YES(O(1), O(n^2)) 7.09/2.27 YES(O(1), O(n^2)) 7.51/2.33 7.51/2.33 7.51/2.33 7.51/2.33 7.51/2.33 7.51/2.33 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 7.51/2.33 7.51/2.33 7.51/2.33
7.51/2.33
7.51/2.33
7.51/2.33
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
7.51/2.33 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
7.51/2.33 
↳2 CdtProblem
7.51/2.33 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
7.51/2.33 
↳4 CdtProblem
7.51/2.33 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
7.51/2.33 
↳6 CdtProblem
7.51/2.33 
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
7.51/2.33 
↳8 CdtProblem
7.51/2.33 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
7.51/2.33 
↳10 CdtProblem
7.51/2.33 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
7.51/2.33 
↳12 CdtProblem
7.51/2.33 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
7.51/2.33 
↳14 BOUNDS(O(1), O(1))
7.51/2.33
7.51/2.33 7.51/2.33
7.51/2.33
7.51/2.33

(0) Obligation:

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

minus(X, s(Y)) → pred(minus(X, Y)) 7.51/2.33
minus(X, 0) → X 7.51/2.33
pred(s(X)) → X 7.51/2.33
le(s(X), s(Y)) → le(X, Y) 7.51/2.33
le(s(X), 0) → false 7.51/2.33
le(0, Y) → true 7.51/2.33
gcd(0, Y) → 0 7.51/2.33
gcd(s(X), 0) → s(X) 7.51/2.33
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y)) 7.51/2.33
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y)) 7.51/2.33
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Rewrite Strategy: INNERMOST
7.51/2.33
7.51/2.33

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

Converted CpxTRS to CDT
7.51/2.33
7.51/2.33

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.33
minus(z0, 0) → z0 7.51/2.33
pred(s(z0)) → z0 7.51/2.33
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.33
le(s(z0), 0) → false 7.51/2.33
le(0, z0) → true 7.51/2.33
gcd(0, z0) → 0 7.51/2.33
gcd(s(z0), 0) → s(z0) 7.51/2.33
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.33
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.33
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1)) 7.51/2.33
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.33
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.33
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.33
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
S tuples:

MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1)) 7.51/2.33
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.33
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.33
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.33
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

MINUS, LE, GCD, IF

Compound Symbols:

c, c3, c8, c9, c10

7.51/2.33
7.51/2.33

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

Removed 1 trailing tuple parts
7.51/2.33
7.51/2.33

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.33
minus(z0, 0) → z0 7.51/2.33
pred(s(z0)) → z0 7.51/2.33
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.33
le(s(z0), 0) → false 7.51/2.33
le(0, z0) → true 7.51/2.33
gcd(0, z0) → 0 7.51/2.33
gcd(s(z0), 0) → s(z0) 7.51/2.33
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

7.51/2.34
7.51/2.34

(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(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
We considered the (Usable) Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true
And the Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.51/2.34

POL(0) = [3]    7.51/2.34
POL(GCD(x1, x2)) = [4]x1 + [4]x2    7.51/2.34
POL(IF(x1, x2, x3)) = [4]x2 + [4]x3    7.51/2.34
POL(LE(x1, x2)) = 0    7.51/2.34
POL(MINUS(x1, x2)) = 0    7.51/2.34
POL(c(x1)) = x1    7.51/2.34
POL(c10(x1, x2)) = x1 + x2    7.51/2.34
POL(c3(x1)) = x1    7.51/2.34
POL(c8(x1, x2)) = x1 + x2    7.51/2.34
POL(c9(x1, x2)) = x1 + x2    7.51/2.34
POL(false) = 0    7.51/2.34
POL(le(x1, x2)) = 0    7.51/2.34
POL(minus(x1, x2)) = x1    7.51/2.34
POL(pred(x1)) = x1    7.51/2.34
POL(s(x1)) = [4] + x1    7.51/2.34
POL(true) = 0   
7.51/2.34
7.51/2.34

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

7.51/2.34
7.51/2.34

(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(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
7.51/2.34
7.51/2.34

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

7.51/2.34
7.51/2.34

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

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true
And the Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.51/2.34

POL(0) = 0    7.51/2.34
POL(GCD(x1, x2)) = [2] + x1·x2    7.51/2.34
POL(IF(x1, x2, x3)) = x2·x3    7.51/2.34
POL(LE(x1, x2)) = 0    7.51/2.34
POL(MINUS(x1, x2)) = [2]x2    7.51/2.34
POL(c(x1)) = x1    7.51/2.34
POL(c10(x1, x2)) = x1 + x2    7.51/2.34
POL(c3(x1)) = x1    7.51/2.34
POL(c8(x1, x2)) = x1 + x2    7.51/2.34
POL(c9(x1, x2)) = x1 + x2    7.51/2.34
POL(false) = 0    7.51/2.34
POL(le(x1, x2)) = 0    7.51/2.34
POL(minus(x1, x2)) = x1    7.51/2.34
POL(pred(x1)) = x1    7.51/2.34
POL(s(x1)) = [2] + x1    7.51/2.34
POL(true) = 0   
7.51/2.34
7.51/2.34

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

7.51/2.34
7.51/2.34

(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)) → c3(LE(z0, z1))
We considered the (Usable) Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true
And the Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.51/2.34

POL(0) = 0    7.51/2.34
POL(GCD(x1, x2)) = [2]x1 + x22 + x12    7.51/2.34
POL(IF(x1, x2, x3)) = [1] + x32 + x22    7.51/2.34
POL(LE(x1, x2)) = [3] + x2    7.51/2.34
POL(MINUS(x1, x2)) = 0    7.51/2.34
POL(c(x1)) = x1    7.51/2.34
POL(c10(x1, x2)) = x1 + x2    7.51/2.34
POL(c3(x1)) = x1    7.51/2.34
POL(c8(x1, x2)) = x1 + x2    7.51/2.34
POL(c9(x1, x2)) = x1 + x2    7.51/2.34
POL(false) = 0    7.51/2.34
POL(le(x1, x2)) = 0    7.51/2.34
POL(minus(x1, x2)) = x1    7.51/2.34
POL(pred(x1)) = x1    7.51/2.34
POL(s(x1)) = [2] + x1    7.51/2.34
POL(true) = 0   
7.51/2.34
7.51/2.34

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
S tuples:none
K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1)) 7.51/2.34
LE(s(z0), s(z1)) → c3(LE(z0, z1))
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

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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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7.51/2.37 EOF