### (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)
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), 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)
gcd(0, z0) → z0
gcd(s(z0), 0) → s(z0)
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), 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))
GCD(0, z0) → c5
GCD(s(z0), 0) → c6
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
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))
GCD(0, z0) → c5
GCD(s(z0), 0) → c6
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

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

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

Removed 5 trailing nodes:

LE(0, z0) → c
LE(s(z0), 0) → c1
GCD(0, z0) → c5
MINUS(z0, 0) → c3
GCD(s(z0), 0) → c6

### (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)
gcd(0, z0) → z0
gcd(s(z0), 0) → s(z0)
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

gcd(0, z0) → z0
gcd(s(z0), 0) → s(z0)
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), 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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

### (7) 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.

GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(GCD(x1, x2)) = [5] + [4]x1 + [4]x2
POL(IF_GCD(x1, x2, x3)) = [4]x2 + [4]x3
POL(LE(x1, x2)) = [4]
POL(MINUS(x1, x2)) = [2]
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(false) = 0
POL(le(x1, x2)) = 0
POL(minus(x1, x2)) = [2] + x1
POL(s(x1)) = [4] + x1
POL(true) = 0

### (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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:

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

GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

### (9) 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(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(GCD(x1, x2)) = [2] + [2]x1 + [2]x2 + [2]x1·x2
POL(IF_GCD(x1, x2, x3)) = [2]x2 + [2]x3 + [2]x2·x3
POL(LE(x1, x2)) = 0
POL(MINUS(x1, x2)) = [2] + [2]x2
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
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(s(x1)) = [1] + 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(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:

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

GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

### (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(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(GCD(x1, x2)) = [3] + x2 + x1·x2
POL(IF_GCD(x1, x2, x3)) = [3] + x2·x3
POL(LE(x1, x2)) = [1] + x1
POL(MINUS(x1, x2)) = 0
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
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(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(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:none
K tuples:

GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

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

The set S is empty