YES(O(1), O(n^1)) 3.12/1.22 YES(O(1), O(n^1)) 3.12/1.28 3.12/1.28 3.12/1.28 3.12/1.28 3.12/1.28 3.12/1.28 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 3.12/1.28 3.12/1.28 3.12/1.28
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
3.12/1.28 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
3.12/1.28 
↳2 CdtProblem
3.12/1.28 
↳3 CdtUnreachableProof (⇔)
3.12/1.28 
↳4 CdtProblem
3.12/1.28 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
3.12/1.28 
↳6 CdtProblem
3.12/1.28 
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
3.12/1.28 
↳8 CdtProblem
3.12/1.28 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
3.12/1.28 
↳10 CdtProblem
3.12/1.28 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
3.12/1.28 
↳12 CdtProblem
3.12/1.28 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
3.12/1.28 
↳14 CdtProblem
3.12/1.28 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
3.12/1.28 
↳16 CdtProblem
3.12/1.28 
↳17 SIsEmptyProof (BOTH BOUNDS(ID, ID))
3.12/1.28 
↳18 BOUNDS(O(1), O(1))
3.12/1.28
3.12/1.28 3.12/1.28
3.12/1.28
3.12/1.28

(0) Obligation:

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

minus(n__0, Y) → 0 3.12/1.28
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y)) 3.12/1.28
geq(X, n__0) → true 3.12/1.28
geq(n__0, n__s(Y)) → false 3.12/1.28
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y)) 3.12/1.28
div(0, n__s(Y)) → 0 3.12/1.28
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0) 3.12/1.28
if(true, X, Y) → activate(X) 3.12/1.28
if(false, X, Y) → activate(Y) 3.12/1.28
0 → n__0 3.12/1.28
s(X) → n__s(X) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(X)) → s(X) 3.12/1.28
activate(X) → X

Rewrite Strategy: INNERMOST
3.12/1.28
3.12/1.28

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

Converted CpxTRS to CDT
3.12/1.28
3.12/1.28

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__0, z0) → c(0') 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
DIV(0, n__s(z0)) → c5(0') 3.12/1.28
DIV(s(z0), n__s(z1)) → c6(IF(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0), GEQ(z0, activate(z1)), ACTIVATE(z1), DIV(minus(z0, activate(z1)), n__s(activate(z1))), MINUS(z0, activate(z1)), ACTIVATE(z1), ACTIVATE(z1)) 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11(0') 3.12/1.28
ACTIVATE(n__s(z0)) → c12(S(z0))
S tuples:

MINUS(n__0, z0) → c(0') 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
DIV(0, n__s(z0)) → c5(0') 3.12/1.28
DIV(s(z0), n__s(z1)) → c6(IF(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0), GEQ(z0, activate(z1)), ACTIVATE(z1), DIV(minus(z0, activate(z1)), n__s(activate(z1))), MINUS(z0, activate(z1)), ACTIVATE(z1), ACTIVATE(z1)) 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11(0') 3.12/1.28
ACTIVATE(n__s(z0)) → c12(S(z0))
K tuples:none
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, DIV, IF, ACTIVATE

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c11, c12

3.12/1.28
3.12/1.28

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

DIV(0, n__s(z0)) → c5(0') 3.12/1.28
DIV(s(z0), n__s(z1)) → c6(IF(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0), GEQ(z0, activate(z1)), ACTIVATE(z1), DIV(minus(z0, activate(z1)), n__s(activate(z1))), MINUS(z0, activate(z1)), ACTIVATE(z1), ACTIVATE(z1))
3.12/1.28
3.12/1.28

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__0, z0) → c(0') 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11(0') 3.12/1.28
ACTIVATE(n__s(z0)) → c12(S(z0))
S tuples:

MINUS(n__0, z0) → c(0') 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11(0') 3.12/1.28
ACTIVATE(n__s(z0)) → c12(S(z0))
K tuples:none
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, IF, ACTIVATE

Compound Symbols:

c, c1, c4, c7, c8, c11, c12

3.12/1.28
3.12/1.28

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

Removed 3 trailing tuple parts
3.12/1.28
3.12/1.28

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
S tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
K tuples:none
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, IF, ACTIVATE

Compound Symbols:

c1, c4, c7, c8, c, c11, c12

3.12/1.28
3.12/1.28

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

Removed 5 trailing nodes:

IF(false, z0, z1) → c8(ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
IF(true, z0, z1) → c7(ACTIVATE(z0)) 3.12/1.28
ACTIVATE(n__s(z0)) → c12 3.12/1.28
ACTIVATE(n__0) → c11
3.12/1.28
3.12/1.28

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
S tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
K tuples:none
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, ACTIVATE

Compound Symbols:

c1, c4, c, c11, c12

3.12/1.28
3.12/1.28

(9) 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.

MINUS(n__0, z0) → c
We considered the (Usable) Rules:

activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
0 → n__0
And the Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
The order we found is given by the following interpretation:
Polynomial interpretation : 3.12/1.28

POL(0) = [4]    3.12/1.28
POL(ACTIVATE(x1)) = 0    3.12/1.28
POL(GEQ(x1, x2)) = 0    3.12/1.28
POL(MINUS(x1, x2)) = [3]    3.12/1.28
POL(activate(x1)) = [2] + [4]x1    3.12/1.28
POL(c) = 0    3.12/1.28
POL(c1(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(c11) = 0    3.12/1.28
POL(c12) = 0    3.12/1.28
POL(c4(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(n__0) = [2]    3.12/1.28
POL(n__s(x1)) = [2]    3.12/1.28
POL(s(x1)) = [4]   
3.12/1.28
3.12/1.28

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
S tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
K tuples:

MINUS(n__0, z0) → c
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, ACTIVATE

Compound Symbols:

c1, c4, c, c11, c12

3.12/1.28
3.12/1.28

(11) 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.

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
0 → n__0
And the Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
The order we found is given by the following interpretation:
Polynomial interpretation : 3.12/1.28

POL(0) = 0    3.12/1.28
POL(ACTIVATE(x1)) = 0    3.12/1.28
POL(GEQ(x1, x2)) = 0    3.12/1.28
POL(MINUS(x1, x2)) = x1 + x2    3.12/1.28
POL(activate(x1)) = x1    3.12/1.28
POL(c) = 0    3.12/1.28
POL(c1(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(c11) = 0    3.12/1.28
POL(c12) = 0    3.12/1.28
POL(c4(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(n__0) = 0    3.12/1.28
POL(n__s(x1)) = [1] + x1    3.12/1.28
POL(s(x1)) = [1] + x1   
3.12/1.28
3.12/1.28

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
S tuples:

GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
K tuples:

MINUS(n__0, z0) → c 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, ACTIVATE

Compound Symbols:

c1, c4, c, c11, c12

3.12/1.28
3.12/1.28

(13) 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.

ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
We considered the (Usable) Rules:

activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
0 → n__0
And the Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
The order we found is given by the following interpretation:
Polynomial interpretation : 3.12/1.28

POL(0) = 0    3.12/1.28
POL(ACTIVATE(x1)) = [2]    3.12/1.28
POL(GEQ(x1, x2)) = [4]x2    3.12/1.28
POL(MINUS(x1, x2)) = [4]x2    3.12/1.28
POL(activate(x1)) = x1    3.12/1.28
POL(c) = 0    3.12/1.28
POL(c1(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(c11) = 0    3.12/1.28
POL(c12) = 0    3.12/1.28
POL(c4(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(n__0) = 0    3.12/1.28
POL(n__s(x1)) = [1] + x1    3.12/1.28
POL(s(x1)) = [1] + x1   
3.12/1.28
3.12/1.28

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
S tuples:

GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:

MINUS(n__0, z0) → c 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, ACTIVATE

Compound Symbols:

c1, c4, c, c11, c12

3.12/1.28
3.12/1.28

(15) 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.

GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
0 → n__0
And the Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
The order we found is given by the following interpretation:
Polynomial interpretation : 3.12/1.28

POL(0) = 0    3.12/1.28
POL(ACTIVATE(x1)) = 0    3.12/1.28
POL(GEQ(x1, x2)) = [3]x1 + [4]x2    3.12/1.28
POL(MINUS(x1, x2)) = 0    3.12/1.28
POL(activate(x1)) = x1    3.12/1.28
POL(c) = 0    3.12/1.28
POL(c1(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(c11) = 0    3.12/1.28
POL(c12) = 0    3.12/1.28
POL(c4(x1, x2, x3)) = x1 + x2 + x3    3.12/1.28
POL(n__0) = 0    3.12/1.28
POL(n__s(x1)) = [5] + x1    3.12/1.28
POL(s(x1)) = [5] + x1   
3.12/1.28
3.12/1.28

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0 3.12/1.28
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1)) 3.12/1.28
geq(z0, n__0) → true 3.12/1.28
geq(n__0, n__s(z0)) → false 3.12/1.28
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1)) 3.12/1.28
div(0, n__s(z0)) → 0 3.12/1.28
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0) 3.12/1.28
if(true, z0, z1) → activate(z0) 3.12/1.28
if(false, z0, z1) → activate(z1) 3.12/1.28
0 → n__0 3.12/1.28
s(z0) → n__s(z0) 3.12/1.28
activate(n__0) → 0 3.12/1.28
activate(n__s(z0)) → s(z0) 3.12/1.28
activate(z0) → z0
Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
MINUS(n__0, z0) → c 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12
S tuples:none
K tuples:

MINUS(n__0, z0) → c 3.12/1.28
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 3.12/1.28
ACTIVATE(n__0) → c11 3.12/1.28
ACTIVATE(n__s(z0)) → c12 3.12/1.28
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, ACTIVATE

Compound Symbols:

c1, c4, c, c11, c12

3.12/1.28
3.12/1.28

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

The set S is empty
3.12/1.28
3.12/1.28

(18) BOUNDS(O(1), O(1))

3.12/1.28
3.12/1.28
3.56/1.42 EOF