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mv_maximum
| Author | SHA1 | Date | |
|---|---|---|---|
|
|
64af278293 |
@@ -43,7 +43,7 @@ The operations are organized as follow:
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* Logic: `any`, `all`, `or`, and `and`.
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* Zippers: `zipWith`, `zip`, `zipWithAll`, and `unzip`.
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* Ranges and enumeration: `range`, `iota`, `enumFrom`, and `enum`.
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* Minima and maxima: `minimum?` and `maximum?`.
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* Minima and maxima: `min?` and `max?`.
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* Other functions: `intersperse`, `intercalate`, `eraseDups`, `eraseReps`, `span`, `groupBy`,
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`removeAll`
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(currently these functions are mostly only used in meta code,
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@@ -1464,30 +1464,34 @@ def enum : List α → List (Nat × α) := enumFrom 0
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/-! ## Minima and maxima -/
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/-! ### minimum? -/
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/-! ### min? -/
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/--
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Returns the smallest element of the list, if it is not empty.
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* `[].minimum? = none`
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* `[4].minimum? = some 4`
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* `[1, 4, 2, 10, 6].minimum? = some 1`
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* `[].min? = none`
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* `[4].min? = some 4`
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* `[1, 4, 2, 10, 6].min? = some 1`
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-/
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def minimum? [Min α] : List α → Option α
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def min? [Min α] : List α → Option α
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| [] => none
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| a::as => some <| as.foldl min a
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/-! ### maximum? -/
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@[inherit_doc min?, deprecated min? (since := "2024-09-29")] abbrev minimum? := @min?
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/-! ### max? -/
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/--
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Returns the largest element of the list, if it is not empty.
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* `[].maximum? = none`
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* `[4].maximum? = some 4`
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* `[1, 4, 2, 10, 6].maximum? = some 10`
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* `[].max? = none`
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* `[4].max? = some 4`
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* `[1, 4, 2, 10, 6].max? = some 10`
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-/
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def maximum? [Max α] : List α → Option α
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def max? [Max α] : List α → Option α
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| [] => none
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| a::as => some <| as.foldl max a
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@[inherit_doc max?, deprecated max? (since := "2024-09-29")] abbrev maximum? := @max?
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/-! ## Other list operations
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The functions are currently mostly used in meta code,
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@@ -31,7 +31,7 @@ The following operations are still missing `@[csimp]` replacements:
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The following operations are not recursive to begin with
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(or are defined in terms of recursive primitives):
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`isEmpty`, `isSuffixOf`, `isSuffixOf?`, `rotateLeft`, `rotateRight`, `insert`, `zip`, `enum`,
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`minimum?`, `maximum?`, and `removeAll`.
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`min?`, `max?`, and `removeAll`.
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The following operations were already given `@[csimp]` replacements in `Init/Data/List/Basic.lean`:
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`length`, `map`, `filter`, `replicate`, `leftPad`, `unzip`, `range'`, `iota`, `intersperse`.
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@@ -55,7 +55,7 @@ See also
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* `Init.Data.List.Erase` for lemmas about `List.eraseP` and `List.erase`.
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* `Init.Data.List.Find` for lemmas about `List.find?`, `List.findSome?`, `List.findIdx`,
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`List.findIdx?`, and `List.indexOf`
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* `Init.Data.List.MinMax` for lemmas about `List.minimum?` and `List.maximum?`.
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* `Init.Data.List.MinMax` for lemmas about `List.min?` and `List.max?`.
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* `Init.Data.List.Pairwise` for lemmas about `List.Pairwise` and `List.Nodup`.
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* `Init.Data.List.Sublist` for lemmas about `List.Subset`, `List.Sublist`, `List.IsPrefix`,
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`List.IsSuffix`, and `List.IsInfix`.
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@@ -7,7 +7,7 @@ prelude
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import Init.Data.List.Lemmas
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/-!
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# Lemmas about `List.minimum?` and `List.maximum?.
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# Lemmas about `List.min?` and `List.max?.
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-/
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namespace List
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@@ -16,24 +16,24 @@ open Nat
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/-! ## Minima and maxima -/
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/-! ### minimum? -/
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/-! ### min? -/
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@[simp] theorem minimum?_nil [Min α] : ([] : List α).minimum? = none := rfl
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@[simp] theorem min?_nil [Min α] : ([] : List α).min? = none := rfl
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-- We don't put `@[simp]` on `minimum?_cons`,
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-- We don't put `@[simp]` on `min?_cons`,
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-- because the definition in terms of `foldl` is not useful for proofs.
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theorem minimum?_cons [Min α] {xs : List α} : (x :: xs).minimum? = foldl min x xs := rfl
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theorem min?_cons [Min α] {xs : List α} : (x :: xs).min? = foldl min x xs := rfl
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@[simp] theorem minimum?_eq_none_iff {xs : List α} [Min α] : xs.minimum? = none ↔ xs = [] := by
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cases xs <;> simp [minimum?]
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@[simp] theorem min?_eq_none_iff {xs : List α} [Min α] : xs.min? = none ↔ xs = [] := by
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cases xs <;> simp [min?]
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theorem minimum?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
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{xs : List α} → xs.minimum? = some a → a ∈ xs := by
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theorem min?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
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{xs : List α} → xs.min? = some a → a ∈ xs := by
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intro xs
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match xs with
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| nil => simp
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| x :: xs =>
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simp only [minimum?_cons, Option.some.injEq, List.mem_cons]
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simp only [min?_cons, Option.some.injEq, List.mem_cons]
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intro eq
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induction xs generalizing x with
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| nil =>
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@@ -49,12 +49,12 @@ theorem minimum?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b
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-- See also `Init.Data.List.Nat.Basic` for specialisations of the next two results to `Nat`.
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theorem le_minimum?_iff [Min α] [LE α]
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theorem le_min?_iff [Min α] [LE α]
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(le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) :
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{xs : List α} → xs.minimum? = some a → ∀ {x}, x ≤ a ↔ ∀ b, b ∈ xs → x ≤ b
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{xs : List α} → xs.min? = some a → ∀ {x}, x ≤ a ↔ ∀ b, b ∈ xs → x ≤ b
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| nil => by simp
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| cons x xs => by
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rw [minimum?]
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rw [min?]
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intro eq y
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simp only [Option.some.injEq] at eq
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induction xs generalizing x with
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@@ -67,46 +67,46 @@ theorem le_minimum?_iff [Min α] [LE α]
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-- This could be refactored by designing appropriate typeclasses to replace `le_refl`, `min_eq_or`,
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-- and `le_min_iff`.
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theorem minimum?_eq_some_iff [Min α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
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theorem min?_eq_some_iff [Min α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
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(le_refl : ∀ a : α, a ≤ a)
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(min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b)
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(le_min_iff : ∀ a b c : α, a ≤ min b c ↔ a ≤ b ∧ a ≤ c) {xs : List α} :
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xs.minimum? = some a ↔ a ∈ xs ∧ ∀ b, b ∈ xs → a ≤ b := by
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refine ⟨fun h => ⟨minimum?_mem min_eq_or h, (le_minimum?_iff le_min_iff h).1 (le_refl _)⟩, ?_⟩
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xs.min? = some a ↔ a ∈ xs ∧ ∀ b, b ∈ xs → a ≤ b := by
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refine ⟨fun h => ⟨min?_mem min_eq_or h, (le_min?_iff le_min_iff h).1 (le_refl _)⟩, ?_⟩
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intro ⟨h₁, h₂⟩
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cases xs with
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| nil => simp at h₁
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| cons x xs =>
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exact congrArg some <| anti.1
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((le_minimum?_iff le_min_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
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(h₂ _ (minimum?_mem min_eq_or (xs := x::xs) rfl))
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((le_min?_iff le_min_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
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(h₂ _ (min?_mem min_eq_or (xs := x::xs) rfl))
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theorem minimum?_replicate [Min α] {n : Nat} {a : α} (w : min a a = a) :
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(replicate n a).minimum? = if n = 0 then none else some a := by
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theorem min?_replicate [Min α] {n : Nat} {a : α} (w : min a a = a) :
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(replicate n a).min? = if n = 0 then none else some a := by
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induction n with
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| zero => rfl
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| succ n ih => cases n <;> simp_all [replicate_succ, minimum?_cons]
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| succ n ih => cases n <;> simp_all [replicate_succ, min?_cons]
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@[simp] theorem minimum?_replicate_of_pos [Min α] {n : Nat} {a : α} (w : min a a = a) (h : 0 < n) :
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(replicate n a).minimum? = some a := by
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simp [minimum?_replicate, Nat.ne_of_gt h, w]
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@[simp] theorem min?_replicate_of_pos [Min α] {n : Nat} {a : α} (w : min a a = a) (h : 0 < n) :
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(replicate n a).min? = some a := by
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simp [min?_replicate, Nat.ne_of_gt h, w]
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/-! ### maximum? -/
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/-! ### max? -/
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@[simp] theorem maximum?_nil [Max α] : ([] : List α).maximum? = none := rfl
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@[simp] theorem max?_nil [Max α] : ([] : List α).max? = none := rfl
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-- We don't put `@[simp]` on `maximum?_cons`,
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-- We don't put `@[simp]` on `max?_cons`,
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-- because the definition in terms of `foldl` is not useful for proofs.
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theorem maximum?_cons [Max α] {xs : List α} : (x :: xs).maximum? = foldl max x xs := rfl
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theorem max?_cons [Max α] {xs : List α} : (x :: xs).max? = foldl max x xs := rfl
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@[simp] theorem maximum?_eq_none_iff {xs : List α} [Max α] : xs.maximum? = none ↔ xs = [] := by
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cases xs <;> simp [maximum?]
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@[simp] theorem max?_eq_none_iff {xs : List α} [Max α] : xs.max? = none ↔ xs = [] := by
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cases xs <;> simp [max?]
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theorem maximum?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
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{xs : List α} → xs.maximum? = some a → a ∈ xs
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theorem max?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
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{xs : List α} → xs.max? = some a → a ∈ xs
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| nil => by simp
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| cons x xs => by
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rw [maximum?]; rintro ⟨⟩
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rw [max?]; rintro ⟨⟩
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induction xs generalizing x with simp at *
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| cons y xs ih =>
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rcases ih (max x y) with h | h <;> simp [h]
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@@ -114,40 +114,57 @@ theorem maximum?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b
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-- See also `Init.Data.List.Nat.Basic` for specialisations of the next two results to `Nat`.
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theorem maximum?_le_iff [Max α] [LE α]
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theorem max?_le_iff [Max α] [LE α]
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(max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) :
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{xs : List α} → xs.maximum? = some a → ∀ {x}, a ≤ x ↔ ∀ b ∈ xs, b ≤ x
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{xs : List α} → xs.max? = some a → ∀ {x}, a ≤ x ↔ ∀ b ∈ xs, b ≤ x
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| nil => by simp
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| cons x xs => by
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rw [maximum?]; rintro ⟨⟩ y
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rw [max?]; rintro ⟨⟩ y
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induction xs generalizing x with
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| nil => simp
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| cons y xs ih => simp [ih, max_le_iff, and_assoc]
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-- This could be refactored by designing appropriate typeclasses to replace `le_refl`, `max_eq_or`,
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-- and `le_min_iff`.
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theorem maximum?_eq_some_iff [Max α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
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theorem max?_eq_some_iff [Max α] [LE α] [anti : Antisymm ((· : α) ≤ ·)]
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(le_refl : ∀ a : α, a ≤ a)
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(max_eq_or : ∀ a b : α, max a b = a ∨ max a b = b)
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(max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) {xs : List α} :
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xs.maximum? = some a ↔ a ∈ xs ∧ ∀ b ∈ xs, b ≤ a := by
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refine ⟨fun h => ⟨maximum?_mem max_eq_or h, (maximum?_le_iff max_le_iff h).1 (le_refl _)⟩, ?_⟩
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xs.max? = some a ↔ a ∈ xs ∧ ∀ b ∈ xs, b ≤ a := by
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refine ⟨fun h => ⟨max?_mem max_eq_or h, (max?_le_iff max_le_iff h).1 (le_refl _)⟩, ?_⟩
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intro ⟨h₁, h₂⟩
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cases xs with
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| nil => simp at h₁
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| cons x xs =>
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exact congrArg some <| anti.1
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(h₂ _ (maximum?_mem max_eq_or (xs := x::xs) rfl))
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((maximum?_le_iff max_le_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
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(h₂ _ (max?_mem max_eq_or (xs := x::xs) rfl))
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((max?_le_iff max_le_iff (xs := x::xs) rfl).1 (le_refl _) _ h₁)
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theorem maximum?_replicate [Max α] {n : Nat} {a : α} (w : max a a = a) :
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(replicate n a).maximum? = if n = 0 then none else some a := by
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theorem max?_replicate [Max α] {n : Nat} {a : α} (w : max a a = a) :
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(replicate n a).max? = if n = 0 then none else some a := by
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induction n with
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| zero => rfl
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| succ n ih => cases n <;> simp_all [replicate_succ, maximum?_cons]
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| succ n ih => cases n <;> simp_all [replicate_succ, max?_cons]
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@[simp] theorem maximum?_replicate_of_pos [Max α] {n : Nat} {a : α} (w : max a a = a) (h : 0 < n) :
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(replicate n a).maximum? = some a := by
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simp [maximum?_replicate, Nat.ne_of_gt h, w]
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@[simp] theorem max?_replicate_of_pos [Max α] {n : Nat} {a : α} (w : max a a = a) (h : 0 < n) :
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(replicate n a).max? = some a := by
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simp [max?_replicate, Nat.ne_of_gt h, w]
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@[deprecated min?_nil (since := "2024-09-29")] abbrev minimum?_nil := @min?_nil
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@[deprecated min?_cons (since := "2024-09-29")] abbrev minimum?_cons := @min?_cons
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@[deprecated min?_eq_none_iff (since := "2024-09-29")] abbrev mininmum?_eq_none_iff := @min?_eq_none_iff
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@[deprecated min?_mem (since := "2024-09-29")] abbrev minimum?_mem := @min?_mem
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@[deprecated le_min?_iff (since := "2024-09-29")] abbrev le_minimum?_iff := @le_min?_iff
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@[deprecated min?_eq_some_iff (since := "2024-09-29")] abbrev minimum?_eq_some_iff := @min?_eq_some_iff
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@[deprecated min?_replicate (since := "2024-09-29")] abbrev minimum?_replicate := @min?_replicate
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@[deprecated min?_replicate_of_pos (since := "2024-09-29")] abbrev minimum?_replicate_of_pos := @min?_replicate_of_pos
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@[deprecated max?_nil (since := "2024-09-29")] abbrev maximum?_nil := @max?_nil
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@[deprecated max?_cons (since := "2024-09-29")] abbrev maximum?_cons := @max?_cons
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@[deprecated max?_eq_none_iff (since := "2024-09-29")] abbrev maximum?_eq_none_iff := @max?_eq_none_iff
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@[deprecated max?_mem (since := "2024-09-29")] abbrev maximum?_mem := @max?_mem
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@[deprecated max?_le_iff (since := "2024-09-29")] abbrev maximum?_le_iff := @max?_le_iff
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@[deprecated max?_eq_some_iff (since := "2024-09-29")] abbrev maximum?_eq_some_iff := @max?_eq_some_iff
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@[deprecated max?_replicate (since := "2024-09-29")] abbrev maximum?_replicate := @max?_replicate
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@[deprecated max?_replicate_of_pos (since := "2024-09-29")] abbrev maximum?_replicate_of_pos := @max?_replicate_of_pos
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end List
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@@ -86,26 +86,26 @@ theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃
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obtain ⟨h', -⟩ := getElem?_eq_some_iff.1 h
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exact ⟨h', h⟩
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/-! ### minimum? -/
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/-! ### min? -/
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-- A specialization of `minimum?_eq_some_iff` to Nat.
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theorem minimum?_eq_some_iff' {xs : List Nat} :
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xs.minimum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
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minimum?_eq_some_iff
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-- A specialization of `min?_eq_some_iff` to Nat.
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theorem min?_eq_some_iff' {xs : List Nat} :
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xs.min? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
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min?_eq_some_iff
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(le_refl := Nat.le_refl)
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(min_eq_or := fun _ _ => by omega)
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(le_min_iff := fun _ _ _ => by omega)
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(min_eq_or := fun _ _ => Nat.min_def .. ▸ by split <;> simp)
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(le_min_iff := fun _ _ _ => Nat.le_min)
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-- This could be generalized,
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-- but will first require further work on order typeclasses in the core repository.
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theorem minimum?_cons' {a : Nat} {l : List Nat} :
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(a :: l).minimum? = some (match l.minimum? with
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theorem min?_cons' {a : Nat} {l : List Nat} :
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(a :: l).min? = some (match l.min? with
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| none => a
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| some m => min a m) := by
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rw [minimum?_eq_some_iff']
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rw [min?_eq_some_iff']
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split <;> rename_i h m
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· simp_all
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· rw [minimum?_eq_some_iff'] at m
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· rw [min?_eq_some_iff'] at m
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obtain ⟨m, le⟩ := m
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rw [Nat.min_def]
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constructor
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@@ -122,11 +122,11 @@ theorem minimum?_cons' {a : Nat} {l : List Nat} :
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theorem foldl_min
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{α : Type _} [Min α] [Std.IdempotentOp (min : α → α → α)] [Std.Associative (min : α → α → α)]
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{l : List α} {a : α} :
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l.foldl (init := a) min = min a (l.minimum?.getD a) := by
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l.foldl (init := a) min = min a (l.min?.getD a) := by
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cases l with
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||||
| nil => simp [Std.IdempotentOp.idempotent]
|
||||
| cons b l =>
|
||||
simp only [minimum?]
|
||||
simp only [min?]
|
||||
induction l generalizing a b with
|
||||
| nil => simp
|
||||
| cons c l ih => simp [ih, Std.Associative.assoc]
|
||||
@@ -134,7 +134,7 @@ theorem foldl_min
|
||||
theorem foldl_min_right {α β : Type _}
|
||||
[Min β] [Std.IdempotentOp (min : β → β → β)] [Std.Associative (min : β → β → β)]
|
||||
{l : List α} {b : β} {f : α → β} :
|
||||
(l.foldl (init := b) fun acc a => min acc (f a)) = min b ((l.map f).minimum?.getD b) := by
|
||||
(l.foldl (init := b) fun acc a => min acc (f a)) = min b ((l.map f).min?.getD b) := by
|
||||
rw [← foldl_map, foldl_min]
|
||||
|
||||
theorem foldl_min_le {l : List Nat} {a : Nat} : l.foldl (init := a) min ≤ a := by
|
||||
@@ -148,12 +148,12 @@ theorem foldl_min_min_of_le {l : List Nat} {a b : Nat} (h : a ≤ b) :
|
||||
l.foldl (init := a) min ≤ b :=
|
||||
Nat.le_trans (foldl_min_le) h
|
||||
|
||||
theorem minimum?_getD_le_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
l.minimum?.getD k ≤ a := by
|
||||
theorem min?_getD_le_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
l.min?.getD k ≤ a := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
| cons b l =>
|
||||
simp [minimum?_cons]
|
||||
simp [min?_cons]
|
||||
simp at h
|
||||
rcases h with (rfl | h)
|
||||
· exact foldl_min_le
|
||||
@@ -166,26 +166,26 @@ theorem minimum?_getD_le_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
· exact foldl_min_min_of_le (Nat.min_le_right _ _)
|
||||
· exact ih _ h
|
||||
|
||||
/-! ### maximum? -/
|
||||
/-! ### max? -/
|
||||
|
||||
-- A specialization of `maximum?_eq_some_iff` to Nat.
|
||||
theorem maximum?_eq_some_iff' {xs : List Nat} :
|
||||
xs.maximum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, b ≤ a) :=
|
||||
maximum?_eq_some_iff
|
||||
-- A specialization of `max?_eq_some_iff` to Nat.
|
||||
theorem max?_eq_some_iff' {xs : List Nat} :
|
||||
xs.max? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, b ≤ a) :=
|
||||
max?_eq_some_iff
|
||||
(le_refl := Nat.le_refl)
|
||||
(max_eq_or := fun _ _ => by omega)
|
||||
(max_le_iff := fun _ _ _ => by omega)
|
||||
(max_eq_or := fun _ _ => Nat.max_def .. ▸ by split <;> simp)
|
||||
(max_le_iff := fun _ _ _ => Nat.max_le)
|
||||
|
||||
-- This could be generalized,
|
||||
-- but will first require further work on order typeclasses in the core repository.
|
||||
theorem maximum?_cons' {a : Nat} {l : List Nat} :
|
||||
(a :: l).maximum? = some (match l.maximum? with
|
||||
theorem max?_cons' {a : Nat} {l : List Nat} :
|
||||
(a :: l).max? = some (match l.max? with
|
||||
| none => a
|
||||
| some m => max a m) := by
|
||||
rw [maximum?_eq_some_iff']
|
||||
rw [max?_eq_some_iff']
|
||||
split <;> rename_i h m
|
||||
· simp_all
|
||||
· rw [maximum?_eq_some_iff'] at m
|
||||
· rw [max?_eq_some_iff'] at m
|
||||
obtain ⟨m, le⟩ := m
|
||||
rw [Nat.max_def]
|
||||
constructor
|
||||
@@ -202,11 +202,11 @@ theorem maximum?_cons' {a : Nat} {l : List Nat} :
|
||||
theorem foldl_max
|
||||
{α : Type _} [Max α] [Std.IdempotentOp (max : α → α → α)] [Std.Associative (max : α → α → α)]
|
||||
{l : List α} {a : α} :
|
||||
l.foldl (init := a) max = max a (l.maximum?.getD a) := by
|
||||
l.foldl (init := a) max = max a (l.max?.getD a) := by
|
||||
cases l with
|
||||
| nil => simp [Std.IdempotentOp.idempotent]
|
||||
| cons b l =>
|
||||
simp only [maximum?]
|
||||
simp only [max?]
|
||||
induction l generalizing a b with
|
||||
| nil => simp
|
||||
| cons c l ih => simp [ih, Std.Associative.assoc]
|
||||
@@ -214,7 +214,7 @@ theorem foldl_max
|
||||
theorem foldl_max_right {α β : Type _}
|
||||
[Max β] [Std.IdempotentOp (max : β → β → β)] [Std.Associative (max : β → β → β)]
|
||||
{l : List α} {b : β} {f : α → β} :
|
||||
(l.foldl (init := b) fun acc a => max acc (f a)) = max b ((l.map f).maximum?.getD b) := by
|
||||
(l.foldl (init := b) fun acc a => max acc (f a)) = max b ((l.map f).max?.getD b) := by
|
||||
rw [← foldl_map, foldl_max]
|
||||
|
||||
theorem le_foldl_max {l : List Nat} {a : Nat} : a ≤ l.foldl (init := a) max := by
|
||||
@@ -228,12 +228,12 @@ theorem le_foldl_max_of_le {l : List Nat} {a b : Nat} (h : a ≤ b) :
|
||||
a ≤ l.foldl (init := b) max :=
|
||||
Nat.le_trans h (le_foldl_max)
|
||||
|
||||
theorem le_maximum?_getD_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
a ≤ l.maximum?.getD k := by
|
||||
theorem le_max?_getD_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
a ≤ l.max?.getD k := by
|
||||
cases l with
|
||||
| nil => simp at h
|
||||
| cons b l =>
|
||||
simp [maximum?_cons]
|
||||
simp [max?_cons]
|
||||
simp at h
|
||||
rcases h with (rfl | h)
|
||||
· exact le_foldl_max
|
||||
@@ -246,4 +246,11 @@ theorem le_maximum?_getD_of_mem {l : List Nat} {a k : Nat} (h : a ∈ l) :
|
||||
· exact le_foldl_max_of_le (Nat.le_max_right b a)
|
||||
· exact ih _ h
|
||||
|
||||
@[deprecated min?_eq_some_iff' (since := "2024-09-29")] abbrev minimum?_eq_some_iff' := @min?_eq_some_iff'
|
||||
@[deprecated min?_cons' (since := "2024-09-29")] abbrev minimum?_cons' := @min?_cons'
|
||||
@[deprecated min?_getD_le_of_mem (since := "2024-09-29")] abbrev minimum?_getD_le_of_mem := @min?_getD_le_of_mem
|
||||
@[deprecated max?_eq_some_iff' (since := "2024-09-29")] abbrev maximum?_eq_some_iff' := @max?_eq_some_iff'
|
||||
@[deprecated max?_cons' (since := "2024-09-29")] abbrev maximum?_cons' := @max?_cons'
|
||||
@[deprecated le_max?_getD_of_mem (since := "2024-09-29")] abbrev le_maximum?_getD_of_mem := @le_max?_getD_of_mem
|
||||
|
||||
end List
|
||||
|
||||
@@ -29,12 +29,14 @@ The minimum non-zero entry in a list of natural numbers, or zero if all entries
|
||||
We completely characterize the function via
|
||||
`nonzeroMinimum_eq_zero_iff` and `nonzeroMinimum_eq_nonzero_iff` below.
|
||||
-/
|
||||
def nonzeroMinimum (xs : List Nat) : Nat := xs.filter (· ≠ 0) |>.minimum? |>.getD 0
|
||||
def nonzeroMinimum (xs : List Nat) : Nat := xs.filter (· ≠ 0) |>.min? |>.getD 0
|
||||
|
||||
-- A specialization of `minimum?_eq_some_iff` to Nat.
|
||||
theorem minimum?_eq_some_iff' {xs : List Nat} :
|
||||
xs.minimum? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
|
||||
minimum?_eq_some_iff
|
||||
-- This is a duplicate `min?_eq_some_iff'` proved in `Init.Data.List.Nat.Basic`,
|
||||
-- and could be deduplicated but the import hierarchy is awkward.
|
||||
theorem min?_eq_some_iff'' {xs : List Nat} :
|
||||
xs.min? = some a ↔ (a ∈ xs ∧ ∀ b ∈ xs, a ≤ b) :=
|
||||
min?_eq_some_iff
|
||||
(le_refl := Nat.le_refl)
|
||||
(min_eq_or := fun _ _ => Nat.min_def .. ▸ by split <;> simp)
|
||||
(le_min_iff := fun _ _ _ => Nat.le_min)
|
||||
@@ -42,27 +44,27 @@ theorem minimum?_eq_some_iff' {xs : List Nat} :
|
||||
open Classical in
|
||||
@[simp] theorem nonzeroMinimum_eq_zero_iff {xs : List Nat} :
|
||||
xs.nonzeroMinimum = 0 ↔ ∀ x ∈ xs, x = 0 := by
|
||||
simp [nonzeroMinimum, Option.getD_eq_iff, minimum?_eq_none_iff, minimum?_eq_some_iff',
|
||||
simp [nonzeroMinimum, Option.getD_eq_iff, min?_eq_none_iff, min?_eq_some_iff'',
|
||||
filter_eq_nil_iff, mem_filter]
|
||||
|
||||
theorem nonzeroMinimum_mem {xs : List Nat} (w : xs.nonzeroMinimum ≠ 0) :
|
||||
xs.nonzeroMinimum ∈ xs := by
|
||||
dsimp [nonzeroMinimum] at *
|
||||
generalize h : (xs.filter (· ≠ 0) |>.minimum?) = m at *
|
||||
generalize h : (xs.filter (· ≠ 0) |>.min?) = m at *
|
||||
match m, w with
|
||||
| some (m+1), _ => simp_all [minimum?_eq_some_iff', mem_filter]
|
||||
| some (m+1), _ => simp_all [min?_eq_some_iff'', mem_filter]
|
||||
|
||||
theorem nonzeroMinimum_pos {xs : List Nat} (m : a ∈ xs) (h : a ≠ 0) : 0 < xs.nonzeroMinimum :=
|
||||
Nat.pos_iff_ne_zero.mpr fun w => h (nonzeroMinimum_eq_zero_iff.mp w _ m)
|
||||
|
||||
theorem nonzeroMinimum_le {xs : List Nat} (m : a ∈ xs) (h : a ≠ 0) : xs.nonzeroMinimum ≤ a := by
|
||||
have : (xs.filter (· ≠ 0) |>.minimum?) = some xs.nonzeroMinimum := by
|
||||
have : (xs.filter (· ≠ 0) |>.min?) = some xs.nonzeroMinimum := by
|
||||
have w := nonzeroMinimum_pos m h
|
||||
dsimp [nonzeroMinimum] at *
|
||||
generalize h : (xs.filter (· ≠ 0) |>.minimum?) = m? at *
|
||||
generalize h : (xs.filter (· ≠ 0) |>.min?) = m? at *
|
||||
match m?, w with
|
||||
| some m?, _ => rfl
|
||||
rw [minimum?_eq_some_iff'] at this
|
||||
rw [min?_eq_some_iff''] at this
|
||||
apply this.2
|
||||
simp [List.mem_filter]
|
||||
exact ⟨m, h⟩
|
||||
@@ -142,4 +144,4 @@ theorem minNatAbs_eq_nonzero_iff (xs : List Int) (w : z ≠ 0) :
|
||||
@[simp] theorem minNatAbs_nil : ([] : List Int).minNatAbs = 0 := rfl
|
||||
|
||||
/-- The maximum absolute value in a list of integers. -/
|
||||
def maxNatAbs (xs : List Int) : Nat := xs.map Int.natAbs |>.maximum? |>.getD 0
|
||||
def maxNatAbs (xs : List Int) : Nat := xs.map Int.natAbs |>.max? |>.getD 0
|
||||
|
||||
@@ -237,7 +237,7 @@ partial def InfoTree.hoverableInfoAt? (t : InfoTree) (hoverPos : String.Pos) (in
|
||||
let _ := @lexOrd
|
||||
let _ := @leOfOrd.{0}
|
||||
let _ := @maxOfLe
|
||||
results.map (·.1) |>.maximum?
|
||||
results.map (·.1) |>.max?
|
||||
let res? := results.find? (·.1 == maxPrio?) |>.map (·.2)
|
||||
if let some i := res? then
|
||||
if let .ofTermInfo ti := i.info then
|
||||
@@ -380,7 +380,7 @@ partial def InfoTree.goalsAt? (text : FileMap) (t : InfoTree) (hoverPos : String
|
||||
priority := if hoverPos.byteIdx == tailPos.byteIdx + trailSize then 0 else 1
|
||||
}]
|
||||
return gs
|
||||
let maxPrio? := gs.map (·.priority) |>.maximum?
|
||||
let maxPrio? := gs.map (·.priority) |>.max?
|
||||
gs.filter (some ·.priority == maxPrio?)
|
||||
where
|
||||
hasNestedTactic (pos tailPos) : InfoTree → Bool
|
||||
|
||||
@@ -15,12 +15,12 @@ namespace CNF
|
||||
/--
|
||||
Obtain the literal with the largest identifier in `c`.
|
||||
-/
|
||||
def Clause.maxLiteral (c : Clause Nat) : Option Nat := (c.map (·.1)) |>.maximum?
|
||||
def Clause.maxLiteral (c : Clause Nat) : Option Nat := (c.map (·.1)) |>.max?
|
||||
|
||||
theorem Clause.of_maxLiteral_eq_some (c : Clause Nat) (h : c.maxLiteral = some maxLit) :
|
||||
∀ lit, Mem lit c → lit ≤ maxLit := by
|
||||
intro lit hlit
|
||||
simp only [maxLiteral, List.maximum?_eq_some_iff', List.mem_map, forall_exists_index, and_imp,
|
||||
simp only [maxLiteral, List.max?_eq_some_iff', List.mem_map, forall_exists_index, and_imp,
|
||||
forall_apply_eq_imp_iff₂] at h
|
||||
simp only [Mem] at hlit
|
||||
rcases h with ⟨_, hbar⟩
|
||||
@@ -35,25 +35,25 @@ theorem Clause.maxLiteral_eq_some_of_mem (c : Clause Nat) (h : Mem l c) :
|
||||
cases h <;> rename_i h
|
||||
all_goals
|
||||
have h1 := List.ne_nil_of_mem h
|
||||
have h2 := not_congr <| @List.maximum?_eq_none_iff _ (c.map (·.1)) _
|
||||
have h2 := not_congr <| @List.max?_eq_none_iff _ (c.map (·.1)) _
|
||||
simp [← Option.ne_none_iff_exists', h1, h2, maxLiteral]
|
||||
|
||||
theorem Clause.of_maxLiteral_eq_none (c : Clause Nat) (h : c.maxLiteral = none) :
|
||||
∀ lit, ¬Mem lit c := by
|
||||
intro lit hlit
|
||||
simp only [maxLiteral, List.maximum?_eq_none_iff, List.map_eq_nil_iff] at h
|
||||
simp only [maxLiteral, List.max?_eq_none_iff, List.map_eq_nil_iff] at h
|
||||
simp only [h, not_mem_nil] at hlit
|
||||
|
||||
/--
|
||||
Obtain the literal with the largest identifier in `f`.
|
||||
-/
|
||||
def maxLiteral (f : CNF Nat) : Option Nat :=
|
||||
List.filterMap Clause.maxLiteral f |>.maximum?
|
||||
List.filterMap Clause.maxLiteral f |>.max?
|
||||
|
||||
theorem of_maxLiteral_eq_some' (f : CNF Nat) (h : f.maxLiteral = some maxLit) :
|
||||
∀ clause, clause ∈ f → clause.maxLiteral = some localMax → localMax ≤ maxLit := by
|
||||
intro clause hclause1 hclause2
|
||||
simp [maxLiteral, List.maximum?_eq_some_iff'] at h
|
||||
simp [maxLiteral, List.max?_eq_some_iff'] at h
|
||||
rcases h with ⟨_, hclause3⟩
|
||||
apply hclause3 localMax clause hclause1 hclause2
|
||||
|
||||
@@ -70,7 +70,7 @@ theorem of_maxLiteral_eq_some (f : CNF Nat) (h : f.maxLiteral = some maxLit) :
|
||||
theorem of_maxLiteral_eq_none (f : CNF Nat) (h : f.maxLiteral = none) :
|
||||
∀ lit, ¬Mem lit f := by
|
||||
intro lit hlit
|
||||
simp only [maxLiteral, List.maximum?_eq_none_iff] at h
|
||||
simp only [maxLiteral, List.max?_eq_none_iff] at h
|
||||
dsimp [Mem] at hlit
|
||||
rcases hlit with ⟨clause, ⟨hclause1, hclause2⟩⟩
|
||||
have := Clause.of_maxLiteral_eq_none clause (List.forall_none_of_filterMap_eq_nil h clause hclause1) lit
|
||||
|
||||
@@ -336,21 +336,21 @@ variable (h : n ≤ m) in
|
||||
-- unzip
|
||||
#check_simp unzip (replicate n (x, y)) ~> (replicate n x, replicate n y)
|
||||
|
||||
-- minimum?
|
||||
-- min?
|
||||
|
||||
-- Note this relies on the fact that we do not have `replicate_succ` as a `@[simp]` lemma
|
||||
#check_simp (replicate (n+1) 7).minimum? ~> some 7
|
||||
#check_simp (replicate (n+1) 7).min? ~> some 7
|
||||
|
||||
variable (h : 0 < n) in
|
||||
#check_tactic (replicate n 7).minimum? ~> some 7 by simp [h]
|
||||
#check_tactic (replicate n 7).min? ~> some 7 by simp [h]
|
||||
|
||||
-- maximum?
|
||||
-- max?
|
||||
|
||||
-- Note this relies on the fact that we do not have `replicate_succ` as a `@[simp]` lemma
|
||||
#check_simp (replicate (n+1) 7).maximum? ~> some 7
|
||||
#check_simp (replicate (n+1) 7).max? ~> some 7
|
||||
|
||||
variable (h : 0 < n) in
|
||||
#check_tactic (replicate n 7).maximum? ~> some 7 by simp [h]
|
||||
#check_tactic (replicate n 7).max? ~> some 7 by simp [h]
|
||||
|
||||
end
|
||||
|
||||
@@ -456,9 +456,9 @@ end Pairwise
|
||||
|
||||
/-! ### enumFrom -/
|
||||
|
||||
/-! ### minimum? -/
|
||||
/-! ### min? -/
|
||||
|
||||
/-! ### maximum? -/
|
||||
/-! ### max? -/
|
||||
|
||||
/-! ## Monadic operations -/
|
||||
|
||||
|
||||
Reference in New Issue
Block a user