HepLean Documentation

Mathlib.Data.List.InsertIdx

insertIdx #

Proves various lemmas about List.insertIdx.

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@[simp]
theorem List.insertIdx_zero {α : Type u} (s : List α) (x : α) :
List.insertIdx 0 x s = x :: s
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@[simp]
theorem List.insertIdx_succ_nil {α : Type u} (n : ℕ) (a : α) :
List.insertIdx (n + 1) a [] = []
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@[simp]
theorem List.insertIdx_succ_cons {α : Type u} (s : List α) (hd : α) (x : α) (n : ℕ) :
List.insertIdx (n + 1) x (hd :: s) = hd :: List.insertIdx n x s
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theorem List.length_insertIdx {α : Type u} {a : α} (n : ℕ) (as : List α) :
n ≤ as.length → (List.insertIdx n a as).length = as.length + 1
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theorem List.eraseIdx_insertIdx {α : Type u} {a : α} (n : ℕ) (l : List α) :
(List.insertIdx n a l).eraseIdx n = l
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theorem List.insertIdx_eraseIdx_of_ge {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) :
n < as.length → n ≤ m → List.insertIdx m a (as.eraseIdx n) = (List.insertIdx (m + 1) a as).eraseIdx n
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theorem List.insertIdx_eraseIdx_of_le {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) :
n < as.length → m ≤ n → List.insertIdx m a (as.eraseIdx n) = (List.insertIdx m a as).eraseIdx (n + 1)
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theorem List.insertIdx_comm {α : Type u} (a : α) (b : α) (i : ℕ) (j : ℕ) (l : List α) :
i ≤ j → j ≤ l.length → List.insertIdx (j + 1) b (List.insertIdx i a l) = List.insertIdx i a (List.insertIdx j b l)
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theorem List.mem_insertIdx {α : Type u} {a : α} {b : α} {n : ℕ} {l : List α} :
n ≤ l.length → (a ∈ List.insertIdx n b l ↔ a = b ∨ a ∈ l)
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theorem List.insertIdx_of_length_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
List.insertIdx n x l = l
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@[simp]
theorem List.insertIdx_length_self {α : Type u} (l : List α) (x : α) :
List.insertIdx l.length x l = l ++ [x]
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theorem List.length_le_length_insertIdx {α : Type u} (l : List α) (x : α) (n : ℕ) :
l.length ≤ (List.insertIdx n x l).length
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theorem List.length_insertIdx_le_succ {α : Type u} (l : List α) (x : α) (n : ℕ) :
(List.insertIdx n x l).length ≤ l.length + 1
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theorem List.getElem_insertIdx_of_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : optParam (k < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l)[k] = l[k]
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theorem List.get_insertIdx_of_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : optParam (k < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l).get ⟨k, hk'⟩ = l.get ⟨k, hk⟩
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@[simp]
theorem List.getElem_insertIdx_self {α : Type u} (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : optParam (n < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l)[n] = x
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theorem List.get_insertIdx_self {α : Type u} (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : optParam (n < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l).get ⟨n, hn'⟩ = x
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theorem List.getElem_insertIdx_add_succ {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hk' : n + k < l.length) (hk : optParam (n + k + 1 < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l)[n + k + 1] = l[n + k]
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theorem List.get_insertIdx_add_succ {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hk' : n + k < l.length) (hk : optParam (n + k + 1 < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l).get ⟨n + k + 1, hk⟩ = l.get ⟨n + k, hk'⟩
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theorem List.insertIdx_injective {α : Type u} (n : ℕ) (x : α) :
Function.Injective (List.insertIdx n x)
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@[deprecated List.insertIdx_zero]
theorem List.insertNth_zero {α : Type u} (s : List α) (x : α) :
List.insertIdx 0 x s = x :: s

Alias of List.insertIdx_zero.

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@[deprecated List.insertIdx_succ_nil]
theorem List.insertNth_succ_nil {α : Type u} (n : ℕ) (a : α) :
List.insertIdx (n + 1) a [] = []

Alias of List.insertIdx_succ_nil.

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@[deprecated List.insertIdx_succ_cons]
theorem List.insertNth_succ_cons {α : Type u} (s : List α) (hd : α) (x : α) (n : ℕ) :
List.insertIdx (n + 1) x (hd :: s) = hd :: List.insertIdx n x s

Alias of List.insertIdx_succ_cons.

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@[deprecated List.length_insertIdx]
theorem List.length_insertNth {α : Type u} {a : α} (n : ℕ) (as : List α) :
n ≤ as.length → (List.insertIdx n a as).length = as.length + 1

Alias of List.length_insertIdx.

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@[deprecated List.eraseIdx_insertIdx]
theorem List.removeNth_insertIdx {α : Type u} {a : α} (n : ℕ) (l : List α) :
(List.insertIdx n a l).eraseIdx n = l

Alias of List.eraseIdx_insertIdx.

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@[deprecated List.eraseIdx_insertIdx]
theorem List.removeNth_insertNth {α : Type u} {a : α} (n : ℕ) (l : List α) :
(List.insertIdx n a l).eraseIdx n = l

Alias of List.eraseIdx_insertIdx.

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@[deprecated List.insertIdx_eraseIdx_of_ge]
theorem List.insertNth_eraseIdx_of_ge {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) :
n < as.length → n ≤ m → List.insertIdx m a (as.eraseIdx n) = (List.insertIdx (m + 1) a as).eraseIdx n

Alias of List.insertIdx_eraseIdx_of_ge.

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@[deprecated List.insertIdx_eraseIdx_of_ge]
theorem List.insertNth_removeNth_of_ge {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) :
n < as.length → n ≤ m → List.insertIdx m a (as.eraseIdx n) = (List.insertIdx (m + 1) a as).eraseIdx n

Alias of List.insertIdx_eraseIdx_of_ge.

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@[deprecated List.insertIdx_eraseIdx_of_le]
theorem List.insertNth_eraseIdx_of_le {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) :
n < as.length → m ≤ n → List.insertIdx m a (as.eraseIdx n) = (List.insertIdx m a as).eraseIdx (n + 1)

Alias of List.insertIdx_eraseIdx_of_le.

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@[deprecated List.insertIdx_eraseIdx_of_le]
theorem List.insertIdx_removeNth_of_le {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) :
n < as.length → m ≤ n → List.insertIdx m a (as.eraseIdx n) = (List.insertIdx m a as).eraseIdx (n + 1)

Alias of List.insertIdx_eraseIdx_of_le.

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@[deprecated List.insertIdx_comm]
theorem List.insertNth_comm {α : Type u} (a : α) (b : α) (i : ℕ) (j : ℕ) (l : List α) :
i ≤ j → j ≤ l.length → List.insertIdx (j + 1) b (List.insertIdx i a l) = List.insertIdx i a (List.insertIdx j b l)

Alias of List.insertIdx_comm.

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@[deprecated List.mem_insertIdx]
theorem List.mem_insertNth {α : Type u} {a : α} {b : α} {n : ℕ} {l : List α} :
n ≤ l.length → (a ∈ List.insertIdx n b l ↔ a = b ∨ a ∈ l)

Alias of List.mem_insertIdx.

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@[deprecated List.insertIdx_of_length_lt]
theorem List.insertNth_of_length_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
List.insertIdx n x l = l

Alias of List.insertIdx_of_length_lt.

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@[deprecated List.insertIdx_length_self]
theorem List.insertNth_length_self {α : Type u} (l : List α) (x : α) :
List.insertIdx l.length x l = l ++ [x]

Alias of List.insertIdx_length_self.

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@[deprecated List.length_le_length_insertIdx]
theorem List.length_le_length_insertNth {α : Type u} (l : List α) (x : α) (n : ℕ) :
l.length ≤ (List.insertIdx n x l).length

Alias of List.length_le_length_insertIdx.

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@[deprecated List.length_insertIdx_le_succ]
theorem List.length_insertNth_le_succ {α : Type u} (l : List α) (x : α) (n : ℕ) :
(List.insertIdx n x l).length ≤ l.length + 1

Alias of List.length_insertIdx_le_succ.

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@[deprecated List.getElem_insertIdx_of_lt]
theorem List.getElem_insertNth_of_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : optParam (k < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l)[k] = l[k]

Alias of List.getElem_insertIdx_of_lt.

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@[deprecated List.get_insertIdx_of_lt]
theorem List.get_insertNth_of_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : optParam (k < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l).get ⟨k, hk'⟩ = l.get ⟨k, hk⟩

Alias of List.get_insertIdx_of_lt.

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@[deprecated List.getElem_insertIdx_self]
theorem List.getElem_insertNth_self {α : Type u} (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : optParam (n < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l)[n] = x

Alias of List.getElem_insertIdx_self.

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@[deprecated List.get_insertIdx_self]
theorem List.get_insertNth_self {α : Type u} (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : optParam (n < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l).get ⟨n, hn'⟩ = x

Alias of List.get_insertIdx_self.

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@[deprecated List.getElem_insertIdx_add_succ]
theorem List.getElem_insertNth_add_succ {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hk' : n + k < l.length) (hk : optParam (n + k + 1 < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l)[n + k + 1] = l[n + k]

Alias of List.getElem_insertIdx_add_succ.

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@[deprecated List.get_insertIdx_add_succ]
theorem List.get_insertNth_add_succ {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hk' : n + k < l.length) (hk : optParam (n + k + 1 < (List.insertIdx n x l).length) ⋯) :
(List.insertIdx n x l).get ⟨n + k + 1, hk⟩ = l.get ⟨n + k, hk'⟩

Alias of List.get_insertIdx_add_succ.

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@[deprecated List.insertIdx_injective]
theorem List.insertNth_injective {α : Type u} (n : ℕ) (x : α) :
Function.Injective (List.insertIdx n x)

Alias of List.insertIdx_injective.