### (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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
lt(s(X), s(Y)) →+ lt(X, Y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X / s(X), Y / s(Y)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative 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)))

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
f_1/3
f_2/1
f_2/3
f_3/1

### (6) Obligation:

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

### (9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
lt, append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort

### (10) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

Generator Equations:

The following defined symbols remain to be analysed:
lt, append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort

### (11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

Induction Base:
true

Induction Step:
true

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (13) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

Lemmas:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

Generator Equations:

The following defined symbols remain to be analysed:
append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort

### (14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol append.

### (15) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

Lemmas:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

Generator Equations:

The following defined symbols remain to be analysed:
split, qsort

They will be analysed ascendingly in the following order:
split < qsort

### (16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol split.

### (17) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

Lemmas:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

Generator Equations:

The following defined symbols remain to be analysed:
qsort

### (18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol qsort.

### (19) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

Lemmas:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

Generator Equations:

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

### (22) Obligation:

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

Types:
lt :: 0':s:nil:add → 0':s:nil:add → true:false
true :: true:false
false :: true:false
split :: 0':s:nil:add → 0':s:nil:add → pair
pair :: 0':s:nil:add → 0':s:nil:add → pair
f_1 :: pair → 0':s:nil:add → 0':s:nil:add → pair
f_2 :: true:false → 0':s:nil:add → 0':s:nil:add → 0':s:nil:add → pair
f_3 :: pair → 0':s:nil:add → 0':s:nil:add
hole_true:false1_0 :: true:false
hole_pair3_0 :: pair

Lemmas:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) → true, rt ∈ Ω(1 + n60)

Generator Equations: