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neg_max_eq_min_neg🔗

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Minimal Lean file

neg_max_eq_min_neg🔗

Lemmaneg_max_eq_min_neg
Details

No docstring.

🔗theorem
neg_max_eq_min_neg.{u_1, u_2} {ι : Type u_1} {α : Type u_2}
  [LinearOrder α] [Fintype ι] [Nonempty ι] (f : ι → α) [AddGroup α]
  [AddLeftMono α] [AddRightMono α] : -Function.max f = Function.min (-f)
neg_max_eq_min_neg.{u_1, u_2}
  {ι : Type u_1} {α : Type u_2}
  [LinearOrder α] [Fintype ι] [Nonempty ι]
  (f : ι → α) [AddGroup α] [AddLeftMono α]
  [AddRightMono α] :
  -Function.max f = Function.min (-f)

Code

lemma neg_max_eq_min_neg [AddGroup α] [AddLeftMono α] [AddRightMono α] : -f.max = (-f).min
Type uses (2)
Body uses (2)

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Proof
by
  refine le_antisymm ?_ ?_
  · simp; grind
  · simp only [inf'_le_iff, mem_univ, Pi.neg_apply, neg_le_neg_iff, sup'_le_iff, forall_const,
      true_and]
    exact ⟨argmax f, isMaxOn_argmax f⟩

Dependency graph

Type dependencies (2)

max🔗

DefinitionFunction.max
Details

The maximum value of a tuple.

🔗def
Function.max.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α
Function.max.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α

Code

abbrev max : α := univ.sup' univ_nonempty f
Used by (8)

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min🔗

DefinitionFunction.min
Details

The minimum value of a tuple.

🔗def
Function.min.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α
Function.min.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α

Code

def min : {ι : Type u_1} → {α : Type u_2} → [LinearOrder α] → [Fintype ι] → [Nonempty ι] → (ι → α) → α
Used by (8)

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