0 CpxTRS
↳1 RcToIrcProof (BOTH BOUNDS(ID, ID), 22 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 91 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 63 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 BOUNDS(1, 1)
and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0) → 0
x(N, s(M)) → plus(x(N, M), N)
activate(X) → X
The duplicating contexts are:
x([], s(M))
The defined contexts are:
plus([], x1)
As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0) → 0
x(N, s(M)) → plus(x(N, M), N)
activate(X) → X
Tuples:
and(tt, z0) → activate(z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
activate(z0) → z0
S tuples:
AND(tt, z0) → c(ACTIVATE(z0))
PLUS(z0, 0) → c1
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, 0) → c3
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) → c5
K tuples:none
AND(tt, z0) → c(ACTIVATE(z0))
PLUS(z0, 0) → c1
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, 0) → c3
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) → c5
and, plus, x, activate
AND, PLUS, X, ACTIVATE
c, c1, c2, c3, c4, c5
X(z0, 0) → c3
ACTIVATE(z0) → c5
PLUS(z0, 0) → c1
AND(tt, z0) → c(ACTIVATE(z0))
Tuples:
and(tt, z0) → activate(z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
activate(z0) → z0
S tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
and, plus, x, activate
PLUS, X
c2, c4
and(tt, z0) → activate(z0)
activate(z0) → z0
Tuples:
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
S tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
x, plus
PLUS, X
c2, c4
We considered the (Usable) Rules:none
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
The order we found is given by the following interpretation:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
POL(0) = 0
POL(PLUS(x1, x2)) = 0
POL(X(x1, x2)) = x2
POL(c2(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(plus(x1, x2)) = 0
POL(s(x1)) = [1] + x1
POL(x(x1, x2)) = 0
Tuples:
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
S tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
Defined Rule Symbols:
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
x, plus
PLUS, X
c2, c4
We considered the (Usable) Rules:none
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
The order we found is given by the following interpretation:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
POL(0) = [1]
POL(PLUS(x1, x2)) = [2] + x2
POL(X(x1, x2)) = x2 + x1·x2
POL(c2(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(plus(x1, x2)) = [2]x1 + x2 + x22 + [2]x1·x2 + x12
POL(s(x1)) = [2] + x1
POL(x(x1, x2)) = [1] + x1 + x2 + x22 + [2]x1·x2
Tuples:
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
S tuples:none
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
Defined Rule Symbols:
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
x, plus
PLUS, X
c2, c4