Runtime Complexity (innermost) proof of /tmp/tmpiZGfoD/thiemann40.xml

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 20 ms)

↳14 CdtProblem

↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CdtProblem

↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 16 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 24 ms)

↳28 CdtProblem

↳29 CdtKnowledgeProof (⇔, 0 ms)

↳30 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

nonZero(0) → false
nonZero(s(x)) → true
p(0) → 0
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0)
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(y))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

RANDOM(z0) → c6(RAND(z0, 0))
RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))

S tuples:

RANDOM(z0) → c6(RAND(z0, 0))
RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))

K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RANDOM, RAND, IF

Compound Symbols:

c6, c7, c9

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

RANDOM(z0) → c6(RAND(z0, 0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))

S tuples:

RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))

K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c9

(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace RAND(z0, z1) → c7(IF(nonZero(z0), z0, z1), NONZERO(z0)) by

RAND(0, x1) → c7(IF(false, 0, x1), NONZERO(0))
RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
RAND(x0, x1) → c7

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))
RAND(0, x1) → c7(IF(false, 0, x1), NONZERO(0))
RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
RAND(x0, x1) → c7

S tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))
RAND(0, x1) → c7(IF(false, 0, x1), NONZERO(0))
RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
RAND(x0, x1) → c7

K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7, c7

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

RAND(x0, x1) → c7
RAND(0, x1) → c7(IF(false, 0, x1), NONZERO(0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))
RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))

S tuples:

IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1))
RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))

K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IF(true, z0, z1) → c9(RAND(p(z0), id_inc(z1)), P(z0), ID_INC(z1)) by

IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c9, c9

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

IF(true, x0, x1) → c9
IF(true, 0, x1) → c9(RAND(0, id_inc(x1)), P(0), ID_INC(x1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

K tuples:none
Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c9

(13) 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.

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

We considered the (Usable) Rules:

id_inc(z0) → z0
id_inc(z0) → s(z0)
p(0) → 0
p(s(z0)) → z0

And the Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(ID_INC(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [2]x₂
POL(NONZERO(x₁)) = 0
POL(P(x₁)) = 0
POL(RAND(x₁, x₂)) = [2]x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(id_inc(x₁)) = [5] + [2]x₁
POL(p(x₁)) = x₁
POL(s(x₁)) = [4] + x₁
POL(true) = [2]

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

S tuples:

RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0)))
IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

RAND, IF

Compound Symbols:

c7, c9

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace RAND(s(z0), x1) → c7(IF(true, s(z0), x1), NONZERO(s(z0))) by

RAND(s(x0), x1) → c7(IF(true, s(x0), x1))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))

S tuples:

IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0))
IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IF(true, x0, z0) → c9(RAND(p(x0), z0), P(x0), ID_INC(z0)) by

IF(true, 0, x1) → c9(RAND(0, x1), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, 0, x1) → c9(RAND(0, x1), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

S tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, 0, x1) → c9(RAND(0, x1), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7, c9

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

IF(true, x0, x1) → c9
IF(true, 0, x1) → c9(RAND(0, x1), P(0), ID_INC(x1))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

S tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

(21) 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.

IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

We considered the (Usable) Rules:

id_inc(z0) → z0
id_inc(z0) → s(z0)
p(0) → 0
p(s(z0)) → z0

And the Tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(ID_INC(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [5]x₁ + [4]x₂
POL(P(x₁)) = 0
POL(RAND(x₁, x₂)) = [4]x₁
POL(c7(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(id_inc(x₁)) = [5] + [2]x₁
POL(p(x₁)) = x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

S tuples:

IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IF(true, x0, z0) → c9(RAND(p(x0), s(z0)), P(x0), ID_INC(z0)) by

IF(true, 0, x1) → c9(RAND(0, s(x1)), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, 0, x1) → c9(RAND(0, s(x1)), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

S tuples:

RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, 0, x1) → c9(RAND(0, s(x1)), P(0), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))
IF(true, x0, x1) → c9

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7, c9

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

IF(true, 0, x1) → c9(RAND(0, s(x1)), P(0), ID_INC(x1))
IF(true, x0, x1) → c9

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

S tuples:

RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

(27) 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.

IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

We considered the (Usable) Rules:

id_inc(z0) → z0
id_inc(z0) → s(z0)

And the Tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ID_INC(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [4]x₁ + [4]x₂
POL(P(x₁)) = 0
POL(RAND(x₁, x₂)) = [4]x₁
POL(c7(x₁)) = x₁
POL(c9(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(id_inc(x₁)) = [4]x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

nonZero(0) → false
nonZero(s(z0)) → true
p(0) → 0
p(s(z0)) → z0
id_inc(z0) → z0
id_inc(z0) → s(z0)
random(z0) → rand(z0, 0)
rand(z0, z1) → if(nonZero(z0), z0, z1)
if(false, z0, z1) → z1
if(true, z0, z1) → rand(p(z0), id_inc(z1))

Tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

S tuples:

RAND(s(x0), x1) → c7(IF(true, s(x0), x1))

K tuples:

IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

Defined Rule Symbols:

nonZero, p, id_inc, random, rand, if

Defined Pair Symbols:

IF, RAND

Compound Symbols:

c9, c7

(29) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

RAND(s(x0), x1) → c7(IF(true, s(x0), x1))
IF(true, s(z0), x1) → c9(RAND(z0, id_inc(x1)), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, x1), P(s(z0)), ID_INC(x1))
IF(true, s(z0), x1) → c9(RAND(z0, s(x1)), P(s(z0)), ID_INC(x1))

Now S is empty

(0) Obligation:

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(10) Obligation:

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(14) Obligation:

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(16) Obligation:

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(18) Obligation:

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(20) Obligation:

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(22) Obligation:

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(24) Obligation:

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(26) Obligation:

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(28) Obligation:

(29) CdtKnowledgeProof (EQUIVALENT transformation)

(30) BOUNDS(O(1), O(1))