTIVE(X2)
ACTIVE(leq(X1, X2)) -> ACTIVE(X1)
POL(ACTIVE(x1)) = x1 POL(leq(x1, x2)) = 1 + x1 + x2 POL(s(x1)) = x1 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 21
↳Polo
...
→DP Problem 23
↳Polynomial Ordering
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(p(X)) -> ACTIVE(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
POL(ACTIVE(x1)) = x1 POL(s(x1)) = 1 + x1 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 21
↳Polo
...
→DP Problem 24
↳Polynomial Ordering
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
ACTIVE(p(X)) -> ACTIVE(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
ACTIVE(p(X)) -> ACTIVE(X)
POL(ACTIVE(x1)) = x1 POL(p(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 21
↳Polo
...
→DP Problem 25
↳Dependency Graph
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polynomial Ordering
→DP Problem 8
↳Nar
PROPER(diff(X1, X2)) -> PROPER(X2)
PROPER(diff(X1, X2)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(leq(X1, X2)) -> PROPER(X2)
PROPER(leq(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(p(X)) -> PROPER(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
PROPER(diff(X1, X2)) -> PROPER(X2)
PROPER(diff(X1, X2)) -> PROPER(X1)
POL(if(x1, x2, x3)) = x1 + x2 + x3 POL(PROPER(x1)) = x1 POL(leq(x1, x2)) = x1 + x2 POL(s(x1)) = x1 POL(diff(x1, x2)) = 1 + x1 + x2 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 26
↳Polynomial Ordering
→DP Problem 8
↳Nar
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(leq(X1, X2)) -> PROPER(X2)
PROPER(leq(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(p(X)) -> PROPER(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
POL(if(x1, x2, x3)) = 1 + x1 + x2 + x3 POL(PROPER(x1)) = x1 POL(leq(x1, x2)) = x1 + x2 POL(s(x1)) = x1 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 26
↳Polo
...
→DP Problem 27
↳Polynomial Ordering
→DP Problem 8
↳Nar
PROPER(leq(X1, X2)) -> PROPER(X2)
PROPER(leq(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(p(X)) -> PROPER(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
PROPER(leq(X1, X2)) -> PROPER(X2)
PROPER(leq(X1, X2)) -> PROPER(X1)
POL(PROPER(x1)) = x1 POL(leq(x1, x2)) = 1 + x1 + x2 POL(s(x1)) = x1 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 26
↳Polo
...
→DP Problem 28
↳Polynomial Ordering
→DP Problem 8
↳Nar
PROPER(s(X)) -> PROPER(X)
PROPER(p(X)) -> PROPER(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
PROPER(s(X)) -> PROPER(X)
POL(PROPER(x1)) = x1 POL(s(x1)) = 1 + x1 POL(p(x1)) = x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 26
↳Polo
...
→DP Problem 29
↳Polynomial Ordering
→DP Problem 8
↳Nar
PROPER(p(X)) -> PROPER(X)
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
PROPER(p(X)) -> PROPER(X)
POL(PROPER(x1)) = x1 POL(p(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 26
↳Polo
...
→DP Problem 30
↳Dependency Graph
→DP Problem 8
↳Nar
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Narrowing Transformation
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
eight new Dependency Pairs are created:
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(leq(X1', X2'))) -> TOP(leq(proper(X1'), proper(X2')))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(false)) -> TOP(ok(false))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(diff(X1', X2'))) -> TOP(diff(proper(X1'), proper(X2')))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
→DP Problem 31
↳Narrowing Transformation
TOP(mark(diff(X1', X2'))) -> TOP(diff(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(false)) -> TOP(ok(false))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(leq(X1', X2'))) -> TOP(leq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(ok(X)) -> TOP(active(X))
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
15 new Dependency Pairs are created:
TOP(ok(X)) -> TOP(active(X))
TOP(ok(p(0))) -> TOP(mark(0))
TOP(ok(p(s(X'')))) -> TOP(mark(X''))
TOP(ok(leq(0, Y'))) -> TOP(mark(true))
TOP(ok(leq(s(X''), 0))) -> TOP(mark(false))
TOP(ok(leq(s(X''), s(Y')))) -> TOP(mark(leq(X'', Y')))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(diff(X'', Y'))) -> TOP(mark(if(leq(X'', Y'), 0, s(diff(p(X''), Y')))))
TOP(ok(p(X''))) -> TOP(p(active(X'')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(leq(X1', X2'))) -> TOP(leq(active(X1'), X2'))
TOP(ok(leq(X1', X2'))) -> TOP(leq(X1', active(X2')))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(diff(X1', X2'))) -> TOP(diff(active(X1'), X2'))
TOP(ok(diff(X1', X2'))) -> TOP(diff(X1', active(X2')))
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
→DP Problem 31
↳Nar
...
→DP Problem 32
↳Polynomial Ordering
TOP(ok(diff(X1', X2'))) -> TOP(diff(X1', active(X2')))
TOP(ok(diff(X1', X2'))) -> TOP(diff(active(X1'), X2'))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(leq(X1', X2'))) -> TOP(leq(X1', active(X2')))
TOP(ok(leq(X1', X2'))) -> TOP(leq(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(p(X''))) -> TOP(p(active(X'')))
TOP(ok(diff(X'', Y'))) -> TOP(mark(if(leq(X'', Y'), 0, s(diff(p(X''), Y')))))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(leq(s(X''), s(Y')))) -> TOP(mark(leq(X'', Y')))
TOP(ok(p(s(X'')))) -> TOP(mark(X''))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(leq(X1', X2'))) -> TOP(leq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(diff(X1', X2'))) -> TOP(diff(proper(X1'), proper(X2')))
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
TOP(ok(p(s(X'')))) -> TOP(mark(X''))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
POL(proper(x1)) = x1 POL(false) = 0 POL(true) = 0 POL(mark(x1)) = x1 POL(ok(x1)) = x1 POL(TOP(x1)) = 1 + x1 POL(active(x1)) = x1 POL(if(x1, x2, x3)) = x2 + x3 POL(0) = 0 POL(leq(x1, x2)) = 0 POL(s(x1)) = x1 POL(diff(x1, x2)) = 0 POL(p(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
→DP Problem 3
↳Polo
→DP Problem 4
↳Polo
→DP Problem 5
↳Polo
→DP Problem 6
↳Polo
→DP Problem 7
↳Polo
→DP Problem 8
↳Nar
→DP Problem 31
↳Nar
...
→DP Problem 33
↳Remaining Obligation(s)
TOP(ok(diff(X1', X2'))) -> TOP(diff(X1', active(X2')))
TOP(ok(diff(X1', X2'))) -> TOP(diff(active(X1'), X2'))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(leq(X1', X2'))) -> TOP(leq(X1', active(X2')))
TOP(ok(leq(X1', X2'))) -> TOP(leq(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(p(X''))) -> TOP(p(active(X'')))
TOP(ok(diff(X'', Y'))) -> TOP(mark(if(leq(X'', Y'), 0, s(diff(p(X''), Y')))))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(leq(s(X''), s(Y')))) -> TOP(mark(leq(X'', Y')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(leq(X1', X2'))) -> TOP(leq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(diff(X1', X2'))) -> TOP(diff(proper(X1'), proper(X2')))
active(p(0)) -> mark(0)
active(p(s(X))) -> mark(X)
active(leq(0, Y)) -> mark(true)
active(leq(s(X), 0)) -> mark(false)
active(leq(s(X), s(Y))) -> mark(leq(X, Y))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) -> p(active(X))
active(s(X)) -> s(active(X))
active(leq(X1, X2)) -> leq(active(X1), X2)
active(leq(X1, X2)) -> leq(X1, active(X2))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(diff(X1, X2)) -> diff(active(X1), X2)
active(diff(X1, X2)) -> diff(X1, active(X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
leq(mark(X1), X2) -> mark(leq(X1, X2))
leq(X1, mark(X2)) -> mark(leq(X1, X2))
leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
diff(mark(X1), X2) -> mark(diff(X1, X2))
diff(X1, mark(X2)) -> mark(diff(X1, X2))
diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
proper(p(X)) -> p(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))