The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

(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(0, y) → 0
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0
if_minus(false, s(x), y) → s(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 CpxTRS 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(0, z0) → 0
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) → 0
if_minus(false, s(z0), z1) → s(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), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

K tuples:none
Defined Rule Symbols:

le, minus, if_minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c6, c9, c10, c11

(3) CdtPolyRedPairProof (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)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

We considered the (Usable) Rules:

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

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(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)) = [1] + [4]x₁ + [4]x₂   
POL(IF_GCD(x₁, x₂, x₃)) = [4]x₂ + [4]x₃   
POL(IF_MINUS(x₁, x₂, x₃)) = 0   
POL(LE(x₁, x₂)) = 0   
POL(MINUS(x₁, x₂)) = 0   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(false) = 0   
POL(if_minus(x₁, x₂, x₃)) = x₂   
POL(le(x₁, x₂)) = 0   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = [4] + x₁   
POL(true) = 0   

(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(0, z0) → 0
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) → 0
if_minus(false, s(z0), z1) → s(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), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))

K tuples:

GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

Defined Rule Symbols:

le, minus, if_minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c6, c9, c10, c11

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(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)) = x₂² + x₁²   
POL(IF_GCD(x₁, x₂, x₃)) = x₃² + x₂²   
POL(IF_MINUS(x₁, x₂, x₃)) = x₂   
POL(LE(x₁, x₂)) = 0   
POL(MINUS(x₁, x₂)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(false) = 0   
POL(if_minus(x₁, x₂, x₃)) = x₂   
POL(le(x₁, x₂)) = 0   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = [2] + x₁   
POL(true) = 0   

(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(0, z0) → 0
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) → 0
if_minus(false, s(z0), z1) → s(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), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))

K tuples:

GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))

Defined Rule Symbols:

le, minus, if_minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c6, c9, c10, c11

(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, 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(0, z0) → 0
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) → 0
if_minus(false, s(z0), z1) → s(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), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(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)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))

Defined Rule Symbols:

le, minus, if_minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c6, c9, c10, c11

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) 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(0, z0) → 0
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1)
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
le(0, z0) → true
if_minus(true, s(z0), z1) → 0
if_minus(false, s(z0), z1) → s(minus(z0, z1))

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(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)) = [1] + x₁ + x₂ + x₂² + x₁·x₂ + x₁² + x₁³ + x₂³   
POL(IF_GCD(x₁, x₂, x₃)) = x₂ + x₃² + x₂·x₃ + x₂² + x₂³ + x₃³   
POL(IF_MINUS(x₁, x₂, x₃)) = x₂²   
POL(LE(x₁, x₂)) = x₁   
POL(MINUS(x₁, x₂)) = x₁ + x₁²   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(false) = 0   
POL(if_minus(x₁, x₂, x₃)) = x₂   
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(0, z0) → 0
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1)
if_minus(true, s(z0), z1) → 0
if_minus(false, s(z0), z1) → s(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), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:none
K tuples:

GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
LE(s(z0), s(z1)) → c2(LE(z0, z1))

Defined Rule Symbols:

le, minus, if_minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, IF_MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c6, c9, c10, c11

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))