The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(2^n, INF).

0 CpxRelTRS
↳1 DecreasingLoopProof (⇔, 791 ms)
↳2 BOUNDS(2^n, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 80 ms)
    ↳10 typed CpxTrs
    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 487 ms)
    ↳12 typed CpxTrs

(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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
disj(And(And(a1430_0, a2431_0), a2)) →+ and(and(disj(a1430_0), conj(a1430_0)), and(disj(a1430_0), conj(a1430_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [a1430_0 / And(And(a1430_0, a2431_0), a2)].
The result substitution is [ ].

The rewrite sequence
disj(And(And(a1430_0, a2431_0), a2)) →+ and(and(disj(a1430_0), conj(a1430_0)), and(disj(a1430_0), conj(a1430_0)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
The pumping substitution is [a1430_0 / And(And(a1430_0, a2431_0), a2)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)

(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:

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

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

Infered types.

(6) Obligation:

Innermost TRS:
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)
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 → a → T:F:Or:And
False :: True:False
And :: T:F:Or:And → b → 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
hole_a3_0 :: a
hole_b4_0 :: b
gen_T:F:Or:And5_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, 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)
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 → a → T:F:Or:And
False :: True:False
And :: T:F:Or:And → b → 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
hole_a3_0 :: a
hole_b4_0 :: b
gen_T:F:Or:And5_0 :: Nat → T:F:Or:And

Generator Equations:
gen_T:F:Or:And5_0(0) ⇔ T
gen_T:F:Or:And5_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And5_0(x), hole_a3_0)

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, 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)
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 → a → T:F:Or:And
False :: True:False
And :: T:F:Or:And → b → 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
hole_a3_0 :: a
hole_b4_0 :: b
gen_T:F:Or:And5_0 :: Nat → T:F:Or:And

Generator Equations:
gen_T:F:Or:And5_0(0) ⇔ T
gen_T:F:Or:And5_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And5_0(x), hole_a3_0)

The following defined symbols remain to be analysed:
disj

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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

Innermost TRS:
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)
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 → a → T:F:Or:And
False :: True:False
And :: T:F:Or:And → b → 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
hole_a3_0 :: a
hole_b4_0 :: b
gen_T:F:Or:And5_0 :: Nat → T:F:Or:And

Generator Equations:
gen_T:F:Or:And5_0(0) ⇔ T
gen_T:F:Or:And5_0(+(x, 1)) ⇔ Or(gen_T:F:Or:And5_0(x), hole_a3_0)

No more defined symbols left to analyse.