YES(O(1), O(n^2)) 6.69/2.13 YES(O(1), O(n^2)) 6.69/2.18 6.69/2.18 6.69/2.18 6.69/2.18 6.69/2.18 6.69/2.18 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.69/2.18 6.69/2.18 6.69/2.18
6.69/2.18 6.69/2.18 6.69/2.18
6.69/2.18
6.69/2.18

(0) Obligation:

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

le(0, y) → true 6.69/2.18
le(s(x), 0) → false 6.69/2.18
le(s(x), s(y)) → le(x, y) 6.69/2.18
pred(s(x)) → x 6.69/2.18
minus(x, 0) → x 6.69/2.18
minus(x, s(y)) → pred(minus(x, y)) 6.69/2.18
gcd(0, y) → y 6.69/2.18
gcd(s(x), 0) → s(x) 6.69/2.18
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) 6.69/2.18
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) 6.69/2.18
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x))

Rewrite Strategy: INNERMOST
6.69/2.18
6.69/2.18

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

Converted CpxTRS to CDT
6.69/2.18
6.69/2.18

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.69/2.18
le(s(z0), 0) → false 6.69/2.18
le(s(z0), s(z1)) → le(z0, z1) 6.69/2.18
pred(s(z0)) → z0 6.69/2.18
minus(z0, 0) → z0 6.69/2.18
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.18
gcd(0, z0) → z0 6.69/2.18
gcd(s(z0), 0) → s(z0) 6.69/2.18
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.69/2.19
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.69/2.19
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.19
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1)) 6.69/2.19
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.19
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.19
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.19
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1)) 6.69/2.19
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.19
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.19
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:

le, pred, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c5, c8, c9, c10

6.69/2.19
6.69/2.19

(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
6.69/2.19
6.69/2.19

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.69/2.19
le(s(z0), 0) → false 6.69/2.19
le(s(z0), s(z1)) → le(z0, z1) 6.69/2.19
pred(s(z0)) → z0 6.69/2.19
minus(z0, 0) → z0 6.69/2.19
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.19
gcd(0, z0) → z0 6.69/2.19
gcd(s(z0), 0) → s(z0) 6.69/2.19
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.69/2.19
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.69/2.19
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.19
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.19
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.19
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.19
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.19
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.19
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.19
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.19
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

le, pred, minus, gcd, if_gcd

Defined Pair Symbols:

LE, GCD, IF_GCD, MINUS

Compound Symbols:

c2, c8, c9, c10, c5

6.69/2.19
6.69/2.19

(5) 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)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.19
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
We considered the (Usable) Rules:

minus(z0, 0) → z0 6.69/2.19
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.19
pred(s(z0)) → z0 6.69/2.19
le(0, z0) → true 6.69/2.19
le(s(z0), 0) → false 6.69/2.19
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.19
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.19
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.19
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.19
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.69/2.19

POL(0) = 0    6.69/2.19
POL(GCD(x1, x2)) = x1 + x2    6.69/2.19
POL(IF_GCD(x1, x2, x3)) = x2 + x3    6.69/2.19
POL(LE(x1, x2)) = 0    6.69/2.19
POL(MINUS(x1, x2)) = 0    6.69/2.19
POL(c10(x1, x2)) = x1 + x2    6.69/2.19
POL(c2(x1)) = x1    6.69/2.19
POL(c5(x1)) = x1    6.69/2.19
POL(c8(x1, x2)) = x1 + x2    6.69/2.19
POL(c9(x1, x2)) = x1 + x2    6.69/2.19
POL(false) = 0    6.69/2.19
POL(le(x1, x2)) = 0    6.69/2.19
POL(minus(x1, x2)) = x1    6.69/2.19
POL(pred(x1)) = x1    6.69/2.19
POL(s(x1)) = [1] + x1    6.69/2.19
POL(true) = 0   
6.69/2.19
6.69/2.19

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.69/2.19
le(s(z0), 0) → false 6.69/2.19
le(s(z0), s(z1)) → le(z0, z1) 6.69/2.19
pred(s(z0)) → z0 6.69/2.19
minus(z0, 0) → z0 6.69/2.19
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.19
gcd(0, z0) → z0 6.69/2.19
gcd(s(z0), 0) → s(z0) 6.69/2.19
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.69/2.19
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.69/2.19
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
Defined Rule Symbols:

le, pred, minus, gcd, if_gcd

Defined Pair Symbols:

LE, GCD, IF_GCD, MINUS

Compound Symbols:

c2, c8, c9, c10, c5

6.69/2.20
6.69/2.20

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

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

GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
6.69/2.20
6.69/2.20

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.69/2.20
le(s(z0), 0) → false 6.69/2.20
le(s(z0), s(z1)) → le(z0, z1) 6.69/2.20
pred(s(z0)) → z0 6.69/2.20
minus(z0, 0) → z0 6.69/2.20
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.20
gcd(0, z0) → z0 6.69/2.20
gcd(s(z0), 0) → s(z0) 6.69/2.20
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.69/2.20
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.69/2.20
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
Defined Rule Symbols:

le, pred, minus, gcd, if_gcd

Defined Pair Symbols:

LE, GCD, IF_GCD, MINUS

Compound Symbols:

c2, c8, c9, c10, c5

6.69/2.20
6.69/2.20

(9) 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.

MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0 6.69/2.20
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.20
pred(s(z0)) → z0 6.69/2.20
le(0, z0) → true 6.69/2.20
le(s(z0), 0) → false 6.69/2.20
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.69/2.20

POL(0) = 0    6.69/2.20
POL(GCD(x1, x2)) = x1·x2    6.69/2.20
POL(IF_GCD(x1, x2, x3)) = x2·x3    6.69/2.20
POL(LE(x1, x2)) = 0    6.69/2.20
POL(MINUS(x1, x2)) = [2] + [2]x2    6.69/2.20
POL(c10(x1, x2)) = x1 + x2    6.69/2.20
POL(c2(x1)) = x1    6.69/2.20
POL(c5(x1)) = x1    6.69/2.20
POL(c8(x1, x2)) = x1 + x2    6.69/2.20
POL(c9(x1, x2)) = x1 + x2    6.69/2.20
POL(false) = 0    6.69/2.20
POL(le(x1, x2)) = 0    6.69/2.20
POL(minus(x1, x2)) = x1    6.69/2.20
POL(pred(x1)) = x1    6.69/2.20
POL(s(x1)) = [2] + x1    6.69/2.20
POL(true) = 0   
6.69/2.20
6.69/2.20

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.69/2.20
le(s(z0), 0) → false 6.69/2.20
le(s(z0), s(z1)) → le(z0, z1) 6.69/2.20
pred(s(z0)) → z0 6.69/2.20
minus(z0, 0) → z0 6.69/2.20
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.20
gcd(0, z0) → z0 6.69/2.20
gcd(s(z0), 0) → s(z0) 6.69/2.20
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.69/2.20
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.69/2.20
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:

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

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:

le, pred, minus, gcd, if_gcd

Defined Pair Symbols:

LE, GCD, IF_GCD, MINUS

Compound Symbols:

c2, c8, c9, c10, c5

6.69/2.20
6.69/2.20

(11) 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.

LE(s(z0), s(z1)) → c2(LE(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0 6.69/2.20
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.20
pred(s(z0)) → z0 6.69/2.20
le(0, z0) → true 6.69/2.20
le(s(z0), 0) → false 6.69/2.20
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.69/2.20

POL(0) = 0    6.69/2.20
POL(GCD(x1, x2)) = [2]x1 + [2]x2 + x22 + [2]x1·x2 + x12    6.69/2.20
POL(IF_GCD(x1, x2, x3)) = x32 + [2]x2·x3 + x1·x3 + x22    6.69/2.20
POL(LE(x1, x2)) = [2] + x1    6.69/2.20
POL(MINUS(x1, x2)) = [2]x2    6.69/2.20
POL(c10(x1, x2)) = x1 + x2    6.69/2.20
POL(c2(x1)) = x1    6.69/2.20
POL(c5(x1)) = x1    6.69/2.20
POL(c8(x1, x2)) = x1 + x2    6.69/2.20
POL(c9(x1, x2)) = x1 + x2    6.69/2.20
POL(false) = 0    6.69/2.20
POL(le(x1, x2)) = [1]    6.69/2.20
POL(minus(x1, x2)) = x1    6.69/2.20
POL(pred(x1)) = x1    6.69/2.20
POL(s(x1)) = [2] + x1    6.69/2.20
POL(true) = 0   
6.69/2.20
6.69/2.20

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.69/2.20
le(s(z0), 0) → false 6.69/2.20
le(s(z0), s(z1)) → le(z0, z1) 6.69/2.20
pred(s(z0)) → z0 6.69/2.20
minus(z0, 0) → z0 6.69/2.20
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.69/2.20
gcd(0, z0) → z0 6.69/2.20
gcd(s(z0), 0) → s(z0) 6.69/2.20
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.69/2.20
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.69/2.20
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
S tuples:none
K tuples:

IF_GCD(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.69/2.20
IF_GCD(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.69/2.20
GCD(s(z0), s(z1)) → c8(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.69/2.20
MINUS(z0, s(z1)) → c5(MINUS(z0, z1)) 6.69/2.20
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, pred, minus, gcd, if_gcd

Defined Pair Symbols:

LE, GCD, IF_GCD, MINUS

Compound Symbols:

c2, c8, c9, c10, c5

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

The set S is empty
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(14) BOUNDS(O(1), O(1))

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7.07/2.23 EOF