YES(O(1), O(n^2)) 20.13/6.14 YES(O(1), O(n^2)) 20.13/6.17 20.13/6.17 20.13/6.17 20.13/6.17 20.13/6.17 20.13/6.17 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 20.13/6.17 20.13/6.17 20.13/6.17
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20.13/6.17
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
20.13/6.17 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
20.13/6.17 
↳2 CdtProblem
20.13/6.17 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
20.13/6.17 
↳4 CdtProblem
20.13/6.17 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
20.13/6.17 
↳6 CdtProblem
20.13/6.17 
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
20.13/6.17 
↳8 CdtProblem
20.13/6.17 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
20.13/6.17 
↳10 CdtProblem
20.13/6.17 
↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
20.13/6.17 
↳12 CdtProblem
20.13/6.17 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
20.13/6.17 
↳14 CdtProblem
20.13/6.17 
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID))
20.13/6.17 
↳16 BOUNDS(O(1), O(1))
20.13/6.17
20.13/6.17 20.13/6.17
20.13/6.17
20.13/6.17

(0) Obligation:

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

lt(0, s(X)) → true 20.13/6.17
lt(s(X), 0) → false 20.13/6.17
lt(s(X), s(Y)) → lt(X, Y) 20.13/6.17
append(nil, Y) → Y 20.13/6.17
append(add(N, X), Y) → add(N, append(X, Y)) 20.13/6.17
split(N, nil) → pair(nil, nil) 20.13/6.17
split(N, add(M, Y)) → f_1(split(N, Y), N, M, Y) 20.13/6.17
f_1(pair(X, Z), N, M, Y) → f_2(lt(N, M), N, M, Y, X, Z) 20.13/6.17
f_2(true, N, M, Y, X, Z) → pair(X, add(M, Z)) 20.13/6.17
f_2(false, N, M, Y, X, Z) → pair(add(M, X), Z) 20.13/6.17
qsort(nil) → nil 20.13/6.17
qsort(add(N, X)) → f_3(split(N, X), N, X) 20.13/6.17
f_3(pair(Y, Z), N, X) → append(qsort(Y), add(X, qsort(Z)))

Rewrite Strategy: INNERMOST
20.13/6.17
20.13/6.17

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

Converted CpxTRS to CDT
20.13/6.17
20.13/6.17

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.17
lt(s(z0), 0) → false 20.13/6.17
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.17
append(nil, z0) → z0 20.13/6.17
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.17
split(z0, nil) → pair(nil, nil) 20.13/6.17
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.17
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.17
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.17
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.17
qsort(nil) → nil 20.13/6.17
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.17
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.17
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.17
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.17
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3)) 20.13/6.17
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.17
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.17
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.17
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.17
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3)) 20.13/6.17
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.17
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
K tuples:none
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, F_1, QSORT, F_3

Compound Symbols:

c2, c4, c6, c7, c11, c12

20.13/6.17
20.13/6.17

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

Removed 1 trailing tuple parts
20.13/6.17
20.13/6.17

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.17
lt(s(z0), 0) → false 20.13/6.17
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.17
append(nil, z0) → z0 20.13/6.17
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.17
split(z0, nil) → pair(nil, nil) 20.13/6.17
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.17
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.17
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.17
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.17
qsort(nil) → nil 20.13/6.17
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.17
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.17
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.17
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.17
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.17
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.17
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
K tuples:none
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c7

20.13/6.18
20.13/6.18

(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.

QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
We considered the (Usable) Rules:

qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1))) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 20.13/6.18

POL(0) = 0    20.13/6.18
POL(APPEND(x1, x2)) = 0    20.13/6.18
POL(F_1(x1, x2, x3, x4)) = 0    20.13/6.18
POL(F_3(x1, x2, x3)) = [4]x1    20.13/6.18
POL(LT(x1, x2)) = 0    20.13/6.18
POL(QSORT(x1)) = [4]x1    20.13/6.18
POL(SPLIT(x1, x2)) = [3]    20.13/6.18
POL(add(x1, x2)) = [1] + x2    20.13/6.18
POL(append(x1, x2)) = [3]    20.13/6.18
POL(c11(x1, x2)) = x1 + x2    20.13/6.18
POL(c12(x1, x2, x3)) = x1 + x2 + x3    20.13/6.18
POL(c2(x1)) = x1    20.13/6.18
POL(c4(x1)) = x1    20.13/6.18
POL(c6(x1, x2)) = x1 + x2    20.13/6.18
POL(c7(x1)) = x1    20.13/6.18
POL(f_1(x1, x2, x3, x4)) = [1] + x1    20.13/6.18
POL(f_2(x1, x2, x3, x4, x5, x6)) = x1 + x5 + x6    20.13/6.18
POL(f_3(x1, x2, x3)) = [3] + [3]x2 + [3]x3    20.13/6.18
POL(false) = [1]    20.13/6.18
POL(lt(x1, x2)) = [1]    20.13/6.18
POL(nil) = 0    20.13/6.18
POL(pair(x1, x2)) = x1 + x2    20.13/6.18
POL(qsort(x1)) = 0    20.13/6.18
POL(s(x1)) = x1    20.13/6.18
POL(split(x1, x2)) = x2    20.13/6.18
POL(true) = [1]   
20.13/6.18
20.13/6.18

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.18
qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
K tuples:

QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c7

20.13/6.18
20.13/6.18

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

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

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
20.13/6.18
20.13/6.18

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.18
qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
K tuples:

QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c7

20.13/6.18
20.13/6.18

(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.

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
We considered the (Usable) Rules:

qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1))) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 20.13/6.18

POL(0) = [3]    20.13/6.18
POL(APPEND(x1, x2)) = 0    20.13/6.18
POL(F_1(x1, x2, x3, x4)) = x3    20.13/6.18
POL(F_3(x1, x2, x3)) = [3]x1 + [2]x12    20.13/6.18
POL(LT(x1, x2)) = x2    20.13/6.18
POL(QSORT(x1)) = [2]x12    20.13/6.18
POL(SPLIT(x1, x2)) = [3] + x2    20.13/6.18
POL(add(x1, x2)) = [2] + x1 + x2    20.13/6.18
POL(append(x1, x2)) = [3] + [3]x22 + [3]x1·x2 + [3]x12    20.13/6.18
POL(c11(x1, x2)) = x1 + x2    20.13/6.18
POL(c12(x1, x2, x3)) = x1 + x2 + x3    20.13/6.18
POL(c2(x1)) = x1    20.13/6.18
POL(c4(x1)) = x1    20.13/6.18
POL(c6(x1, x2)) = x1 + x2    20.13/6.18
POL(c7(x1)) = x1    20.13/6.18
POL(f_1(x1, x2, x3, x4)) = [2] + x1 + x3    20.13/6.18
POL(f_2(x1, x2, x3, x4, x5, x6)) = [2] + x3 + x5 + x6    20.13/6.18
POL(f_3(x1, x2, x3)) = [3] + [3]x2 + [3]x3 + [3]x32 + [3]x2·x3 + [3]x22    20.13/6.18
POL(false) = 0    20.13/6.18
POL(lt(x1, x2)) = 0    20.13/6.18
POL(nil) = 0    20.13/6.18
POL(pair(x1, x2)) = x1 + x2    20.13/6.18
POL(qsort(x1)) = 0    20.13/6.18
POL(s(x1)) = [2] + x1    20.13/6.18
POL(split(x1, x2)) = [1] + x2    20.13/6.18
POL(true) = 0   
20.13/6.18
20.13/6.18

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.18
qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
S tuples:

APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
K tuples:

QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c7

20.13/6.18
20.13/6.18

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

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

F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3)) 20.13/6.18
LT(s(z0), s(z1)) → c2(LT(z0, z1))
20.13/6.18
20.13/6.18

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.18
qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
S tuples:

APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
K tuples:

QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c7

20.13/6.18
20.13/6.18

(13) 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.

APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
We considered the (Usable) Rules:

qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1))) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4)
And the Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 20.13/6.18

POL(0) = [3]    20.13/6.18
POL(APPEND(x1, x2)) = [2]x1    20.13/6.18
POL(F_1(x1, x2, x3, x4)) = 0    20.13/6.18
POL(F_3(x1, x2, x3)) = [3] + x1 + x3 + [3]x12    20.13/6.18
POL(LT(x1, x2)) = 0    20.13/6.18
POL(QSORT(x1)) = [3] + [2]x1 + [3]x12    20.13/6.18
POL(SPLIT(x1, x2)) = [1]    20.13/6.18
POL(add(x1, x2)) = [1] + x2    20.13/6.18
POL(append(x1, x2)) = x1 + x2    20.13/6.18
POL(c11(x1, x2)) = x1 + x2    20.13/6.18
POL(c12(x1, x2, x3)) = x1 + x2 + x3    20.13/6.18
POL(c2(x1)) = x1    20.13/6.18
POL(c4(x1)) = x1    20.13/6.18
POL(c6(x1, x2)) = x1 + x2    20.13/6.18
POL(c7(x1)) = x1    20.13/6.18
POL(f_1(x1, x2, x3, x4)) = [1] + x1    20.13/6.18
POL(f_2(x1, x2, x3, x4, x5, x6)) = [2] + x5 + x6    20.13/6.18
POL(f_3(x1, x2, x3)) = [2]x1    20.13/6.18
POL(false) = 0    20.13/6.18
POL(lt(x1, x2)) = 0    20.13/6.18
POL(nil) = 0    20.13/6.18
POL(pair(x1, x2)) = [1] + x1 + x2    20.13/6.18
POL(qsort(x1)) = [2]x1    20.13/6.18
POL(s(x1)) = 0    20.13/6.18
POL(split(x1, x2)) = [1] + x2    20.13/6.18
POL(true) = 0   
20.13/6.18
20.13/6.18

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true 20.13/6.18
lt(s(z0), 0) → false 20.13/6.18
lt(s(z0), s(z1)) → lt(z0, z1) 20.13/6.18
append(nil, z0) → z0 20.13/6.18
append(add(z0, z1), z2) → add(z0, append(z1, z2)) 20.13/6.18
split(z0, nil) → pair(nil, nil) 20.13/6.18
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2) 20.13/6.18
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1) 20.13/6.18
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4)) 20.13/6.18
f_2(false, z0, z1, z2, z3, z4) → pair(add(z1, z3), z4) 20.13/6.18
qsort(nil) → nil 20.13/6.18
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1) 20.13/6.18
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3))
S tuples:none
K tuples:

QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1)) 20.13/6.18
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1)) 20.13/6.18
LT(s(z0), s(z1)) → c2(LT(z0, z1)) 20.13/6.18
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) 20.13/6.18
F_1(pair(z0, z1), z2, z3, z4) → c7(LT(z2, z3)) 20.13/6.18
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
Defined Rule Symbols:

lt, append, split, f_1, f_2, qsort, f_3

Defined Pair Symbols:

LT, APPEND, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c7

20.13/6.18
20.13/6.18

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

The set S is empty
20.13/6.18
20.13/6.18

(16) BOUNDS(O(1), O(1))

20.13/6.18
20.13/6.18
20.89/6.24 EOF