### (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)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(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, s(x), s(y)) → mod(minus(x, y), s(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)
pred(s(z0)) → z0
minus(z0, 0) → z0
minus(z0, s(z1)) → pred(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, s(z0), s(z1)) → mod(minus(z0, z1), s(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))
PRED(s(z0)) → c3
MINUS(z0, 0) → c4
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0, z0) → c6
MOD(s(z0), 0) → c7
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) → c10
S tuples:

LE(0, z0) → c
LE(s(z0), 0) → c1
LE(s(z0), s(z1)) → c2(LE(z0, z1))
PRED(s(z0)) → c3
MINUS(z0, 0) → c4
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(0, z0) → c6
MOD(s(z0), 0) → c7
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) → c10
K tuples:none
Defined Rule Symbols:

le, pred, minus, mod, if_mod

Defined Pair Symbols:

LE, PRED, MINUS, MOD, IF_MOD

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

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

Removed 7 trailing nodes:

IF_MOD(false, s(z0), s(z1)) → c10
LE(0, z0) → c
LE(s(z0), 0) → c1
PRED(s(z0)) → c3
MINUS(z0, 0) → c4
MOD(0, z0) → c6
MOD(s(z0), 0) → c7

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

le, pred, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c5, c8, c9

### (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

le, pred, minus, mod, if_mod

Defined Pair Symbols:

LE, MOD, IF_MOD, MINUS

Compound Symbols:

c2, c8, c9, c5

### (7) 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, s(z0), s(z1)) → mod(minus(z0, z1), s(z1))
if_mod(false, s(z0), s(z1)) → s(z0)

### (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(z0, s(z1)) → pred(minus(z0, z1))
pred(s(z0)) → z0
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, MOD, IF_MOD, MINUS

Compound Symbols:

c2, c8, c9, c5

### (9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
We considered the (Usable) Rules:

minus(z0, s(z1)) → pred(minus(z0, z1))
minus(z0, 0) → z0
pred(s(z0)) → z0
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]
POL(IF_MOD(x1, x2, x3)) = [5]x2 + [3]x3
POL(LE(x1, x2)) = 0
POL(MINUS(x1, x2)) = [1]
POL(MOD(x1, x2)) = [3] + [5]x1 + [3]x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(false) = [2]
POL(le(x1, x2)) = x2
POL(minus(x1, x2)) = [3] + x1
POL(pred(x1)) = x1
POL(s(x1)) = [4] + x1
POL(true) = 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(z0, s(z1)) → pred(minus(z0, z1))
pred(s(z0)) → z0
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:

MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, MOD, IF_MOD, MINUS

Compound Symbols:

c2, c8, c9, c5

### (11) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

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

minus(z0, s(z1)) → pred(minus(z0, z1))
minus(z0, 0) → z0
pred(s(z0)) → z0
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(IF_MOD(x1, x2, x3)) = [1] + x22
POL(LE(x1, x2)) = x2
POL(MINUS(x1, x2)) = 0
POL(MOD(x1, x2)) = x1 + x12
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(false) = 0
POL(le(x1, x2)) = 0
POL(minus(x1, x2)) = x1
POL(pred(x1)) = x1
POL(s(x1)) = [1] + x1
POL(true) = 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(z0, s(z1)) → pred(minus(z0, z1))
pred(s(z0)) → z0
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:

MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, MOD, IF_MOD, MINUS

Compound Symbols:

c2, c8, c9, c5

### (13) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(z0, s(z1)) → pred(minus(z0, z1))
minus(z0, 0) → z0
pred(s(z0)) → z0
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(IF_MOD(x1, x2, x3)) = x2·x3
POL(LE(x1, x2)) = 0
POL(MINUS(x1, x2)) = x2
POL(MOD(x1, x2)) = x1·x2
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(false) = 0
POL(le(x1, x2)) = 0
POL(minus(x1, x2)) = x1
POL(pred(x1)) = x1
POL(s(x1)) = [1] + 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(z0, s(z1)) → pred(minus(z0, z1))
pred(s(z0)) → z0
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:none
K tuples:

MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, MOD, IF_MOD, MINUS

Compound Symbols:

c2, c8, c9, c5

### (15) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty