(0) Obligation:

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

disj(T) → True
disj(F) → True
conj(Or(o1, o2)) → False
conj(T) → True
conj(F) → True
disj(And(a1, a2)) → conj(And(a1, a2))
disj(Or(t1, t2)) → and(conj(t1), disj(t1))
conj(And(t1, t2)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1, a2)) → False
bool(Or(o1, o2)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1, y2)) → False
isConsTerm(T, Or(x1, x2)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1, y2)) → False
isConsTerm(F, Or(x1, x2)) → False
isConsTerm(And(a1, a2), T) → False
isConsTerm(And(a1, a2), F) → False
isConsTerm(And(a1, a2), And(y1, y2)) → True
isConsTerm(And(a1, a2), Or(x1, x2)) → False
isConsTerm(Or(o1, o2), T) → False
isConsTerm(Or(o1, o2), F) → False
isConsTerm(Or(o1, o2), And(y1, y2)) → False
isConsTerm(Or(o1, o2), Or(x1, x2)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1, t2)) → True
isOp(Or(t1, t2)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1, t2)) → True
isAnd(Or(t1, t2)) → False
getSecond(And(t1, t2)) → t1
getSecond(Or(t1, t2)) → t1
getFirst(And(t1, t2)) → t1
getFirst(Or(t1, t2)) → t1
disjconj(p) → disj(p)

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

disj(T) → True
disj(F) → True
conj(Or(o1, o2)) → False
conj(T) → True
conj(F) → True
disj(And(a1, a2)) → conj(And(a1, a2))
disj(Or(t1, t2)) → and(conj(t1), disj(t1))
conj(And(t1, t2)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1, a2)) → False
bool(Or(o1, o2)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1, y2)) → False
isConsTerm(T, Or(x1, x2)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1, y2)) → False
isConsTerm(F, Or(x1, x2)) → False
isConsTerm(And(a1, a2), T) → False
isConsTerm(And(a1, a2), F) → False
isConsTerm(And(a1, a2), And(y1, y2)) → True
isConsTerm(And(a1, a2), Or(x1, x2)) → False
isConsTerm(Or(o1, o2), T) → False
isConsTerm(Or(o1, o2), F) → False
isConsTerm(Or(o1, o2), And(y1, y2)) → False
isConsTerm(Or(o1, o2), Or(x1, x2)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1, t2)) → True
isOp(Or(t1, t2)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1, t2)) → True
isAnd(Or(t1, t2)) → False
getSecond(And(t1, t2)) → t1
getSecond(Or(t1, t2)) → t1
getFirst(And(t1, t2)) → t1
getFirst(Or(t1, t2)) → t1
disjconj(p) → disj(p)

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Or/1
And/1

(4) Obligation:

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

disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
disj :: T:F:Or:And → True:False
T :: T:F:Or:And
True :: True:False
F :: T:F:Or:And
conj :: T:F:Or:And → True:False
Or :: T:F:Or:And → T:F:Or:And
False :: True:False
And :: T:F:Or:And → T:F:Or:And
and :: True:False → True:False → True:False
bool :: T:F:Or:And → True:False
isConsTerm :: T:F:Or:And → T:F:Or:And → True:False
isOp :: T:F:Or:And → True:False
isAnd :: T:F:Or:And → True:False
getSecond :: T:F:Or:And → T:F:Or:And
getFirst :: T:F:Or:And → T:F:Or:And
disjconj :: T:F:Or:And → True:False
hole_True:False1_0 :: True:False
hole_T:F:Or:And2_0 :: T:F:Or:And
gen_T:F:Or:And3_0 :: Nat → T:F:Or:And

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

Heuristically decided to analyse the following defined symbols:
disj, conj

They will be analysed ascendingly in the following order:
disj = conj

(8) Obligation:

Innermost TRS:
Rules:
disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
disj :: T:F:Or:And → True:False
T :: T:F:Or:And
True :: True:False
F :: T:F:Or:And
conj :: T:F:Or:And → True:False
Or :: T:F:Or:And → T:F:Or:And
False :: True:False
And :: T:F:Or:And → T:F:Or:And
and :: True:False → True:False → True:False
bool :: T:F:Or:And → True:False
isConsTerm :: T:F:Or:And → T:F:Or:And → True:False
isOp :: T:F:Or:And → True:False
isAnd :: T:F:Or:And → True:False
getSecond :: T:F:Or:And → T:F:Or:And
getFirst :: T:F:Or:And → T:F:Or:And
disjconj :: T:F:Or:And → True:False
hole_True:False1_0 :: True:False
hole_T:F:Or:And2_0 :: T:F:Or:And
gen_T:F:Or:And3_0 :: Nat → T:F:Or:And

Generator Equations:
gen_T:F:Or:And3_0(0) ⇔ T
gen_T:F:Or:And3_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And3_0(x))

The following defined symbols remain to be analysed:
conj, disj

They will be analysed ascendingly in the following order:
disj = conj

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

Innermost TRS:
Rules:
disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
disj :: T:F:Or:And → True:False
T :: T:F:Or:And
True :: True:False
F :: T:F:Or:And
conj :: T:F:Or:And → True:False
Or :: T:F:Or:And → T:F:Or:And
False :: True:False
And :: T:F:Or:And → T:F:Or:And
and :: True:False → True:False → True:False
bool :: T:F:Or:And → True:False
isConsTerm :: T:F:Or:And → T:F:Or:And → True:False
isOp :: T:F:Or:And → True:False
isAnd :: T:F:Or:And → True:False
getSecond :: T:F:Or:And → T:F:Or:And
getFirst :: T:F:Or:And → T:F:Or:And
disjconj :: T:F:Or:And → True:False
hole_True:False1_0 :: True:False
hole_T:F:Or:And2_0 :: T:F:Or:And
gen_T:F:Or:And3_0 :: Nat → T:F:Or:And

Generator Equations:
gen_T:F:Or:And3_0(0) ⇔ T
gen_T:F:Or:And3_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And3_0(x))

The following defined symbols remain to be analysed:
disj

They will be analysed ascendingly in the following order:
disj = conj

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

Proved the following rewrite lemma:
disj(gen_T:F:Or:And3_0(+(1, n25_0))) → *4_0, rt ∈ Ω(n250)

Induction Base:
disj(gen_T:F:Or:And3_0(+(1, 0)))

Induction Step:
disj(gen_T:F:Or:And3_0(+(1, +(n25_0, 1)))) →RΩ(1)
and(conj(gen_T:F:Or:And3_0(+(1, n25_0))), disj(gen_T:F:Or:And3_0(+(1, n25_0)))) →RΩ(1)
and(False, disj(gen_T:F:Or:And3_0(+(1, n25_0)))) →IH
and(False, *4_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
disj :: T:F:Or:And → True:False
T :: T:F:Or:And
True :: True:False
F :: T:F:Or:And
conj :: T:F:Or:And → True:False
Or :: T:F:Or:And → T:F:Or:And
False :: True:False
And :: T:F:Or:And → T:F:Or:And
and :: True:False → True:False → True:False
bool :: T:F:Or:And → True:False
isConsTerm :: T:F:Or:And → T:F:Or:And → True:False
isOp :: T:F:Or:And → True:False
isAnd :: T:F:Or:And → True:False
getSecond :: T:F:Or:And → T:F:Or:And
getFirst :: T:F:Or:And → T:F:Or:And
disjconj :: T:F:Or:And → True:False
hole_True:False1_0 :: True:False
hole_T:F:Or:And2_0 :: T:F:Or:And
gen_T:F:Or:And3_0 :: Nat → T:F:Or:And

Lemmas:
disj(gen_T:F:Or:And3_0(+(1, n25_0))) → *4_0, rt ∈ Ω(n250)

Generator Equations:
gen_T:F:Or:And3_0(0) ⇔ T
gen_T:F:Or:And3_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And3_0(x))

The following defined symbols remain to be analysed:
conj

They will be analysed ascendingly in the following order:
disj = conj

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

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

(15) Obligation:

Innermost TRS:
Rules:
disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
disj :: T:F:Or:And → True:False
T :: T:F:Or:And
True :: True:False
F :: T:F:Or:And
conj :: T:F:Or:And → True:False
Or :: T:F:Or:And → T:F:Or:And
False :: True:False
And :: T:F:Or:And → T:F:Or:And
and :: True:False → True:False → True:False
bool :: T:F:Or:And → True:False
isConsTerm :: T:F:Or:And → T:F:Or:And → True:False
isOp :: T:F:Or:And → True:False
isAnd :: T:F:Or:And → True:False
getSecond :: T:F:Or:And → T:F:Or:And
getFirst :: T:F:Or:And → T:F:Or:And
disjconj :: T:F:Or:And → True:False
hole_True:False1_0 :: True:False
hole_T:F:Or:And2_0 :: T:F:Or:And
gen_T:F:Or:And3_0 :: Nat → T:F:Or:And

Lemmas:
disj(gen_T:F:Or:And3_0(+(1, n25_0))) → *4_0, rt ∈ Ω(n250)

Generator Equations:
gen_T:F:Or:And3_0(0) ⇔ T
gen_T:F:Or:And3_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And3_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
disj(gen_T:F:Or:And3_0(+(1, n25_0))) → *4_0, rt ∈ Ω(n250)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
disj(T) → True
disj(F) → True
conj(Or(o1)) → False
conj(T) → True
conj(F) → True
disj(And(a1)) → conj(And(a1))
disj(Or(t1)) → and(conj(t1), disj(t1))
conj(And(t1)) → and(disj(t1), conj(t1))
bool(T) → True
bool(F) → True
bool(And(a1)) → False
bool(Or(o1)) → False
isConsTerm(T, T) → True
isConsTerm(T, F) → False
isConsTerm(T, And(y1)) → False
isConsTerm(T, Or(x1)) → False
isConsTerm(F, T) → False
isConsTerm(F, F) → True
isConsTerm(F, And(y1)) → False
isConsTerm(F, Or(x1)) → False
isConsTerm(And(a1), T) → False
isConsTerm(And(a1), F) → False
isConsTerm(And(a1), And(y1)) → True
isConsTerm(And(a1), Or(x1)) → False
isConsTerm(Or(o1), T) → False
isConsTerm(Or(o1), F) → False
isConsTerm(Or(o1), And(y1)) → False
isConsTerm(Or(o1), Or(x1)) → True
isOp(T) → False
isOp(F) → False
isOp(And(t1)) → True
isOp(Or(t1)) → True
isAnd(T) → False
isAnd(F) → False
isAnd(And(t1)) → True
isAnd(Or(t1)) → False
getSecond(And(t1)) → t1
getSecond(Or(t1)) → t1
getFirst(And(t1)) → t1
getFirst(Or(t1)) → t1
disjconj(p) → disj(p)
and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Types:
disj :: T:F:Or:And → True:False
T :: T:F:Or:And
True :: True:False
F :: T:F:Or:And
conj :: T:F:Or:And → True:False
Or :: T:F:Or:And → T:F:Or:And
False :: True:False
And :: T:F:Or:And → T:F:Or:And
and :: True:False → True:False → True:False
bool :: T:F:Or:And → True:False
isConsTerm :: T:F:Or:And → T:F:Or:And → True:False
isOp :: T:F:Or:And → True:False
isAnd :: T:F:Or:And → True:False
getSecond :: T:F:Or:And → T:F:Or:And
getFirst :: T:F:Or:And → T:F:Or:And
disjconj :: T:F:Or:And → True:False
hole_True:False1_0 :: True:False
hole_T:F:Or:And2_0 :: T:F:Or:And
gen_T:F:Or:And3_0 :: Nat → T:F:Or:And

Lemmas:
disj(gen_T:F:Or:And3_0(+(1, n25_0))) → *4_0, rt ∈ Ω(n250)

Generator Equations:
gen_T:F:Or:And3_0(0) ⇔ T
gen_T:F:Or:And3_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
disj(gen_T:F:Or:And3_0(+(1, n25_0))) → *4_0, rt ∈ Ω(n250)

(20) BOUNDS(n^1, INF)