(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,
conjThey will be analysed ascendingly in the following order:
disj = conj
(8) Obligation:
Innermost TRS:
Rules:
disj(
T) →
Truedisj(
F) →
Trueconj(
Or(
o1)) →
Falseconj(
T) →
Trueconj(
F) →
Truedisj(
And(
a1)) →
conj(
And(
a1))
disj(
Or(
t1)) →
and(
conj(
t1),
disj(
t1))
conj(
And(
t1)) →
and(
disj(
t1),
conj(
t1))
bool(
T) →
Truebool(
F) →
Truebool(
And(
a1)) →
Falsebool(
Or(
o1)) →
FalseisConsTerm(
T,
T) →
TrueisConsTerm(
T,
F) →
FalseisConsTerm(
T,
And(
y1)) →
FalseisConsTerm(
T,
Or(
x1)) →
FalseisConsTerm(
F,
T) →
FalseisConsTerm(
F,
F) →
TrueisConsTerm(
F,
And(
y1)) →
FalseisConsTerm(
F,
Or(
x1)) →
FalseisConsTerm(
And(
a1),
T) →
FalseisConsTerm(
And(
a1),
F) →
FalseisConsTerm(
And(
a1),
And(
y1)) →
TrueisConsTerm(
And(
a1),
Or(
x1)) →
FalseisConsTerm(
Or(
o1),
T) →
FalseisConsTerm(
Or(
o1),
F) →
FalseisConsTerm(
Or(
o1),
And(
y1)) →
FalseisConsTerm(
Or(
o1),
Or(
x1)) →
TrueisOp(
T) →
FalseisOp(
F) →
FalseisOp(
And(
t1)) →
TrueisOp(
Or(
t1)) →
TrueisAnd(
T) →
FalseisAnd(
F) →
FalseisAnd(
And(
t1)) →
TrueisAnd(
Or(
t1)) →
FalsegetSecond(
And(
t1)) →
t1getSecond(
Or(
t1)) →
t1getFirst(
And(
t1)) →
t1getFirst(
Or(
t1)) →
t1disjconj(
p) →
disj(
p)
and(
False,
False) →
Falseand(
True,
False) →
Falseand(
False,
True) →
Falseand(
True,
True) →
TrueTypes:
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) →
Truedisj(
F) →
Trueconj(
Or(
o1)) →
Falseconj(
T) →
Trueconj(
F) →
Truedisj(
And(
a1)) →
conj(
And(
a1))
disj(
Or(
t1)) →
and(
conj(
t1),
disj(
t1))
conj(
And(
t1)) →
and(
disj(
t1),
conj(
t1))
bool(
T) →
Truebool(
F) →
Truebool(
And(
a1)) →
Falsebool(
Or(
o1)) →
FalseisConsTerm(
T,
T) →
TrueisConsTerm(
T,
F) →
FalseisConsTerm(
T,
And(
y1)) →
FalseisConsTerm(
T,
Or(
x1)) →
FalseisConsTerm(
F,
T) →
FalseisConsTerm(
F,
F) →
TrueisConsTerm(
F,
And(
y1)) →
FalseisConsTerm(
F,
Or(
x1)) →
FalseisConsTerm(
And(
a1),
T) →
FalseisConsTerm(
And(
a1),
F) →
FalseisConsTerm(
And(
a1),
And(
y1)) →
TrueisConsTerm(
And(
a1),
Or(
x1)) →
FalseisConsTerm(
Or(
o1),
T) →
FalseisConsTerm(
Or(
o1),
F) →
FalseisConsTerm(
Or(
o1),
And(
y1)) →
FalseisConsTerm(
Or(
o1),
Or(
x1)) →
TrueisOp(
T) →
FalseisOp(
F) →
FalseisOp(
And(
t1)) →
TrueisOp(
Or(
t1)) →
TrueisAnd(
T) →
FalseisAnd(
F) →
FalseisAnd(
And(
t1)) →
TrueisAnd(
Or(
t1)) →
FalsegetSecond(
And(
t1)) →
t1getSecond(
Or(
t1)) →
t1getFirst(
And(
t1)) →
t1getFirst(
Or(
t1)) →
t1disjconj(
p) →
disj(
p)
and(
False,
False) →
Falseand(
True,
False) →
Falseand(
False,
True) →
Falseand(
True,
True) →
TrueTypes:
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(n25
0)
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) →
Truedisj(
F) →
Trueconj(
Or(
o1)) →
Falseconj(
T) →
Trueconj(
F) →
Truedisj(
And(
a1)) →
conj(
And(
a1))
disj(
Or(
t1)) →
and(
conj(
t1),
disj(
t1))
conj(
And(
t1)) →
and(
disj(
t1),
conj(
t1))
bool(
T) →
Truebool(
F) →
Truebool(
And(
a1)) →
Falsebool(
Or(
o1)) →
FalseisConsTerm(
T,
T) →
TrueisConsTerm(
T,
F) →
FalseisConsTerm(
T,
And(
y1)) →
FalseisConsTerm(
T,
Or(
x1)) →
FalseisConsTerm(
F,
T) →
FalseisConsTerm(
F,
F) →
TrueisConsTerm(
F,
And(
y1)) →
FalseisConsTerm(
F,
Or(
x1)) →
FalseisConsTerm(
And(
a1),
T) →
FalseisConsTerm(
And(
a1),
F) →
FalseisConsTerm(
And(
a1),
And(
y1)) →
TrueisConsTerm(
And(
a1),
Or(
x1)) →
FalseisConsTerm(
Or(
o1),
T) →
FalseisConsTerm(
Or(
o1),
F) →
FalseisConsTerm(
Or(
o1),
And(
y1)) →
FalseisConsTerm(
Or(
o1),
Or(
x1)) →
TrueisOp(
T) →
FalseisOp(
F) →
FalseisOp(
And(
t1)) →
TrueisOp(
Or(
t1)) →
TrueisAnd(
T) →
FalseisAnd(
F) →
FalseisAnd(
And(
t1)) →
TrueisAnd(
Or(
t1)) →
FalsegetSecond(
And(
t1)) →
t1getSecond(
Or(
t1)) →
t1getFirst(
And(
t1)) →
t1getFirst(
Or(
t1)) →
t1disjconj(
p) →
disj(
p)
and(
False,
False) →
Falseand(
True,
False) →
Falseand(
False,
True) →
Falseand(
True,
True) →
TrueTypes:
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) →
Truedisj(
F) →
Trueconj(
Or(
o1)) →
Falseconj(
T) →
Trueconj(
F) →
Truedisj(
And(
a1)) →
conj(
And(
a1))
disj(
Or(
t1)) →
and(
conj(
t1),
disj(
t1))
conj(
And(
t1)) →
and(
disj(
t1),
conj(
t1))
bool(
T) →
Truebool(
F) →
Truebool(
And(
a1)) →
Falsebool(
Or(
o1)) →
FalseisConsTerm(
T,
T) →
TrueisConsTerm(
T,
F) →
FalseisConsTerm(
T,
And(
y1)) →
FalseisConsTerm(
T,
Or(
x1)) →
FalseisConsTerm(
F,
T) →
FalseisConsTerm(
F,
F) →
TrueisConsTerm(
F,
And(
y1)) →
FalseisConsTerm(
F,
Or(
x1)) →
FalseisConsTerm(
And(
a1),
T) →
FalseisConsTerm(
And(
a1),
F) →
FalseisConsTerm(
And(
a1),
And(
y1)) →
TrueisConsTerm(
And(
a1),
Or(
x1)) →
FalseisConsTerm(
Or(
o1),
T) →
FalseisConsTerm(
Or(
o1),
F) →
FalseisConsTerm(
Or(
o1),
And(
y1)) →
FalseisConsTerm(
Or(
o1),
Or(
x1)) →
TrueisOp(
T) →
FalseisOp(
F) →
FalseisOp(
And(
t1)) →
TrueisOp(
Or(
t1)) →
TrueisAnd(
T) →
FalseisAnd(
F) →
FalseisAnd(
And(
t1)) →
TrueisAnd(
Or(
t1)) →
FalsegetSecond(
And(
t1)) →
t1getSecond(
Or(
t1)) →
t1getFirst(
And(
t1)) →
t1getFirst(
Or(
t1)) →
t1disjconj(
p) →
disj(
p)
and(
False,
False) →
Falseand(
True,
False) →
Falseand(
False,
True) →
Falseand(
True,
True) →
TrueTypes:
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) →
Truedisj(
F) →
Trueconj(
Or(
o1)) →
Falseconj(
T) →
Trueconj(
F) →
Truedisj(
And(
a1)) →
conj(
And(
a1))
disj(
Or(
t1)) →
and(
conj(
t1),
disj(
t1))
conj(
And(
t1)) →
and(
disj(
t1),
conj(
t1))
bool(
T) →
Truebool(
F) →
Truebool(
And(
a1)) →
Falsebool(
Or(
o1)) →
FalseisConsTerm(
T,
T) →
TrueisConsTerm(
T,
F) →
FalseisConsTerm(
T,
And(
y1)) →
FalseisConsTerm(
T,
Or(
x1)) →
FalseisConsTerm(
F,
T) →
FalseisConsTerm(
F,
F) →
TrueisConsTerm(
F,
And(
y1)) →
FalseisConsTerm(
F,
Or(
x1)) →
FalseisConsTerm(
And(
a1),
T) →
FalseisConsTerm(
And(
a1),
F) →
FalseisConsTerm(
And(
a1),
And(
y1)) →
TrueisConsTerm(
And(
a1),
Or(
x1)) →
FalseisConsTerm(
Or(
o1),
T) →
FalseisConsTerm(
Or(
o1),
F) →
FalseisConsTerm(
Or(
o1),
And(
y1)) →
FalseisConsTerm(
Or(
o1),
Or(
x1)) →
TrueisOp(
T) →
FalseisOp(
F) →
FalseisOp(
And(
t1)) →
TrueisOp(
Or(
t1)) →
TrueisAnd(
T) →
FalseisAnd(
F) →
FalseisAnd(
And(
t1)) →
TrueisAnd(
Or(
t1)) →
FalsegetSecond(
And(
t1)) →
t1getSecond(
Or(
t1)) →
t1getFirst(
And(
t1)) →
t1getFirst(
Or(
t1)) →
t1disjconj(
p) →
disj(
p)
and(
False,
False) →
Falseand(
True,
False) →
Falseand(
False,
True) →
Falseand(
True,
True) →
TrueTypes:
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)