YES(O(1), O(n^2)) 6.26/2.09 YES(O(1), O(n^2)) 6.73/2.14 6.73/2.14 6.73/2.14 6.73/2.14 6.73/2.14 6.73/2.14 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.73/2.14 6.73/2.14 6.73/2.14
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6.73/2.14
6.73/2.14

(0) Obligation:

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

le(0, y) → true 6.73/2.14
le(s(x), 0) → false 6.73/2.14
le(s(x), s(y)) → le(x, y) 6.73/2.14
minus(x, 0) → x 6.73/2.14
minus(s(x), s(y)) → minus(x, y) 6.73/2.14
gcd(0, y) → y 6.73/2.14
gcd(s(x), 0) → s(x) 6.73/2.14
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) 6.73/2.14
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) 6.73/2.14
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x))

Rewrite Strategy: INNERMOST
6.73/2.14
6.73/2.14

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

Converted CpxTRS to CDT
6.73/2.14
6.73/2.14

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.73/2.14
le(s(z0), 0) → false 6.73/2.14
le(s(z0), s(z1)) → le(z0, z1) 6.73/2.14
minus(z0, 0) → z0 6.73/2.14
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.14
gcd(0, z0) → z0 6.73/2.14
gcd(s(z0), 0) → s(z0) 6.73/2.14
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.73/2.14
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.73/2.14
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.14
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.14
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.14
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.14
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)) 6.73/2.14
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.14
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.14
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.14
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

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6.73/2.14

(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)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.14
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.14
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
We considered the (Usable) Rules:

minus(z0, 0) → z0 6.73/2.14
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.14
le(0, z0) → true 6.73/2.14
le(s(z0), 0) → false 6.73/2.14
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.14
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.14
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.14
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.14
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 : 6.73/2.14

POL(0) = 0    6.73/2.14
POL(GCD(x1, x2)) = [2] + [2]x1 + [2]x2    6.73/2.14
POL(IF_GCD(x1, x2, x3)) = [2]x2 + [2]x3    6.73/2.14
POL(LE(x1, x2)) = 0    6.73/2.14
POL(MINUS(x1, x2)) = [3]    6.73/2.14
POL(c2(x1)) = x1    6.73/2.14
POL(c4(x1)) = x1    6.73/2.14
POL(c7(x1, x2)) = x1 + x2    6.73/2.14
POL(c8(x1, x2)) = x1 + x2    6.73/2.14
POL(c9(x1, x2)) = x1 + x2    6.73/2.14
POL(false) = 0    6.73/2.14
POL(le(x1, x2)) = 0    6.73/2.14
POL(minus(x1, x2)) = [1] + x1    6.73/2.14
POL(s(x1)) = [4] + x1    6.73/2.14
POL(true) = 0   
6.73/2.14
6.73/2.14

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.73/2.14
le(s(z0), 0) → false 6.73/2.14
le(s(z0), s(z1)) → le(z0, z1) 6.73/2.14
minus(z0, 0) → z0 6.73/2.14
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.14
gcd(0, z0) → z0 6.73/2.14
gcd(s(z0), 0) → s(z0) 6.73/2.14
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.73/2.14
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.73/2.14
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.14
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.14
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.14
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.14
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)) 6.73/2.14
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)) 6.73/2.14
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.14
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

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6.73/2.14

(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.

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(z0, 0) → z0 6.73/2.14
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.14
le(0, z0) → true 6.73/2.14
le(s(z0), 0) → false 6.73/2.14
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.14
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.14
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.15
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.15
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 : 6.73/2.15

POL(0) = 0    6.73/2.15
POL(GCD(x1, x2)) = [2] + [2]x1 + [3]x2 + x1·x2    6.73/2.15
POL(IF_GCD(x1, x2, x3)) = [2]x2 + [2]x3 + x2·x3    6.73/2.15
POL(LE(x1, x2)) = 0    6.73/2.15
POL(MINUS(x1, x2)) = x2    6.73/2.15
POL(c2(x1)) = x1    6.73/2.15
POL(c4(x1)) = x1    6.73/2.15
POL(c7(x1, x2)) = x1 + x2    6.73/2.15
POL(c8(x1, x2)) = x1 + x2    6.73/2.15
POL(c9(x1, x2)) = x1 + x2    6.73/2.15
POL(false) = 0    6.73/2.15
POL(le(x1, x2)) = 0    6.73/2.15
POL(minus(x1, x2)) = x1    6.73/2.15
POL(s(x1)) = [2] + x1    6.73/2.15
POL(true) = 0   
6.73/2.15
6.73/2.15

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.73/2.15
le(s(z0), 0) → false 6.73/2.15
le(s(z0), s(z1)) → le(z0, z1) 6.73/2.15
minus(z0, 0) → z0 6.73/2.15
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.15
gcd(0, z0) → z0 6.73/2.15
gcd(s(z0), 0) → s(z0) 6.73/2.15
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.73/2.15
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.73/2.15
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.15
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.15
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.15
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.15
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)) 6.73/2.15
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.15
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.73/2.15
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

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6.73/2.15

(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.

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

minus(z0, 0) → z0 6.73/2.15
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.15
le(0, z0) → true 6.73/2.15
le(s(z0), 0) → false 6.73/2.15
le(s(z0), s(z1)) → le(z0, z1)
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.15
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.15
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.15
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.15
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 : 6.73/2.15

POL(0) = 0    6.73/2.15
POL(GCD(x1, x2)) = [1] + x1 + [2]x2 + x1·x2    6.73/2.15
POL(IF_GCD(x1, x2, x3)) = x2 + x2·x3 + x1·x3    6.73/2.15
POL(LE(x1, x2)) = x1    6.73/2.15
POL(MINUS(x1, x2)) = 0    6.73/2.15
POL(c2(x1)) = x1    6.73/2.15
POL(c4(x1)) = x1    6.73/2.15
POL(c7(x1, x2)) = x1 + x2    6.73/2.15
POL(c8(x1, x2)) = x1 + x2    6.73/2.15
POL(c9(x1, x2)) = x1 + x2    6.73/2.15
POL(false) = [1]    6.73/2.15
POL(le(x1, x2)) = [1]    6.73/2.15
POL(minus(x1, x2)) = x1    6.73/2.15
POL(s(x1)) = [1] + x1    6.73/2.15
POL(true) = [1]   
6.73/2.15
6.73/2.15

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 6.73/2.15
le(s(z0), 0) → false 6.73/2.15
le(s(z0), s(z1)) → le(z0, z1) 6.73/2.15
minus(z0, 0) → z0 6.73/2.15
minus(s(z0), s(z1)) → minus(z0, z1) 6.73/2.15
gcd(0, z0) → z0 6.73/2.15
gcd(s(z0), 0) → s(z0) 6.73/2.15
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1)) 6.73/2.15
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 6.73/2.15
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.73/2.15
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.15
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.73/2.15
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.15
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)) 6.73/2.15
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.73/2.15
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 6.73/2.15
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1)) 6.73/2.15
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

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6.73/2.15

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

The set S is empty
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6.73/2.15

(10) BOUNDS(O(1), O(1))

6.73/2.15
6.73/2.15
6.73/2.19 EOF