### (0) Obligation:

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

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: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

The duplicating contexts are:
mod(s(x), s([]))
mod(s([]), s(y))

The defined contexts are:
if([], x1, s(x2))
if(x0, [], s(x2))
mod([], s(x1))
leq(x0, [])
-(s([]), s(x1))
if(x0, x1, s([]))
-([], x1)

[] just represents basic- or constructor-terms in the following defined contexts:
if([], x1, s(x2))
mod([], s(x1))

As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.

### (2) Obligation:

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

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

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

Converted Cpx (relative) TRS to CDT

### (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(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

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

Removed 7 trailing nodes:

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

### (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(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

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

Removed 1 trailing tuple parts

### (8) 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

### (9) 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))

### (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(-(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

### (11) 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)))

### (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))
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

### (13) 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(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)) = 0
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)) = [1] + 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:

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

### (15) CdtRuleRemovalProof (UPPER BOUND(ADD(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
POL(0) = 0
POL(LEQ(x1, x2)) = [1] + [2]x2
POL(MOD(x1, x2)) = [2]x1·x2 + x12
POL(c2(x1)) = x1
POL(c6(x1)) = x1
POL(c9(x1, x2, x3)) = x1 + x2 + x3
POL(s(x1)) = [2] + x1

### (16) 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

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

The set S is empty