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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
cond1(true, x, y) → cond2(gr(x, 0), x, y)
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cond2(true, x, y) → cond1(gr(add(x, y), 0), p(x), y)
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cond2(false, x, y) → cond3(gr(y, 0), x, y)
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cond3(true, x, y) → cond1(gr(add(x, y), 0), x, p(y))
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cond3(false, x, y) → cond1(gr(add(x, y), 0), x, y)
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gr(0, x) → false
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gr(s(x), 0) → true
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gr(s(x), s(y)) → gr(x, y)
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add(0, x) → x
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add(s(x), y) → s(add(x, y))
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p(0) → 0
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p(s(x)) → x
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
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(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1), GR(z0, 0))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), GR(add(z0, z1), 0), ADD(z0, z1), P(z0))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1), GR(z1, 0))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), GR(add(z0, z1), 0), ADD(z0, z1), P(z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), GR(add(z0, z1), 0), ADD(z0, z1))
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GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
S tuples:
COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1), GR(z0, 0))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), GR(add(z0, z1), 0), ADD(z0, z1), P(z0))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1), GR(z1, 0))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), GR(add(z0, z1), 0), ADD(z0, z1), P(z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), GR(add(z0, z1), 0), ADD(z0, z1))
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GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
K tuples:none
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
COND1, COND2, COND3, GR, ADD
Compound Symbols:
c, c1, c2, c3, c4, c7, c9
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(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing tuple parts
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
S tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
K tuples:none
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c2, c3, c4
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(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
GR(s(z0), s(z1)) → c7(GR(z0, z1))
We considered the (Usable) Rules:
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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gr(0, z0) → false
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gr(s(z0), 0) → true
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p(0) → 0
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p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = [2]
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POL(ADD(x1, x2)) = 0
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POL(COND1(x1, x2, x3)) = 0
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POL(COND2(x1, x2, x3)) = 0
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POL(COND3(x1, x2, x3)) = 0
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POL(GR(x1, x2)) = [2]x2
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POL(add(x1, x2)) = [2] + [4]x1 + x2
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POL(c(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c3(x1, x2)) = x1 + x2
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POL(c4(x1, x2)) = x1 + x2
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POL(c7(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = 0
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POL(gr(x1, x2)) = 0
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POL(p(x1)) = [2] + x1
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POL(s(x1)) = [2] + x1
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POL(true) = 0
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, z0, z1) → c(COND2(gr(z0, 0), z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c2, c3, c4
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(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND1(
true,
z0,
z1) →
c(
COND2(
gr(
z0,
0),
z0,
z1)) by
COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND2(true, z0, z1) → c1(COND1(gr(add(z0, z1), 0), p(z0), z1), ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND2, COND3, COND1
Compound Symbols:
c7, c9, c1, c2, c3, c4, c
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(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
z0,
z1) →
c1(
COND1(
gr(
add(
z0,
z1),
0),
p(
z0),
z1),
ADD(
z0,
z1)) by
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1), ADD(0, x1))
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COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0), ADD(0, z0))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
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(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1), ADD(0, x1))
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COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0), ADD(0, z0))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
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COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
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COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1), ADD(0, x1))
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COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0), ADD(0, z0))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND2, COND3, COND1
Compound Symbols:
c7, c9, c2, c3, c4, c, c1
701.97/218.04
701.97/218.04
(11) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
701.97/218.04
701.97/218.04
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
701.97/218.04
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
701.97/218.04
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
701.97/218.04
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
701.97/218.04
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
701.97/218.04
gr(0, z0) → false
701.97/218.04
gr(s(z0), 0) → true
701.97/218.04
gr(s(z0), s(z1)) → gr(z0, z1)
701.97/218.04
add(0, z0) → z0
701.97/218.04
add(s(z0), z1) → s(add(z0, z1))
701.97/218.04
p(0) → 0
701.97/218.04
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND2, COND3, COND1
Compound Symbols:
c7, c9, c2, c3, c4, c, c1, c1
701.97/218.04
701.97/218.04
(13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
701.97/218.04
701.97/218.04
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
701.97/218.04
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
701.97/218.04
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
701.97/218.04
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
701.97/218.04
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
701.97/218.04
gr(0, z0) → false
701.97/218.04
gr(s(z0), 0) → true
701.97/218.04
gr(s(z0), s(z1)) → gr(z0, z1)
701.97/218.04
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
701.97/218.04
p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND2, COND3, COND1
Compound Symbols:
c7, c9, c2, c3, c4, c, c1, c1
701.97/218.04
701.97/218.04
(15) CdtPolyRedPairProof (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, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
We considered the (Usable) Rules:
gr(0, z0) → false
701.97/218.04
gr(s(z0), 0) → true
701.97/218.04
p(0) → 0
701.97/218.04
p(s(z0)) → z0
701.97/218.04
add(0, z0) → z0
701.97/218.04
add(s(z0), z1) → s(add(z0, z1))
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
701.97/218.04
POL(0) = 0
701.97/218.04
POL(ADD(x1, x2)) = 0
701.97/218.04
POL(COND1(x1, x2, x3)) = [4]x2
701.97/218.04
POL(COND2(x1, x2, x3)) = [4]x2
701.97/218.04
POL(COND3(x1, x2, x3)) = [4]x2
701.97/218.04
POL(GR(x1, x2)) = [5]x1 + [5]x2
701.97/218.04
POL(add(x1, x2)) = 0
701.97/218.04
POL(c(x1)) = x1
701.97/218.04
POL(c1(x1)) = x1
701.97/218.04
POL(c1(x1, x2)) = x1 + x2
701.97/218.04
POL(c2(x1)) = x1
701.97/218.04
POL(c3(x1, x2)) = x1 + x2
701.97/218.04
POL(c4(x1, x2)) = x1 + x2
701.97/218.04
POL(c7(x1)) = x1
701.97/218.04
POL(c9(x1)) = x1
701.97/218.04
POL(false) = 0
701.97/218.04
POL(gr(x1, x2)) = 0
701.97/218.04
POL(p(x1)) = x1
701.97/218.04
POL(s(x1)) = [4] + x1
701.97/218.04
POL(true) = 0
701.97/218.04
701.97/218.04
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
701.97/218.04
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
701.97/218.04
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
701.97/218.04
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
701.97/218.04
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
701.97/218.04
gr(0, z0) → false
701.97/218.04
gr(s(z0), 0) → true
701.97/218.04
gr(s(z0), s(z1)) → gr(z0, z1)
701.97/218.04
add(0, z0) → z0
701.97/218.04
add(s(z0), z1) → s(add(z0, z1))
701.97/218.04
p(0) → 0
701.97/218.04
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND2(false, z0, z1) → c2(COND3(gr(z1, 0), z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND2, COND3, COND1
Compound Symbols:
c7, c9, c2, c3, c4, c, c1, c1
701.97/218.04
701.97/218.04
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
false,
z0,
z1) →
c2(
COND3(
gr(
z1,
0),
z0,
z1)) by
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
701.97/218.04
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
701.97/218.04
701.97/218.04
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
701.97/218.04
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
701.97/218.04
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
701.97/218.04
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
701.97/218.04
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
701.97/218.04
gr(0, z0) → false
701.97/218.04
gr(s(z0), 0) → true
701.97/218.04
gr(s(z0), s(z1)) → gr(z0, z1)
701.97/218.04
add(0, z0) → z0
701.97/218.04
add(s(z0), z1) → s(add(z0, z1))
701.97/218.04
p(0) → 0
701.97/218.04
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
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COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
701.97/218.04
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
701.97/218.04
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND3(true, z0, z1) → c3(COND1(gr(add(z0, z1), 0), z0, p(z1)), ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
701.97/218.04
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
701.97/218.04
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c3, c4, c, c1, c1, c2
701.97/218.04
701.97/218.04
(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
true,
z0,
z1) →
c3(
COND1(
gr(
add(
z0,
z1),
0),
z0,
p(
z1)),
ADD(
z0,
z1)) by
COND3(true, x0, 0) → c3(COND1(gr(add(x0, 0), 0), x0, 0), ADD(x0, 0))
701.97/218.04
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
701.97/218.04
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)), ADD(0, z0))
701.97/218.04
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
701.97/218.04
701.97/218.04
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
701.97/218.04
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
701.97/218.04
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
701.97/218.04
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
701.97/218.04
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
701.97/218.04
gr(0, z0) → false
701.97/218.04
gr(s(z0), 0) → true
701.97/218.04
gr(s(z0), s(z1)) → gr(z0, z1)
701.97/218.04
add(0, z0) → z0
701.97/218.04
add(s(z0), z1) → s(add(z0, z1))
701.97/218.04
p(0) → 0
701.97/218.04
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
701.97/218.04
ADD(s(z0), z1) → c9(ADD(z0, z1))
701.97/218.04
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
701.97/218.04
COND1(true, 0, x1) → c(COND2(false, 0, x1))
701.97/218.04
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
701.97/218.04
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
701.97/218.04
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
701.97/218.04
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, 0) → c3(COND1(gr(add(x0, 0), 0), x0, 0), ADD(x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)), ADD(0, z0))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, 0) → c3(COND1(gr(add(x0, 0), 0), x0, 0), ADD(x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)), ADD(0, z0))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3
702.70/218.21
702.70/218.21
(21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.21
702.70/218.21
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, 0) → c3(COND1(gr(add(x0, 0), 0), x0, 0), ADD(x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, 0) → c3(COND1(gr(add(x0, 0), 0), x0, 0), ADD(x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3, c3
702.70/218.21
702.70/218.21
(23) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
702.70/218.21
702.70/218.21
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, 0) → c5(ADD(x0, 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, 0) → c5(ADD(x0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3, c3, c5
702.70/218.21
702.70/218.21
(25) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND3(true, x0, 0) → c5(ADD(x0, 0))
702.70/218.21
702.70/218.21
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3, c3, c5
702.70/218.21
702.70/218.21
(27) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
702.70/218.21
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3, c3, c5
702.70/218.21
702.70/218.21
(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
We considered the (Usable) Rules:
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.21
POL(0) = 0
702.70/218.21
POL(ADD(x1, x2)) = 0
702.70/218.21
POL(COND1(x1, x2, x3)) = [4]x3
702.70/218.21
POL(COND2(x1, x2, x3)) = [4]x3
702.70/218.21
POL(COND3(x1, x2, x3)) = [4]x3
702.70/218.21
POL(GR(x1, x2)) = [5]x1 + [5]x2
702.70/218.21
POL(add(x1, x2)) = [2]x1
702.70/218.21
POL(c(x1)) = x1
702.70/218.21
POL(c1(x1)) = x1
702.70/218.21
POL(c1(x1, x2)) = x1 + x2
702.70/218.21
POL(c2(x1)) = x1
702.70/218.21
POL(c3(x1)) = x1
702.70/218.21
POL(c3(x1, x2)) = x1 + x2
702.70/218.21
POL(c4(x1, x2)) = x1 + x2
702.70/218.21
POL(c5(x1)) = x1
702.70/218.21
POL(c7(x1)) = x1
702.70/218.21
POL(c9(x1)) = x1
702.70/218.21
POL(false) = 0
702.70/218.21
POL(gr(x1, x2)) = 0
702.70/218.21
POL(p(x1)) = x1
702.70/218.21
POL(s(x1)) = [4] + x1
702.70/218.21
POL(true) = 0
702.70/218.21
702.70/218.21
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3, c3, c5
702.70/218.21
702.70/218.21
(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
We considered the (Usable) Rules:
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.21
POL(0) = 0
702.70/218.21
POL(ADD(x1, x2)) = 0
702.70/218.21
POL(COND1(x1, x2, x3)) = 0
702.70/218.21
POL(COND2(x1, x2, x3)) = x1·x2
702.70/218.21
POL(COND3(x1, x2, x3)) = [2]x2
702.70/218.21
POL(GR(x1, x2)) = 0
702.70/218.21
POL(add(x1, x2)) = 0
702.70/218.21
POL(c(x1)) = x1
702.70/218.21
POL(c1(x1)) = x1
702.70/218.21
POL(c1(x1, x2)) = x1 + x2
702.70/218.21
POL(c2(x1)) = x1
702.70/218.21
POL(c3(x1)) = x1
702.70/218.21
POL(c3(x1, x2)) = x1 + x2
702.70/218.21
POL(c4(x1, x2)) = x1 + x2
702.70/218.21
POL(c5(x1)) = x1
702.70/218.21
POL(c7(x1)) = x1
702.70/218.21
POL(c9(x1)) = x1
702.70/218.21
POL(false) = [2]
702.70/218.21
POL(gr(x1, x2)) = 0
702.70/218.21
POL(p(x1)) = [2]
702.70/218.21
POL(s(x1)) = [1]
702.70/218.21
POL(true) = 0
702.70/218.21
702.70/218.21
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND3(false, z0, z1) → c4(COND1(gr(add(z0, z1), 0), z0, z1), ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND3, COND1, COND2
Compound Symbols:
c7, c9, c4, c, c1, c1, c2, c3, c3, c5
702.70/218.21
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(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
false,
z0,
z1) →
c4(
COND1(
gr(
add(
z0,
z1),
0),
z0,
z1),
ADD(
z0,
z1)) by
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0), ADD(0, z0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
702.70/218.21
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0), ADD(0, z0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0), ADD(0, z0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c1, c2, c3, c3, c5, c4
702.70/218.21
702.70/218.21
(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.21
702.70/218.21
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
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gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
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COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
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COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c1, c2, c3, c3, c5, c4, c4
702.70/218.21
702.70/218.21
(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
We considered the (Usable) Rules:
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.21
POL(0) = 0
702.70/218.21
POL(ADD(x1, x2)) = 0
702.70/218.21
POL(COND1(x1, x2, x3)) = 0
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POL(COND2(x1, x2, x3)) = [2]x1·x2
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POL(COND3(x1, x2, x3)) = [2]x2
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POL(GR(x1, x2)) = 0
702.70/218.21
POL(add(x1, x2)) = 0
702.70/218.21
POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
702.70/218.21
POL(c2(x1)) = x1
702.70/218.21
POL(c3(x1)) = x1
702.70/218.21
POL(c3(x1, x2)) = x1 + x2
702.70/218.21
POL(c4(x1)) = x1
702.70/218.21
POL(c4(x1, x2)) = x1 + x2
702.70/218.21
POL(c5(x1)) = x1
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POL(c7(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = [2]
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POL(gr(x1, x2)) = 0
702.70/218.21
POL(p(x1)) = 0
702.70/218.21
POL(s(x1)) = [2]
702.70/218.21
POL(true) = 0
702.70/218.21
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(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
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COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
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COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
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COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c1, c2, c3, c3, c5, c4, c4
702.70/218.21
702.70/218.21
(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
s(
z0),
x1) →
c1(
COND1(
gr(
add(
s(
z0),
x1),
0),
z0,
x1),
ADD(
s(
z0),
x1)) by
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
702.70/218.21
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), p(s(z0)), z1), ADD(s(z0), z1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c1, c2, c3, c3, c5, c4, c4
702.70/218.21
702.70/218.21
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
s(
z0),
z1) →
c1(
COND1(
gr(
s(
add(
z0,
z1)),
0),
p(
s(
z0)),
z1),
ADD(
s(
z0),
z1)) by
COND2(true, s(z0), x1) → c1(COND1(gr(s(add(z0, x1)), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
702.70/218.21
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(s(add(z0, x1)), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c2, c3, c3, c5, c4, c4, c1
702.70/218.21
702.70/218.21
(43) CdtPolyRedPairProof (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, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
We considered the (Usable) Rules:
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(0, z0) → false
702.70/218.21
p(s(z0)) → z0
702.70/218.21
p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.21
POL(0) = 0
702.70/218.21
POL(ADD(x1, x2)) = 0
702.70/218.21
POL(COND1(x1, x2, x3)) = [4]x2
702.70/218.21
POL(COND2(x1, x2, x3)) = [4]x2
702.70/218.21
POL(COND3(x1, x2, x3)) = [4]x2
702.70/218.21
POL(GR(x1, x2)) = [5]x1 + [5]x2
702.70/218.21
POL(add(x1, x2)) = 0
702.70/218.21
POL(c(x1)) = x1
702.70/218.21
POL(c1(x1)) = x1
702.70/218.21
POL(c1(x1, x2)) = x1 + x2
702.70/218.21
POL(c2(x1)) = x1
702.70/218.21
POL(c3(x1)) = x1
702.70/218.21
POL(c3(x1, x2)) = x1 + x2
702.70/218.21
POL(c4(x1)) = x1
702.70/218.21
POL(c4(x1, x2)) = x1 + x2
702.70/218.21
POL(c5(x1)) = x1
702.70/218.21
POL(c7(x1)) = x1
702.70/218.21
POL(c9(x1)) = x1
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POL(false) = 0
702.70/218.21
POL(gr(x1, x2)) = 0
702.70/218.21
POL(p(x1)) = x1
702.70/218.21
POL(s(x1)) = [2] + x1
702.70/218.21
POL(true) = 0
702.70/218.21
702.70/218.21
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c2, c3, c3, c5, c4, c4, c1
702.70/218.21
702.70/218.21
(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
0,
x1) →
c1(
COND1(
gr(
add(
0,
x1),
0),
0,
x1)) by
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.21
702.70/218.21
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c2, c3, c3, c5, c4, c4, c1
702.70/218.21
702.70/218.21
(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND2(
true,
0,
z0) →
c1(
COND1(
gr(
z0,
0),
p(
0),
z0)) by
COND2(true, 0, x0) → c1(COND1(gr(x0, 0), 0, x0))
702.70/218.21
COND2(true, 0, 0) → c1(COND1(false, p(0), 0))
702.70/218.21
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.21
702.70/218.21
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, 0, 0) → c1(COND1(false, p(0), 0))
702.70/218.21
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c3, c5, c4, c4, c1, c1
702.70/218.21
702.70/218.21
(49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.21
702.70/218.21
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.21
COND2(true, 0, 0) → c1
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND2(true, 0, x1) → c1(COND1(gr(add(0, x1), 0), 0, x1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), p(0), z0))
702.70/218.21
COND2(true, s(z0), x1) → c1(COND1(gr(add(s(z0), x1), 0), z0, x1), ADD(s(z0), x1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c3, c5, c4, c4, c1, c1, c1
702.70/218.21
702.70/218.21
(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND2(true, 0, 0) → c1
702.70/218.21
702.70/218.21
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c3, c5, c4, c4, c1, c1
702.70/218.21
702.70/218.21
(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
true,
x0,
s(
z0)) →
c3(
COND1(
gr(
add(
x0,
s(
z0)),
0),
x0,
z0),
ADD(
x0,
s(
z0))) by
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1), ADD(0, s(x1)))
702.70/218.21
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.21
702.70/218.21
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.21
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1), ADD(0, s(x1)))
702.70/218.21
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c3, c5, c4, c4, c1, c1
702.70/218.21
702.70/218.21
(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.21
702.70/218.21
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.21
p(0) → 0
702.70/218.21
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.21
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.21
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.21
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.21
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.21
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.21
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.21
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.21
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.21
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.21
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.21
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.21
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.21
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.21
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.21
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.21
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.21
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c3, c5, c4, c4, c1, c1
702.70/218.21
702.70/218.21
(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
true,
s(
z0),
z1) →
c3(
COND1(
gr(
s(
add(
z0,
z1)),
0),
s(
z0),
p(
z1)),
ADD(
s(
z0),
z1)) by
COND3(true, s(x0), 0) → c3(COND1(gr(s(add(x0, 0)), 0), s(x0), 0), ADD(s(x0), 0))
702.70/218.21
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.21
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.21
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.21
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.21
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.21
702.70/218.21
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.21
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.21
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.21
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.21
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.21
gr(0, z0) → false
702.70/218.21
gr(s(z0), 0) → true
702.70/218.21
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.21
add(0, z0) → z0
702.70/218.21
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), 0) → c3(COND1(gr(s(add(x0, 0)), 0), s(x0), 0), ADD(s(x0), 0))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.22
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c5, c4, c4, c1, c1, c3
702.70/218.22
702.70/218.22
(59) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
702.70/218.22
702.70/218.22
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, s(x0), 0) → c6(ADD(s(x0), 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(true, x0, s(z0)) → c3(COND1(gr(add(x0, s(z0)), 0), x0, z0), ADD(x0, s(z0)))
702.70/218.22
COND3(true, s(z0), z1) → c3(COND1(gr(s(add(z0, z1)), 0), s(z0), p(z1)), ADD(s(z0), z1))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c5, c4, c4, c1, c1, c3, c6
702.70/218.22
702.70/218.22
(61) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND3(true, s(x0), 0) → c6(ADD(s(x0), 0))
702.70/218.22
702.70/218.22
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, 0, z0) → c3(COND1(gr(z0, 0), 0, p(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c3, c5, c4, c4, c1, c1, c3, c6
702.70/218.22
702.70/218.22
(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
true,
0,
z0) →
c3(
COND1(
gr(
z0,
0),
0,
p(
z0))) by
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(false, 0, p(0)))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
702.70/218.22
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(false, 0, p(0)))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(false, 0, p(0)))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c5, c4, c4, c1, c1, c3, c3, c6
702.70/218.22
702.70/218.22
(65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.22
702.70/218.22
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c3
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c3
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c5, c4, c4, c1, c1, c3, c3, c6, c3
702.70/218.22
702.70/218.22
(67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(true, 0, 0) → c3
702.70/218.22
702.70/218.22
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c5, c4, c4, c1, c1, c3, c3, c6
702.70/218.22
702.70/218.22
(69) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)
The following tuples could be moved from S to K by knowledge propagation:
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
702.70/218.22
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c5, c4, c4, c1, c1, c3, c3, c6
702.70/218.22
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(71) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
We considered the (Usable) Rules:
p(s(z0)) → z0
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p(0) → 0
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gr(0, z0) → false
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gr(s(z0), 0) → true
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(ADD(x1, x2)) = 0
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POL(COND1(x1, x2, x3)) = x3
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POL(COND2(x1, x2, x3)) = x3
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POL(COND3(x1, x2, x3)) = x3
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POL(GR(x1, x2)) = [5]x1 + [5]x2
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POL(add(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c3(x1)) = x1
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POL(c3(x1, x2)) = x1 + x2
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POL(c4(x1)) = x1
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POL(c4(x1, x2)) = x1 + x2
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POL(c5(x1)) = x1
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POL(c6(x1)) = x1
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POL(c7(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = 0
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POL(gr(x1, x2)) = [2]x1
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POL(p(x1)) = x1
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POL(s(x1)) = [1] + x1
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POL(true) = 0
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(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c5, c4, c4, c1, c1, c3, c3, c6
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(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
true,
x0,
0) →
c5(
COND1(
gr(
add(
x0,
0),
0),
x0,
0)) by
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
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(74) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c4, c4, c1, c1, c3, c3, c6, c5
702.70/218.22
702.70/218.22
(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
false,
s(
z0),
z1) →
c4(
COND1(
gr(
s(
add(
z0,
z1)),
0),
s(
z0),
z1),
ADD(
s(
z0),
z1)) by
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
702.70/218.22
(76) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND3(false, 0, z0) → c4(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c4, c1, c1, c3, c3, c6, c5, c4
702.70/218.22
702.70/218.22
(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
COND3(
false,
0,
z0) →
c4(
COND1(
gr(
z0,
0),
0,
z0)) by
COND3(false, 0, 0) → c4(COND1(false, 0, 0))
702.70/218.22
COND3(false, 0, s(z0)) → c4(COND1(true, 0, s(z0)))
702.70/218.22
702.70/218.22
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4(COND1(false, 0, 0))
702.70/218.22
COND3(false, 0, s(z0)) → c4(COND1(true, 0, s(z0)))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(false, 0, 0) → c4(COND1(false, 0, 0))
702.70/218.22
COND3(false, 0, s(z0)) → c4(COND1(true, 0, s(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
702.70/218.22
COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4
702.70/218.22
702.70/218.22
(79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.22
702.70/218.22
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, s(z0)) → c4(COND1(true, 0, s(z0)))
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COND3(false, 0, 0) → c4
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(false, 0, s(z0)) → c4(COND1(true, 0, s(z0)))
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COND3(false, 0, 0) → c4
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, x0, 0) → c5(COND1(gr(add(x0, 0), 0), x0, 0))
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COND3(false, s(z0), z1) → c4(COND1(gr(s(add(z0, z1)), 0), s(z0), z1), ADD(s(z0), z1))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
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(81) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND3(false, 0, s(z0)) → c4(COND1(true, 0, s(z0)))
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
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(82) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(false, 0, 0) → c4
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
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(83) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND3(false, 0, 0) → c4
We considered the (Usable) Rules:
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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gr(s(z0), 0) → true
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gr(0, z0) → false
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p(s(z0)) → z0
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p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = [4]
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POL(ADD(x1, x2)) = 0
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POL(COND1(x1, x2, x3)) = [4]
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POL(COND2(x1, x2, x3)) = x2
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POL(COND3(x1, x2, x3)) = x2
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POL(GR(x1, x2)) = 0
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POL(add(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c3(x1)) = x1
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POL(c3(x1, x2)) = x1 + x2
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POL(c4) = 0
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POL(c4(x1)) = x1
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POL(c4(x1, x2)) = x1 + x2
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POL(c5(x1)) = x1
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POL(c6(x1)) = x1
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POL(c7(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = [2]
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POL(gr(x1, x2)) = [2]x1
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POL(p(x1)) = 0
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POL(s(x1)) = [4]
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POL(true) = 0
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(84) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
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COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(85) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1)) by COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.22
702.70/218.22
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.22
702.70/218.22
(88) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(z0), x1) → c1(COND1(gr(s(add(z0, x1)), 0), z0, x1), ADD(s(z0), x1))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(89) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, s(z0), x1) → c1(COND1(gr(s(add(z0, x1)), 0), z0, x1), ADD(s(z0), x1)) by COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.22
702.70/218.22
(90) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(gr(s(add(z0, z1)), 0), z0, z1), ADD(s(z0), z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.22
702.70/218.22
(92) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(93) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, s(x0), x1) → c1(COND1(true, p(s(x0)), x1), ADD(s(x0), x1)) by COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.22
702.70/218.22
(94) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.22
702.70/218.22
(96) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(false, 0, 0) → c4
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.22
702.70/218.22
(97) CdtPolyRedPairProof (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, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
We considered the (Usable) Rules:
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(0, z0) → false
702.70/218.22
p(s(z0)) → z0
702.70/218.22
p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.22
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.22
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.22
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.22
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.22
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.22
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.22
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.22
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.22
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.22
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.22
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.22
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.22
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.22
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.22
COND3(false, 0, 0) → c4
702.70/218.22
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.22
POL(0) = [4]
702.70/218.22
POL(ADD(x1, x2)) = 0
702.70/218.22
POL(COND1(x1, x2, x3)) = [2]x2
702.70/218.22
POL(COND2(x1, x2, x3)) = [2]x2
702.70/218.22
POL(COND3(x1, x2, x3)) = [2]x2
702.70/218.22
POL(GR(x1, x2)) = [5]x1 + [5]x2
702.70/218.22
POL(add(x1, x2)) = 0
702.70/218.22
POL(c(x1)) = x1
702.70/218.22
POL(c1(x1)) = x1
702.70/218.22
POL(c1(x1, x2)) = x1 + x2
702.70/218.22
POL(c2(x1)) = x1
702.70/218.22
POL(c3(x1)) = x1
702.70/218.22
POL(c3(x1, x2)) = x1 + x2
702.70/218.22
POL(c4) = 0
702.70/218.22
POL(c4(x1)) = x1
702.70/218.22
POL(c4(x1, x2)) = x1 + x2
702.70/218.22
POL(c5(x1)) = x1
702.70/218.22
POL(c6(x1)) = x1
702.70/218.22
POL(c7(x1)) = x1
702.70/218.22
POL(c9(x1)) = x1
702.70/218.22
POL(false) = 0
702.70/218.22
POL(gr(x1, x2)) = 0
702.70/218.22
POL(p(x1)) = x1
702.70/218.22
POL(s(x1)) = [4] + x1
702.70/218.22
POL(true) = 0
702.70/218.22
702.70/218.22
(98) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.22
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.22
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.22
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.22
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.22
gr(0, z0) → false
702.70/218.22
gr(s(z0), 0) → true
702.70/218.22
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.22
add(0, z0) → z0
702.70/218.22
add(s(z0), z1) → s(add(z0, z1))
702.70/218.22
p(0) → 0
702.70/218.22
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.22
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.22
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.22
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.22
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.22
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.22
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.22
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.22
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.23
702.70/218.23
(99) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), p(s(0)), z0), ADD(s(0), z0)) by COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
702.70/218.23
(100) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.23
702.70/218.23
(101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.23
702.70/218.23
(102) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.23
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(103) CdtPolyRedPairProof (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, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
We considered the (Usable) Rules:
gr(s(z0), 0) → true
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gr(0, z0) → false
702.70/218.23
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(s(z0)) → z0
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p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(ADD(x1, x2)) = 0
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POL(COND1(x1, x2, x3)) = [4]x2
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POL(COND2(x1, x2, x3)) = [4]x2
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POL(COND3(x1, x2, x3)) = [4]x2
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POL(GR(x1, x2)) = [5]x1 + [5]x2
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POL(add(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c3(x1)) = x1
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POL(c3(x1, x2)) = x1 + x2
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POL(c4) = 0
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POL(c4(x1)) = x1
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POL(c4(x1, x2)) = x1 + x2
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POL(c5(x1)) = x1
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POL(c6(x1)) = x1
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POL(c7(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = 0
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POL(gr(x1, x2)) = 0
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POL(p(x1)) = x1
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POL(s(x1)) = [4] + x1
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POL(true) = 0
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(104) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c1, c3, c3, c6, c5, c4, c4, c4
702.70/218.23
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(105) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), p(s(s(z0))), z1), ADD(s(s(z0)), z1)) by COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
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(106) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
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(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
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(108) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
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(109) CdtPolyRedPairProof (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, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
We considered the (Usable) Rules:
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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gr(s(z0), 0) → true
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gr(0, z0) → false
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p(s(z0)) → z0
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p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(ADD(x1, x2)) = 0
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POL(COND1(x1, x2, x3)) = x2
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POL(COND2(x1, x2, x3)) = x2
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POL(COND3(x1, x2, x3)) = x2
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POL(GR(x1, x2)) = [3]x1 + [3]x2
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POL(add(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c3(x1)) = x1
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POL(c3(x1, x2)) = x1 + x2
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POL(c4) = 0
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POL(c4(x1)) = x1
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POL(c4(x1, x2)) = x1 + x2
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POL(c5(x1)) = x1
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POL(c6(x1)) = x1
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POL(c7(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = 0
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POL(gr(x1, x2)) = 0
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POL(p(x1)) = 0
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POL(s(x1)) = [2] + x1
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POL(true) = 0
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(110) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
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COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
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COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
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COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
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COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
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COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
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COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
S tuples:
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
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(111) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
ADD(s(z0), z1) → c9(ADD(z0, z1))
We considered the (Usable) Rules:
add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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gr(s(z0), 0) → true
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gr(0, z0) → false
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p(s(z0)) → z0
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p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(false, x0, 0) → c2(COND3(false, x0, 0))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
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COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
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COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
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COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.23
POL(0) = 0
702.70/218.23
POL(ADD(x1, x2)) = [2]x1
702.70/218.23
POL(COND1(x1, x2, x3)) = x22
702.70/218.23
POL(COND2(x1, x2, x3)) = [2]x1·x2 + x22
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POL(COND3(x1, x2, x3)) = [2]x2 + x22
702.70/218.23
POL(GR(x1, x2)) = x1 + x2
702.70/218.23
POL(add(x1, x2)) = 0
702.70/218.23
POL(c(x1)) = x1
702.70/218.23
POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
702.70/218.23
POL(c2(x1)) = x1
702.70/218.23
POL(c3(x1)) = x1
702.70/218.23
POL(c3(x1, x2)) = x1 + x2
702.70/218.23
POL(c4) = 0
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POL(c4(x1)) = x1
702.70/218.23
POL(c4(x1, x2)) = x1 + x2
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POL(c5(x1)) = x1
702.70/218.23
POL(c6(x1)) = x1
702.70/218.23
POL(c7(x1)) = x1
702.70/218.23
POL(c9(x1)) = x1
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POL(false) = [2]
702.70/218.23
POL(gr(x1, x2)) = 0
702.70/218.23
POL(p(x1)) = 0
702.70/218.23
POL(s(x1)) = [2] + x1
702.70/218.23
POL(true) = 0
702.70/218.23
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(112) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(113) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
We considered the (Usable) Rules:
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(0, z0) → false
702.70/218.23
p(s(z0)) → z0
702.70/218.23
p(0) → 0
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.23
POL(0) = 0
702.70/218.23
POL(ADD(x1, x2)) = 0
702.70/218.23
POL(COND1(x1, x2, x3)) = x2 + [2]x22
702.70/218.23
POL(COND2(x1, x2, x3)) = x1·x2 + [2]x22
702.70/218.23
POL(COND3(x1, x2, x3)) = [2]x2 + [2]x22
702.70/218.23
POL(GR(x1, x2)) = x1 + x2
702.70/218.23
POL(add(x1, x2)) = 0
702.70/218.23
POL(c(x1)) = x1
702.70/218.23
POL(c1(x1)) = x1
702.70/218.23
POL(c1(x1, x2)) = x1 + x2
702.70/218.23
POL(c2(x1)) = x1
702.70/218.23
POL(c3(x1)) = x1
702.70/218.23
POL(c3(x1, x2)) = x1 + x2
702.70/218.23
POL(c4) = 0
702.70/218.23
POL(c4(x1)) = x1
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POL(c4(x1, x2)) = x1 + x2
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POL(c5(x1)) = x1
702.70/218.23
POL(c6(x1)) = x1
702.70/218.23
POL(c7(x1)) = x1
702.70/218.23
POL(c9(x1)) = x1
702.70/218.23
POL(false) = [2]
702.70/218.23
POL(gr(x1, x2)) = 0
702.70/218.23
POL(p(x1)) = 0
702.70/218.23
POL(s(x1)) = [2] + x1
702.70/218.23
POL(true) = 0
702.70/218.23
702.70/218.23
(114) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0)))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(115) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND2(true, 0, s(z0)) → c1(COND1(true, p(0), s(z0))) by COND2(true, 0, s(z0)) → c1(COND1(true, 0, s(z0)))
702.70/218.23
702.70/218.23
(116) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND2(true, 0, s(z0)) → c1(COND1(true, 0, s(z0)))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(117) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND2(true, 0, s(z0)) → c1(COND1(true, 0, s(z0)))
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.23
702.70/218.23
(118) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(119) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1))) by COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
702.70/218.23
(120) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(z0), s(x1)) → c3(COND1(gr(s(add(z0, s(x1))), 0), s(z0), x1), ADD(s(z0), s(x1)))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.23
702.70/218.23
(122) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(123) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1)) by COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
702.70/218.23
(124) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
COND3(false, 0, 0) → c4
702.70/218.23
702.70/218.23
(126) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, 0) → c2(COND3(false, x0, 0))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(127) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND2(
false,
x0,
0) →
c2(
COND3(
false,
x0,
0)) by
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
702.70/218.23
(128) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(COND1(true, s(x0), x1), ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c4(COND1(gr(s(z0), 0), s(0), z0), ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c4(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c4, c1
702.70/218.23
702.70/218.23
(129) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
702.70/218.23
702.70/218.23
(130) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, s(x0), x1) → c8(COND1(true, s(x0), x1))
702.70/218.23
COND3(false, s(x0), x1) → c8(ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.23
COND3(false, s(0), z0) → c8(ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c4, c1, c8
702.70/218.23
702.70/218.23
(131) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 5 leading nodes:
COND3(false, s(x0), x1) → c8(COND1(true, s(x0), x1))
702.70/218.23
COND3(false, s(x0), x1) → c4(ADD(s(x0), x1))
702.70/218.23
COND3(false, s(x0), x1) → c8(ADD(s(x0), x1))
702.70/218.23
COND3(false, s(0), z0) → c8(ADD(s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(ADD(s(s(z0)), z1))
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
702.70/218.23
(132) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0)))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.23
702.70/218.23
(133) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, s(x0), s(z0)) → c3(COND1(gr(s(add(x0, s(z0))), 0), s(x0), z0), ADD(s(x0), s(z0))) by COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
702.70/218.23
(134) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.23
702.70/218.23
(135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
702.70/218.23
(136) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.23
702.70/218.23
(137) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, s(0), z0) → c3(COND1(gr(s(z0), 0), s(0), p(z0)), ADD(s(0), z0)) by COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.23
702.70/218.23
(138) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.23
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.23
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.23
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.23
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.23
gr(0, z0) → false
702.70/218.23
gr(s(z0), 0) → true
702.70/218.23
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.23
add(0, z0) → z0
702.70/218.23
add(s(z0), z1) → s(add(z0, z1))
702.70/218.23
p(0) → 0
702.70/218.23
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.23
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.23
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.23
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.23
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.23
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.23
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.23
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.23
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.23
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.23
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.23
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.23
COND3(false, 0, 0) → c4
702.70/218.23
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.23
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.23
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.23
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.23
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
702.70/218.24
(140) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(141) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, s(s(z0)), z1) → c3(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), p(z1)), ADD(s(s(z0)), z1)) by COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
702.70/218.24
(142) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
702.70/218.24
(144) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c6, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(145) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, s(x0), 0) → c6(COND1(gr(s(add(x0, 0)), 0), s(x0), 0)) by COND3(true, s(z0), 0) → c6(COND1(true, s(z0), 0))
702.70/218.24
702.70/218.24
(146) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), 0) → c6(COND1(true, s(z0), 0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8, c6
702.70/218.24
702.70/218.24
(147) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
COND3(true, s(z0), 0) → c6(COND1(true, s(z0), 0))
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
702.70/218.24
(148) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(149) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0)) by COND3(true, 0, 0) → c3(COND1(false, 0, 0))
702.70/218.24
702.70/218.24
(150) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(false, 0, 0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.24
702.70/218.24
(152) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, 0) → c3
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, 0) → c3(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8, c3
702.70/218.24
702.70/218.24
(153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(true, 0, 0) → c3
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
702.70/218.24
(154) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(155) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, 0, s(z0)) → c3(COND1(gr(s(z0), 0), 0, z0)) by COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
702.70/218.24
(156) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(true, 0, s(x1)) → c3(COND1(gr(s(x1), 0), 0, x1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
702.70/218.24
(158) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0))))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(159) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, 0, s(z0)) → c3(COND1(true, 0, p(s(z0)))) by COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
702.70/218.24
(160) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
702.70/218.24
(162) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(163) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
We considered the (Usable) Rules:
p(0) → 0
702.70/218.24
p(s(z0)) → z0
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(0, z0) → false
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.24
POL(0) = 0
702.70/218.24
POL(ADD(x1, x2)) = 0
702.70/218.24
POL(COND1(x1, x2, x3)) = [2]x3
702.70/218.24
POL(COND2(x1, x2, x3)) = [2]x3
702.70/218.24
POL(COND3(x1, x2, x3)) = [2]x3
702.70/218.24
POL(GR(x1, x2)) = [5]x1 + [5]x2
702.70/218.24
POL(add(x1, x2)) = 0
702.70/218.24
POL(c(x1)) = x1
702.70/218.24
POL(c1(x1)) = x1
702.70/218.24
POL(c1(x1, x2)) = x1 + x2
702.70/218.24
POL(c2(x1)) = x1
702.70/218.24
POL(c3(x1)) = x1
702.70/218.24
POL(c3(x1, x2)) = x1 + x2
702.70/218.24
POL(c4) = 0
702.70/218.24
POL(c5(x1)) = x1
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POL(c7(x1)) = x1
702.70/218.24
POL(c8(x1)) = x1
702.70/218.24
POL(c9(x1)) = x1
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POL(false) = 0
702.70/218.24
POL(gr(x1, x2)) = 0
702.70/218.24
POL(p(x1)) = x1
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POL(s(x1)) = [1] + x1
702.70/218.24
POL(true) = 0
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(164) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
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(165) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)
Used rewriting to replace COND3(true, 0, 0) → c5(COND1(gr(0, 0), 0, 0)) by COND3(true, 0, 0) → c5(COND1(false, 0, 0))
702.70/218.24
702.70/218.24
(166) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, 0) → c5(COND1(false, 0, 0))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
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(167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
702.70/218.24
702.70/218.24
(168) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, 0) → c5
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8, c5
702.70/218.24
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(169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(true, 0, 0) → c5
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
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(170) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(171) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
We considered the (Usable) Rules:
p(0) → 0
702.70/218.24
p(s(z0)) → z0
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(0, z0) → false
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.24
POL(0) = 0
702.70/218.24
POL(ADD(x1, x2)) = 0
702.70/218.24
POL(COND1(x1, x2, x3)) = [1]
702.70/218.24
POL(COND2(x1, x2, x3)) = [1] + [2]x1·x2
702.70/218.24
POL(COND3(x1, x2, x3)) = x2 + x3
702.70/218.24
POL(GR(x1, x2)) = 0
702.70/218.24
POL(add(x1, x2)) = 0
702.70/218.24
POL(c(x1)) = x1
702.70/218.24
POL(c1(x1)) = x1
702.70/218.24
POL(c1(x1, x2)) = x1 + x2
702.70/218.24
POL(c2(x1)) = x1
702.70/218.24
POL(c3(x1)) = x1
702.70/218.24
POL(c3(x1, x2)) = x1 + x2
702.70/218.24
POL(c4) = 0
702.70/218.24
POL(c5(x1)) = x1
702.70/218.24
POL(c7(x1)) = x1
702.70/218.24
POL(c8(x1)) = x1
702.70/218.24
POL(c9(x1)) = x1
702.70/218.24
POL(false) = [2]
702.70/218.24
POL(gr(x1, x2)) = 0
702.70/218.24
POL(p(x1)) = 0
702.70/218.24
POL(s(x1)) = [1]
702.70/218.24
POL(true) = 0
702.70/218.24
702.70/218.24
(172) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(173) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
We considered the (Usable) Rules:
p(0) → 0
702.70/218.24
p(s(z0)) → z0
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(0, z0) → false
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
702.70/218.24
POL(0) = 0
702.70/218.24
POL(ADD(x1, x2)) = 0
702.70/218.24
POL(COND1(x1, x2, x3)) = [1] + [2]x3
702.70/218.24
POL(COND2(x1, x2, x3)) = [1] + [2]x3 + x1·x2
702.70/218.24
POL(COND3(x1, x2, x3)) = [2]x2 + [2]x3
702.70/218.24
POL(GR(x1, x2)) = x1 + x2
702.70/218.24
POL(add(x1, x2)) = 0
702.70/218.24
POL(c(x1)) = x1
702.70/218.24
POL(c1(x1)) = x1
702.70/218.24
POL(c1(x1, x2)) = x1 + x2
702.70/218.24
POL(c2(x1)) = x1
702.70/218.24
POL(c3(x1)) = x1
702.70/218.24
POL(c3(x1, x2)) = x1 + x2
702.70/218.24
POL(c4) = 0
702.70/218.24
POL(c5(x1)) = x1
702.70/218.24
POL(c7(x1)) = x1
702.70/218.24
POL(c8(x1)) = x1
702.70/218.24
POL(c9(x1)) = x1
702.70/218.24
POL(false) = [2]
702.70/218.24
POL(gr(x1, x2)) = 0
702.70/218.24
POL(p(x1)) = x1
702.70/218.24
POL(s(x1)) = [2] + x1
702.70/218.24
POL(true) = 0
702.70/218.24
702.70/218.24
(174) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND2(false, x0, s(z0)) → c2(COND3(true, x0, s(z0)))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c2, c1, c3, c3, c5, c4, c1, c8
702.70/218.24
702.70/218.24
(175) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
COND2(
false,
x0,
s(
z0)) →
c2(
COND3(
true,
x0,
s(
z0))) by
COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
702.70/218.24
702.70/218.24
(176) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(COND1(true, s(x0), p(x1)), ADD(s(x0), x1))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c3(COND1(true, s(z0), z1), ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
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COND2(false, 0, 0) → c2(COND3(false, 0, 0))
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COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND3(true, s(0), z0) → c3(COND1(true, s(0), p(z0)), ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c3(COND1(true, s(s(z0)), p(z1)), ADD(s(s(z0)), z1))
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COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(0), z0) → c1(COND1(gr(s(z0), 0), 0, z0), ADD(s(0), z0))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
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COND2(false, 0, 0) → c2(COND3(false, 0, 0))
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COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c3, c3, c5, c4, c1, c2, c8
702.70/218.24
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(177) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
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(178) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
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COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
702.70/218.24
COND3(true, s(x0), x1) → c6(COND1(true, s(x0), p(x1)))
702.70/218.24
COND3(true, s(x0), x1) → c6(ADD(s(x0), x1))
702.70/218.24
COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND2(true, s(0), z0) → c6(ADD(s(0), z0))
702.70/218.24
COND3(true, s(z0), s(z1)) → c6(COND1(true, s(z0), z1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c6(ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, s(0), z0) → c6(COND1(true, s(0), p(z0)))
702.70/218.24
COND3(true, s(0), z0) → c6(ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c6(COND1(true, s(s(z0)), p(z1)))
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COND3(true, s(s(z0)), z1) → c6(ADD(s(s(z0)), z1))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
702.70/218.24
COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND2(true, s(0), z0) → c6(ADD(s(0), z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c3, c5, c4, c1, c2, c8, c6
702.70/218.24
702.70/218.24
(179) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 6 leading nodes:
COND3(true, s(x0), x1) → c3(ADD(s(x0), x1))
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COND3(true, s(z0), s(z1)) → c6(COND1(true, s(z0), z1))
702.70/218.24
COND3(true, s(x0), x1) → c6(ADD(s(x0), x1))
702.70/218.24
COND3(true, s(z0), s(z1)) → c6(ADD(s(z0), s(z1)))
702.70/218.24
COND3(true, s(0), z0) → c6(ADD(s(0), z0))
702.70/218.24
COND3(true, s(s(z0)), z1) → c6(ADD(s(s(z0)), z1))
Removed 2 trailing nodes:
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, 0, 0) → c4
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(180) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
702.70/218.24
cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
702.70/218.24
cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
702.70/218.24
cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
702.70/218.24
cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
702.70/218.24
gr(0, z0) → false
702.70/218.24
gr(s(z0), 0) → true
702.70/218.24
gr(s(z0), s(z1)) → gr(z0, z1)
702.70/218.24
add(0, z0) → z0
702.70/218.24
add(s(z0), z1) → s(add(z0, z1))
702.70/218.24
p(0) → 0
702.70/218.24
p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
702.70/218.24
COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
702.70/218.24
COND3(true, s(x0), x1) → c6(COND1(true, s(x0), p(x1)))
702.70/218.24
COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND2(true, s(0), z0) → c6(ADD(s(0), z0))
702.70/218.24
COND3(true, s(0), z0) → c6(COND1(true, s(0), p(z0)))
702.70/218.24
COND3(true, s(s(z0)), z1) → c6(COND1(true, s(s(z0)), p(z1)))
S tuples:
COND1(true, 0, x1) → c(COND2(false, 0, x1))
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
702.70/218.24
COND3(false, 0, 0) → c4
702.70/218.24
COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
702.70/218.24
COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
702.70/218.24
COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
702.70/218.24
COND2(true, s(0), z0) → c6(ADD(s(0), z0))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c5, c4, c1, c3, c2, c8, c6
702.70/218.24
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(181) CdtPolyRedPairProof (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, 0, x1) → c(COND2(false, 0, x1))
We considered the (Usable) Rules:
p(0) → 0
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p(s(z0)) → z0
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gr(s(z0), 0) → true
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gr(0, z0) → false
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
And the Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
702.70/218.24
COND1(true, 0, x1) → c(COND2(false, 0, x1))
702.70/218.24
COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
702.70/218.24
COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
702.70/218.24
COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
702.70/218.24
COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
702.70/218.24
COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
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COND2(false, 0, 0) → c2(COND3(false, 0, 0))
702.70/218.24
COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
702.70/218.24
COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
702.70/218.24
COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
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COND3(true, s(x0), x1) → c6(COND1(true, s(x0), p(x1)))
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COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
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COND2(true, s(0), z0) → c6(ADD(s(0), z0))
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COND3(true, s(0), z0) → c6(COND1(true, s(0), p(z0)))
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COND3(true, s(s(z0)), z1) → c6(COND1(true, s(s(z0)), p(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = 0
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POL(ADD(x1, x2)) = 0
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POL(COND1(x1, x2, x3)) = [1] + x3
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POL(COND2(x1, x2, x3)) = x1 + x3
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POL(COND3(x1, x2, x3)) = x2 + x3
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POL(GR(x1, x2)) = [5]x1 + [5]x2
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POL(add(x1, x2)) = 0
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POL(c(x1)) = x1
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POL(c1(x1)) = x1
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POL(c1(x1, x2)) = x1 + x2
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POL(c2(x1)) = x1
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POL(c3(x1)) = x1
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POL(c4) = 0
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POL(c5(x1)) = x1
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POL(c6(x1)) = x1
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POL(c7(x1)) = x1
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POL(c8(x1)) = x1
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POL(c9(x1)) = x1
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POL(false) = 0
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POL(gr(x1, x2)) = [4]
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POL(p(x1)) = x1
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POL(s(x1)) = [2] + x1
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POL(true) = [1]
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(182) Obligation:
Complexity Dependency Tuples Problem
Rules:
cond1(true, z0, z1) → cond2(gr(z0, 0), z0, z1)
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cond2(true, z0, z1) → cond1(gr(add(z0, z1), 0), p(z0), z1)
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cond2(false, z0, z1) → cond3(gr(z1, 0), z0, z1)
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cond3(true, z0, z1) → cond1(gr(add(z0, z1), 0), z0, p(z1))
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cond3(false, z0, z1) → cond1(gr(add(z0, z1), 0), z0, z1)
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gr(0, z0) → false
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gr(s(z0), 0) → true
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gr(s(z0), s(z1)) → gr(z0, z1)
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add(0, z0) → z0
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add(s(z0), z1) → s(add(z0, z1))
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p(0) → 0
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p(s(z0)) → z0
Tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND2(true, 0, z0) → c1(COND1(gr(z0, 0), 0, z0))
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COND3(true, s(z0), 0) → c5(COND1(gr(s(add(z0, 0)), 0), s(z0), 0))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
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COND2(false, 0, 0) → c2(COND3(false, 0, 0))
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COND3(false, s(0), z0) → c8(COND1(gr(s(z0), 0), s(0), z0))
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COND3(false, s(s(z0)), z1) → c8(COND1(gr(s(s(add(z0, z1))), 0), s(s(z0)), z1))
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COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
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COND3(true, s(x0), x1) → c6(COND1(true, s(x0), p(x1)))
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COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
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COND2(true, s(0), z0) → c6(ADD(s(0), z0))
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COND3(true, s(0), z0) → c6(COND1(true, s(0), p(z0)))
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COND3(true, s(s(z0)), z1) → c6(COND1(true, s(s(z0)), p(z1)))
S tuples:none
K tuples:
GR(s(z0), s(z1)) → c7(GR(z0, z1))
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COND3(false, 0, 0) → c4
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COND2(true, s(z0), z1) → c1(COND1(true, z0, z1), ADD(s(z0), z1))
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COND2(true, s(s(z0)), z1) → c1(COND1(gr(s(s(add(z0, z1))), 0), s(z0), z1), ADD(s(s(z0)), z1))
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ADD(s(z0), z1) → c9(ADD(z0, z1))
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COND1(true, s(z0), x1) → c(COND2(true, s(z0), x1))
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COND3(true, 0, s(z0)) → c3(COND1(true, 0, z0))
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COND2(false, 0, 0) → c2(COND3(false, 0, 0))
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COND2(false, 0, s(z1)) → c2(COND3(true, 0, s(z1)))
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COND2(true, s(0), z0) → c6(COND1(gr(s(z0), 0), 0, z0))
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COND2(true, s(0), z0) → c6(ADD(s(0), z0))
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COND1(true, 0, x1) → c(COND2(false, 0, x1))
Defined Rule Symbols:
cond1, cond2, cond3, gr, add, p
Defined Pair Symbols:
GR, ADD, COND1, COND2, COND3
Compound Symbols:
c7, c9, c, c1, c5, c4, c1, c3, c2, c8, c6
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(183) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(184) BOUNDS(O(1), O(1))
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