Operation on tuples

We interpret maps ∀ i : Fin n, α i as n-tuples of elements of possibly varying type α i, (α 0, …, α (n-1)). A particular case is Fin n → α of elements with all the same type. In this case when α i is a constant map, then tuples are isomorphic (but not definitionally equal) to Vectors.

Main declarations

There are three (main) ways to consider Fin n as a subtype of Fin (n + 1), hence three (main) ways to move between tuples of length n and of length n + 1 by adding/removing an entry.

Adding at the start

• Fin.succ: Send i : Fin n to i + 1 : Fin (n + 1). This is defined in Core.
• Fin.cases: Induction/recursion principle for Fin: To prove a property/define a function for all Fin (n + 1), it is enough to prove/define it for 0 and for i.succ for all i : Fin n. This is defined in Core.
• Fin.cons: Turn a tuple f : Fin n → α and an entry a : α into a tuple Fin.cons a f : Fin (n + 1) → α by adding a at the start. In general, tuples can be dependent functions, in which case f : ∀ i : Fin n, α i.succ and a : α 0. This is a special case of Fin.cases.
• Fin.tail: Turn a tuple f : Fin (n + 1) → α into a tuple Fin.tail f : Fin n → α by forgetting the start. In general, tuples can be dependent functions, in which case Fin.tail f : ∀ i : Fin n, α i.succ.

Adding at the end

• Fin.castSucc: Send i : Fin n to i : Fin (n + 1). This is defined in Core.
• Fin.lastCases: Induction/recursion principle for Fin: To prove a property/define a function for all Fin (n + 1), it is enough to prove/define it for last n and for i.castSucc for all i : Fin n. This is defined in Core.
• Fin.snoc: Turn a tuple f : Fin n → α and an entry a : α into a tuple Fin.snoc f a : Fin (n + 1) → α by adding a at the end. In general, tuples can be dependent functions, in which case f : ∀ i : Fin n, α i.castSucc and a : α (last n). This is a special case of Fin.lastCases.
• Fin.init: Turn a tuple f : Fin (n + 1) → α into a tuple Fin.init f : Fin n → α by forgetting the start. In general, tuples can be dependent functions, in which case Fin.init f : ∀ i : Fin n, α i.castSucc.

Adding in the middle

For a pivot p : Fin (n + 1),

• Fin.succAbove: Send i : Fin n to
  • i : Fin (n + 1) if i < p,
  • i + 1 : Fin (n + 1) if p ≤ i.
• Fin.succAboveCases: Induction/recursion principle for Fin: To prove a property/define a function for all Fin (n + 1), it is enough to prove/define it for p and for p.succAbove i for all i : Fin n.
• Fin.insertNth: Turn a tuple f : Fin n → α and an entry a : α into a tuple Fin.insertNth f a : Fin (n + 1) → α by adding a in position p. In general, tuples can be dependent functions, in which case f : ∀ i : Fin n, α (p.succAbove i) and a : α p. This is a special case of Fin.succAboveCases.
• Fin.removeNth: Turn a tuple f : Fin (n + 1) → α into a tuple Fin.removeNth p f : Fin n → α by forgetting the p-th value. In general, tuples can be dependent functions, in which case Fin.removeNth f : ∀ i : Fin n, α (succAbove p i).

p = 0 means we add at the start. p = last n means we add at the end.

Miscellaneous

• Fin.find p : returns the first index n where p n is satisfied, and none if it is never satisfied.
• Fin.append a b : append two tuples.
• Fin.repeat n a : repeat a tuple n times.

theorem Fin.tuple0_le {α : Fin 0 → Type u_1} [(i : Fin 0) → Preorder (α i)] (f g : (i : Fin 0) → α i) : f ≤ g

def Fin.tail {n : ℕ} {α : Fin (n + 1) → Sort u} (q : (i : Fin (n + 1)) → α i) (i : Fin n) : α i.succ

The tail of an n+1 tuple, i.e., its last n entries.

Equations

• Fin.tail q i = q i.succ

theorem Fin.tail_def {n : ℕ} {α : Fin (n + 1) → Sort u_1} {q : (i : Fin (n + 1)) → α i} : (tail fun (k : Fin (n + 1)) => q k) = fun (k : Fin n) => q k.succ

def Fin.cons {n : ℕ} {α : Fin (n + 1) → Sort u} (x : α 0) (p : (i : Fin n) → α i.succ) (i : Fin (n + 1)) : α i

Adding an element at the beginning of an n-tuple, to get an n+1-tuple.

Equations

• Fin.cons x p j = Fin.cases x p j

@[simp]

theorem Fin.tail_cons {n : ℕ} {α : Fin (n + 1) → Sort u} (x : α 0) (p : (i : Fin n) → α i.succ) : tail (cons x p) = p

@[simp]

theorem Fin.cons_succ {n : ℕ} {α : Fin (n + 1) → Sort u} (x : α 0) (p : (i : Fin n) → α i.succ) (i : Fin n) : cons x p i.succ = p i

@[simp]

theorem Fin.cons_zero {n : ℕ} {α : Fin (n + 1) → Sort u} (x : α 0) (p : (i : Fin n) → α i.succ) : cons x p 0 = x

@[simp]

theorem Fin.cons_one {n : ℕ} {α : Fin (n + 2) → Sort u_1} (x : α 0) (p : (i : Fin n.succ) → α i.succ) : cons x p 1 = p 0

@[simp]

theorem Fin.cons_update {n : ℕ} {α : Fin (n + 1) → Sort u} (x : α 0) (p : (i : Fin n) → α i.succ) (i : Fin n) (y : α i.succ) : cons x (Function.update p i y) = Function.update (cons x p) i.succ y

Updating a tuple and adding an element at the beginning commute.

theorem Fin.cons_injective2 {n : ℕ} {α : Fin (n + 1) → Sort u} : Function.Injective2 cons

As a binary function, Fin.cons is injective.

@[simp]

theorem Fin.cons_inj {n : ℕ} {α : Fin (n + 1) → Sort u} {x₀ y₀ : α 0} {x y : (i : Fin n) → α i.succ} : cons x₀ x = cons y₀ y ↔ x₀ = y₀ ∧ x = y

theorem Fin.cons_left_injective {n : ℕ} {α : Fin (n + 1) → Sort u} (x : (i : Fin n) → α i.succ) : Function.Injective fun (x₀ : α 0) => cons x₀ x

theorem Fin.cons_right_injective {n : ℕ} {α : Fin (n + 1) → Sort u} (x₀ : α 0) : Function.Injective (cons x₀)

theorem Fin.update_cons_zero {n : ℕ} {α : Fin (n + 1) → Sort u} (x : α 0) (p : (i : Fin n) → α i.succ) (z : α 0) : Function.update (cons x p) 0 z = cons z p

Adding an element at the beginning of a tuple and then updating it amounts to adding it directly.

@[simp]

theorem Fin.cons_self_tail {n : ℕ} {α : Fin (n + 1) → Sort u} (q : (i : Fin (n + 1)) → α i) : cons (q 0) (tail q) = q

Concatenating the first element of a tuple with its tail gives back the original tuple

def Fin.consEquiv {n : ℕ} (α : Fin (n + 1) → Type u_1) : α 0 × ((i : Fin n) → α i.succ) ≃ ((i : Fin (n + 1)) → α i)

Equivalence between tuples of length n + 1 and pairs of an element and a tuple of length n given by separating out the first element of the tuple.

This is Fin.cons as an Equiv.

Equations

• Fin.consEquiv α = { toFun := fun (f : α 0 × ((i : Fin n) → α i.succ)) => Fin.cons f.1 f.2, invFun := fun (f : (i : Fin (n + 1)) → α i) => (f 0, Fin.tail f), left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Fin.consEquiv_symm_apply {n : ℕ} (α : Fin (n + 1) → Type u_1) (f : (i : Fin (n + 1)) → α i) : (consEquiv α).symm f = (f 0, tail f)

@[simp]

theorem Fin.consEquiv_apply {n : ℕ} (α : Fin (n + 1) → Type u_1) (f : α 0 × ((i : Fin n) → α i.succ)) (i : Fin (n + 1)) : (consEquiv α) f i = cons f.1 f.2 i

def Fin.consCases {n : ℕ} {α : Fin (n + 1) → Sort u} {P : ((i : Fin n.succ) → α i) → Sort v} (h : (x₀ : α 0) → (x : (i : Fin n) → α i.succ) → P (cons x₀ x)) (x : (i : Fin n.succ) → α i) : P x

Recurse on an n+1-tuple by splitting it into a single element and an n-tuple.

Equations

• Fin.consCases h x = cast ⋯ (h (x 0) (Fin.tail x))

@[simp]

theorem Fin.consCases_cons {n : ℕ} {α : Fin (n + 1) → Sort u} {P : ((i : Fin n.succ) → α i) → Sort v} (h : (x₀ : α 0) → (x : (i : Fin n) → α i.succ) → P (cons x₀ x)) (x₀ : α 0) (x : (i : Fin n) → α i.succ) : consCases h (cons x₀ x) = h x₀ x

def Fin.consInduction {α : Sort u_1} {P : {n : ℕ} → (Fin n → α) → Sort v} (h0 : P elim0) (h : {n : ℕ} → (x₀ : α) → (x : Fin n → α) → P x → P (cons x₀ x)) {n : ℕ} (x : Fin n → α) : P x

Recurse on a tuple by splitting into Fin.elim0 and Fin.cons.

Equations

• Fin.consInduction h0 h x_2 = ⋯.mpr h0
• Fin.consInduction h0 h x_2 = Fin.consCases (fun (x : α) (x_1 : Fin n → α) => h x x_1 (Fin.consInduction h0 (fun {n : ℕ} => h) x_1)) x_2

theorem Fin.cons_injective_of_injective {n : ℕ} {α : Type u_1} {x₀ : α} {x : Fin n → α} (hx₀ : x₀ ∉ Set.range x) (hx : Function.Injective x) : Function.Injective (cons x₀ x)

theorem Fin.cons_injective_iff {n : ℕ} {α : Type u_1} {x₀ : α} {x : Fin n → α} : Function.Injective (cons x₀ x) ↔ x₀ ∉ Set.range x ∧ Function.Injective x

@[simp]

theorem Fin.forall_fin_zero_pi {α : Fin 0 → Sort u_1} {P : ((i : Fin 0) → α i) → Prop} : (∀ (x : (i : Fin 0) → α i), P x) ↔ P finZeroElim

@[simp]

theorem Fin.exists_fin_zero_pi {α : Fin 0 → Sort u_1} {P : ((i : Fin 0) → α i) → Prop} : (∃ (x : (i : Fin 0) → α i), P x) ↔ P finZeroElim

theorem Fin.forall_fin_succ_pi {n : ℕ} {α : Fin (n + 1) → Sort u} {P : ((i : Fin (n + 1)) → α i) → Prop} : (∀ (x : (i : Fin (n + 1)) → α i), P x) ↔ ∀ (a : α 0) (v : (i : Fin n) → α i.succ), P (cons a v)

theorem Fin.exists_fin_succ_pi {n : ℕ} {α : Fin (n + 1) → Sort u} {P : ((i : Fin (n + 1)) → α i) → Prop} : (∃ (x : (i : Fin (n + 1)) → α i), P x) ↔ ∃ (a : α 0) (v : (i : Fin n) → α i.succ), P (cons a v)

@[simp]

theorem Fin.tail_update_zero {n : ℕ} {α : Fin (n + 1) → Sort u} (q : (i : Fin (n + 1)) → α i) (z : α 0) : tail (Function.update q 0 z) = tail q

Updating the first element of a tuple does not change the tail.

@[simp]

theorem Fin.tail_update_succ {n : ℕ} {α : Fin (n + 1) → Sort u} (q : (i : Fin (n + 1)) → α i) (i : Fin n) (y : α i.succ) : tail (Function.update q i.succ y) = Function.update (tail q) i y

Updating a nonzero element and taking the tail commute.

theorem Fin.comp_cons {n : ℕ} {α : Sort u_1} {β : Sort u_2} (g : α → β) (y : α) (q : Fin n → α) : g ∘ cons y q = cons (g y) (g ∘ q)

theorem Fin.comp_tail {n : ℕ} {α : Sort u_1} {β : Sort u_2} (g : α → β) (q : Fin n.succ → α) : g ∘ tail q = tail (g ∘ q)

theorem Fin.le_cons {n : ℕ} {α : Fin (n + 1) → Type u_1} [(i : Fin (n + 1)) → Preorder (α i)] {x : α 0} {q : (i : Fin (n + 1)) → α i} {p : (i : Fin n) → α i.succ} : q ≤ cons x p ↔ q 0 ≤ x ∧ tail q ≤ p

theorem Fin.cons_le {n : ℕ} {α : Fin (n + 1) → Type u_1} [(i : Fin (n + 1)) → Preorder (α i)] {x : α 0} {q : (i : Fin (n + 1)) → α i} {p : (i : Fin n) → α i.succ} : cons x p ≤ q ↔ x ≤ q 0 ∧ p ≤ tail q

theorem Fin.cons_le_cons {n : ℕ} {α : Fin (n + 1) → Type u_1} [(i : Fin (n + 1)) → Preorder (α i)] {x₀ y₀ : α 0} {x y : (i : Fin n) → α i.succ} : cons x₀ x ≤ cons y₀ y ↔ x₀ ≤ y₀ ∧ x ≤ y

theorem Fin.range_fin_succ {n : ℕ} {α : Type u_1} (f : Fin (n + 1) → α) : Set.range f = insert (f 0) (Set.range (tail f))

@[simp]

theorem Fin.range_cons {α : Type u_1} {n : ℕ} (x : α) (b : Fin n → α) : Set.range (cons x b) = insert x (Set.range b)

def Fin.append {m n : ℕ} {α : Sort u_1} (a : Fin m → α) (b : Fin n → α) : Fin (m + n) → α

Append a tuple of length m to a tuple of length n to get a tuple of length m + n. This is a non-dependent version of Fin.add_cases.

Equations

• Fin.append a b = Fin.addCases a b

@[simp]

theorem Fin.append_left {m n : ℕ} {α : Sort u_1} (u : Fin m → α) (v : Fin n → α) (i : Fin m) : append u v (castAdd n i) = u i

@[simp]

theorem Fin.append_right {m n : ℕ} {α : Sort u_1} (u : Fin m → α) (v : Fin n → α) (i : Fin n) : append u v (natAdd m i) = v i

theorem Fin.append_right_nil {m n : ℕ} {α : Sort u_1} (u : Fin m → α) (v : Fin n → α) (hv : n = 0) : append u v = u ∘ Fin.cast ⋯

@[simp]

theorem Fin.append_elim0 {m : ℕ} {α : Sort u_1} (u : Fin m → α) : append u elim0 = u ∘ Fin.cast ⋯

theorem Fin.append_left_nil {m n : ℕ} {α : Sort u_1} (u : Fin m → α) (v : Fin n → α) (hu : m = 0) : append u v = v ∘ Fin.cast ⋯

@[simp]

theorem Fin.elim0_append {n : ℕ} {α : Sort u_1} (v : Fin n → α) : append elim0 v = v ∘ Fin.cast ⋯

theorem Fin.append_assoc {m n : ℕ} {α : Sort u_1} {p : ℕ} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) : append (append a b) c = append a (append b c) ∘ Fin.cast ⋯

theorem Fin.append_left_eq_cons {α : Sort u_1} {n : ℕ} (x₀ : Fin 1 → α) (x : Fin n → α) : append x₀ x = cons (x₀ 0) x ∘ Fin.cast ⋯

Appending a one-tuple to the left is the same as Fin.cons.

theorem Fin.cons_eq_append {n : ℕ} {α : Sort u_1} (x : α) (xs : Fin n → α) : cons x xs = append (cons x elim0) xs ∘ Fin.cast ⋯

Fin.cons is the same as appending a one-tuple to the left.

@[simp]

theorem Fin.append_cast_left {α : Sort u_1} {n m : ℕ} (xs : Fin n → α) (ys : Fin m → α) (n' : ℕ) (h : n' = n) : append (xs ∘ Fin.cast h) ys = append xs ys ∘ Fin.cast ⋯

@[simp]

theorem Fin.append_cast_right {α : Sort u_1} {n m : ℕ} (xs : Fin n → α) (ys : Fin m → α) (m' : ℕ) (h : m' = m) : append xs (ys ∘ Fin.cast h) = append xs ys ∘ Fin.cast ⋯

theorem Fin.append_rev {α : Sort u_1} {m n : ℕ} (xs : Fin m → α) (ys : Fin n → α) (i : Fin (m + n)) : append xs ys i.rev = append (ys ∘ rev) (xs ∘ rev) (Fin.cast ⋯ i)

theorem Fin.append_comp_rev {α : Sort u_1} {m n : ℕ} (xs : Fin m → α) (ys : Fin n → α) : append xs ys ∘ rev = append (ys ∘ rev) (xs ∘ rev) ∘ Fin.cast ⋯

theorem Fin.append_castAdd_natAdd {m n : ℕ} {α : Sort u_1} {f : Fin (m + n) → α} : (append (fun (i : Fin m) => f (castAdd n i)) fun (i : Fin n) => f (natAdd m i)) = f

def Fin.repeat {n : ℕ} {α : Sort u_1} (m : ℕ) (a : Fin n → α) : Fin (m * n) → α

Repeat a m times. For example Fin.repeat 2 ![0, 3, 7] = ![0, 3, 7, 0, 3, 7].

Equations

• Fin.repeat m a x✝ = a x✝.modNat

@[simp]

theorem Fin.repeat_apply {m n : ℕ} {α : Sort u_1} (a : Fin n → α) (i : Fin (m * n)) : «repeat» m a i = a i.modNat

@[simp]

theorem Fin.repeat_zero {n : ℕ} {α : Sort u_1} (a : Fin n → α) : «repeat» 0 a = elim0 ∘ Fin.cast ⋯

@[simp]

theorem Fin.repeat_one {n : ℕ} {α : Sort u_1} (a : Fin n → α) : «repeat» 1 a = a ∘ Fin.cast ⋯

theorem Fin.repeat_succ {n : ℕ} {α : Sort u_1} (a : Fin n → α) (m : ℕ) : «repeat» m.succ a = append a («repeat» m a) ∘ Fin.cast ⋯

@[simp]

theorem Fin.repeat_add {n : ℕ} {α : Sort u_1} (a : Fin n → α) (m₁ m₂ : ℕ) : «repeat» (m₁ + m₂) a = append («repeat» m₁ a) («repeat» m₂ a) ∘ Fin.cast ⋯

theorem Fin.repeat_rev {m n : ℕ} {α : Sort u_1} (a : Fin n → α) (k : Fin (m * n)) : «repeat» m a k.rev = «repeat» m (a ∘ rev) k

theorem Fin.repeat_comp_rev {m n : ℕ} {α : Sort u_1} (a : Fin n → α) : «repeat» m a ∘ rev = «repeat» m (a ∘ rev)

In the previous section, we have discussed inserting or removing elements on the left of a tuple. In this section, we do the same on the right. A difference is that Fin (n+1) is constructed inductively from Fin n starting from the left, not from the right. This implies that Lean needs more help to realize that elements belong to the right types, i.e., we need to insert casts at several places.

def Fin.init {n : ℕ} {α : Fin (n + 1) → Sort u_1} (q : (i : Fin (n + 1)) → α i) (i : Fin n) : α i.castSucc

The beginning of an n+1 tuple, i.e., its first n entries

Equations

• Fin.init q i = q i.castSucc

theorem Fin.init_def {n : ℕ} {α : Fin (n + 1) → Sort u_1} {q : (i : Fin (n + 1)) → α i} : (init fun (k : Fin (n + 1)) => q k) = fun (k : Fin n) => q k.castSucc

def Fin.snoc {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : (i : Fin n) → α i.castSucc) (x : α (last n)) (i : Fin (n + 1)) : α i

Adding an element at the end of an n-tuple, to get an n+1-tuple. The name snoc comes from cons (i.e., adding an element to the left of a tuple) read in reverse order.

Equations

• Fin.snoc p x i = if h : ↑i < n then cast ⋯ (p (i.castLT h)) else cast ⋯ x

@[simp]

theorem Fin.init_snoc {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (last n)) (p : (i : Fin n) → α i.castSucc) : init (snoc p x) = p

@[simp]

theorem Fin.snoc_castSucc {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (last n)) (p : (i : Fin n) → α i.castSucc) (i : Fin n) : snoc p x i.castSucc = p i

@[simp]

theorem Fin.snoc_comp_castSucc {n : ℕ} {α : Sort u_2} {a : α} {f : Fin n → α} : snoc f a ∘ castSucc = f

@[simp]

theorem Fin.snoc_last {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (last n)) (p : (i : Fin n) → α i.castSucc) : snoc p x (last n) = x

theorem Fin.snoc_zero {α : Sort u_2} (p : Fin 0 → α) (x : α) : snoc p x = fun (x_1 : Fin (0 + 1)) => x

@[simp]

theorem Fin.snoc_comp_nat_add {n m : ℕ} {α : Sort u_2} (f : Fin (m + n) → α) (a : α) : snoc f a ∘ natAdd m = snoc (f ∘ natAdd m) a

@[simp]

theorem Fin.snoc_cast_add {m n : ℕ} {α : Fin (n + m + 1) → Sort u_2} (f : (i : Fin (n + m)) → α i.castSucc) (a : α (last (n + m))) (i : Fin n) : snoc f a (castAdd (m + 1) i) = f (castAdd m i)

@[simp]

theorem Fin.snoc_comp_cast_add {n m : ℕ} {α : Sort u_2} (f : Fin (n + m) → α) (a : α) : snoc f a ∘ castAdd (m + 1) = f ∘ castAdd m

@[simp]

theorem Fin.snoc_update {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (last n)) (p : (i : Fin n) → α i.castSucc) (i : Fin n) (y : α i.castSucc) : snoc (Function.update p i y) x = Function.update (snoc p x) i.castSucc y

Updating a tuple and adding an element at the end commute.

theorem Fin.update_snoc_last {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (last n)) (p : (i : Fin n) → α i.castSucc) (z : α (last n)) : Function.update (snoc p x) (last n) z = snoc p z

Adding an element at the beginning of a tuple and then updating it amounts to adding it directly.

theorem Fin.snoc_injective2 {n : ℕ} {α : Fin (n + 1) → Sort u_1} : Function.Injective2 snoc

As a binary function, Fin.snoc is injective.

@[simp]

theorem Fin.snoc_inj {n : ℕ} {α : Fin (n + 1) → Sort u_1} {x y : (i : Fin n) → α i.castSucc} {xₙ yₙ : α (last n)} : snoc x xₙ = snoc y yₙ ↔ x = y ∧ xₙ = yₙ

theorem Fin.snoc_right_injective {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : (i : Fin n) → α i.castSucc) : Function.Injective (snoc x)

theorem Fin.snoc_left_injective {n : ℕ} {α : Fin (n + 1) → Sort u_1} (xₙ : α (last n)) : Function.Injective fun (x : (i : Fin n) → α i.castSucc) => snoc x xₙ

@[simp]

theorem Fin.snoc_init_self {n : ℕ} {α : Fin (n + 1) → Sort u_1} (q : (i : Fin (n + 1)) → α i) : snoc (init q) (q (last n)) = q

Concatenating the first element of a tuple with its tail gives back the original tuple

@[simp]

theorem Fin.init_update_last {n : ℕ} {α : Fin (n + 1) → Sort u_1} (q : (i : Fin (n + 1)) → α i) (z : α (last n)) : init (Function.update q (last n) z) = init q

Updating the last element of a tuple does not change the beginning.

@[simp]

theorem Fin.init_update_castSucc {n : ℕ} {α : Fin (n + 1) → Sort u_1} (q : (i : Fin (n + 1)) → α i) (i : Fin n) (y : α i.castSucc) : init (Function.update q i.castSucc y) = Function.update (init q) i y

Updating an element and taking the beginning commute.

theorem Fin.tail_init_eq_init_tail {n : ℕ} {β : Sort u_2} (q : Fin (n + 2) → β) : tail (init q) = init (tail q)

tail and init commute. We state this lemma in a non-dependent setting, as otherwise it would involve a cast to convince Lean that the two types are equal, making it harder to use.

theorem Fin.cons_snoc_eq_snoc_cons {n : ℕ} {β : Sort u_2} (a : β) (q : Fin n → β) (b : β) : cons a (snoc q b) = snoc (cons a q) b

cons and snoc commute. We state this lemma in a non-dependent setting, as otherwise it would involve a cast to convince Lean that the two types are equal, making it harder to use.

theorem Fin.comp_snoc {n : ℕ} {α : Sort u_2} {β : Sort u_3} (g : α → β) (q : Fin n → α) (y : α) : g ∘ snoc q y = snoc (g ∘ q) (g y)

theorem Fin.append_right_eq_snoc {α : Sort u_2} {n : ℕ} (x : Fin n → α) (x₀ : Fin 1 → α) : append x x₀ = snoc x (x₀ 0)

Appending a one-tuple to the right is the same as Fin.snoc.

theorem Fin.snoc_eq_append {n : ℕ} {α : Sort u_2} (xs : Fin n → α) (x : α) : snoc xs x = append xs (cons x elim0)

Fin.snoc is the same as appending a one-tuple

theorem Fin.append_left_snoc {n m : ℕ} {α : Sort u_2} (xs : Fin n → α) (x : α) (ys : Fin m → α) : append (snoc xs x) ys = append xs (cons x ys) ∘ Fin.cast ⋯

theorem Fin.append_right_cons {n m : ℕ} {α : Sort u_2} (xs : Fin n → α) (y : α) (ys : Fin m → α) : append xs (cons y ys) = append (snoc xs y) ys ∘ Fin.cast ⋯

theorem Fin.append_cons {m n : ℕ} {α : Sort u_2} (a : α) (as : Fin n → α) (bs : Fin m → α) : append (cons a as) bs = cons a (append as bs) ∘ Fin.cast ⋯

theorem Fin.append_snoc {m n : ℕ} {α : Sort u_2} (as : Fin n → α) (bs : Fin m → α) (b : α) : append as (snoc bs b) = snoc (append as bs) b

theorem Fin.comp_init {n : ℕ} {α : Sort u_2} {β : Sort u_3} (g : α → β) (q : Fin n.succ → α) : g ∘ init q = init (g ∘ q)

def Fin.snocEquiv {n : ℕ} (α : Fin (n + 1) → Type u_2) : α (last n) × ((i : Fin n) → α i.castSucc) ≃ ((i : Fin (n + 1)) → α i)

Equivalence between tuples of length n + 1 and pairs of an element and a tuple of length n given by separating out the last element of the tuple.

This is Fin.snoc as an Equiv.

Equations

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

@[simp]

theorem Fin.snocEquiv_apply {n : ℕ} (α : Fin (n + 1) → Type u_2) (f : α (last n) × ((i : Fin n) → α i.castSucc)) (x✝ : Fin (n + 1)) : (snocEquiv α) f x✝ = snoc f.2 f.1 x✝

@[simp]

theorem Fin.snocEquiv_symm_apply {n : ℕ} (α : Fin (n + 1) → Type u_2) (f : (i : Fin (n + 1)) → α i) : (snocEquiv α).symm f = (f (last n), init f)

@[inline]

def Fin.snocCases {n : ℕ} {α : Fin (n + 1) → Sort u_1} {P : ((i : Fin n.succ) → α i) → Sort u_2} (h : (xs : (i : Fin n) → α i.castSucc) → (x : α (last n)) → P (snoc xs x)) (x : (i : Fin n.succ) → α i) : P x

Recurse on an n+1-tuple by splitting it its initial n-tuple and its last element.

Equations

• Fin.snocCases h x = cast ⋯ (h (Fin.init x) (x (Fin.last n)))

@[simp]

theorem Fin.snocCases_snoc {n : ℕ} {α : Fin (n + 1) → Sort u_1} {P : ((i : Fin (n + 1)) → α i) → Sort u_2} (h : (x : (i : Fin n) → α i.castSucc) → (x₀ : α (last n)) → P (snoc x x₀)) (x : (i : Fin n) → init α i) (x₀ : α (last n)) : snocCases h (snoc x x₀) = h x x₀

def Fin.snocInduction {α : Sort u_2} {P : {n : ℕ} → (Fin n → α) → Sort u_3} (h0 : P elim0) (h : {n : ℕ} → (x : Fin n → α) → (x₀ : α) → P x → P (snoc x x₀)) {n : ℕ} (x : Fin n → α) : P x

Recurse on a tuple by splitting into Fin.elim0 and Fin.snoc.

Equations

• Fin.snocInduction h0 h x_2 = ⋯.mpr h0
• Fin.snocInduction h0 h x_2 = Fin.snocCases (fun (x : Fin n → α) (x_1 : α) => h x x_1 (Fin.snocInduction h0 (fun {n : ℕ} => h) x)) x_2

def Fin.succAboveCases {n : ℕ} {α : Fin (n + 1) → Sort u} (i : Fin (n + 1)) (x : α i) (p : (j : Fin n) → α (i.succAbove j)) (j : Fin (n + 1)) : α j

Define a function on Fin (n + 1) from a value on i : Fin (n + 1) and values on each Fin.succAbove i j, j : Fin n. This version is elaborated as eliminator and works for propositions, see also Fin.insertNth for a version without an @[elab_as_elim] attribute.

Equations

• i.succAboveCases x p j = if hj : j = i then ⋯ ▸ x else if hlt : j < i then Eq.recOn ⋯ (p (j.castPred ⋯)) else Eq.recOn ⋯ (p (j.pred ⋯))

theorem Fin.forall_iff_succ {n : ℕ} {P : Fin (n + 1) → Prop} : (∀ (i : Fin (n + 1)), P i) ↔ P 0 ∧ ∀ (i : Fin n), P i.succ

Alias of Fin.forall_fin_succ.

theorem Fin.exists_iff_succ {n : ℕ} {P : Fin (n + 1) → Prop} : (∃ (i : Fin (n + 1)), P i) ↔ P 0 ∨ ∃ (i : Fin n), P i.succ

Alias of Fin.exists_fin_succ.

theorem Fin.forall_iff_castSucc {n : ℕ} {P : Fin (n + 1) → Prop} : (∀ (i : Fin (n + 1)), P i) ↔ P (last n) ∧ ∀ (i : Fin n), P i.castSucc

theorem Fin.exists_iff_castSucc {n : ℕ} {P : Fin (n + 1) → Prop} : (∃ (i : Fin (n + 1)), P i) ↔ P (last n) ∨ ∃ (i : Fin n), P i.castSucc

theorem Fin.forall_iff_succAbove {n : ℕ} {P : Fin (n + 1) → Prop} (p : Fin (n + 1)) : (∀ (i : Fin (n + 1)), P i) ↔ P p ∧ ∀ (i : Fin n), P (p.succAbove i)

theorem Fin.exists_iff_succAbove {n : ℕ} {P : Fin (n + 1) → Prop} (p : Fin (n + 1)) : (∃ (i : Fin (n + 1)), P i) ↔ P p ∨ ∃ (i : Fin n), P (p.succAbove i)

theorem Fin.eq_self_or_eq_succAbove {n : ℕ} (p i : Fin (n + 1)) : i = p ∨ ∃ (j : Fin n), i = p.succAbove j

Analogue of Fin.eq_zero_or_eq_succ for succAbove.

def Fin.removeNth {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : Fin (n + 1)) (f : (i : Fin (n + 1)) → α i) (i : Fin n) : α (p.succAbove i)

Remove the p-th entry of a tuple.

Equations

• p.removeNth f i = f (p.succAbove i)

def Fin.insertNth {n : ℕ} {α : Fin (n + 1) → Sort u_1} (i : Fin (n + 1)) (x : α i) (p : (j : Fin n) → α (i.succAbove j)) (j : Fin (n + 1)) : α j

Insert an element into a tuple at a given position. For i = 0 see Fin.cons, for i = Fin.last n see Fin.snoc. See also Fin.succAboveCases for a version elaborated as an eliminator.

Equations

• i.insertNth x p j = i.succAboveCases x p j

@[simp]

theorem Fin.insertNth_apply_same {n : ℕ} {α : Fin (n + 1) → Sort u_1} (i : Fin (n + 1)) (x : α i) (p : (j : Fin n) → α (i.succAbove j)) : i.insertNth x p i = x

@[simp]

theorem Fin.insertNth_apply_succAbove {n : ℕ} {α : Fin (n + 1) → Sort u_1} (i : Fin (n + 1)) (x : α i) (p : (j : Fin n) → α (i.succAbove j)) (j : Fin n) : i.insertNth x p (i.succAbove j) = p j

@[simp]

theorem Fin.succAbove_cases_eq_insertNth : @succAboveCases = @insertNth

@[simp]

theorem Fin.removeNth_insertNth {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : Fin (n + 1)) (a : α p) (f : (i : Fin n) → α (p.succAbove i)) : p.removeNth (p.insertNth a f) = f

@[simp]

theorem Fin.removeNth_zero {n : ℕ} {α : Fin (n + 1) → Sort u_1} (f : (i : Fin (n + 1)) → α i) : removeNth 0 f = tail f

@[simp]

theorem Fin.removeNth_last {n : ℕ} {α : Type u_3} (f : Fin (n + 1) → α) : (last n).removeNth f = init f

@[simp]

theorem Fin.insertNth_comp_succAbove {n : ℕ} {β : Sort u_2} (i : Fin (n + 1)) (x : β) (p : Fin n → β) : i.insertNth x p ∘ i.succAbove = p

theorem Fin.insertNth_eq_iff {n : ℕ} {α : Fin (n + 1) → Sort u_1} {p : Fin (n + 1)} {a : α p} {f : (i : Fin n) → α (p.succAbove i)} {g : (j : Fin (n + 1)) → α j} : p.insertNth a f = g ↔ a = g p ∧ f = p.removeNth g

theorem Fin.eq_insertNth_iff {n : ℕ} {α : Fin (n + 1) → Sort u_1} {p : Fin (n + 1)} {a : α p} {f : (i : Fin n) → α (p.succAbove i)} {g : (j : Fin (n + 1)) → α j} : g = p.insertNth a f ↔ g p = a ∧ p.removeNth g = f

theorem Fin.insertNth_injective2 {n : ℕ} {α : Fin (n + 1) → Sort u_1} {p : Fin (n + 1)} : Function.Injective2 p.insertNth

As a binary function, Fin.insertNth is injective.

@[simp]

theorem Fin.insertNth_inj {n : ℕ} {α : Fin (n + 1) → Sort u_1} {p : Fin (n + 1)} {x y : (i : Fin n) → α (p.succAbove i)} {xₚ yₚ : α p} : p.insertNth xₚ x = p.insertNth yₚ y ↔ xₚ = yₚ ∧ x = y

theorem Fin.insertNth_left_injective {n : ℕ} {α : Fin (n + 1) → Sort u_1} {p : Fin (n + 1)} (x : (i : Fin n) → α (p.succAbove i)) : Function.Injective fun (x_1 : α p) => p.insertNth x_1 x

theorem Fin.insertNth_right_injective {n : ℕ} {α : Fin (n + 1) → Sort u_1} {p : Fin (n + 1)} (x : α p) : Function.Injective (p.insertNth x)

theorem Fin.insertNth_apply_below {n : ℕ} {α : Fin (n + 1) → Sort u_1} {i j : Fin (n + 1)} (h : j < i) (x : α i) (p : (k : Fin n) → α (i.succAbove k)) : i.insertNth x p j = Eq.recOn ⋯ (p (j.castPred ⋯))

theorem Fin.insertNth_apply_above {n : ℕ} {α : Fin (n + 1) → Sort u_1} {i j : Fin (n + 1)} (h : i < j) (x : α i) (p : (k : Fin n) → α (i.succAbove k)) : i.insertNth x p j = Eq.recOn ⋯ (p (j.pred ⋯))

theorem Fin.insertNth_zero {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α 0) (p : (j : Fin n) → α (succAbove 0 j)) : insertNth 0 x p = cons x fun (j : Fin n) => cast ⋯ (p j)

@[simp]

theorem Fin.insertNth_zero' {n : ℕ} {β : Sort u_2} (x : β) (p : Fin n → β) : insertNth 0 x p = cons x p

theorem Fin.insertNth_last {n : ℕ} {α : Fin (n + 1) → Sort u_1} (x : α (last n)) (p : (j : Fin n) → α ((last n).succAbove j)) : (last n).insertNth x p = snoc (fun (j : Fin n) => cast ⋯ (p j)) x

@[simp]

theorem Fin.insertNth_last' {n : ℕ} {β : Sort u_2} (x : β) (p : Fin n → β) : (last n).insertNth x p = snoc p x

theorem Fin.insertNth_rev {n : ℕ} {α : Sort u_3} (i : Fin (n + 1)) (a : α) (f : Fin n → α) (j : Fin (n + 1)) : i.insertNth a f j.rev = i.rev.insertNth a (f ∘ rev) j

theorem Fin.insertNth_comp_rev {n : ℕ} {α : Sort u_3} (i : Fin (n + 1)) (x : α) (p : Fin n → α) : i.insertNth x p ∘ rev = i.rev.insertNth x (p ∘ rev)

theorem Fin.cons_rev {α : Sort u_3} {n : ℕ} (a : α) (f : Fin n → α) (i : Fin (n + 1)) : cons a f i.rev = snoc (f ∘ rev) a i

theorem Fin.cons_comp_rev {α : Sort u_3} {n : ℕ} (a : α) (f : Fin n → α) : cons a f ∘ rev = snoc (f ∘ rev) a

theorem Fin.snoc_rev {α : Sort u_3} {n : ℕ} (a : α) (f : Fin n → α) (i : Fin (n + 1)) : snoc f a i.rev = cons a (f ∘ rev) i

theorem Fin.snoc_comp_rev {α : Sort u_3} {n : ℕ} (a : α) (f : Fin n → α) : snoc f a ∘ rev = cons a (f ∘ rev)

theorem Fin.insertNth_binop {n : ℕ} {α : Fin (n + 1) → Sort u_1} (op : (j : Fin (n + 1)) → α j → α j → α j) (i : Fin (n + 1)) (x y : α i) (p q : (j : Fin n) → α (i.succAbove j)) : (i.insertNth (op i x y) fun (j : Fin n) => op (i.succAbove j) (p j) (q j)) = fun (j : Fin (n + 1)) => op j (i.insertNth x p j) (i.insertNth y q j)

theorem Fin.insertNth_le_iff {n : ℕ} {α : Fin (n + 1) → Type u_3} [(i : Fin (n + 1)) → Preorder (α i)] {i : Fin (n + 1)} {x : α i} {p : (j : Fin n) → α (i.succAbove j)} {q : (j : Fin (n + 1)) → α j} : i.insertNth x p ≤ q ↔ x ≤ q i ∧ p ≤ fun (j : Fin n) => q (i.succAbove j)

theorem Fin.le_insertNth_iff {n : ℕ} {α : Fin (n + 1) → Type u_3} [(i : Fin (n + 1)) → Preorder (α i)] {i : Fin (n + 1)} {x : α i} {p : (j : Fin n) → α (i.succAbove j)} {q : (j : Fin (n + 1)) → α j} : q ≤ i.insertNth x p ↔ q i ≤ x ∧ (fun (j : Fin n) => q (i.succAbove j)) ≤ p

@[simp]

theorem Fin.removeNth_update {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : Fin (n + 1)) (x : α p) (f : (j : Fin (n + 1)) → α j) : p.removeNth (Function.update f p x) = p.removeNth f

@[simp]

theorem Fin.insertNth_removeNth {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : Fin (n + 1)) (x : α p) (f : (j : Fin (n + 1)) → α j) : p.insertNth x (p.removeNth f) = Function.update f p x

theorem Fin.insertNth_self_removeNth {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : Fin (n + 1)) (f : (j : Fin (n + 1)) → α j) : p.insertNth (f p) (p.removeNth f) = f

@[simp]

theorem Fin.update_insertNth {n : ℕ} {α : Fin (n + 1) → Sort u_1} (p : Fin (n + 1)) (x y : α p) (f : (i : Fin n) → α (p.succAbove i)) : Function.update (p.insertNth x f) p y = p.insertNth y f

def Fin.insertNthEquiv {n : ℕ} (α : Fin (n + 1) → Type u) (p : Fin (n + 1)) : α p × ((i : Fin n) → α (p.succAbove i)) ≃ ((i : Fin (n + 1)) → α i)

Equivalence between tuples of length n + 1 and pairs of an element and a tuple of length n given by separating out the p-th element of the tuple.

This is Fin.insertNth as an Equiv.

Equations

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

@[simp]

theorem Fin.insertNthEquiv_apply {n : ℕ} (α : Fin (n + 1) → Type u) (p : Fin (n + 1)) (f : α p × ((i : Fin n) → α (p.succAbove i))) (j : Fin (n + 1)) : (insertNthEquiv α p) f j = p.insertNth f.1 f.2 j

@[simp]

theorem Fin.insertNthEquiv_symm_apply {n : ℕ} (α : Fin (n + 1) → Type u) (p : Fin (n + 1)) (f : (i : Fin (n + 1)) → α i) : (insertNthEquiv α p).symm f = (f p, p.removeNth f)

@[simp]

theorem Fin.insertNthEquiv_zero {n : ℕ} (α : Fin (n + 1) → Type u_3) : insertNthEquiv α 0 = consEquiv α

@[simp]

theorem Fin.insertNthEquiv_last (n : ℕ) (α : Type u_3) : insertNthEquiv (fun (x : Fin (n + 1)) => α) (last n) = snocEquiv fun (x : Fin (n + 1)) => α

Note this lemma can only be written about non-dependent tuples as insertNth (last n) = snoc is not a definitional equality.

def Fin.find {n : ℕ} (p : Fin n → Prop) [DecidablePred p] : Option (Fin n)

find p returns the first index n where p n is satisfied, and none if it is never satisfied.

Equations

• Fin.find _p = none
• Fin.find p = Option.casesOn (Fin.find fun (i : Fin n) => p (i.castLT ⋯)) (if x : p (Fin.last n) then some (Fin.last n) else none) fun (i : Fin n) => some (i.castLT ⋯)

theorem Fin.find_spec {n : ℕ} (p : Fin n → Prop) [DecidablePred p] {i : Fin n} : i ∈ find p → p i

If find p = some i, then p i holds

theorem Fin.isSome_find_iff {n : ℕ} {p : Fin n → Prop} [DecidablePred p] : (find p).isSome = true ↔ ∃ (i : Fin n), p i

find p does not return none if and only if p i holds at some index i.

theorem Fin.find_eq_none_iff {n : ℕ} {p : Fin n → Prop} [DecidablePred p] : find p = none ↔ ∀ (i : Fin n), ¬p i

find p returns none if and only if p i never holds.

theorem Fin.find_min {n : ℕ} {p : Fin n → Prop} [DecidablePred p] {i : Fin n} : i ∈ find p → ∀ {j : Fin n}, j < i → ¬p j

If find p returns some i, then p j does not hold for j < i, i.e., i is minimal among the indices where p holds.

theorem Fin.find_min' {n : ℕ} {p : Fin n → Prop} [DecidablePred p] {i : Fin n} (h : i ∈ find p) {j : Fin n} (hj : p j) : i ≤ j

theorem Fin.nat_find_mem_find {n : ℕ} {p : Fin n → Prop} [DecidablePred p] (h : ∃ (i : ℕ) (hin : i < n), p ⟨i, hin⟩) : ⟨Nat.find h, ⋯⟩ ∈ find p

theorem Fin.mem_find_iff {n : ℕ} {p : Fin n → Prop} [DecidablePred p] {i : Fin n} : i ∈ find p ↔ p i ∧ ∀ (j : Fin n), p j → i ≤ j

theorem Fin.find_eq_some_iff {n : ℕ} {p : Fin n → Prop} [DecidablePred p] {i : Fin n} : find p = some i ↔ p i ∧ ∀ (j : Fin n), p j → i ≤ j

theorem Fin.mem_find_of_unique {n : ℕ} {p : Fin n → Prop} [DecidablePred p] (h : ∀ (i j : Fin n), p i → p j → i = j) {i : Fin n} (hi : p i) : i ∈ find p

def Fin.contractNth {n : ℕ} {α : Sort u_1} (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n) : α

Sends (g₀, ..., gₙ) to (g₀, ..., op gⱼ gⱼ₊₁, ..., gₙ).

Equations

• j.contractNth op g k = if ↑k < ↑j then g k.castSucc else if ↑k = ↑j then op (g k.castSucc) (g k.succ) else g k.succ

theorem Fin.contractNth_apply_of_lt {n : ℕ} {α : Sort u_1} (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n) (h : ↑k < ↑j) : j.contractNth op g k = g k.castSucc

theorem Fin.contractNth_apply_of_eq {n : ℕ} {α : Sort u_1} (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n) (h : ↑k = ↑j) : j.contractNth op g k = op (g k.castSucc) (g k.succ)

theorem Fin.contractNth_apply_of_gt {n : ℕ} {α : Sort u_1} (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n) (h : ↑j < ↑k) : j.contractNth op g k = g k.succ

theorem Fin.contractNth_apply_of_ne {n : ℕ} {α : Sort u_1} (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n) (hjk : ↑j ≠ ↑k) : j.contractNth op g k = g (j.succAbove k)

theorem Fin.comp_contractNth {n : ℕ} {α : Sort u_1} {β : Sort u_2} (opα : α → α → α) (opβ : β → β → β) {f : α → β} (hf : ∀ (x y : α), f (opα x y) = opβ (f x) (f y)) (j : Fin (n + 1)) (g : Fin (n + 1) → α) : f ∘ j.contractNth opα g = j.contractNth opβ (f ∘ g)

theorem Fin.sigma_eq_of_eq_comp_cast {α : Type u_1} {a b : (ii : ℕ) × (Fin ii → α)} (h : a.fst = b.fst) : a.snd = b.snd ∘ Fin.cast h → a = b

To show two sigma pairs of tuples agree, it to show the second elements are related via Fin.cast.

theorem Fin.sigma_eq_iff_eq_comp_cast {α : Type u_1} {a b : (ii : ℕ) × (Fin ii → α)} : a = b ↔ ∃ (h : a.fst = b.fst), a.snd = b.snd ∘ Fin.cast h

Fin.sigma_eq_of_eq_comp_cast as an iff.

def piFinTwoEquiv (α : Fin 2 → Type u) : ((i : Fin 2) → α i) ≃ α 0 × α 1

Π i : Fin 2, α i is equivalent to α 0 × α 1. See also finTwoArrowEquiv for a non-dependent version and prodEquivPiFinTwo for a version with inputs α β : Type u.

Equations

• piFinTwoEquiv α = { toFun := fun (f : (i : Fin 2) → α i) => (f 0, f 1), invFun := fun (p : α 0 × α 1) => Fin.cons p.1 (Fin.cons p.2 finZeroElim), left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem piFinTwoEquiv_apply (α : Fin 2 → Type u) : ⇑(piFinTwoEquiv α) = fun (f : (i : Fin 2) → α i) => (f 0, f 1)

@[simp]

theorem piFinTwoEquiv_symm_apply (α : Fin 2 → Type u) : ⇑(piFinTwoEquiv α).symm = fun (p : α 0 × α 1) => Fin.cons p.1 (Fin.cons p.2 finZeroElim)