### (0) Obligation:

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

null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / add(n, x)].
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:

null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

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

Heuristically decided to analyse the following defined symbols:
app, reverse, shuff

They will be analysed ascendingly in the following order:
app < reverse
app < shuff
reverse < shuff

### (8) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Generator Equations:

The following defined symbols remain to be analysed:
app, reverse, shuff

They will be analysed ascendingly in the following order:
app < reverse
app < shuff
reverse < shuff

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

Proved the following rewrite lemma:

Induction Base:

Induction Step:

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

### (11) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
reverse, shuff

They will be analysed ascendingly in the following order:
reverse < shuff

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

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (14) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
shuff

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

Proved the following rewrite lemma:
shuff(gen_nil:add4_0(n836_0), gen_nil:add4_0(b)) → *5_0, rt ∈ Ω(b·n8360 + n8360 + n83602 + n83603)

Induction Base:

Induction Step:
*5_0

We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).

### (17) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Lemmas:
shuff(gen_nil:add4_0(n836_0), gen_nil:add4_0(b)) → *5_0, rt ∈ Ω(b·n8360 + n8360 + n83602 + n83603)

Generator Equations:

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuff(gen_nil:add4_0(n836_0), gen_nil:add4_0(b)) → *5_0, rt ∈ Ω(b·n8360 + n8360 + n83602 + n83603)

### (20) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Lemmas:
shuff(gen_nil:add4_0(n836_0), gen_nil:add4_0(b)) → *5_0, rt ∈ Ω(b·n8360 + n8360 + n83602 + n83603)

Generator Equations:

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuff(gen_nil:add4_0(n836_0), gen_nil:add4_0(b)) → *5_0, rt ∈ Ω(b·n8360 + n8360 + n83602 + n83603)

### (23) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (26) Obligation:

TRS:
Rules:
null(nil) → true
tail(nil) → nil
app(nil, y) → y
reverse(nil) → nil
shuffle(x) → shuff(x, nil)
if(true, x, y, z) → y
if(false, x, y, z) → shuff(reverse(tail(x)), z)

Types:
true :: true:false
false :: true:false
hole_true:false1_0 :: true:false

Lemmas: