YES(O(1), O(n^3)) 15.15/5.17 YES(O(1), O(n^3)) 15.59/5.20 15.59/5.20 15.59/5.20 15.59/5.20 15.59/5.20 15.59/5.20 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 15.59/5.20 15.59/5.20 15.59/5.20
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
15.59/5.20 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
15.59/5.20 
↳2 CdtProblem
15.59/5.20 
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
15.59/5.20 
↳4 CdtProblem
15.59/5.20 
↳5 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
15.59/5.20 
↳6 CdtProblem
15.59/5.20 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
15.59/5.20 
↳8 CdtProblem
15.59/5.20 
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
15.59/5.20 
↳10 CdtProblem
15.59/5.20 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
15.59/5.20 
↳12 CdtProblem
15.59/5.20 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
15.59/5.20 
↳14 BOUNDS(O(1), O(1))
15.59/5.20
15.59/5.20 15.59/5.20
15.59/5.20
15.59/5.20

(0) Obligation:

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

le(0, y) → true 15.59/5.20
le(s(x), 0) → false 15.59/5.20
le(s(x), s(y)) → le(x, y) 15.59/5.20
minus(0, y) → 0 15.59/5.20
minus(s(x), y) → if_minus(le(s(x), y), s(x), y) 15.59/5.20
if_minus(true, s(x), y) → 0 15.59/5.20
if_minus(false, s(x), y) → s(minus(x, y)) 15.59/5.20
gcd(0, y) → y 15.59/5.20
gcd(s(x), 0) → s(x) 15.59/5.20
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) 15.59/5.20
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) 15.59/5.20
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x))

Rewrite Strategy: INNERMOST
15.59/5.20
15.59/5.20

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
15.59/5.20
15.59/5.20

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 15.59/5.20
le(s(z0), 0) → false 15.59/5.20
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.20
minus(0, z0) → 0 15.59/5.20
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.20
if_minus(true, s(z0), z1) → 0 15.59/5.20
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 15.59/5.20
gcd(0, z0) → z0 15.59/5.20
gcd(s(z0), 0) → s(z0) 15.59/5.20
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 15.59/5.20
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 15.59/5.20
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.20
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.20
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.20
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.20
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
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)) 15.59/5.20
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.20
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.20
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.20
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
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

15.59/5.20
15.59/5.20

(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.

IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
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 15.59/5.20
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.20
le(s(z0), 0) → false 15.59/5.20
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.20
le(0, z0) → true 15.59/5.20
if_minus(true, s(z0), z1) → 0 15.59/5.20
if_minus(false, s(z0), z1) → s(minus(z0, z1))
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.20
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.20
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.20
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.20
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
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 : 15.59/5.20

POL(0) = 0    15.59/5.20
POL(GCD(x1, x2)) = [2]x1 + [2]x2    15.59/5.20
POL(IF_GCD(x1, x2, x3)) = [2]x2 + [2]x3    15.59/5.20
POL(IF_MINUS(x1, x2, x3)) = 0    15.59/5.20
POL(LE(x1, x2)) = 0    15.59/5.20
POL(MINUS(x1, x2)) = 0    15.59/5.20
POL(c10(x1, x2)) = x1 + x2    15.59/5.20
POL(c11(x1, x2)) = x1 + x2    15.59/5.20
POL(c2(x1)) = x1    15.59/5.20
POL(c4(x1, x2)) = x1 + x2    15.59/5.20
POL(c6(x1)) = x1    15.59/5.20
POL(c9(x1, x2)) = x1 + x2    15.59/5.20
POL(false) = 0    15.59/5.20
POL(if_minus(x1, x2, x3)) = x2    15.59/5.20
POL(le(x1, x2)) = 0    15.59/5.20
POL(minus(x1, x2)) = x1    15.59/5.20
POL(s(x1)) = [4] + x1    15.59/5.20
POL(true) = 0   
15.59/5.20
15.59/5.20

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 15.59/5.20
le(s(z0), 0) → false 15.59/5.20
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.20
minus(0, z0) → 0 15.59/5.20
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.20
if_minus(true, s(z0), z1) → 0 15.59/5.20
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 15.59/5.20
gcd(0, z0) → z0 15.59/5.20
gcd(s(z0), 0) → s(z0) 15.59/5.20
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 15.59/5.20
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 15.59/5.20
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.20
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.20
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.20
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.20
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
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)) 15.59/5.20
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.20
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.20
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
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

15.59/5.20
15.59/5.20

(5) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

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

GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.20
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.20
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
15.59/5.20
15.59/5.20

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 15.59/5.20
le(s(z0), 0) → false 15.59/5.20
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.20
minus(0, z0) → 0 15.59/5.20
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.20
if_minus(true, s(z0), z1) → 0 15.59/5.20
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 15.59/5.20
gcd(0, z0) → z0 15.59/5.20
gcd(s(z0), 0) → s(z0) 15.59/5.20
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 15.59/5.20
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 15.59/5.20
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.20
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
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)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(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

15.59/5.21
15.59/5.21

(7) 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 15.59/5.21
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.21
le(s(z0), 0) → false 15.59/5.21
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.21
le(0, z0) → true 15.59/5.21
if_minus(true, s(z0), z1) → 0 15.59/5.21
if_minus(false, s(z0), z1) → s(minus(z0, z1))
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
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 : 15.59/5.21

POL(0) = 0    15.59/5.21
POL(GCD(x1, x2)) = [1] + x1 + x22 + x12    15.59/5.21
POL(IF_GCD(x1, x2, x3)) = x32 + x22    15.59/5.21
POL(IF_MINUS(x1, x2, x3)) = x1·x2    15.59/5.21
POL(LE(x1, x2)) = 0    15.59/5.21
POL(MINUS(x1, x2)) = x1    15.59/5.21
POL(c10(x1, x2)) = x1 + x2    15.59/5.21
POL(c11(x1, x2)) = x1 + x2    15.59/5.21
POL(c2(x1)) = x1    15.59/5.21
POL(c4(x1, x2)) = x1 + x2    15.59/5.21
POL(c6(x1)) = x1    15.59/5.21
POL(c9(x1, x2)) = x1 + x2    15.59/5.21
POL(false) = [1]    15.59/5.21
POL(if_minus(x1, x2, x3)) = x1·x2    15.59/5.21
POL(le(x1, x2)) = [1]    15.59/5.21
POL(minus(x1, x2)) = x1    15.59/5.21
POL(s(x1)) = [1] + x1    15.59/5.21
POL(true) = 0   
15.59/5.21
15.59/5.21

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 15.59/5.21
le(s(z0), 0) → false 15.59/5.21
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.21
minus(0, z0) → 0 15.59/5.21
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.21
if_minus(true, s(z0), z1) → 0 15.59/5.21
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 15.59/5.21
gcd(0, z0) → z0 15.59/5.21
gcd(s(z0), 0) → s(z0) 15.59/5.21
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 15.59/5.21
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 15.59/5.21
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
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)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
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

15.59/5.21
15.59/5.21

(9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) 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)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
15.59/5.21
15.59/5.21

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 15.59/5.21
le(s(z0), 0) → false 15.59/5.21
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.21
minus(0, z0) → 0 15.59/5.21
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.21
if_minus(true, s(z0), z1) → 0 15.59/5.21
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 15.59/5.21
gcd(0, z0) → z0 15.59/5.21
gcd(s(z0), 0) → s(z0) 15.59/5.21
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 15.59/5.21
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 15.59/5.21
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
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:

IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
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

15.59/5.21
15.59/5.21

(11) 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 15.59/5.21
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.21
le(s(z0), 0) → false 15.59/5.21
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.21
le(0, z0) → true 15.59/5.21
if_minus(true, s(z0), z1) → 0 15.59/5.21
if_minus(false, s(z0), z1) → s(minus(z0, z1))
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
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 : 15.59/5.21

POL(0) = 0    15.59/5.21
POL(GCD(x1, x2)) = [1] + x1 + x2 + x1·x2 + x13 + x23    15.59/5.21
POL(IF_GCD(x1, x2, x3)) = x2·x3 + x23 + x33    15.59/5.21
POL(IF_MINUS(x1, x2, x3)) = x22    15.59/5.21
POL(LE(x1, x2)) = x1    15.59/5.21
POL(MINUS(x1, x2)) = x1 + x12    15.59/5.21
POL(c10(x1, x2)) = x1 + x2    15.59/5.21
POL(c11(x1, x2)) = x1 + x2    15.59/5.21
POL(c2(x1)) = x1    15.59/5.21
POL(c4(x1, x2)) = x1 + x2    15.59/5.21
POL(c6(x1)) = x1    15.59/5.21
POL(c9(x1, x2)) = x1 + x2    15.59/5.21
POL(false) = 0    15.59/5.21
POL(if_minus(x1, x2, x3)) = x2    15.59/5.21
POL(le(x1, x2)) = 0    15.59/5.21
POL(minus(x1, x2)) = x1    15.59/5.21
POL(s(x1)) = [1] + x1    15.59/5.21
POL(true) = 0   
15.59/5.21
15.59/5.21

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 15.59/5.21
le(s(z0), 0) → false 15.59/5.21
le(s(z0), s(z1)) → le(z0, z1) 15.59/5.21
minus(0, z0) → 0 15.59/5.21
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 15.59/5.21
if_minus(true, s(z0), z1) → 0 15.59/5.21
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 15.59/5.21
gcd(0, z0) → z0 15.59/5.21
gcd(s(z0), 0) → s(z0) 15.59/5.21
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 15.59/5.21
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 15.59/5.21
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:none
K tuples:

IF_GCD(true, s(z0), s(z1)) → c10(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 15.59/5.21
IF_GCD(false, s(z0), s(z1)) → c11(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 15.59/5.21
GCD(s(z0), s(z1)) → c9(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 15.59/5.21
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 15.59/5.21
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 15.59/5.21
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

15.59/5.21
15.59/5.21

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

The set S is empty
15.59/5.21
15.59/5.21

(14) BOUNDS(O(1), O(1))

15.59/5.21
15.59/5.21
15.59/5.29 EOF