The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳6 CdtProblem
↳7 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳12 CdtProblem
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳14 CdtProblem
↳15 CdtKnowledgeProof (⇔)
↳16 CdtProblem
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳18 CdtProblem
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳20 CdtProblem
↳21 SIsEmptyProof (⇔)
↳22 BOUNDS(O(1), O(1))

(0) Obligation:

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) → mod(minus(x, y), y)
if_mod(false, s(x), s(y)) → s(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c8

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

Use narrowing to replace MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) by

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 6 dangling nodes:

MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7

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

Removed 1 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7, c7

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

Use narrowing to replace IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1)) by

IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(11) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 6 dangling nodes:

IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(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_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
We considered the (Usable) Rules:

minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(IF_MOD(x1, x2, x3)) = [2]x2 + [4]x3   
POL(LE(x1, x2)) = 0   
POL(MINUS(x1, x2)) = [3]   
POL(MOD(x1, x2)) = [4] + [2]x1 + [4]x2   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c7(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(le(x1, x2)) = [3] + [4]x1 + [2]x2   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = [4] + x1   
POL(true) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(15) CdtKnowledgeProof (EQUIVALENT transformation)

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

MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(17) 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.

LE(s(z0), s(z1)) → c2(LE(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF_MOD(x1, x2, x3)) = [2]x1 + x2·x3   
POL(LE(x1, x2)) = x1   
POL(MINUS(x1, x2)) = 0   
POL(MOD(x1, x2)) = [2] + x2 + x1·x2   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c7(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(false) = [2]   
POL(le(x1, x2)) = [2]   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = [2] + x1   
POL(true) = [2]   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(19) 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.

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF_MOD(x1, x2, x3)) = [3] + [2]x2 + [3]x22   
POL(LE(x1, x2)) = [1]   
POL(MINUS(x1, x2)) = [1] + x1   
POL(MOD(x1, x2)) = [2] + [3]x1 + [3]x12   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c7(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(le(x1, x2)) = 0   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
S tuples:none
K tuples:

IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))