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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.

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instance PLift.instUnique {α : Sort u} [Unique α] :
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  • PLift.instUnique = Equiv.plift.unique
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  • PLift.instDecidableEq_mathlib = Equiv.plift.decidableEq
instance PLift.instIsEmpty {α : Sort u} [IsEmpty α] :
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@[simp]
theorem PLift.up_inj {α : Sort u} {x : α} {y : α} :
{ down := x } = { down := y } x = y
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} :
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instance ULift.instUnique {α : Type u} [Unique α] :
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  • ULift.instUnique = Equiv.ulift.unique
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  • ULift.instDecidableEq_mathlib = Equiv.ulift.decidableEq
instance ULift.instIsEmpty {α : Type u} [IsEmpty α] :
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@[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_iff {α : Type u} {x : ULift.{u_1, u} α} {y : ULift.{u_1, u} α} :
x = y x.down = y.down
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