YES(O(1), O(n^1)) 2.69/1.18 YES(O(1), O(n^1)) 2.69/1.19 2.69/1.19 2.69/1.19 2.69/1.19 2.69/1.19 2.69/1.19 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.69/1.19 2.69/1.19 2.69/1.19
2.69/1.19
2.69/1.19
2.69/1.19
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
2.69/1.19 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳2 CdtProblem
2.69/1.19 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳4 CdtProblem
2.69/1.19 
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳6 CdtProblem
2.69/1.19 
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳8 CdtProblem
2.69/1.19 
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳10 CdtProblem
2.69/1.19 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.69/1.19 
↳12 CdtProblem
2.69/1.19 
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳14 CdtProblem
2.69/1.19 
↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
2.69/1.19 
↳16 CdtProblem
2.69/1.19 
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.69/1.19 
↳18 CdtProblem
2.69/1.19 
↳19 CdtKnowledgeProof (⇔)
2.69/1.19 
↳20 BOUNDS(O(1), O(1))
2.69/1.19
2.69/1.19 2.69/1.19
2.69/1.19
2.69/1.19

(0) Obligation:

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

is_empty(nil) → true 2.69/1.19
is_empty(cons(x, l)) → false 2.69/1.19
hd(cons(x, l)) → x 2.69/1.19
tl(cons(x, l)) → l 2.69/1.19
append(l1, l2) → ifappend(l1, l2, is_empty(l1)) 2.69/1.19
ifappend(l1, l2, true) → l2 2.69/1.19
ifappend(l1, l2, false) → cons(hd(l1), append(tl(l1), l2))

Rewrite Strategy: INNERMOST
2.69/1.19
2.69/1.19

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

Converted CpxTRS to CDT
2.69/1.19
2.69/1.19

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 2.69/1.19
is_empty(cons(z0, z1)) → false 2.69/1.19
hd(cons(z0, z1)) → z0 2.69/1.19
tl(cons(z0, z1)) → z1 2.69/1.19
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 2.69/1.19
ifappend(z0, z1, true) → z1 2.69/1.19
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0)) 2.69/1.19
IFAPPEND(z0, z1, false) → c6(HD(z0), APPEND(tl(z0), z1), TL(z0))
S tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0)) 2.69/1.20
IFAPPEND(z0, z1, false) → c6(HD(z0), APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

2.69/1.20
2.69/1.20

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

Removed 3 trailing tuple parts
2.69/1.20
2.69/1.20

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 2.69/1.20
is_empty(cons(z0, z1)) → false 2.69/1.20
hd(cons(z0, z1)) → z0 2.69/1.20
tl(cons(z0, z1)) → z1 2.69/1.20
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 2.69/1.20
ifappend(z0, z1, true) → z1 2.69/1.20
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0))) 2.69/1.20
IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1))
S tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0))) 2.69/1.20
IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

2.69/1.20
2.69/1.20

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

Use narrowing to replace APPEND(z0, z1) → c4(IFAPPEND(z0, z1, is_empty(z0))) by

APPEND(nil, x1) → c4(IFAPPEND(nil, x1, true)) 2.69/1.20
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
2.69/1.20
2.69/1.20

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 2.69/1.20
is_empty(cons(z0, z1)) → false 2.69/1.20
hd(cons(z0, z1)) → z0 2.69/1.20
tl(cons(z0, z1)) → z1 2.69/1.20
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 2.69/1.20
ifappend(z0, z1, true) → z1 2.69/1.20
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 2.69/1.20
APPEND(nil, x1) → c4(IFAPPEND(nil, x1, true)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(nil, x1) → c4(IFAPPEND(nil, x1, true)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4

3.03/1.21
3.03/1.21

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

Removed 1 trailing tuple parts
3.03/1.21
3.03/1.21

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 3.03/1.21
is_empty(cons(z0, z1)) → false 3.03/1.21
hd(cons(z0, z1)) → z0 3.03/1.21
tl(cons(z0, z1)) → z1 3.03/1.21
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 3.03/1.21
ifappend(z0, z1, true) → z1 3.03/1.21
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4, c4

3.03/1.21
3.03/1.21

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

Removed 1 trailing nodes:

APPEND(nil, x1) → c4
3.03/1.21
3.03/1.21

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 3.03/1.21
is_empty(cons(z0, z1)) → false 3.03/1.21
hd(cons(z0, z1)) → z0 3.03/1.21
tl(cons(z0, z1)) → z1 3.03/1.21
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 3.03/1.21
ifappend(z0, z1, true) → z1 3.03/1.21
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4
K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4, c4

3.03/1.21
3.03/1.21

(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.

APPEND(nil, x1) → c4
We considered the (Usable) Rules:

tl(cons(z0, z1)) → z1
And the Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 3.03/1.21

POL(APPEND(x1, x2)) = [1]    3.03/1.21
POL(IFAPPEND(x1, x2, x3)) = [1]    3.03/1.21
POL(c4) = 0    3.03/1.21
POL(c4(x1)) = x1    3.03/1.21
POL(c6(x1)) = x1    3.03/1.21
POL(cons(x1, x2)) = x1    3.03/1.21
POL(false) = [1]    3.03/1.21
POL(nil) = [1]    3.03/1.21
POL(tl(x1)) = [1]   
3.03/1.21
3.03/1.21

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 3.03/1.21
is_empty(cons(z0, z1)) → false 3.03/1.21
hd(cons(z0, z1)) → z0 3.03/1.21
tl(cons(z0, z1)) → z1 3.03/1.21
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 3.03/1.21
ifappend(z0, z1, true) → z1 3.03/1.21
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4
S tuples:

IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) 3.03/1.21
APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:

APPEND(nil, x1) → c4
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

IFAPPEND, APPEND

Compound Symbols:

c6, c4, c4

3.03/1.21
3.03/1.21

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

Use narrowing to replace IFAPPEND(z0, z1, false) → c6(APPEND(tl(z0), z1)) by

IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
3.03/1.21
3.03/1.21

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 3.03/1.21
is_empty(cons(z0, z1)) → false 3.03/1.21
hd(cons(z0, z1)) → z0 3.03/1.21
tl(cons(z0, z1)) → z1 3.03/1.21
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 3.03/1.21
ifappend(z0, z1, true) → z1 3.03/1.21
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
S tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
K tuples:

APPEND(nil, x1) → c4
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c4, c6

3.03/1.21
3.03/1.21

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

Removed 1 trailing nodes:

APPEND(nil, x1) → c4
3.03/1.21
3.03/1.21

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 3.03/1.21
is_empty(cons(z0, z1)) → false 3.03/1.21
hd(cons(z0, z1)) → z0 3.03/1.21
tl(cons(z0, z1)) → z1 3.03/1.21
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 3.03/1.21
ifappend(z0, z1, true) → z1 3.03/1.21
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
S tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
K tuples:

APPEND(nil, x1) → c4
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c4, c6

3.03/1.21
3.03/1.21

(17) 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.

IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
We considered the (Usable) Rules:none
And the Tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
The order we found is given by the following interpretation:
Polynomial interpretation : 3.03/1.21

POL(APPEND(x1, x2)) = x1    3.03/1.21
POL(IFAPPEND(x1, x2, x3)) = x1    3.03/1.21
POL(c4) = 0    3.03/1.21
POL(c4(x1)) = x1    3.03/1.21
POL(c6(x1)) = x1    3.03/1.21
POL(cons(x1, x2)) = [1] + x2    3.03/1.21
POL(false) = [1]    3.03/1.21
POL(nil) = 0   
3.03/1.21
3.03/1.21

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true 3.03/1.21
is_empty(cons(z0, z1)) → false 3.03/1.21
hd(cons(z0, z1)) → z0 3.03/1.21
tl(cons(z0, z1)) → z1 3.03/1.21
append(z0, z1) → ifappend(z0, z1, is_empty(z0)) 3.03/1.21
ifappend(z0, z1, true) → z1 3.03/1.21
ifappend(z0, z1, false) → cons(hd(z0), append(tl(z0), z1))
Tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
APPEND(nil, x1) → c4 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
S tuples:

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:

APPEND(nil, x1) → c4 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c4, c6

3.03/1.21
3.03/1.21

(19) CdtKnowledgeProof (EQUIVALENT transformation)

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

APPEND(cons(z0, z1), x1) → c4(IFAPPEND(cons(z0, z1), x1, false)) 3.03/1.21
IFAPPEND(cons(z0, z1), x1, false) → c6(APPEND(z1, x1))
Now S is empty
3.03/1.21
3.03/1.21

(20) BOUNDS(O(1), O(1))

3.03/1.21
3.03/1.21
3.03/1.30 EOF