245 lines
4.4 KiB
Plaintext
245 lines
4.4 KiB
Plaintext
/-! # Smullyan.Aviary
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A collection of combinator birds representable in Lean. Certain duplicators,
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e.g. mockingbirds, are not directly expressible since they would require
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encoding a signature in which an argument has types `α` *and* `α → α`.
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Duplicators that are included, e.g. the warbler, are not exactly correct
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considering they still have the same limitation described above during actual
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use. Their inclusion here serves more as pseudo-documentation than anything.
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-/
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/-- #### Bald Eagle
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`E'xy₁y₂y₃z₁z₂z₃ = x(y₁y₂y₃)(z₁z₂z₃)`
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-/
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def E' (x : α → β → γ)
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(y₁ : δ → ε → α) (y₂ : δ) (y₃ : ε)
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(z₁ : ζ → η → β) (z₂ : ζ) (z₃ : η) := x (y₁ y₂ y₃) (z₁ z₂ z₃)
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/-- #### Becard
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`B₃xyzw = x(y(zw))`
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-/
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def B₃ (x : α → ε) (y : β → α) (z : γ → β) (w : γ) := x (y (z w))
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/-- #### Blackbird
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`B₁xyzw = x(yzw)`
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-/
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def B₁ (x : α → ε) (y : β → γ → α) (z : β) (w : γ) := x (y z w)
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/-- #### Bluebird
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`Bxyz = x(yz)`
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-/
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def B (x : α → γ) (y : β → α) (z : β) := x (y z)
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/-- #### Bunting
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`B₂xyzwv = x(yzwv)`
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-/
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def B₂ (x : α → ζ) (y : β → γ → ε → α) (z : β) (w : γ) (v : ε) := x (y z w v)
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/-- #### Cardinal Once Removed
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`C*xyzw = xywz`
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-/
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def C_star (x : α → β → γ → δ) (y : α) (z : γ) (w : β) := x y w z
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notation "C*" => C_star
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/-- #### Cardinal
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`Cxyz = xzy`
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-/
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def C (x : α → β → δ) (y : β) (z : α) := x z y
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/-- #### Converse Warbler
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`W'xy = yxx`
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-/
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def W' (x : α) (y : α → α → β) := y x x
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/-- #### Dickcissel
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`D₁xyzwv = xyz(wv)`
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-/
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def D₁ (x : α → β → δ → ε) (y : α) (z : β) (w : γ → δ) (v : γ) := x y z (w v)
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/-! #### Double Mockingbird
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`M₂xy = xy(xy)`
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-/
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/-- #### Dove
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`Dxyzw = xy(zw)`
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-/
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def D (x : α → γ → δ) (y : α) (z : β → γ) (w : β) := x y (z w)
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/-- #### Dovekie
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`D₂xyzwv = x(yz)(wv)`
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-/
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def D₂ (x : α → δ → ε) (y : β → α) (z : β) (w : γ → δ) (v : γ) := x (y z) (w v)
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/-- #### Eagle
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`Exyzwv = xy(zwv)`
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-/
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def E (x : α → δ → ε) (y : α) (z : β → γ → δ) (w : β) (v : γ) := x y (z w v)
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/-- #### Finch Once Removed
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`F*xyzw = xwzy`
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-/
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def F_star (x : α → β → γ → δ) (y : γ) (z : β) (w : α) := x w z y
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notation "F*" => F_star
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/-- #### Finch
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`Fxyz = zyx`
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-/
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def F (x : α) (y : β) (z : β → α → γ) := z y x
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/-- #### Goldfinch
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`Gxyzw = xw(yz)`
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-/
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def G (x : α → γ → δ) (y : β → γ) (z : β) (w : α) := x w (y z)
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/-- #### Hummingbird
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`Hxyz = xyzy`
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-/
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def H (x : α → β → α → γ) (y : α) (z : β) := x y z y
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/-- #### Identity Bird
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`Ix = x`
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-/
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def I (x : α) : α := x
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/-- #### Kestrel
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`Kxy = x`
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-/
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def K (x : α) (_ : β) := x
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/-! #### Lark
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`Lxy = x(yy)`
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-/
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/-! #### Mockingbird
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`Mx = xx`
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-/
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/-- #### Owl
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`Oxy = y(xy)`
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-/
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def O (x : (α → β) → α) (y : α → β) := y (x y)
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/-- #### Phoenix
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`Φxyzw = x(yw)(zw)`
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-/
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def Φ (x : β → γ → δ) (y : α → β) (z : α → γ) (w : α) := x (y w) (z w)
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/-- #### Psi Bird
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`Ψxyzw = x(yz)(yw)`
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-/
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def Ψ (x : α → α → γ) (y : β → α) (z : β) (w : β) := x (y z) (y w)
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/-- #### Quacky Bird
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`Q₄xyz = z(yx)`
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-/
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def Q₄ (x : α) (y : α → β) (z : β → γ) := z (y x)
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/-- #### Queer Bird
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`Qxyz = y(xz)`
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-/
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def Q (x : α → β) (y : β → γ) (z : α) := y (x z)
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/-- #### Quirky Bird
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`Q₃xyz = z(xy)`
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-/
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def Q₃ (x : α → β) (y : α) (z : β → γ) := z (x y)
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/-- #### Quixotic Bird
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`Q₁xyz = x(zy)`
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-/
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def Q₁ (x : α → γ) (y : β) (z : β → α) := x (z y)
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/-- #### Quizzical Bird
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`Q₂xyz = y(zx)`
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-/
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def Q₂ (x : α) (y : β → γ) (z : α → β) := y (z x)
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/-- #### Robin Once Removed
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`R*xyzw = xzwy`
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-/
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def R_star (x : α → β → γ → δ) (y : γ) (z : α) (w : β) := x z w y
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notation "R*" => R_star
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/-- #### Robin
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`Rxyz = yzx`
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-/
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def R (x : α) (y : β → α → γ) (z : β) := y z x
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/-- #### Sage Bird
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`Θx = x(Θx)`
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-/
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partial def Θ [Inhabited α] (x : α → α) := x (Θ x)
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/-- #### Starling
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`Sxyz = xz(yz)`
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-/
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def S (x : α → β → γ) (y : α → β) (z : α) := x z (y z)
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/-- #### Thrush
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`Txy = yx`
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-/
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def T (x : α) (y : α → β) := y x
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/-! #### Turing Bird
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`Uxy = y(xxy)`
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-/
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/-- #### Vireo Once Removed
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`V*xyzw = xwyz`
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-/
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def V_star (x : α → β → γ → δ) (y : β) (z : γ) (w : α) := x w y z
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notation "V*" => V_star
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/-- #### Vireo
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`Vxyz = zxy`
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-/
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def V (x : α) (y : β) (z : α → β → γ) := z x y
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/-- #### Warbler
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`Wxy = xyy`
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-/
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def W (x : α → α → β) (y : α) := x y y
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