Documentation

Mathlib.Logic.Equiv.Option

Equivalences for Option α #

We define

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def Equiv.optionCongr {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
Option α ≃ Option β

A universe-polymorphic version of EquivFunctor.mapEquiv Option e.

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      @[simp]
      theorem Equiv.optionCongr_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (a✝ : Option α) :
      e.optionCongr a✝ = Option.map (⇑e) a✝
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      @[simp]
      theorem Equiv.optionCongr_refl {α : Type u_1} :
      (Equiv.refl α).optionCongr = Equiv.refl (Option α)
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      @[simp]
      theorem Equiv.optionCongr_symm {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
      e.optionCongr.symm = e.symm.optionCongr
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      @[simp]
      theorem Equiv.optionCongr_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e₁ : α ≃ β) (e₂ : β ≃ γ) :
      e₁.optionCongr.trans e₂.optionCongr = (e₁.trans e₂).optionCongr
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      theorem Equiv.optionCongr_eq_equivFunctor_mapEquiv {α β : Type u} (e : α ≃ β) :
      e.optionCongr = EquivFunctor.mapEquiv Option e

      When α and β are in the same universe, this is the same as the result of EquivFunctor.mapEquiv.

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      def Equiv.removeNone_aux {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) (x : α) :
      β

      If we have a value on one side of an Equiv of Option we also have a value on the other side of the equivalence

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          theorem Equiv.removeNone_aux_some {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) {x : α} (h : ∃ (x' : β), e (some x) = some x') :
          some (e.removeNone_aux x) = e (some x)
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          theorem Equiv.removeNone_aux_none {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) {x : α} (h : e (some x) = none) :
          some (e.removeNone_aux x) = e none
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          theorem Equiv.removeNone_aux_inv {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) (x : α) :
          e.symm.removeNone_aux (e.removeNone_aux x) = x
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          def Equiv.removeNone {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) :
          α ≃ β

          Given an equivalence between two Option types, eliminate none from that equivalence by mapping e.symm none to e none.

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              @[simp]
              theorem Equiv.removeNone_symm {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) :
              e.removeNone.symm = e.symm.removeNone
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              theorem Equiv.removeNone_some {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) {x : α} (h : ∃ (x' : β), e (some x) = some x') :
              some (e.removeNone x) = e (some x)
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              theorem Equiv.removeNone_none {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) {x : α} (h : e (some x) = none) :
              some (e.removeNone x) = e none
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              @[simp]
              theorem Equiv.option_symm_apply_none_iff {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) :
              e.symm none = none ↔ e none = none
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              theorem Equiv.some_removeNone_iff {α : Type u_1} {β : Type u_2} (e : Option α ≃ Option β) {x : α} :
              some (e.removeNone x) = e none ↔ e.symm none = some x
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              @[simp]
              theorem Equiv.removeNone_optionCongr {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
              e.optionCongr.removeNone = e
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              theorem Equiv.optionCongr_injective {α : Type u_1} {β : Type u_2} :
              Function.Injective optionCongr
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              def Equiv.optionSubtype {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) :
              { e : Option α ≃ β // e none = x } ≃ (α ≃ { y : β // y ≠ x })

              Equivalences between Option α and β that send none to x are equivalent to equivalences between α and {y : β // y ≠ x}.

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                  @[simp]
                  theorem Equiv.optionSubtype_apply_apply {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : { e : Option α ≃ β // e none = x }) (a : α) (h : ↑e (some a) ≠ x) :
                  ((optionSubtype x) e) a = ⟨↑e (some a), h⟩
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                  @[simp]
                  theorem Equiv.coe_optionSubtype_apply_apply {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : { e : Option α ≃ β // e none = x }) (a : α) :
                  ↑(((optionSubtype x) e) a) = ↑e (some a)
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                  @[simp]
                  theorem Equiv.optionSubtype_apply_symm_apply {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : { e : Option α ≃ β // e none = x }) (b : { y : β // y ≠ x }) :
                  some (((optionSubtype x) e).symm b) = (↑e).symm ↑b
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                  @[simp]
                  theorem Equiv.optionSubtype_symm_apply_apply_coe {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : α ≃ { y : β // y ≠ x }) (a : α) :
                  ↑((optionSubtype x).symm e) (some a) = ↑(e a)
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                  @[simp]
                  theorem Equiv.optionSubtype_symm_apply_apply_some {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : α ≃ { y : β // y ≠ x }) (a : α) :
                  ↑((optionSubtype x).symm e) (some a) = ↑(e a)
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                  @[simp]
                  theorem Equiv.optionSubtype_symm_apply_apply_none {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : α ≃ { y : β // y ≠ x }) :
                  ↑((optionSubtype x).symm e) none = x
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                  @[simp]
                  theorem Equiv.optionSubtype_symm_apply_symm_apply {α : Type u_1} {β : Type u_2} [DecidableEq β] (x : β) (e : α ≃ { y : β // y ≠ x }) (b : { y : β // y ≠ x }) :
                  (↑((optionSubtype x).symm e)).symm ↑b = some (e.symm b)
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                  def Equiv.optionSubtypeNe {α : Type u_1} [DecidableEq α] (a : α) :
                  Option { b : α // b ≠ a } ≃ α

                  Any type with a distinguished element is equivalent to an Option type on the subtype excluding that element.

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                      @[simp]
                      theorem Equiv.optionSubtypeNe_symm_apply {α : Type u_1} [DecidableEq α] (a b : α) :
                      (optionSubtypeNe a).symm b = if h : b = a then none else some ⟨b, h⟩
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                      @[simp]
                      theorem Equiv.optionSubtypeNe_apply {α : Type u_1} [DecidableEq α] (a : α) (a✝ : Option { y : α // y ≠ a }) :
                      (optionSubtypeNe a) a✝ = a✝.casesOn' a Subtype.val
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                      theorem Equiv.optionSubtypeNe_symm_self {α : Type u_1} [DecidableEq α] (a : α) :
                      (optionSubtypeNe a).symm a = none
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                      theorem Equiv.optionSubtypeNe_symm_of_ne {α : Type u_1} [DecidableEq α] {a b : α} (hba : b ≠ a) :
                      (optionSubtypeNe a).symm b = some ⟨b, hba⟩
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                      @[simp]
                      theorem Equiv.optionSubtypeNe_none {α : Type u_1} [DecidableEq α] (a : α) :
                      (optionSubtypeNe a) none = a
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                      @[simp]
                      theorem Equiv.optionSubtypeNe_some {α : Type u_1} [DecidableEq α] (a : α) (b : { b : α // b ≠ a }) :
                      (optionSubtypeNe a) (some b) = ↑b
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                      def Equiv.optionEquivSumPUnit (α : Type w) :
                      Option α ≃ α ⊕ PUnit.{v + 1}

                      Option α is equivalent to α ⊕ PUnit

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                          @[simp]
                          theorem Equiv.optionEquivSumPUnit_none {α : Type u_4} :
                          (optionEquivSumPUnit α) none = Sum.inr PUnit.unit
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                          @[simp]
                          theorem Equiv.optionEquivSumPUnit_some {α : Type u_4} (a : α) :
                          (optionEquivSumPUnit α) (some a) = Sum.inl a
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                          @[simp]
                          theorem Equiv.optionEquivSumPUnit_coe {α : Type u_4} (a : α) :
                          (optionEquivSumPUnit α) (some a) = Sum.inl a
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                          @[simp]
                          theorem Equiv.optionEquivSumPUnit_symm_inl {α : Type u_4} (a : α) :
                          (optionEquivSumPUnit α).symm (Sum.inl a) = some a
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                          @[simp]
                          theorem Equiv.optionEquivSumPUnit_symm_inr {α : Type u_4} (a : PUnit.{u_5 + 1}) :
                          (optionEquivSumPUnit α).symm (Sum.inr a) = none
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                          def Equiv.optionIsSomeEquiv (α : Type u_4) :
                          { x : Option α // x.isSome = true } ≃ α

                          The set of x : Option α such that isSome x is equivalent to α.

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                              @[simp]
                              theorem Equiv.optionIsSomeEquiv_symm_apply_coe (α : Type u_4) (x : α) :
                              ↑((optionIsSomeEquiv α).symm x) = some x
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                              @[simp]
                              theorem Equiv.optionIsSomeEquiv_apply (α : Type u_4) (o : { x : Option α // x.isSome = true }) :
                              (optionIsSomeEquiv α) o = (↑o).get ⋯