### (0) Obligation:

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

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
a__and(true, and(true, X216_4)) →+ a__and(true, X216_4)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X216_4 / and(true, X216_4)].
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:

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0', X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0') → 0'
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
s/0
cons/0
cons/1
a__from/0
from/0

### (6) Obligation:

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

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0', X) → nil
a__first(s, cons) → cons
a__fromcons
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from) → a__from
mark(true) → true
mark(false) → false
mark(0') → 0'
mark(s) → s
mark(nil) → nil
mark(cons) → cons
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__fromfrom

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0', X) → nil
a__first(s, cons) → cons
a__fromcons
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from) → a__from
mark(true) → true
mark(false) → false
mark(0') → 0'
mark(s) → s
mark(nil) → nil
mark(cons) → cons
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__fromfrom

Types:

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

Heuristically decided to analyse the following defined symbols:
mark

### (10) Obligation:

TRS:
Rules:
a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0', X) → nil
a__first(s, cons) → cons
a__fromcons
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from) → a__from
mark(true) → true
mark(false) → false
mark(0') → 0'
mark(s) → s
mark(nil) → nil
mark(cons) → cons
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__fromfrom

Types:

Generator Equations:

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:

Induction Base:
true

Induction Step:
mark(true) →RΩ(1)
true

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

### (13) Obligation:

TRS:
Rules:
a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0', X) → nil
a__first(s, cons) → cons
a__fromcons
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from) → a__from
mark(true) → true
mark(false) → false
mark(0') → 0'
mark(s) → s
mark(nil) → nil
mark(cons) → cons
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__fromfrom

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:

### (16) Obligation:

TRS:
Rules:
a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__first(0', X) → nil
a__first(s, cons) → cons
a__fromcons
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from) → a__from
mark(true) → true
mark(false) → false
mark(0') → 0'
mark(s) → s
mark(nil) → nil
mark(cons) → cons
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__first(X1, X2) → first(X1, X2)
a__fromfrom

Types:

Lemmas: