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nat.dist
nat.dist.def
nat.dist.triangle_inequality
nat.dist_add_add_left
nat.dist_add_add_right
nat.dist_comm
nat.dist_eq_intro
nat.dist_eq_sub_of_le
nat.dist_eq_sub_of_le_right
nat.dist_eq_zero
nat.dist_mul_left
nat.dist_mul_right
nat.dist_pos_of_ne
nat.dist_self
nat.dist_succ_succ
nat.dist_zero_left
nat.dist_zero_right
nat.eq_of_dist_eq_zero
nat.sub_lt_sub_add_sub
source
def
nat.dist
:
ℕ
→
ℕ
→
ℕ
Distance (absolute value of difference) between natural numbers.
Equations
n.
dist
m
=
n
-
m
+
(m
-
n)
source
theorem
nat.dist.def
(n m :
ℕ
)
:
n.
dist
m
=
n
-
m
+
(m
-
n)
source
theorem
nat.dist_comm
(n m :
ℕ
)
:
n.
dist
m
=
m.
dist
n
source
@[simp]
theorem
nat.dist_self
(n :
ℕ
)
:
n.
dist
n
=
0
source
theorem
nat.eq_of_dist_eq_zero
{n m :
ℕ
}
:
n.
dist
m
=
0
→
n
=
m
source
theorem
nat.dist_eq_zero
{n m :
ℕ
}
:
n
=
m
→
n.
dist
m
=
0
source
theorem
nat.dist_eq_sub_of_le
{n m :
ℕ
}
:
n
≤
m
→
n.
dist
m
=
m
-
n
source
theorem
nat.dist_eq_sub_of_le_right
{n m :
ℕ
}
:
m
≤
n
→
n.
dist
m
=
n
-
m
source
theorem
nat.dist_zero_right
(n :
ℕ
)
:
n.
dist
0
=
n
source
theorem
nat.dist_zero_left
(n :
ℕ
)
:
0.
dist
n
=
n
source
theorem
nat.dist_add_add_right
(n k m :
ℕ
)
:
(n
+
k)
.
dist
(m
+
k)
=
n.
dist
m
source
theorem
nat.dist_add_add_left
(k n m :
ℕ
)
:
(k
+
n)
.
dist
(k
+
m)
=
n.
dist
m
source
theorem
nat.dist_eq_intro
{n m k l :
ℕ
}
:
n
+
m
=
k
+
l
→
n.
dist
k
=
l.
dist
m
source
theorem
nat.sub_lt_sub_add_sub
(n m k :
ℕ
)
:
n
-
k
≤
n
-
m
+
(m
-
k)
source
theorem
nat.dist.triangle_inequality
(n m k :
ℕ
)
:
n.
dist
k
≤
n.
dist
m
+
m.
dist
k
source
theorem
nat.dist_mul_right
(n k m :
ℕ
)
:
(n
*
k)
.
dist
(m
*
k)
=
n.
dist
m
*
k
source
theorem
nat.dist_mul_left
(k n m :
ℕ
)
:
(k
*
n)
.
dist
(k
*
m)
=
k
*
n.
dist
m
source
theorem
nat.dist_succ_succ
{i j :
ℕ
}
:
i.
succ
.
dist
j.
succ
=
i.
dist
j
source
theorem
nat.dist_pos_of_ne
{i j :
ℕ
}
:
i
≠
j
→
0
<
i.
dist
j
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