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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtUsableRulesProof (⇔, 0 ms)
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 546 ms)
↳18 CdtProblem
↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 207 ms)
↳20 CdtProblem
↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 100 ms)
↳22 CdtProblem
↳23 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳24 BOUNDS(1, 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 Cpx (relative) TRS 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(0, z0) → c
LE(s(z0), 0) → c1
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, 0) → c3
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(0, z0) → c5
MOD(s(z0), 0) → c6
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))
IF_MOD(false, s(z0), s(z1)) → c9
S tuples:

LE(0, z0) → c
LE(s(z0), 0) → c1
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, 0) → c3
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(0, z0) → c5
MOD(s(z0), 0) → c6
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))
IF_MOD(false, s(z0), s(z1)) → c9
K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Removed 6 trailing nodes:

MOD(0, z0) → c5
IF_MOD(false, s(z0), s(z1)) → c9
MOD(s(z0), 0) → c6
MINUS(z0, 0) → c3
LE(0, z0) → c
LE(s(z0), 0) → c1

(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))
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

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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)

(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)
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

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c8

(7) 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)))

(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)
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

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7

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

Removed 1 trailing nodes:

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

(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)
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

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7

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

Removed 1 trailing tuple parts

(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)
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

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7, c7

(13) 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)))

(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)
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

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

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

Removed 1 trailing nodes:

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

(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)
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

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

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)))
We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
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   

(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)
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:

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)))
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(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:

le(s(z0), s(z1)) → le(z0, z1)
le(s(z0), 0) → false
le(0, z0) → true
minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
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]   

(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)
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:

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)))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(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(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0
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   

(22) 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)
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:

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)))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

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

The set S is empty

(24) BOUNDS(1, 1)