### (0) Obligation:

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

leq(0, y) → true
leq(s(x), 0) → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), 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:

leq(0, z0) → true
leq(s(z0), 0) → false
leq(s(z0), s(z1)) → leq(z0, z1)
if(true, z0, z1) → z0
if(false, z0, z1) → z1
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(0, z0) → c
LEQ(s(z0), 0) → c1
LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
IF(true, z0, z1) → c3
IF(false, z0, z1) → c4
-'(z0, 0) → c5
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(0, z0) → c7
MOD(s(z0), 0) → c8
MOD(s(z0), s(z1)) → c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(0, z0) → c
LEQ(s(z0), 0) → c1
LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
IF(true, z0, z1) → c3
IF(false, z0, z1) → c4
-'(z0, 0) → c5
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(0, z0) → c7
MOD(s(z0), 0) → c8
MOD(s(z0), s(z1)) → c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, IF, -', MOD

Compound Symbols:

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

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

Removed 7 trailing nodes:

LEQ(0, z0) → c
LEQ(s(z0), 0) → c1
-'(z0, 0) → c5
IF(true, z0, z1) → c3
IF(false, z0, z1) → c4
MOD(0, z0) → c7
MOD(s(z0), 0) → c8

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true
leq(s(z0), 0) → false
leq(s(z0), s(z1)) → leq(z0, z1)
if(true, z0, z1) → z0
if(false, z0, z1) → z1
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

leq(0, z0) → true
leq(s(z0), 0) → false
leq(s(z0), s(z1)) → leq(z0, z1)
if(true, z0, z1) → z0
if(false, z0, z1) → z1
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

leq, if, -, mod

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

leq(0, z0) → true
leq(s(z0), 0) → false
leq(s(z0), s(z1)) → leq(z0, z1)
if(true, z0, z1) → z0
if(false, z0, z1) → z1
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(s(z0), s(z1)) → -(z0, z1)
-(z0, 0) → z0
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

-

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

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

Use narrowing to replace MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) by

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(s(z0), s(z1)) → -(z0, z1)
-(z0, 0) → z0
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

-

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

### (11) 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)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
We considered the (Usable) Rules:

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

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(-(x1, x2)) = x1
POL(-'(x1, x2)) = [3]
POL(0) = 0
POL(LEQ(x1, x2)) = 0
POL(MOD(x1, x2)) = x1
POL(c2(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(s(x1)) = [4] + x1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(s(z0), s(z1)) → -(z0, z1)
-(z0, 0) → z0
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
K tuples:

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
Defined Rule Symbols:

-

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

### (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.

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
We considered the (Usable) Rules:

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

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(-(x1, x2)) = x1
POL(-'(x1, x2)) = [2]x1 + [2]x2
POL(0) = 0
POL(LEQ(x1, x2)) = [1] + [2]x2
POL(MOD(x1, x2)) = x1 + x1·x2 + x12
POL(c2(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(s(x1)) = [2] + x1

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(s(z0), s(z1)) → -(z0, z1)
-(z0, 0) → z0
Tuples:

LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
S tuples:none
K tuples:

MOD(s(z0), s(z1)) → c9(LEQ(z1, z0), MOD(-(z0, z1), s(z1)), -'(s(z0), s(z1)))
LEQ(s(z0), s(z1)) → c2(LEQ(z0, z1))
-'(s(z0), s(z1)) → c6(-'(z0, z1))
Defined Rule Symbols:

-

Defined Pair Symbols:

LEQ, -', MOD

Compound Symbols:

c2, c6, c9

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

The set S is empty