Lemmas about List.range and List.enum

Ranges and enumeration

range'

theorem List.range'_succ (s : Nat) (n : Nat) (step : Nat) : List.range' s (n + 1) step = s :: List.range' (s + step) n step

@[simp]

theorem List.range'_one {s : Nat} {step : Nat} : List.range' s 1 step = [s]

@[simp]

theorem List.length_range' (s : Nat) (step : Nat) (n : Nat) : (List.range' s n step).length = n

@[simp]

theorem List.range'_eq_nil {s : Nat} {n : Nat} {step : Nat} : List.range' s n step = [] ↔ n = 0

theorem List.mem_range' {m : Nat} {s : Nat} {step : Nat} {n : Nat} : m ∈ List.range' s n step ↔ ∃ (i : Nat), i < n ∧ m = s + step * i

@[simp]

theorem List.mem_range'_1 {m : Nat} {s : Nat} {n : Nat} : m ∈ List.range' s n ↔ s ≤ m ∧ m < s + n

theorem List.pairwise_lt_range' (s : Nat) (n : Nat) (step : optParam Nat 1) (pos : autoParam (0 < step) _auto✝) : List.Pairwise (fun (x x_1 : Nat) => x < x_1) (List.range' s n step)

theorem List.pairwise_le_range' (s : Nat) (n : Nat) (step : optParam Nat 1) : List.Pairwise (fun (x x_1 : Nat) => x ≤ x_1) (List.range' s n step)

theorem List.nodup_range' (s : Nat) (n : Nat) (step : optParam Nat 1) (h : autoParam (0 < step) _auto✝) : (List.range' s n step).Nodup

@[simp]

theorem List.map_add_range' (a : Nat) (s : Nat) (n : Nat) (step : Nat) : List.map (fun (x : Nat) => a + x) (List.range' s n step) = List.range' (a + s) n step

theorem List.map_sub_range' {step : Nat} (a : Nat) (s : Nat) (n : Nat) (h : a ≤ s) : List.map (fun (x : Nat) => x - a) (List.range' s n step) = List.range' (s - a) n step

theorem List.range'_append (s : Nat) (m : Nat) (n : Nat) (step : Nat) : List.range' s m step ++ List.range' (s + step * m) n step = List.range' s (n + m) step

@[simp]

theorem List.range'_append_1 (s : Nat) (m : Nat) (n : Nat) : List.range' s m ++ List.range' (s + m) n = List.range' s (n + m)

theorem List.range'_sublist_right {step : Nat} {s : Nat} {m : Nat} {n : Nat} : (List.range' s m step).Sublist (List.range' s n step) ↔ m ≤ n

theorem List.range'_subset_right {step : Nat} {s : Nat} {m : Nat} {n : Nat} (step0 : 0 < step) : List.range' s m step ⊆ List.range' s n step ↔ m ≤ n

theorem List.range'_subset_right_1 {s : Nat} {m : Nat} {n : Nat} : List.range' s m ⊆ List.range' s n ↔ m ≤ n

theorem List.getElem?_range' (s : Nat) (step : Nat) {m : Nat} {n : Nat} : m < n → (List.range' s n step)[m]? = some (s + step * m)

@[simp]

theorem List.getElem_range' {n : Nat} {m : Nat} {step : Nat} (i : Nat) (H : i < (List.range' n m step).length) : (List.range' n m step)[i] = n + step * i

theorem List.range'_concat {step : Nat} (s : Nat) (n : Nat) : List.range' s (n + 1) step = List.range' s n step ++ [s + step * n]

theorem List.range'_1_concat (s : Nat) (n : Nat) : List.range' s (n + 1) = List.range' s n ++ [s + n]

range

theorem List.range_loop_range' (s : Nat) (n : Nat) : List.range.loop s (List.range' s n) = List.range' 0 (n + s)

theorem List.range_eq_range' (n : Nat) : List.range n = List.range' 0 n

theorem List.range_succ_eq_map (n : Nat) : List.range (n + 1) = 0 :: List.map Nat.succ (List.range n)

theorem List.reverse_range' (s : Nat) (n : Nat) : (List.range' s n).reverse = List.map (fun (x : Nat) => s + n - 1 - x) (List.range n)

theorem List.range'_eq_map_range (s : Nat) (n : Nat) : List.range' s n = List.map (fun (x : Nat) => s + x) (List.range n)

@[simp]

theorem List.length_range (n : Nat) : (List.range n).length = n

@[simp]

theorem List.range_eq_nil {n : Nat} : List.range n = [] ↔ n = 0

@[simp]

theorem List.range_sublist {m : Nat} {n : Nat} : (List.range m).Sublist (List.range n) ↔ m ≤ n

@[simp]

theorem List.range_subset {m : Nat} {n : Nat} : List.range m ⊆ List.range n ↔ m ≤ n

@[simp]

theorem List.mem_range {m : Nat} {n : Nat} : m ∈ List.range n ↔ m < n

theorem List.not_mem_range_self {n : Nat} : ¬n ∈ List.range n

theorem List.self_mem_range_succ (n : Nat) : n ∈ List.range (n + 1)

theorem List.pairwise_lt_range (n : Nat) : List.Pairwise (fun (x x_1 : Nat) => x < x_1) (List.range n)

theorem List.pairwise_le_range (n : Nat) : List.Pairwise (fun (x x_1 : Nat) => x ≤ x_1) (List.range n)

theorem List.getElem?_range {m : Nat} {n : Nat} (h : m < n) : (List.range n)[m]? = some m

@[simp]

theorem List.getElem_range {n : Nat} (m : Nat) (h : m < (List.range n).length) : (List.range n)[m] = m

theorem List.range_succ (n : Nat) : List.range n.succ = List.range n ++ [n]

theorem List.range_add (a : Nat) (b : Nat) : List.range (a + b) = List.range a ++ List.map (fun (x : Nat) => a + x) (List.range b)

theorem List.take_range (m : Nat) (n : Nat) : List.take m (List.range n) = List.range (min m n)

theorem List.nodup_range (n : Nat) : (List.range n).Nodup

iota

theorem List.iota_eq_reverse_range' (n : Nat) : List.iota n = (List.range' 1 n).reverse

@[simp]

theorem List.length_iota (n : Nat) : (List.iota n).length = n

@[simp]

theorem List.mem_iota {m : Nat} {n : Nat} : m ∈ List.iota n ↔ 1 ≤ m ∧ m ≤ n

theorem List.pairwise_gt_iota (n : Nat) : List.Pairwise (fun (x x_1 : Nat) => x > x_1) (List.iota n)

theorem List.nodup_iota (n : Nat) : (List.iota n).Nodup

enumFrom

@[simp]

theorem List.enumFrom_singleton {α : Type u_1} (x : α) (n : Nat) : List.enumFrom n [x] = [(n, x)]

@[simp]

theorem List.enumFrom_eq_nil {α : Type u_1} {n : Nat} {l : List α} : List.enumFrom n l = [] ↔ l = []

@[simp]

theorem List.enumFrom_length {α : Type u_1} {n : Nat} {l : List α} : (List.enumFrom n l).length = l.length

@[simp]

theorem List.getElem?_enumFrom {α : Type u_1} (n : Nat) (l : List α) (m : Nat) : (List.enumFrom n l)[m]? = Option.map (fun (a : α) => (n + m, a)) l[m]?

@[simp]

theorem List.getElem_enumFrom {α : Type u_1} (l : List α) (n : Nat) (i : Nat) (h : i < (List.enumFrom n l).length) : (List.enumFrom n l)[i] = (n + i, l[i])

theorem List.mk_add_mem_enumFrom_iff_getElem? {α : Type u_1} {n : Nat} {i : Nat} {x : α} {l : List α} : (n + i, x) ∈ List.enumFrom n l ↔ l[i]? = some x

theorem List.mk_mem_enumFrom_iff_le_and_getElem?_sub {α : Type u_1} {n : Nat} {i : Nat} {x : α} {l : List α} : (i, x) ∈ List.enumFrom n l ↔ n ≤ i ∧ l[i - n]? = some x

theorem List.le_fst_of_mem_enumFrom {α : Type u_1} {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ List.enumFrom n l) : n ≤ x.fst

theorem List.fst_lt_add_of_mem_enumFrom {α : Type u_1} {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ List.enumFrom n l) : x.fst < n + l.length

theorem List.map_enumFrom {α : Type u_1} {β : Type u_2} (f : α → β) (n : Nat) (l : List α) : List.map (Prod.map id f) (List.enumFrom n l) = List.enumFrom n (List.map f l)

@[simp]

theorem List.enumFrom_map_fst {α : Type u_1} (n : Nat) (l : List α) : List.map Prod.fst (List.enumFrom n l) = List.range' n l.length

@[simp]

theorem List.enumFrom_map_snd {α : Type u_1} (n : Nat) (l : List α) : List.map Prod.snd (List.enumFrom n l) = l

theorem List.snd_mem_of_mem_enumFrom {α : Type u_1} {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ List.enumFrom n l) : x.snd ∈ l

theorem List.mem_enumFrom {α : Type u_1} {x : α} {i : Nat} {j : Nat} (xs : List α) (h : (i, x) ∈ List.enumFrom j xs) : j ≤ i ∧ i < j + xs.length ∧ x ∈ xs

theorem List.map_fst_add_enumFrom_eq_enumFrom {α : Type u_1} (l : List α) (n : Nat) (k : Nat) : List.map (Prod.map (fun (x : Nat) => x + n) id) (List.enumFrom k l) = List.enumFrom (n + k) l

theorem List.map_fst_add_enum_eq_enumFrom {α : Type u_1} (l : List α) (n : Nat) : List.map (Prod.map (fun (x : Nat) => x + n) id) l.enum = List.enumFrom n l

theorem List.enumFrom_cons' {α : Type u_1} (n : Nat) (x : α) (xs : List α) : List.enumFrom n (x :: xs) = (n, x) :: List.map (Prod.map (fun (x : Nat) => x + 1) id) (List.enumFrom n xs)

theorem List.enumFrom_map {α : Type u_1} {β : Type u_2} (n : Nat) (l : List α) (f : α → β) : List.enumFrom n (List.map f l) = List.map (Prod.map id f) (List.enumFrom n l)

theorem List.enumFrom_append {α : Type u_1} (xs : List α) (ys : List α) (n : Nat) : List.enumFrom n (xs ++ ys) = List.enumFrom n xs ++ List.enumFrom (n + xs.length) ys

theorem List.enumFrom_eq_zip_range' {α : Type u_1} (l : List α) {n : Nat} : List.enumFrom n l = (List.range' n l.length).zip l

@[simp]

theorem List.unzip_enumFrom_eq_prod {α : Type u_1} (l : List α) {n : Nat} : (List.enumFrom n l).unzip = (List.range' n l.length, l)

enum

theorem List.enum_cons : ∀ {α : Type u_1} {a : α} {as : List α}, (a :: as).enum = (0, a) :: List.enumFrom 1 as

theorem List.enum_cons' {α : Type u_1} (x : α) (xs : List α) : (x :: xs).enum = (0, x) :: List.map (Prod.map (fun (x : Nat) => x + 1) id) xs.enum

@[simp]

theorem List.enum_eq_nil {α : Type u_1} {l : List α} : l.enum = [] ↔ l = []

@[simp]

theorem List.enum_singleton {α : Type u_1} (x : α) : [x].enum = [(0, x)]

@[simp]

theorem List.enum_length : ∀ {α : Type u_1} {l : List α}, l.enum.length = l.length

@[simp]

theorem List.getElem?_enum {α : Type u_1} (l : List α) (n : Nat) : l.enum[n]? = Option.map (fun (a : α) => (n, a)) l[n]?

@[simp]

theorem List.getElem_enum {α : Type u_1} (l : List α) (i : Nat) (h : i < l.enum.length) : l.enum[i] = (i, l[i])

theorem List.mk_mem_enum_iff_getElem? {α : Type u_1} {i : Nat} {x : α} {l : List α} : (i, x) ∈ l.enum ↔ l[i]? = some x

theorem List.mem_enum_iff_getElem? {α : Type u_1} {x : Nat × α} {l : List α} : x ∈ l.enum ↔ l[x.fst]? = some x.snd

theorem List.fst_lt_of_mem_enum {α : Type u_1} {x : Nat × α} {l : List α} (h : x ∈ l.enum) : x.fst < l.length

theorem List.snd_mem_of_mem_enum {α : Type u_1} {x : Nat × α} {l : List α} (h : x ∈ l.enum) : x.snd ∈ l

theorem List.map_enum {α : Type u_1} {β : Type u_2} (f : α → β) (l : List α) : List.map (Prod.map id f) l.enum = (List.map f l).enum

@[simp]

theorem List.enum_map_fst {α : Type u_1} (l : List α) : List.map Prod.fst l.enum = List.range l.length

@[simp]

theorem List.enum_map_snd {α : Type u_1} (l : List α) : List.map Prod.snd l.enum = l

theorem List.enum_map {α : Type u_1} {β : Type u_2} (l : List α) (f : α → β) : (List.map f l).enum = List.map (Prod.map id f) l.enum

theorem List.enum_append {α : Type u_1} (xs : List α) (ys : List α) : (xs ++ ys).enum = xs.enum ++ List.enumFrom xs.length ys

theorem List.enum_eq_zip_range {α : Type u_1} (l : List α) : l.enum = (List.range l.length).zip l

@[simp]

theorem List.unzip_enum_eq_prod {α : Type u_1} (l : List α) : l.enum.unzip = (List.range l.length, l)