core / init.data.fin.basic

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structure fin (n : ℕ) :

Type

val : ℕ
property : self.val < n

fin n is the subtype of ℕ consisting of natural numbers strictly smaller than n.

Instances for fin

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@[protected]

def fin.lt {n : ℕ} (a b : fin n) :

Prop

Equations

a.lt b = (a.val < b.val)

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@[protected]

def fin.le {n : ℕ} (a b : fin n) :

Prop

Equations

a.le b = (a.val ≤ b.val)

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@[protected, instance]

def fin.has_lt {n : ℕ} :

has_lt (fin n)

Equations

fin.has_lt = {lt := fin.lt n}

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@[protected, instance]

def fin.has_le {n : ℕ} :

has_le (fin n)

Equations

fin.has_le = {le := fin.le n}

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@[protected, instance]

def fin.decidable_lt {n : ℕ} (a b : fin n) :

decidable (a < b)

Equations

a.decidable_lt b = a.val.decidable_lt b.val

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@[protected, instance]

def fin.decidable_le {n : ℕ} (a b : fin n) :

decidable (a ≤ b)

Equations

a.decidable_le b = a.val.decidable_le b.val

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def fin.elim0 {α : fin 0 → Sort u} (x : fin 0) :

α x

Equations

⟨n, h⟩.elim0 = absurd h _

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theorem fin.eq_of_veq {n : ℕ} {i j : fin n} :

i.val = j.val → i = j

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theorem fin.veq_of_eq {n : ℕ} {i j : fin n} :

i = j → i.val = j.val

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theorem fin.ne_of_vne {n : ℕ} {i j : fin n} (h : i.val ≠ j.val) :

i ≠ j

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theorem fin.vne_of_ne {n : ℕ} {i j : fin n} (h : i ≠ j) :

i.val ≠ j.val

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@[protected, instance]

def fin.decidable_eq (n : ℕ) :

decidable_eq (fin n)

Equations

fin.decidable_eq n = λ (i j : fin n), decidable_of_decidable_of_iff (nat.decidable_eq i.val j.val) _