Runtime Complexity (innermost) proof of /tmp/tmpcFJRNS/enno.xml

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))

↳2 CdtProblem

↳3 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳4 CdtProblem

↳5 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳6 CdtProblem

↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳8 CdtProblem

↳9 CdtKnowledgeProof (⇔)

↳10 CdtProblem

↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳12 CdtProblem

↳13 CdtKnowledgeProof (⇔)

↳14 CdtProblem

↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳16 CdtProblem

↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳18 CdtProblem

↳19 SIsEmptyProof (⇔)

↳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) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(F_2(lt(z2, z3), z2, z3, z4, z0, z1))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))

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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(F_2(lt(z2, z3), z2, z3, z4, z0, z1))
F_1(pair(z0, z1), z2, z3, z4) → c(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, c

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

Removed 1 trailing tuple parts

(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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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, c, c

(7) 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
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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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_3(x₁, x₂, x₃)) = [4]x₁
POL(LT(x₁, x₂)) = 0
POL(QSORT(x₁)) = [4]x₁
POL(SPLIT(x₁, x₂)) = [2]
POL(add(x₁, x₂)) = [2] + x₂
POL(append(x₁, x₂)) = [4]
POL(c) = 0
POL(c(x₁)) = x₁
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(f_1(x₁, x₂, x₃, x₄)) = [2] + x₁
POL(f_2(x₁, x₂, x₃, x₄, x₅, x₆)) = [2] + 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)) = [5]
POL(split(x₁, x₂)) = x₂
POL(true) = 0

(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))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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, c, c

(9) CdtKnowledgeProof (EQUIVALENT 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))
QSORT(add(z0, z1)) → c11(F_3(split(z0, z1), z0, z1), SPLIT(z0, z1))

(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))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

K tuples:

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))

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, c, c

(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.

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) → c

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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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)) = [3]
POL(F_3(x₁, x₂, x₃)) = [2] + [2]x₃ + x₁²
POL(LT(x₁, x₂)) = 0
POL(QSORT(x₁)) = [1] + x₁²
POL(SPLIT(x₁, x₂)) = [2] + [2]x₂
POL(add(x₁, x₂)) = [2] + x₂
POL(append(x₁, x₂)) = 0
POL(c) = 0
POL(c(x₁)) = x₁
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(f_1(x₁, x₂, x₃, x₄)) = [2] + x₁
POL(f_2(x₁, x₂, x₃, x₄, x₅, x₆)) = [2] + 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

(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))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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) → c(LT(z2, z3))

K tuples:

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(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
F_1(pair(z0, z1), z2, z3, z4) → c

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, c, c

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))

(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))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
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))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
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) → c
F_1(pair(z0, z1), z2, z3, z4) → c(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, c, c

(15) 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:

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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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)) = [1] + x₃
POL(F_3(x₁, x₂, x₃)) = [1] + x₂ + x₃ + x₁²
POL(LT(x₁, x₂)) = x₂
POL(QSORT(x₁)) = x₁²
POL(SPLIT(x₁, x₂)) = x₁ + x₂ + x₁²
POL(add(x₁, x₂)) = [1] + x₁ + x₂
POL(append(x₁, x₂)) = 0
POL(c) = 0
POL(c(x₁)) = x₁
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(f_1(x₁, x₂, x₃, x₄)) = [1] + x₁ + x₃
POL(f_2(x₁, x₂, x₃, x₄, x₅, x₆)) = [1] + 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

(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))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

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))
F_3(pair(z0, z1), z2, z3) → c12(APPEND(qsort(z0), add(z3, qsort(z1))), QSORT(z0), QSORT(z1))
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) → c
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
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, SPLIT, QSORT, F_3, F_1

Compound Symbols:

c2, c4, c6, c11, c12, c, c

(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:

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))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(APPEND(x₁, x₂)) = [1] + x₁
POL(F_1(x₁, x₂, x₃, x₄)) = 0
POL(F_3(x₁, x₂, x₃)) = [2] + [2]x₁ + x₁²
POL(LT(x₁, x₂)) = 0
POL(QSORT(x₁)) = x₁²
POL(SPLIT(x₁, x₂)) = [1]
POL(add(x₁, x₂)) = [2] + x₂
POL(append(x₁, x₂)) = x₁ + x₂
POL(c) = 0
POL(c(x₁)) = x₁
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(f_1(x₁, x₂, x₃, x₄)) = [2] + x₁
POL(f_2(x₁, x₂, x₃, x₄, x₅, x₆)) = [2] + 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))
SPLIT(z0, add(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
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))
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
F_1(pair(z0, z1), z2, z3, z4) → c

S tuples:none
K tuples:

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(z1, z2)) → c6(F_1(split(z0, z2), z0, z1, z2), SPLIT(z0, z2))
F_1(pair(z0, z1), z2, z3, z4) → c
F_1(pair(z0, z1), z2, z3, z4) → c(LT(z2, z3))
LT(s(z0), s(z1)) → c2(LT(z0, z1))
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, c, c

(19) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(8) Obligation:

(9) CdtKnowledgeProof (EQUIVALENT transformation)

(10) Obligation:

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(12) Obligation:

(13) CdtKnowledgeProof (EQUIVALENT transformation)

(14) Obligation:

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(16) Obligation:

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(18) Obligation:

(19) SIsEmptyProof (EQUIVALENT transformation)

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