Equivalences for Fin n

Miscellaneous

This is currently not very sorted. PRs welcome!

theorem Fin.preimage_apply_01_prod {α : Fin 2 → Type u} (s : Set (α 0)) (t : Set (α 1)) : (fun (f : (i : Fin 2) → α i) => (f 0, f 1)) ⁻¹' s ×ˢ t = Set.univ.pi (cons s (cons t finZeroElim))

theorem Fin.preimage_apply_01_prod' {α : Type u} (s t : Set α) : (fun (f : Fin 2 → α) => (f 0, f 1)) ⁻¹' s ×ˢ t = Set.univ.pi ![s, t]

def prodEquivPiFinTwo (α β : Type u) : α × β ≃ ((i : Fin 2) → ![α, β] i)

A product space α × β is equivalent to the space Π i : Fin 2, γ i, where γ = Fin.cons α (Fin.cons β finZeroElim). See also piFinTwoEquiv and finTwoArrowEquiv.

Equations

• prodEquivPiFinTwo α β = (piFinTwoEquiv (Fin.cons α (Fin.cons β finZeroElim))).symm

@[simp]

theorem prodEquivPiFinTwo_symm_apply (α β : Type u) : ⇑(prodEquivPiFinTwo α β).symm = fun (f : (i : Fin 2) → Fin.cons α (Fin.cons β finZeroElim) i) => (f 0, f 1)

@[simp]

theorem prodEquivPiFinTwo_apply (α β : Type u) : ⇑(prodEquivPiFinTwo α β) = fun (p : Fin.cons α (Fin.cons β finZeroElim) 0 × Fin.cons α (Fin.cons β finZeroElim) 1) => Fin.cons p.1 (Fin.cons p.2 finZeroElim)

def finTwoArrowEquiv (α : Type u_1) : (Fin 2 → α) ≃ α × α

The space of functions Fin 2 → α is equivalent to α × α. See also piFinTwoEquiv and prodEquivPiFinTwo.

Equations

• finTwoArrowEquiv α = { toFun := (piFinTwoEquiv fun (x : Fin 2) => α).toFun, invFun := fun (x : α × α) => ![x.1, x.2], left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem finTwoArrowEquiv_symm_apply (α : Type u_1) : ⇑(finTwoArrowEquiv α).symm = fun (x : α × α) => ![x.1, x.2]

@[simp]

theorem finTwoArrowEquiv_apply (α : Type u_1) : ⇑(finTwoArrowEquiv α) = (piFinTwoEquiv fun (x : Fin 2) => α).toFun

def finSuccEquiv' {n : ℕ} (i : Fin (n + 1)) : Fin (n + 1) ≃ Option (Fin n)

An equivalence that removes i and maps it to none. This is a version of Fin.predAbove that produces Option (Fin n) instead of mapping both i.castSucc and i.succ to i.

Equations

• finSuccEquiv' i = { toFun := i.insertNth none some, invFun := fun (x : Option (Fin n)) => x.casesOn' i i.succAbove, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem finSuccEquiv'_at {n : ℕ} (i : Fin (n + 1)) : (finSuccEquiv' i) i = none

@[simp]

theorem finSuccEquiv'_succAbove {n : ℕ} (i : Fin (n + 1)) (j : Fin n) : (finSuccEquiv' i) (i.succAbove j) = some j

theorem finSuccEquiv'_below {n : ℕ} {i : Fin (n + 1)} {m : Fin n} (h : m.castSucc < i) : (finSuccEquiv' i) m.castSucc = some m

theorem finSuccEquiv'_above {n : ℕ} {i : Fin (n + 1)} {m : Fin n} (h : i ≤ m.castSucc) : (finSuccEquiv' i) m.succ = some m

@[simp]

theorem finSuccEquiv'_symm_none {n : ℕ} (i : Fin (n + 1)) : (finSuccEquiv' i).symm none = i

@[simp]

theorem finSuccEquiv'_symm_some {n : ℕ} (i : Fin (n + 1)) (j : Fin n) : (finSuccEquiv' i).symm (some j) = i.succAbove j

@[simp]

theorem finSuccEquiv'_eq_some {n : ℕ} {i j : Fin (n + 1)} {k : Fin n} : (finSuccEquiv' i) j = some k ↔ j = i.succAbove k

@[simp]

theorem finSuccEquiv'_eq_none {n : ℕ} {i j : Fin (n + 1)} : (finSuccEquiv' i) j = none ↔ i = j

theorem finSuccEquiv'_symm_some_below {n : ℕ} {i : Fin (n + 1)} {m : Fin n} (h : m.castSucc < i) : (finSuccEquiv' i).symm (some m) = m.castSucc

theorem finSuccEquiv'_symm_some_above {n : ℕ} {i : Fin (n + 1)} {m : Fin n} (h : i ≤ m.castSucc) : (finSuccEquiv' i).symm (some m) = m.succ

theorem finSuccEquiv'_symm_coe_below {n : ℕ} {i : Fin (n + 1)} {m : Fin n} (h : m.castSucc < i) : (finSuccEquiv' i).symm (some m) = m.castSucc

theorem finSuccEquiv'_symm_coe_above {n : ℕ} {i : Fin (n + 1)} {m : Fin n} (h : i ≤ m.castSucc) : (finSuccEquiv' i).symm (some m) = m.succ

def finSuccEquiv (n : ℕ) : Fin (n + 1) ≃ Option (Fin n)

Equivalence between Fin (n + 1) and Option (Fin n). This is a version of Fin.pred that produces Option (Fin n) instead of requiring a proof that the input is not 0.

Equations

• finSuccEquiv n = finSuccEquiv' 0

@[simp]

theorem finSuccEquiv_zero {n : ℕ} : (finSuccEquiv n) 0 = none

@[simp]

theorem finSuccEquiv_succ {n : ℕ} (m : Fin n) : (finSuccEquiv n) m.succ = some m

@[simp]

theorem finSuccEquiv_symm_none {n : ℕ} : (finSuccEquiv n).symm none = 0

@[simp]

theorem finSuccEquiv_symm_some {n : ℕ} (m : Fin n) : (finSuccEquiv n).symm (some m) = m.succ

@[simp]

theorem finSuccEquiv_eq_some {n : ℕ} {i : Fin (n + 1)} {j : Fin n} : (finSuccEquiv n) i = some j ↔ i = j.succ

@[simp]

theorem finSuccEquiv_eq_none {n : ℕ} {i : Fin (n + 1)} : (finSuccEquiv n) i = none ↔ i = 0

theorem finSuccEquiv'_zero {n : ℕ} : finSuccEquiv' 0 = finSuccEquiv n

The equiv version of Fin.predAbove_zero.

theorem finSuccEquiv'_last_apply_castSucc {n : ℕ} (i : Fin n) : (finSuccEquiv' (Fin.last n)) i.castSucc = some i

theorem finSuccEquiv'_last_apply {n : ℕ} {i : Fin (n + 1)} (h : i ≠ Fin.last n) : (finSuccEquiv' (Fin.last n)) i = some (i.castLT ⋯)

theorem finSuccEquiv'_ne_last_apply {n : ℕ} {i j : Fin (n + 1)} (hi : i ≠ Fin.last n) (hj : j ≠ i) : (finSuccEquiv' i) j = some ((i.castLT ⋯).predAbove j)

def finSuccAboveEquiv {n : ℕ} (p : Fin (n + 1)) : Fin n ≃ { x : Fin (n + 1) // x ≠ p }

Fin.succAbove as an order isomorphism between Fin n and {x : Fin (n + 1) // x ≠ p}.

Equations

• finSuccAboveEquiv p = (Equiv.optionSubtype p) ⟨(finSuccEquiv' p).symm, ⋯⟩

theorem finSuccAboveEquiv_apply {n : ℕ} (p : Fin (n + 1)) (i : Fin n) : (finSuccAboveEquiv p) i = ⟨p.succAbove i, ⋯⟩

theorem finSuccAboveEquiv_symm_apply_last {n : ℕ} (x : { x : Fin (n + 1) // x ≠ Fin.last n }) : (finSuccAboveEquiv (Fin.last n)).symm x = (↑x).castLT ⋯

theorem finSuccAboveEquiv_symm_apply_ne_last {n : ℕ} {p : Fin (n + 1)} (h : p ≠ Fin.last n) (x : { x : Fin (n + 1) // x ≠ p }) : (finSuccAboveEquiv p).symm x = (p.castLT ⋯).predAbove ↑x

def finSuccEquivLast {n : ℕ} : Fin (n + 1) ≃ Option (Fin n)

Equiv between Fin (n + 1) and Option (Fin n) sending Fin.last n to none

Equations

• finSuccEquivLast = finSuccEquiv' (Fin.last n)

@[simp]

theorem finSuccEquivLast_castSucc {n : ℕ} (i : Fin n) : finSuccEquivLast i.castSucc = some i

@[simp]

theorem finSuccEquivLast_last {n : ℕ} : finSuccEquivLast (Fin.last n) = none

@[simp]

theorem finSuccEquivLast_symm_some {n : ℕ} (i : Fin n) : finSuccEquivLast.symm (some i) = i.castSucc

@[simp]

theorem finSuccEquivLast_symm_none {n : ℕ} : finSuccEquivLast.symm none = Fin.last n

def Equiv.embeddingFinSucc (n : ℕ) (ι : Type u_1) : (Fin (n + 1) ↪ ι) ≃ (e : Fin n ↪ ι) × { i : ι // i ∉ Set.range ⇑e }

An embedding e : Fin (n+1) ↪ ι corresponds to an embedding f : Fin n ↪ ι (corresponding the last n coordinates of e) together with a value not taken by f (corresponding to e 0).

Equations

• Equiv.embeddingFinSucc n ι = ((finSuccEquiv n).embeddingCongr (Equiv.refl ι)).trans (Function.Embedding.optionEmbeddingEquiv (Fin n) ι)

@[simp]

theorem Equiv.embeddingFinSucc_fst {n : ℕ} {ι : Type u_1} (e : Fin (n + 1) ↪ ι) : ⇑((embeddingFinSucc n ι) e).fst = ⇑e ∘ Fin.succ

@[simp]

theorem Equiv.embeddingFinSucc_snd {n : ℕ} {ι : Type u_1} (e : Fin (n + 1) ↪ ι) : ↑((embeddingFinSucc n ι) e).snd = e 0

@[simp]

theorem Equiv.coe_embeddingFinSucc_symm {n : ℕ} {ι : Type u_1} (f : (e : Fin n ↪ ι) × { i : ι // i ∉ Set.range ⇑e }) : ⇑((embeddingFinSucc n ι).symm f) = Fin.cons ↑f.snd ⇑f.fst

def finSumFinEquiv {m n : ℕ} : Fin m ⊕ Fin n ≃ Fin (m + n)

Equivalence between Fin m ⊕ Fin n and Fin (m + n)

Equations

• finSumFinEquiv = { toFun := Sum.elim (Fin.castAdd n) (Fin.natAdd m), invFun := fun (i : Fin (m + n)) => Fin.addCases Sum.inl Sum.inr i, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem finSumFinEquiv_apply_left {m n : ℕ} (i : Fin m) : finSumFinEquiv (Sum.inl i) = Fin.castAdd n i

@[simp]

theorem finSumFinEquiv_apply_right {m n : ℕ} (i : Fin n) : finSumFinEquiv (Sum.inr i) = Fin.natAdd m i

@[simp]

theorem finSumFinEquiv_symm_apply_castAdd {m n : ℕ} (x : Fin m) : finSumFinEquiv.symm (Fin.castAdd n x) = Sum.inl x

@[simp]

theorem finSumFinEquiv_symm_apply_castSucc {m : ℕ} (x : Fin m) : finSumFinEquiv.symm x.castSucc = Sum.inl x

@[simp]

theorem finSumFinEquiv_symm_apply_natAdd {m n : ℕ} (x : Fin n) : finSumFinEquiv.symm (Fin.natAdd m x) = Sum.inr x

@[simp]

theorem finSumFinEquiv_symm_last {n : ℕ} : finSumFinEquiv.symm (Fin.last n) = Sum.inr 0

def finSumNatEquiv (n : ℕ) : Fin n ⊕ ℕ ≃ ℕ

Equivalence between Fin n ⊕ ℕ and ℕ that sends inl (a : Fin n) to (a : ℕ) and inr a to n + a.

Equations

• finSumNatEquiv n = { toFun := Sum.elim Fin.val fun (x : ℕ) => n + x, invFun := fun (i : ℕ) => if hi : i < n then Sum.inl ⟨i, hi⟩ else Sum.inr (i - n), left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem finSumNatEquiv_apply_left {n : ℕ} (i : Fin n) : (finSumNatEquiv n) (Sum.inl i) = ↑i

@[simp]

theorem finSumNatEquiv_apply_right {n : ℕ} (i : ℕ) : (finSumNatEquiv n) (Sum.inr i) = n + i

@[simp]

theorem finSumNatEquiv_symm_apply_of_lt {n i : ℕ} (hi : i < n) : (finSumNatEquiv n).symm i = Sum.inl ⟨i, hi⟩

@[simp]

theorem finSumNatEquiv_symm_apply_of_ge {n i : ℕ} (hi : n ≤ i) : (finSumNatEquiv n).symm i = Sum.inr (i - n)

theorem finSumNatEquiv_symm_apply_fin {n : ℕ} (i : Fin n) : (finSumNatEquiv n).symm ↑i = Sum.inl i

theorem finSumNatEquiv_symm_apply_add_left {n : ℕ} (i : ℕ) : (finSumNatEquiv n).symm (i + n) = Sum.inr i

theorem finSumNatEquiv_symm_apply_add_right {n : ℕ} (i : ℕ) : (finSumNatEquiv n).symm (n + i) = Sum.inr i

@[simp]

theorem isLeft_finSumNatEquiv_symm_apply {n : ℕ} (i : ℕ) : ((finSumNatEquiv n).symm i).isLeft = decide (i < n)

@[simp]

theorem isRight_finSumNatEquiv_symm_apply {n : ℕ} (i : ℕ) : ((finSumNatEquiv n).symm i).isRight = decide (n ≤ i)

def finAddFlip {m n : ℕ} : Fin (m + n) ≃ Fin (n + m)

The equivalence between Fin (m + n) and Fin (n + m) which rotates by n.

Equations

• finAddFlip = (finSumFinEquiv.symm.trans (Equiv.sumComm (Fin m) (Fin n))).trans finSumFinEquiv

@[simp]

theorem finAddFlip_apply_castAdd {m : ℕ} (k : Fin m) (n : ℕ) : finAddFlip (Fin.castAdd n k) = Fin.natAdd n k

@[simp]

theorem finAddFlip_apply_natAdd {n : ℕ} (k : Fin n) (m : ℕ) : finAddFlip (Fin.natAdd m k) = Fin.castAdd m k

@[simp]

theorem finAddFlip_apply_mk_left {m n k : ℕ} (h : k < m) (hk : k < m + n := ⋯) (hnk : n + k < n + m := ⋯) : finAddFlip ⟨k, hk⟩ = ⟨n + k, hnk⟩

@[simp]

theorem finAddFlip_apply_mk_right {m n k : ℕ} (h₁ : m ≤ k) (h₂ : k < m + n) : finAddFlip ⟨k, h₂⟩ = ⟨k - m, ⋯⟩

def finProdFinEquiv {m n : ℕ} : Fin m × Fin n ≃ Fin (m * n)

Equivalence between Fin m × Fin n and Fin (m * n)

Equations

• finProdFinEquiv = { toFun := fun (x : Fin m × Fin n) => ⟨↑x.2 + n * ↑x.1, ⋯⟩, invFun := fun (x : Fin (m * n)) => (x.divNat, x.modNat), left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem finProdFinEquiv_apply_val {m n : ℕ} (x : Fin m × Fin n) : ↑(finProdFinEquiv x) = ↑x.2 + n * ↑x.1

@[simp]

theorem finProdFinEquiv_symm_apply {m n : ℕ} (x : Fin (m * n)) : finProdFinEquiv.symm x = (x.divNat, x.modNat)

def Nat.divModEquiv (n : ℕ) [NeZero n] : ℕ ≃ ℕ × Fin n

The equivalence induced by a ↦ (a / n, a % n) for nonzero n. This is like finProdFinEquiv.symm but with m infinite. See Nat.div_mod_unique for a similar propositional statement.

Equations

• n.divModEquiv = { toFun := fun (a : ℕ) => (a / n, Fin.ofNat n a), invFun := fun (p : ℕ × Fin n) => p.1 * n + ↑p.2, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Nat.divModEquiv_apply (n : ℕ) [NeZero n] (a : ℕ) : n.divModEquiv a = (a / n, Fin.ofNat n a)

@[simp]

theorem Nat.divModEquiv_symm_apply (n : ℕ) [NeZero n] (p : ℕ × Fin n) : n.divModEquiv.symm p = p.1 * n + ↑p.2

def Int.divModEquiv (n : ℕ) [NeZero n] : ℤ ≃ ℤ × Fin n

The equivalence induced by a ↦ (a / n, a % n) for nonzero n. See Int.ediv_emod_unique for a similar propositional statement.

Equations

• Int.divModEquiv n = { toFun := fun (a : ℤ) => (a / ↑n, Fin.ofNat n (a.natMod ↑n)), invFun := fun (p : ℤ × Fin n) => p.1 * ↑n + ↑↑p.2, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Int.divModEquiv_symm_apply (n : ℕ) [NeZero n] (p : ℤ × Fin n) : (divModEquiv n).symm p = p.1 * ↑n + ↑↑p.2

@[simp]

theorem Int.divModEquiv_apply (n : ℕ) [NeZero n] (a : ℤ) : (divModEquiv n) a = (a / ↑n, Fin.ofNat n (a.natMod ↑n))

def Fin.castLEquiv {n m : ℕ} (h : n ≤ m) : Fin n ≃ { i : Fin m // ↑i < n }

Promote a Fin n into a larger Fin m, as a subtype where the underlying values are retained.

This is the Equiv version of Fin.castLE.

Equations

• Fin.castLEquiv h = { toFun := fun (i : Fin n) => ⟨Fin.castLE h i, ⋯⟩, invFun := fun (i : { i : Fin m // ↑i < n }) => ⟨↑↑i, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Fin.castLEquiv_symm_apply {n m : ℕ} (h : n ≤ m) (i : { i : Fin m // ↑i < n }) : (castLEquiv h).symm i = ⟨↑↑i, ⋯⟩

@[simp]

theorem Fin.castLEquiv_apply {n m : ℕ} (h : n ≤ m) (i : Fin n) : (castLEquiv h) i = ⟨castLE h i, ⋯⟩

@[deprecated Fin.subsingleton_zero (since := "2025-06-03")]

theorem subsingleton_fin_zero : Subsingleton (Fin 0)

@[deprecated Fin.subsingleton_one (since := "2025-06-03")]

theorem subsingleton_fin_one : Subsingleton (Fin 1)

def Fin.appendEquiv {α : Type u_1} (m n : ℕ) : (Fin m → α) × (Fin n → α) ≃ (Fin (m + n) → α)

The natural Equiv between (Fin m → α) × (Fin n → α) and Fin (m + n) → α

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Fin.appendEquiv_apply {α : Type u_1} (m n : ℕ) (fg : (Fin m → α) × (Fin n → α)) (a✝ : Fin (m + n)) : (appendEquiv m n) fg a✝ = append fg.1 fg.2 a✝

@[simp]

theorem Fin.appendEquiv_symm_apply {α : Type u_1} (m n : ℕ) (f : Fin (m + n) → α) : (appendEquiv m n).symm f = (fun (i : Fin m) => f (castAdd n i), fun (i : Fin n) => f (natAdd m i))

def Fin.succFunEquiv (α : Type u_1) (n : ℕ) : (Fin (n + 1) → α) ≃ (Fin n → α) × α

Fin (n + 1) → α and (Fin n → α) × α are equivalent.

Equations

• Fin.succFunEquiv α n = (Fin.appendEquiv n 1).symm.trans (Equiv.prodCongrRight fun (x : Fin n → α) => Equiv.funUnique (Fin 1) α)

@[simp]

theorem Fin.succFunEquiv_apply (α : Type u_1) (n : ℕ) (a✝ : Fin (n + 1) → α) : (succFunEquiv α n) a✝ = (fun (i : Fin n) => a✝ (castAdd 1 i), a✝ (natAdd n 0))

@[simp]

theorem Fin.succFunEquiv_symm_apply (α : Type u_1) (n : ℕ) (a✝ : (Fin n → α) × α) (a✝¹ : Fin (n + 1)) : (succFunEquiv α n).symm a✝ a✝¹ = append ((Equiv.prodCongrRight fun (x : Fin n → α) => Equiv.funUnique (Fin 1) α).symm a✝).1 ((Equiv.prodCongrRight fun (x : Fin n → α) => Equiv.funUnique (Fin 1) α).symm a✝).2 a✝¹