LeanMachineLearning exposition

isMinOn_argmin🔗

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Minimal Lean file

isMinOn_argmin🔗

TheoremisMinOn_argmin
Details

No docstring.

🔗theorem
isMinOn_argmin.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) (x : ι) : f (argmin f) ≤ f x
isMinOn_argmin.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α)
  (x : ι) : f (argmin f) ≤ f x

Code

theorem isMinOn_argmin : ∀ {ι : Type u_1} {α : Type u_2} [inst : LinearOrder α] [inst_1 : Fintype ι] [inst_2 : Nonempty ι] (f : ι → α) (x : ι),
  f (argmin f) ≤ f x
Type uses (1)
Body uses (3)

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Proof
@[to_dual isMinOn_argmin]

Dependency graph

Type dependencies (1)

argmin🔗

Definitionargmin
Details

The index of the minimum value of a tuple.

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

Code

def argmin : {ι : Type u_1} → {α : Type u_2} → [LinearOrder α] → [Fintype ι] → [Nonempty ι] → (ι → α) → ι
Body uses (2)
Used by (3)

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All dependencies, transitively (2)

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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exists_argmin🔗

Theoremexists_argmin
Details

No docstring.

🔗theorem
exists_argmin.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ∃ i, f i = Function.min f
exists_argmin.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) :
  ∃ i, f i = Function.min f

Code

theorem exists_argmin : ∀ {ι : Type u_1} {α : Type u_2} [inst : LinearOrder α] [inst_1 : Fintype ι] [inst_2 : Nonempty ι] (f : ι → α),
  ∃ i, f i = Function.min f
Type uses (1)
Used by (3)

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Proof
@[to_dual exists_argmin]