Theorems · Definition · order theory

Function.Injective.linearOrder

{α : Type u_2} →
  {β : Type u_3} →
    [inst : LinearOrder β] →
      [inst_1 : LE α] →
        [inst_2 : LT α] →
          [inst_3 : Max α] →
            [inst_4 : Min α] →
              [inst_5 : Ord α] →
                [DecidableEq α] →
                  [DecidableLE α] →
                    [DecidableLT α] →
                      (f : α → β) →
                        Function.Injective f →
                          (∀ {x y : α}, f x ≤ f y ↔ x ≤ y) →
                            (∀ {x y : α}, f x < f y ↔ x < y) →
                              (∀ (x y : α), f (x ⊓ y) = min (f x) (f y)) →
                                (∀ (x y : α), f (x ⊔ y) = max (f x) (f y)) →
                                  (∀ (x y : α), compare (f x) (f y) = compare x y) → LinearOrder α

Pull back a LinearOrder instance along an injective function. See note [reducible non-instances].

Defined in

Mathlib.Order.Basic

Cited by

0 results in Mathlib

Foundations

Depth 14 from the axioms · uses propext

Assumes

LinearOrderLELTMaxMinOrdDecidableEqDecidableLEDecidableLT