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nat.decidable_pred
nat.even
nat.even.add
nat.even.sub
nat.even_add
nat.even_bit0
nat.even_div
nat.even_iff
nat.even_mul
nat.even_pow
nat.even_sub
nat.even_succ
nat.even_zero
nat.mod_two_ne_one
nat.mod_two_ne_zero
nat.neg_one_pow_eq_one_iff_even
nat.not_even_bit1
nat.not_even_iff
nat.not_even_one
nat.two_not_dvd_two_mul_add_one
nat.two_not_dvd_two_mul_sub_one
source
@[simp]
theorem
nat.mod_two_ne_one
{n :
ℕ
}
:
¬
n
%
2
=
1
↔
n
%
2
=
0
source
@[simp]
theorem
nat.mod_two_ne_zero
{n :
ℕ
}
:
¬
n
%
2
=
0
↔
n
%
2
=
1
source
def
nat.even
:
ℕ
→ Prop
A natural number
n
is
even
if
2 | n
.
Equations
n.
even
=
(2
∣
n)
source
theorem
nat.even_iff
{n :
ℕ
}
:
n.
even
↔
n
%
2
=
0
source
theorem
nat.not_even_iff
{n :
ℕ
}
:
¬
n.
even
↔
n
%
2
=
1
source
@[instance]
def
nat.decidable_pred
:
decidable_pred
nat.even
Equations
nat.decidable_pred
=
λ (n :
ℕ
),
decidable_of_decidable_of_iff
(
nat.decidable_eq
(n
%
2)
0)
_
source
@[simp]
theorem
nat.even_zero
:
0.
even
source
@[simp]
theorem
nat.not_even_one
:
¬
1.
even
source
@[simp]
theorem
nat.even_bit0
(n :
ℕ
)
:
(
bit0
n)
.
even
source
theorem
nat.even_add
{m n :
ℕ
}
:
(m
+
n)
.
even
↔
(m.
even
↔
n.
even
)
source
theorem
nat.even.add
{m n :
ℕ
}
:
m.
even
→
n.
even
→
(m
+
n)
.
even
source
@[simp]
theorem
nat.not_even_bit1
(n :
ℕ
)
:
¬
(
bit1
n)
.
even
source
theorem
nat.two_not_dvd_two_mul_add_one
(a :
ℕ
)
:
¬
2
∣
2
*
a
+
1
source
theorem
nat.two_not_dvd_two_mul_sub_one
{a :
ℕ
}
:
0
<
a
→
¬
2
∣
2
*
a
-
1
source
theorem
nat.even_sub
{m n :
ℕ
}
:
n
≤
m
→
((m
-
n)
.
even
↔
(m.
even
↔
n.
even
))
source
theorem
nat.even.sub
{m n :
ℕ
}
:
m.
even
→
n.
even
→
(m
-
n)
.
even
source
theorem
nat.even_succ
{n :
ℕ
}
:
n.
succ
.
even
↔
¬
n.
even
source
theorem
nat.even_mul
{m n :
ℕ
}
:
(m
*
n)
.
even
↔
m.
even
∨
n.
even
source
theorem
nat.even_pow
{m n :
ℕ
}
:
(m
^
n)
.
even
↔
m.
even
∧
n
≠
0
source
theorem
nat.even_div
{a b :
ℕ
}
:
(a
/
b)
.
even
↔
a
%
(2
*
b)
/
b
=
0
source
theorem
nat.neg_one_pow_eq_one_iff_even
{α : Type u_1}
[
ring
α]
{n :
ℕ
}
:
-1
≠
1
→
((-1)
^
n
=
1
↔
n.
even
)
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