Documentation

Mathlib.Data.ULift

Extra lemmas about ULift and PLift #

In this file we provide Subsingleton, Unique, DecidableEq, and isEmpty instances for ULift α and PLift α. We also prove ULift.forall, ULift.exists, PLift.forall, and PLift.exists.

Equations
instance PLift.instNonemptyPLift {α : Sort u} [Nonempty α] :
Equations
instance PLift.instUniquePLift {α : Sort u} [Unique α] :
Equations
Equations
instance PLift.instIsEmptyPLift {α : Sort u} [IsEmpty α] :
Equations
@[simp]
theorem PLift.up_inj {α : Sort u} {x : α} {y : α} :
{ down := x } = { down := y } x = y
@[simp]
theorem PLift.forall {α : Sort u} {p : PLift αProp} :
(∀ (x : PLift α), p x) ∀ (x : α), p { down := x }
@[simp]
theorem PLift.exists {α : Sort u} {p : PLift αProp} :
(∃ (x : PLift α), p x) ∃ (x : α), p { down := x }
@[simp]
theorem PLift.map_injective {α : Sort u} {β : Sort v} {f : αβ} :
@[simp]
theorem PLift.map_surjective {α : Sort u} {β : Sort v} {f : αβ} :
@[simp]
theorem PLift.map_bijective {α : Sort u} {β : Sort v} {f : αβ} :
Equations
Equations
@[simp]
theorem ULift.up_inj {α : Type u} {x : α} {y : α} :
{ down := x } = { down := y } x = y
@[simp]
theorem ULift.forall {α : Type u} {p : ULift.{u_1, u} αProp} :
(∀ (x : ULift.{u_1, u} α), p x) ∀ (x : α), p { down := x }
@[simp]
theorem ULift.exists {α : Type u} {p : ULift.{u_1, u} αProp} :
(∃ (x : ULift.{u_1, u} α), p x) ∃ (x : α), p { down := x }
@[simp]
theorem ULift.map_injective {α : Type u} {β : Type v} {f : αβ} :
@[simp]
theorem ULift.map_surjective {α : Type u} {β : Type v} {f : αβ} :
@[simp]
theorem ULift.map_bijective {α : Type u} {β : Type v} {f : αβ} :
theorem ULift.ext {α : Type u} (x : ULift.{u_1, u} α) (y : ULift.{u_1, u} α) (h : x.down = y.down) :
x = y
theorem ULift.ext_iff {α : Type u_1} (x : ULift.{u_2, u_1} α) (y : ULift.{u_2, u_1} α) :
x = y x.down = y.down