mathlib documentation

data.fin.fin2

Inductive type variant of fin #

fin is defined as a subtype of ℕ. This file defines an equivalent type, fin2, which is defined inductively. This is useful for its induction principle and different definitional equalities.

Main declarations #

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inductive fin2  :
ℕ → Type

An alternate definition of fin n defined as an inductive type instead of a subtype of ℕ.

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@[protected]
def fin2.cases' {n : ℕ} {C : fin2 n.succ → Sort u} (H1 : C fin2.fz) (H2 : Π (n_1 : fin2 n), C n_1.fs) (i : fin2 n.succ) :
C i

Define a dependent function on fin2 (succ n) by giving its value at zero (H1) and by giving a dependent function on the rest (H2).

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def fin2.elim0 {C : fin2 0 → Sort u} (i : fin2 0) :
C i

Ex falso. The dependent eliminator for the empty fin2 0 type.

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def fin2.to_nat {n : ℕ} :
fin2 n → ℕ

Converts a fin2 into a natural.

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def fin2.opt_of_nat {n : ℕ} (k : ℕ) :
option (fin2 n)

Converts a natural into a fin2 if it is in range

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def fin2.add {n : ℕ} (i : fin2 n) (k : ℕ) :
fin2 (n + k)

i + k : fin2 (n + k) when i : fin2 n and k : ℕ

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def fin2.left (k : ℕ) {n : ℕ} :
fin2 n → fin2 (k + n)

left k is the embedding fin2 n → fin2 (k + n)

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def fin2.insert_perm {n : ℕ} :
fin2 n → fin2 n → fin2 n

insert_perm a is a permutation of fin2 n with the following properties:

  • insert_perm a i = i+1 if i < a
  • insert_perm a a = 0
  • insert_perm a i = i if i > a
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def fin2.remap_left {m n : ℕ} (f : fin2 m → fin2 n) (k : ℕ) :
fin2 (m + k) → fin2 (n + k)

remap_left f k : fin2 (m + k) → fin2 (n + k) applies the function f : fin2 m → fin2 n to inputs less than m, and leaves the right part on the right (that is, remap_left f k (m + i) = n + i).

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@[class]
structure fin2.is_lt (m n : ℕ) :
Type
  • h : m < n

This is a simple type class inference prover for proof obligations of the form m < n where m n : ℕ.

Instances
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@[protected, instance]
def fin2.is_lt.zero (n : ℕ) :
fin2.is_lt 0 n.succ
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@[protected, instance]
def fin2.is_lt.succ (m n : ℕ) [l : fin2.is_lt m n] :
fin2.is_lt m.succ n.succ
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def fin2.of_nat' {n : ℕ} (m : ℕ) [fin2.is_lt m n] :
fin2 n

Use type class inference to infer the boundedness proof, so that we can directly convert a nat into a fin2 n. This supports notation like &1 : fin 3.

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@[protected, instance]
def fin2.inhabited  :
inhabited (fin2 1)
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