(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
GR(0, z0) → c3
GR(s(z0), 0) → c4
GR(s(z0), s(z1)) → c5(GR(z0, z1))
P(0) → c6
P(s(z0)) → c7
S tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
GR(0, z0) → c3
GR(s(z0), 0) → c4
GR(s(z0), s(z1)) → c5(GR(z0, z1))
P(0) → c6
P(s(z0)) → c7
K tuples:none
Defined Rule Symbols:
cond1, cond2, gr, p
Defined Pair Symbols:
COND1, COND2, GR, P
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
GR(0, z0) → c3
GR(s(z0), 0) → c4
P(0) → c6
P(s(z0)) → c7
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
S tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2), P(z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2), P(z0))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
K tuples:none
Defined Rule Symbols:
cond1, cond2, gr, p
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c5
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
S tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
K tuples:none
Defined Rule Symbols:
cond1, cond2, gr, p
Defined Pair Symbols:
COND1, GR, COND2
Compound Symbols:
c, c5, c1, c2
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
cond1(true, z0, z1, z2) → cond2(gr(z1, z2), z0, z1, z2)
cond2(true, z0, z1, z2) → cond2(gr(z1, z2), z0, p(z1), z2)
cond2(false, z0, z1, z2) → cond1(gr(z0, z2), p(z0), z1, z2)
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
S tuples:
COND1(true, z0, z1, z2) → c(COND2(gr(z1, z2), z0, z1, z2), GR(z1, z2))
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
K tuples:none
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
COND1, GR, COND2
Compound Symbols:
c, c5, c1, c2
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND1(
true,
z0,
z1,
z2) →
c(
COND2(
gr(
z1,
z2),
z0,
z1,
z2),
GR(
z1,
z2)) by
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0), GR(0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0), GR(s(z0), 0))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0), GR(0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0), GR(s(z0), 0))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0), GR(0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0), GR(s(z0), 0))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND2, COND1
Compound Symbols:
c5, c1, c2, c
(11) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(true, z0, z1, z2) → c1(COND2(gr(z1, z2), z0, p(z1), z2), GR(z1, z2))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
K tuples:none
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND2, COND1
Compound Symbols:
c5, c1, c2, c, c
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
z0,
z1,
z2) →
c1(
COND2(
gr(
z1,
z2),
z0,
p(
z1),
z2),
GR(
z1,
z2)) by
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2), GR(0, x2))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0), GR(0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2), GR(0, x2))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0), GR(0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2), GR(0, x2))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0), GR(0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND2, COND1
Compound Symbols:
c5, c2, c, c, c1
(15) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing tuple parts
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
K tuples:none
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND2, COND1
Compound Symbols:
c5, c2, c, c, c1, c1
(17) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
We considered the (Usable) Rules:
p(s(z0)) → z0
p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x3
POL(COND2(x1, x2, x3, x4)) = x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND2(false, z0, z1, z2) → c2(COND1(gr(z0, z2), p(z0), z1, z2), GR(z0, z2))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND2, COND1
Compound Symbols:
c5, c2, c, c, c1, c1
(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
z0,
z1,
z2) →
c2(
COND1(
gr(
z0,
z2),
p(
z0),
z1,
z2),
GR(
z0,
z2)) by
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2
(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND2(false, 0, x1, z0) → c2(COND1(false, p(0), x1, z0), GR(0, z0))
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2), GR(0, x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2
(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(25) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x2
POL(COND2(x1, x2, x3, x4)) = [4]x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, s(z0), s(z1)) → c(COND2(gr(z0, z1), x0, s(z0), s(z1)), GR(s(z0), s(z1)))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND1(
true,
x0,
s(
z0),
s(
z1)) →
c(
COND2(
gr(
z0,
z1),
x0,
s(
z0),
s(
z1)),
GR(
s(
z0),
s(
z1))) by
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c1, c2, c2, c
(29) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
We considered the (Usable) Rules:none
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x4
POL(COND2(x1, x2, x3, x4)) = [4]x4
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = [4]
POL(gr(x1, x2)) = 0
POL(p(x1)) = [2] + [2]x1
POL(s(x1)) = [2]
POL(true) = 0
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c1, c2, c2, c
(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
z0),
x2) →
c1(
COND2(
gr(
s(
z0),
x2),
x0,
z0,
x2),
GR(
s(
z0),
x2)) by
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0), GR(s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c1, c2, c2, c
(33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, p(s(z0)), s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c1, c2, c2, c
(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
z0),
s(
z1)) →
c1(
COND2(
gr(
z0,
z1),
x0,
p(
s(
z0)),
s(
z1)),
GR(
s(
z0),
s(
z1))) by
COND2(true, x0, s(z0), s(x2)) → c1(COND2(gr(z0, x2), x0, z0, s(x2)), GR(s(z0), s(x2)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(z0), s(x2)) → c1(COND2(gr(z0, x2), x0, z0, s(x2)), GR(s(z0), s(x2)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c2, c2, c, c1
(37) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
We considered the (Usable) Rules:
p(s(z0)) → z0
p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x3
POL(COND2(x1, x2, x3, x4)) = x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, x2) → c1(COND2(gr(0, x2), x0, 0, x2))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c2, c2, c, c1
(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
0,
x2) →
c1(
COND2(
gr(
0,
x2),
x0,
0,
x2)) by
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, p(0), z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c2, c2, c, c1
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
0,
z0) →
c1(
COND2(
false,
x0,
p(
0),
z0)) by
COND2(true, x0, 0, x1) → c1(COND2(false, x0, 0, x1))
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
gr(0, z0) → false
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:
gr, p
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c2, c2, c, c1
(43) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
p(0) → 0
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, p(s(z0)), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c1, c2, c2, c, c1
(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
z0),
0) →
c1(
COND2(
true,
x0,
p(
s(
z0)),
0)) by
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c2, c, c1, c1
(47) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
We considered the (Usable) Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [2]x1 + x3
POL(COND2(x1, x2, x3, x4)) = x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c2, c, c1, c1
(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
z0),
x1,
x2) →
c2(
COND1(
gr(
s(
z0),
x2),
z0,
x1,
x2),
GR(
s(
z0),
x2)) by
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0), GR(s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c2, c, c1, c1
(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), p(s(z0)), x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c2, c, c1, c1
(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
z0),
x1,
s(
z1)) →
c2(
COND1(
gr(
z0,
z1),
p(
s(
z0)),
x1,
s(
z1)),
GR(
s(
z0),
s(
z1))) by
COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, p(s(0)), x1, s(z0)), GR(s(0), s(z0)))
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, p(s(0)), x1, s(z0)), GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, p(s(0)), x1, s(z0)), GR(s(0), s(z0)))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c, c1, c1, c2
(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(x2)) → c2(COND1(gr(z0, x2), z0, x1, s(x2)), GR(s(z0), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c, c1, c1, c2
(57) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x2 + x3
POL(COND2(x1, x2, x3, x4)) = x2 + x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [5] + [2]x1 + [3]x2
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = [4]
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, 0, x1, x2) → c2(COND1(gr(0, x2), 0, x1, x2))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c, c1, c1, c2
(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
0,
x1,
x2) →
c2(
COND1(
gr(
0,
x2),
0,
x1,
x2)) by
COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))
K tuples:
COND2(true, x0, s(z0), x2) → c1(COND2(gr(s(z0), x2), x0, z0, x2), GR(s(z0), x2))
COND2(false, s(z0), x1, x2) → c2(COND1(gr(s(z0), x2), z0, x1, x2), GR(s(z0), x2))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c, c1, c1, c2
(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND2(false, 0, x0, z0) → c2(COND1(false, 0, x0, z0))
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, p(s(z0)), x1, 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c2, c, c1, c1, c2
(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
z0),
x1,
0) →
c2(
COND1(
true,
p(
s(
z0)),
x1,
0)) by
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(65) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x2
POL(COND2(x1, x2, x3, x4)) = x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(67) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(s(z1))) → c(COND2(gr(z0, z1), x0, s(s(z0)), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND1(
true,
x0,
s(
s(
z0)),
s(
s(
z1))) →
c(
COND2(
gr(
z0,
z1),
x0,
s(
s(
z0)),
s(
s(
z1))),
GR(
s(
s(
z0)),
s(
s(
z1)))) by
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(71) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
We considered the (Usable) Rules:none
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = [2]
POL(COND1(x1, x2, x3, x4)) = [2] + [2]x4
POL(COND2(x1, x2, x3, x4)) = [2] + [2]x4
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = [2]
POL(gr(x1, x2)) = [2]x1 + x2
POL(p(x1)) = [2] + [2]x1
POL(s(x1)) = 0
POL(true) = 0
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
z0),
s(
z1)) →
c1(
COND2(
gr(
z0,
z1),
x0,
z0,
s(
z1)),
GR(
s(
z0),
s(
z1))) by
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
(74) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, p(s(0)), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
0),
s(
z0)) →
c1(
COND2(
false,
x0,
p(
s(
0)),
s(
z0)),
GR(
s(
0),
s(
z0))) by
COND2(true, x0, s(0), s(x1)) → c1(COND2(false, x0, 0, s(x1)), GR(s(0), s(x1)))
(76) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(0), s(x1)) → c1(COND2(false, x0, 0, s(x1)), GR(s(0), s(x1)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(77) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x3
POL(COND2(x1, x2, x3, x4)) = [4]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, p(s(s(z0))), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
z0)),
s(
0)) →
c1(
COND2(
true,
x0,
p(
s(
s(
z0))),
s(
0)),
GR(
s(
s(
z0)),
s(
0))) by
COND2(true, x0, s(s(x1)), s(0)) → c1(COND2(true, x0, s(x1), s(0)), GR(s(s(x1)), s(0)))
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(0)) → c1(COND2(true, x0, s(x1), s(0)), GR(s(s(x1)), s(0)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(81) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x3
POL(COND2(x1, x2, x3, x4)) = [4]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0
(82) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, p(s(s(z0))), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c1, c2, c2
(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
z0)),
s(
s(
z1))) →
c1(
COND2(
gr(
z0,
z1),
x0,
p(
s(
s(
z0))),
s(
s(
z1))),
GR(
s(
s(
z0)),
s(
s(
z1)))) by
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(COND2(gr(x1, x2), x0, s(x1), s(s(x2))), GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
(84) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(COND2(gr(x1, x2), x0, s(x1), s(s(x2))), GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(85) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [2] + x3
POL(COND2(x1, x2, x3, x4)) = [2] + x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
z0),
x1,
s(
z1)) →
c2(
COND1(
gr(
z0,
z1),
z0,
x1,
s(
z1)),
GR(
s(
z0),
s(
z1))) by
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
(88) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(90) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
z0),
x1,
s(
x2)) →
c2(
COND1(
gr(
z0,
x2),
z0,
x1,
s(
x2)),
GR(
s(
z0),
s(
x2))) by
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
(92) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(0), x1, s(z0)) → c2(COND1(false, 0, x1, s(z0)), GR(s(0), s(z0)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(94) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, p(s(s(z0))), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
s(
z0)),
x1,
s(
0)) →
c2(
COND1(
true,
p(
s(
s(
z0))),
x1,
s(
0)),
GR(
s(
s(
z0)),
s(
0))) by
COND2(false, s(s(x0)), x1, s(0)) → c2(COND1(true, s(x0), x1, s(0)), GR(s(s(x0)), s(0)))
(96) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(0)) → c2(COND1(true, s(x0), x1, s(0)), GR(s(s(x0)), s(0)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(97) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [1] + x2
POL(COND2(x1, x2, x3, x4)) = [1] + x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0
(98) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(99) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(x1)), s(0)) → c1(COND2(true, x0, s(x1), s(0)), GR(s(s(x1)), s(0)))
(100) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), p(s(s(z0))), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c2, c1
(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
s(
s(
z0)),
x1,
s(
s(
z1))) →
c2(
COND1(
gr(
z0,
z1),
p(
s(
s(
z0))),
x1,
s(
s(
z1))),
GR(
s(
s(
z0)),
s(
s(
z1)))) by
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(COND1(false, p(s(s(0))), x1, s(s(z0))), GR(s(s(0)), s(s(z0))))
(102) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(COND1(false, p(s(s(0))), x1, s(s(z0))), GR(s(s(0)), s(s(z0))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(COND1(false, p(s(s(0))), x1, s(s(z0))), GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(104) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(105) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
We considered the (Usable) Rules:none
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = [4]
POL(COND1(x1, x2, x3, x4)) = [4]
POL(COND2(x1, x2, x3, x4)) = [4]
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = [3]
POL(gr(x1, x2)) = [4] + [5]x1 + [3]x2
POL(p(x1)) = [2] + [4]x1
POL(s(x1)) = [5] + x1
POL(true) = [2]
(106) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(107) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x2
POL(COND2(x1, x2, x3, x4)) = x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(108) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(s(z1)))) → c(COND2(gr(z0, z1), x0, s(s(s(z0))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND1(
true,
x0,
s(
s(
s(
z0))),
s(
s(
s(
z1)))) →
c(
COND2(
gr(
z0,
z1),
x0,
s(
s(
s(
z0))),
s(
s(
s(
z1)))),
GR(
s(
s(
s(
z0))),
s(
s(
s(
z1))))) by
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
(110) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(111) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
We considered the (Usable) Rules:none
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = [4]
POL(COND1(x1, x2, x3, x4)) = [2]
POL(COND2(x1, x2, x3, x4)) = [2]
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = [5]
POL(gr(x1, x2)) = [4] + [4]x1 + [3]x2
POL(p(x1)) = [3] + x1
POL(s(x1)) = [2]
POL(true) = [3]
(112) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
z0)),
s(
s(
z1))) →
c1(
COND2(
gr(
z0,
z1),
x0,
s(
z0),
s(
s(
z1))),
GR(
s(
s(
z0)),
s(
s(
z1)))) by
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
(114) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
x1)),
s(
s(
x2))) →
c1(
COND2(
gr(
x1,
x2),
x0,
s(
x1),
s(
s(
x2))),
GR(
s(
s(
x1)),
s(
s(
x2)))) by
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
(116) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, p(s(s(s(z0)))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
s(
z0))),
s(
s(
0))) →
c1(
COND2(
true,
x0,
p(
s(
s(
s(
z0)))),
s(
s(
0))),
GR(
s(
s(
s(
z0))),
s(
s(
0)))) by
COND2(true, x0, s(s(s(x1))), s(s(0))) → c1(COND2(true, x0, s(s(x1)), s(s(0))), GR(s(s(s(x1))), s(s(0))))
(118) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(x1))), s(s(0))) → c1(COND2(true, x0, s(s(x1)), s(s(0))), GR(s(s(s(x1))), s(s(0))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(119) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [2]x3
POL(COND2(x1, x2, x3, x4)) = [2]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(120) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(z0)))), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
s(
z0))),
s(
s(
s(
z1)))) →
c1(
COND2(
gr(
z0,
z1),
x0,
p(
s(
s(
s(
z0)))),
s(
s(
s(
z1)))),
GR(
s(
s(
s(
z0))),
s(
s(
s(
z1))))) by
COND2(true, x0, s(s(s(x1))), s(s(s(x2)))) → c1(COND2(gr(x1, x2), x0, s(s(x1)), s(s(s(x2)))), GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
(122) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(x1))), s(s(s(x2)))) → c1(COND2(gr(x1, x2), x0, s(s(x1)), s(s(s(x2)))), GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(123) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [2]x3
POL(COND2(x1, x2, x3, x4)) = [2]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = [5]
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0
(124) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, p(s(s(0))), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
0)),
s(
s(
z0))) →
c1(
COND2(
false,
x0,
p(
s(
s(
0))),
s(
s(
z0))),
GR(
s(
s(
0)),
s(
s(
z0)))) by
COND2(true, x0, s(s(0)), s(s(x1))) → c1(COND2(false, x0, s(0), s(s(x1))), GR(s(s(0)), s(s(x1))))
(126) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(0)), s(s(x1))) → c1(COND2(false, x0, s(0), s(s(x1))), GR(s(s(0)), s(s(x1))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(127) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x3
POL(COND2(x1, x2, x3, x4)) = [4]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 0
(128) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
S tuples:
GR(s(z0), s(z1)) → c5(GR(z0, z1))
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
GR, COND1, COND2
Compound Symbols:
c5, c, c, c1, c2, c1, c2
(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
GR(
s(
z0),
s(
z1)) →
c5(
GR(
z0,
z1)) by
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
(130) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
S tuples:
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), s(z1)) → c1(COND2(gr(z0, z1), x0, z0, s(z1)), GR(s(z0), s(z1)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, s(z1)) → c2(COND1(gr(z0, z1), z0, x1, s(z1)), GR(s(z0), s(z1)))
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(s(z1))) → c1(COND2(gr(z0, z1), x0, s(z0), s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c1, c2, c1, c2, c5
(131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND2(false, s(0), x1, s(z0)) → c2(GR(s(0), s(z0)))
(132) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
S tuples:
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)), GR(s(0), s(z0)))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)), GR(s(0), s(z0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)), GR(s(s(z0)), s(0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)), GR(s(s(z0)), s(0)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c1, c2, c1, c2, c5
(133) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 9 trailing tuple parts
(134) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
S tuples:
COND1(true, x0, 0, z0) → c(COND2(false, x0, 0, z0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(135) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND1(
true,
x0,
0,
z0) →
c(
COND2(
false,
x0,
0,
z0)) by
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
(136) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
S tuples:
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(137) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND2(false, s(s(x0)), x1, s(0)) → c2(COND1(true, s(x0), x1, s(0)))
(138) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
S tuples:
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
K tuples:
COND1(true, x0, s(x1), s(x2)) → c(GR(s(x1), s(x2)))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
COND1(
true,
x0,
s(
x1),
s(
x2)) →
c(
GR(
s(
x1),
s(
x2))) by
COND1(true, z0, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
(140) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
S tuples:
COND2(true, x0, 0, z0) → c1(COND2(false, x0, 0, z0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(141) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND2(
true,
x0,
0,
z0) →
c1(
COND2(
false,
x0,
0,
z0)) by
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
(142) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(143) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
(144) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c1(COND2(true, x0, p(s(s(s(s(z0))))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
x0,
s(
s(
s(
s(
z0)))),
s(
s(
s(
0)))) →
c1(
COND2(
true,
x0,
p(
s(
s(
s(
s(
z0))))),
s(
s(
s(
0)))),
GR(
s(
s(
s(
s(
z0)))),
s(
s(
s(
0))))) by
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
(146) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(147) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND2(true, x0, 0, s(s(s(0)))) → c1(COND2(false, x0, 0, s(s(s(0)))))
(148) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(149) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
We considered the (Usable) Rules:none
And the Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [1]
POL(COND2(x1, x2, x3, x4)) = [1]
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = [3]
POL(gr(x1, x2)) = [2] + [2]x1 + [5]x2
POL(p(x1)) = [3] + [3]x1
POL(s(x1)) = [2]
POL(true) = 0
(150) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(151) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x3
POL(COND2(x1, x2, x3, x4)) = [4]x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [2] + x1
POL(true) = 0
(152) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(153) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND2(
false,
s(
z0),
x1,
0) →
c2(
COND1(
true,
z0,
x1,
0)) by
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
(154) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND2(false, s(z0), x1, 0) → c2(COND1(true, z0, x1, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c2, c, c2, c1, c5
(155) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND2(
false,
s(
z0),
x1,
0) →
c2(
COND1(
true,
z0,
x1,
0)) by
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
(156) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c1, c, c2, c2, c1, c5
(157) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND1(true, x0, s(z0), 0) → c(COND2(true, x0, s(z0), 0))
(158) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND2, COND1, GR
Compound Symbols:
c1, c, c, c2, c2, c1, c5
(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
COND2(
true,
x0,
s(
z0),
0) →
c1(
COND2(
true,
x0,
z0,
0)) by
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
(160) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND2(true, x0, s(z0), 0) → c1(COND2(true, x0, z0, 0))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND2, COND1, GR
Compound Symbols:
c1, c, c, c2, c2, c1, c5
(161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
COND2(
true,
x0,
s(
z0),
0) →
c1(
COND2(
true,
x0,
z0,
0)) by
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
(162) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(0)), s(s(z0))) → c(COND2(false, x0, s(s(0)), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
K tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c1, c2, c2, c1, c5
(163) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND1(
true,
x0,
s(
s(
0)),
s(
s(
z0))) →
c(
COND2(
false,
x0,
s(
s(
0)),
s(
s(
z0))),
GR(
s(
s(
0)),
s(
s(
z0)))) by
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
(164) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c1, c2, c2, c1, c5
(165) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND1(
true,
x0,
s(
s(
x1)),
s(
s(
x2))) →
c(
GR(
s(
s(
x1)),
s(
s(
x2)))) by
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
(166) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(x1), s(x2)) → c1(GR(s(x1), s(x2)))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c1, c2, c2, c1, c5
(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
COND2(
true,
x0,
s(
x1),
s(
x2)) →
c1(
GR(
s(
x1),
s(
x2))) by
COND2(true, z0, s(s(y0)), s(s(y1))) → c1(GR(s(s(y0)), s(s(y1))))
(168) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c2, c2, c1, c1, c5
(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
COND2(
false,
s(
x0),
x1,
s(
x2)) →
c2(
GR(
s(
x0),
s(
x2))) by
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
(170) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c2, c2, c1, c1, c5
(171) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(false, s(s(s(z0))), x1, s(s(0))) → c2(COND1(true, p(s(s(s(z0)))), x1, s(s(0))), GR(s(s(s(z0))), s(s(0)))) by COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
(172) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c2, c2, c1, c1, c5
(173) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x2
POL(COND2(x1, x2, x3, x4)) = x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(174) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
S tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c2, c2, c1, c1, c5
(175) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(x1))), s(s(0))) → c1(COND2(true, x0, s(s(x1)), s(s(0))), GR(s(s(s(x1))), s(s(0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(x0)), x1, s(s(x2))) → c2(COND1(gr(x0, x2), s(x0), x1, s(s(x2))), GR(s(s(x0)), s(s(x2))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
(176) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
S tuples:
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, x0, s(s(s(z0))), s(s(0))) → c(COND2(true, x0, s(s(s(z0))), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c, c2, c2, c1, c1, c5
(177) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND1(
true,
x0,
s(
s(
s(
z0))),
s(
s(
0))) →
c(
COND2(
true,
x0,
s(
s(
s(
z0))),
s(
s(
0))),
GR(
s(
s(
s(
z0))),
s(
s(
0)))) by
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
(178) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
S tuples:
COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c2, c2, c, c1, c1, c5
(179) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(false, s(s(s(z0))), x1, s(s(s(z1)))) → c2(COND1(gr(z0, z1), p(s(s(s(z0)))), x1, s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1))))) by COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
(180) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
S tuples:
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c2, c2, c, c1, c1, c5
(181) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
We considered the (Usable) Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
And the Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [4]x1 + [4]x2 + x3
POL(COND2(x1, x2, x3, x4)) = [2]x1 + [4]x2 + x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [4]
POL(p(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = [4]
(182) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
S tuples:
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c2, c2, c, c1, c1, c5
(183) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
We considered the (Usable) Rules:
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
And the Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [3] + x1 + [4]x2
POL(COND2(x1, x2, x3, x4)) = [4]x2
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [3]
POL(p(x1)) = [1]
POL(s(x1)) = [4] + x1
POL(true) = 0
(184) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(x0), x1, s(x2)) → c2(GR(s(x0), s(x2)))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
S tuples:
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c2, c2, c, c1, c1, c5
(185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use forward instantiation to replace
COND2(
false,
s(
x0),
x1,
s(
x2)) →
c2(
GR(
s(
x0),
s(
x2))) by
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
(186) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
S tuples:
COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c2, c2, c, c1, c1, c5
(187) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c1(COND2(gr(z0, z1), x0, p(s(s(s(s(z0))))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1)))))) by COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
(188) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND1(true, x0, s(s(x1)), s(s(x2))) → c(GR(s(s(x1)), s(s(x2))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
S tuples:
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND1, COND2, GR
Compound Symbols:
c, c2, c2, c, c1, c1, c5
(189) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND2(true, x0, 0, s(s(s(s(x2))))) → c1(COND2(false, x0, 0, s(s(s(s(x2))))))
(190) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, z0, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
S tuples:
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND2, COND1, GR
Compound Symbols:
c2, c2, c, c, c1, c1, c5
(191) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
We considered the (Usable) Rules:
p(s(z0)) → z0
And the Tuples:
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, z0, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = x3
POL(COND2(x1, x2, x3, x4)) = x3
POL(GR(x1, x2)) = 0
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = 0
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 0
(192) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, z0, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
S tuples:
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND2, COND1, GR
Compound Symbols:
c2, c2, c, c, c1, c1, c5
(193) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
We considered the (Usable) Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
And the Tuples:
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, z0, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(COND1(x1, x2, x3, x4)) = [2] + [2]x1 + [3]x2 + [2]x4 + [2]x3·x4 + [3]x2·x4 + x12 + [2]x1·x3
POL(COND2(x1, x2, x3, x4)) = [3] + [2]x1 + [3]x2 + [2]x3 + [2]x3·x4 + [3]x2·x4 + x1·x4
POL(GR(x1, x2)) = x2
POL(c(x1)) = x1
POL(c(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c1(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c2(x1, x2)) = x1 + x2
POL(c5(x1)) = x1
POL(false) = 0
POL(gr(x1, x2)) = [1]
POL(p(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = [1]
(194) Obligation:
Complexity Dependency Tuples Problem
Rules:
p(s(z0)) → z0
gr(s(z0), 0) → true
gr(s(z0), s(z1)) → gr(z0, z1)
gr(0, z0) → false
Tuples:
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND2(true, x0, s(s(x1)), s(s(x2))) → c1(GR(s(s(x1)), s(s(x2))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND1(true, z0, s(s(y0)), s(s(y1))) → c(GR(s(s(y0)), s(s(y1))))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND2(true, z0, s(s(s(y1))), 0) → c1(COND2(true, z0, s(s(y1)), 0))
COND2(true, z0, s(s(0)), 0) → c1(COND2(true, z0, s(0), 0))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(y0)), z1, s(s(y1))) → c2(GR(s(s(y0)), s(s(y1))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
S tuples:none
K tuples:
COND2(false, s(s(0)), x1, s(s(z0))) → c2(GR(s(s(0)), s(s(z0))))
COND2(false, s(s(z0)), x1, s(s(z1))) → c2(COND1(gr(z0, z1), s(z0), x1, s(s(z1))), GR(s(s(z0)), s(s(z1))))
COND1(true, x0, s(s(s(x1))), s(s(s(x2)))) → c(GR(s(s(s(x1))), s(s(s(x2)))))
COND2(true, x0, s(s(s(z0))), s(s(0))) → c1(COND2(true, x0, s(s(z0)), s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND2(true, x0, s(s(s(z0))), s(s(s(z1)))) → c1(COND2(gr(z0, z1), x0, s(s(z0)), s(s(s(z1)))), GR(s(s(s(z0))), s(s(s(z1)))))
COND2(true, x0, s(s(0)), s(s(z0))) → c1(COND2(false, x0, s(0), s(s(z0))), GR(s(s(0)), s(s(z0))))
COND1(true, x0, s(s(z0)), s(0)) → c(COND2(true, x0, s(s(z0)), s(0)))
COND2(true, x0, s(s(z0)), s(0)) → c1(COND2(true, x0, s(z0), s(0)))
COND2(true, x0, s(0), s(z0)) → c1(COND2(false, x0, 0, s(z0)))
COND2(false, s(s(z0)), x1, s(0)) → c2(COND1(true, s(z0), x1, s(0)))
COND1(true, x0, 0, 0) → c(COND2(false, x0, 0, 0))
COND1(true, s(x0), 0, s(0)) → c(COND2(false, s(x0), 0, s(0)))
COND2(true, x0, 0, 0) → c1(COND2(false, x0, 0, 0))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(true, x0, s(s(s(s(x1)))), s(s(s(0)))) → c1(COND2(true, x0, s(s(s(x1))), s(s(s(0)))), GR(s(s(s(s(x1)))), s(s(s(0)))))
COND2(false, s(z0), 0, 0) → c2(COND1(true, z0, 0, 0))
COND2(true, z0, s(s(y1)), 0) → c1(COND2(true, z0, s(y1), 0))
COND2(true, z0, s(0), 0) → c1(COND2(true, z0, 0, 0))
COND1(true, s(x0), s(s(z1)), s(s(x2))) → c(GR(s(s(z1)), s(s(x2))))
COND1(true, y0, s(s(z1)), s(s(0))) → c(GR(s(s(z1)), s(s(0))))
COND1(true, y1, s(s(z1)), s(s(s(x2)))) → c(GR(s(s(z1)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(0))) → c2(COND1(true, s(s(z0)), z1, s(s(0))), GR(s(s(s(z0))), s(s(0))))
COND1(true, y0, 0, s(s(0))) → c(COND2(false, y0, 0, s(s(0))))
COND1(true, y0, s(s(0)), s(s(0))) → c(COND2(false, y0, s(s(0)), s(s(0))), GR(s(s(0)), s(s(0))))
COND1(true, s(x0), s(s(s(z1))), s(s(0))) → c(COND2(true, s(x0), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, s(s(x0)), s(s(s(z1))), s(s(0))) → c(COND2(true, s(s(x0)), s(s(s(z1))), s(s(0))), GR(s(s(s(z1))), s(s(0))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(0)))) → c(COND2(true, x0, s(s(s(s(z0)))), s(s(s(0)))), GR(s(s(s(s(z0)))), s(s(s(0)))))
COND1(true, x0, s(s(s(s(z0)))), s(s(s(s(z1))))) → c(COND2(gr(z0, z1), x0, s(s(s(s(z0)))), s(s(s(s(z1))))), GR(s(s(s(s(z0)))), s(s(s(s(z1))))))
COND1(true, x0, s(s(s(0))), s(s(s(z0)))) → c(COND2(false, x0, s(s(s(0))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND2(true, x0, s(s(s(0))), s(s(s(z0)))) → c1(COND2(false, x0, p(s(s(s(0)))), s(s(s(z0)))), GR(s(s(s(0))), s(s(s(z0)))))
COND1(true, x0, s(0), s(z0)) → c(COND2(false, x0, s(0), s(z0)))
COND1(true, s(x0), 0, s(s(x2))) → c(COND2(false, s(x0), 0, s(s(x2))))
COND1(true, y1, 0, s(s(s(x2)))) → c(COND2(false, y1, 0, s(s(s(x2)))))
COND1(true, s(x0), s(s(0)), s(s(x2))) → c(COND2(false, s(x0), s(s(0)), s(s(x2))), GR(s(s(0)), s(s(x2))))
COND1(true, y1, s(s(0)), s(s(s(x2)))) → c(COND2(false, y1, s(s(0)), s(s(s(x2)))), GR(s(s(0)), s(s(s(x2)))))
COND2(false, s(s(s(z0))), z1, s(s(s(z2)))) → c2(COND1(gr(z0, z2), s(s(z0)), z1, s(s(s(z2)))), GR(s(s(s(z0))), s(s(s(z2)))))
COND2(true, z0, s(s(s(s(z1)))), s(s(s(s(z2))))) → c1(COND2(gr(z1, z2), z0, s(s(s(z1))), s(s(s(s(z2))))), GR(s(s(s(s(z1)))), s(s(s(s(z2))))))
GR(s(s(y0)), s(s(y1))) → c5(GR(s(y0), s(y1)))
Defined Rule Symbols:
p, gr
Defined Pair Symbols:
COND2, COND1, GR
Compound Symbols:
c2, c2, c, c, c1, c1, c5
(195) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(196) BOUNDS(O(1), O(1))