The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 650 ms)
↳4 CdtProblem
↳5 CdtKnowledgeProof (⇔, 0 ms)
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 11 ms)
↳8 CdtProblem
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 3300 ms)
↳12 CdtProblem
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 3055 ms)
↳14 CdtProblem
↳15 CdtKnowledgeProof (⇔, 0 ms)
↳16 CdtProblem
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 4040 ms)
↳18 CdtProblem
↳19 SIsEmptyProof (⇔, 0 ms)
↳20 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
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))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
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

(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.

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
We considered the (Usable) Rules:

qsort(nil) → nil
qsort(add(z0, z1)) → f_3(split(z0, z1), z0, z1)
split(z0, nil) → pair(nil, nil)
split(z0, add(z1, z2)) → f_1(split(z0, z2), z0, z1, z2)
f_3(pair(z0, z1), z2, z3) → append(qsort(z0), add(z3, qsort(z1)))
append(nil, z0) → z0
append(add(z0, z1), z2) → add(z0, append(z1, z2))
f_1(pair(z0, z1), z2, z3, z4) → f_2(lt(z2, z3), z2, z3, z4, z0, z1)
lt(0, s(z0)) → true
lt(s(z0), 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
f_2(true, z0, z1, z2, z3, z4) → pair(z3, add(z1, z4))
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))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(APPEND(x1, x2)) = 0   
POL(F_1(x1, x2, x3, x4)) = 0   
POL(F_2(x1, x2, x3, x4, x5, x6)) = 0   
POL(F_3(x1, x2, x3)) = [4] + [4]x1   
POL(LT(x1, x2)) = 0   
POL(QSORT(x1)) = [4]x1   
POL(SPLIT(x1, x2)) = 0   
POL(add(x1, x2)) = [1] + x2   
POL(append(x1, x2)) = 0   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2, x3)) = x1 + x2 + x3   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(f_1(x1, x2, x3, x4)) = [1] + x1   
POL(f_2(x1, x2, x3, x4, x5, x6)) = [1] + x5 + x6   
POL(f_3(x1, x2, x3)) = [3] + [2]x2 + [3]x3   
POL(false) = 0   
POL(lt(x1, x2)) = 0   
POL(nil) = 0   
POL(pair(x1, x2)) = x1 + x2   
POL(qsort(x1)) = 0   
POL(s(x1)) = 0   
POL(split(x1, x2)) = x2   
POL(true) = 0   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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, F_1, QSORT, F_3

Compound Symbols:

c2, c4, c6, c7, c11, c12

(5) CdtKnowledgeProof (EQUIVALENT transformation)

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
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, F_1, QSORT, F_3

Compound Symbols:

c2, c4, c6, c7, c11, c12

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

Use narrowing to replace SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2)) by

SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
SPLIT(x0, add(x1, x2)) → c6

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
SPLIT(x0, add(x1, x2)) → c6
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
SPLIT(x0, add(x1, x2)) → c6
K tuples:

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
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, F_1, QSORT, F_3, SPLIT

Compound Symbols:

c2, c4, c7, c11, c12, c6, c6

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

Removed 1 trailing nodes:

SPLIT(x0, add(x1, x2)) → c6

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
K tuples:

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
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, F_1, QSORT, F_3, SPLIT

Compound Symbols:

c2, c4, c7, c11, c12, c6

(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.

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

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(APPEND(x1, x2)) = 0   
POL(F_1(x1, x2, x3, x4)) = x2·x3   
POL(F_2(x1, x2, x3, x4, x5, x6)) = 0   
POL(F_3(x1, x2, x3)) = x12   
POL(LT(x1, x2)) = x1·x2   
POL(QSORT(x1)) = x12   
POL(SPLIT(x1, x2)) = x1·x2   
POL(add(x1, x2)) = x1 + x2   
POL(append(x1, x2)) = 0   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2, x3)) = x1 + x2 + x3   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(f_1(x1, x2, x3, x4)) = x1 + x3   
POL(f_2(x1, x2, x3, x4, x5, x6)) = x3 + x5 + x6   
POL(f_3(x1, x2, x3)) = 0   
POL(false) = 0   
POL(lt(x1, x2)) = 0   
POL(nil) = 0   
POL(pair(x1, x2)) = x1 + x2   
POL(qsort(x1)) = 0   
POL(s(x1)) = [2] + x1   
POL(split(x1, x2)) = x2   
POL(true) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
S tuples:

APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
K tuples:

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

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

Defined Pair Symbols:

LT, APPEND, F_1, QSORT, F_3, SPLIT

Compound Symbols:

c2, c4, c7, c11, c12, c6

(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.

SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
We considered the (Usable) Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(APPEND(x1, x2)) = 0   
POL(F_1(x1, x2, x3, x4)) = 0   
POL(F_2(x1, x2, x3, x4, x5, x6)) = 0   
POL(F_3(x1, x2, x3)) = [1] + x12   
POL(LT(x1, x2)) = 0   
POL(QSORT(x1)) = x12   
POL(SPLIT(x1, x2)) = x2   
POL(add(x1, x2)) = [1] + x2   
POL(append(x1, x2)) = 0   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2, x3)) = x1 + x2 + x3   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(f_1(x1, x2, x3, x4)) = [1] + x1   
POL(f_2(x1, x2, x3, x4, x5, x6)) = [1] + x5 + x6   
POL(f_3(x1, x2, x3)) = 0   
POL(false) = 0   
POL(lt(x1, x2)) = 0   
POL(nil) = 0   
POL(pair(x1, x2)) = x1 + x2   
POL(qsort(x1)) = 0   
POL(s(x1)) = 0   
POL(split(x1, x2)) = x2   
POL(true) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
S tuples:

APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
K tuples:

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
LT(s(z0), s(z1)) → c2(LT(z0, z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
Defined Rule Symbols:

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

Defined Pair Symbols:

LT, APPEND, F_1, QSORT, F_3, SPLIT

Compound Symbols:

c2, c4, c7, c11, c12, c6

(15) CdtKnowledgeProof (EQUIVALENT transformation)

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

F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
LT(s(z0), s(z1)) → c2(LT(z0, z1))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
S tuples:

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

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
LT(s(z0), s(z1)) → c2(LT(z0, z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
Defined Rule Symbols:

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

Defined Pair Symbols:

LT, APPEND, F_1, QSORT, F_3, SPLIT

Compound Symbols:

c2, c4, c7, c11, c12, c6

(17) 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:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(APPEND(x1, x2)) = x1   
POL(F_1(x1, x2, x3, x4)) = 0   
POL(F_2(x1, x2, x3, x4, x5, x6)) = 0   
POL(F_3(x1, x2, x3)) = [2]x1 + [2]x12   
POL(LT(x1, x2)) = 0   
POL(QSORT(x1)) = [2]x12   
POL(SPLIT(x1, x2)) = 0   
POL(add(x1, x2)) = [1] + x2   
POL(append(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2, x3)) = x1 + x2 + x3   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(f_1(x1, x2, x3, x4)) = [1] + x1   
POL(f_2(x1, x2, x3, x4, x5, x6)) = [1] + x5 + x6   
POL(f_3(x1, x2, x3)) = [2] + [2]x1   
POL(false) = 0   
POL(lt(x1, x2)) = 0   
POL(nil) = 0   
POL(pair(x1, x2)) = x1 + x2   
POL(qsort(x1)) = [2]x1   
POL(s(x1)) = 0   
POL(split(x1, x2)) = x2   
POL(true) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
APPEND(add(z0, z1), z2) → c4(APPEND(z1, z2))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
S tuples:none
K tuples:

F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))
LT(s(z0), s(z1)) → c2(LT(z0, z1))
SPLIT(z0, add(x1, nil)) → c6(F_1(pair(nil, nil), z0, x1, nil), SPLIT(z0, nil))
SPLIT(z0, add(x1, add(z1, z2))) → c6(F_1(f_1(split(z0, z2), z0, z1, z2), z0, x1, add(z1, z2)), SPLIT(z0, add(z1, z2)))
F_1(pair(z0, z1), z2, z3, z4) → c7(F_2(lt(z2, z3), z2, z3, z4, z0, z1), LT(z2, z3))
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, F_1, QSORT, F_3, SPLIT

Compound Symbols:

c2, c4, c7, c11, c12, c6

(19) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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