250 lines
7.3 KiB
Text
250 lines
7.3 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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abbreviation id {A : Type} (a : A) := a
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abbreviation compose {A : Type} {B : Type} {C : Type} (g : B → C) (f : A → B) := λ x, g (f x)
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infixl `∘`:60 := compose
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inductive path {A : Type} (a : A) : A → Type :=
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| refl : path a a
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infix `=`:50 := path
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definition transport {A : Type} {a b : A} {P : A → Type} (H1 : a = b) (H2 : P a) : P b
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:= path_rec H2 H1
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namespace logic
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notation p `*(`:75 u `)` := transport p u
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end
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using logic
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definition symm {A : Type} {a b : A} (p : a = b) : b = a
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:= p*(refl a)
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definition trans {A : Type} {a b c : A} (p1 : a = b) (p2 : b = c) : a = c
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:= p2*(p1)
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calc_subst transport
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calc_refl refl
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calc_trans trans
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namespace logic
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postfix `⁻¹`:100 := symm
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infixr `⬝`:75 := trans
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end
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using logic
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theorem trans_refl_right {A : Type} {x y : A} (p : x = y) : p = p ⬝ (refl y)
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:= refl p
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theorem trans_refl_left {A : Type} {x y : A} (p : x = y) : p = (refl x) ⬝ p
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:= path_rec (trans_refl_right (refl x)) p
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theorem refl_symm {A : Type} (x : A) : (refl x)⁻¹ = (refl x)
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:= refl (refl x)
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theorem refl_trans {A : Type} (x : A) : (refl x) ⬝ (refl x) = (refl x)
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:= refl (refl x)
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theorem trans_symm {A : Type} {x y : A} (p : x = y) : p ⬝ p⁻¹ = refl x
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:= have q : (refl x) ⬝ (refl x)⁻¹ = refl x, from
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((refl_symm x)⁻¹)*(refl_trans x),
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path_rec q p
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theorem symm_trans {A : Type} {x y : A} (p : x = y) : p⁻¹ ⬝ p = refl y
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:= have q : (refl x)⁻¹ ⬝ (refl x) = refl x, from
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((refl_symm x)⁻¹)*(refl_trans x),
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path_rec q p
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theorem symm_symm {A : Type} {x y : A} (p : x = y) : (p⁻¹)⁻¹ = p
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:= have q : ((refl x)⁻¹)⁻¹ = refl x, from
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refl (refl x),
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path_rec q p
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theorem trans_assoc {A : Type} {x y z w : A} (p : x = y) (q : y = z) (r : z = w) : p ⬝ (q ⬝ r) = (p ⬝ q) ⬝ r
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:= have e1 : (p ⬝ q) ⬝ (refl z) = p ⬝ q, from
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(trans_refl_right (p ⬝ q))⁻¹,
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have e2 : q ⬝ (refl z) = q, from
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(trans_refl_right q)⁻¹,
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have e3 : p ⬝ (q ⬝ (refl z)) = p ⬝ q, from
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e2*(refl (p ⬝ (q ⬝ (refl z)))),
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path_rec (e3 ⬝ e1⁻¹) r
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definition ap {A : Type} {B : Type} (f : A → B) {a b : A} (p : a = b) : f a = f b
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:= p*(refl (f a))
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theorem ap_refl {A : Type} {B : Type} (f : A → B) (a : A) : ap f (refl a) = refl (f a)
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:= refl (refl (f a))
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section
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parameters {A : Type} {B : Type} {C : Type}
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parameters (f : A → B) (g : B → C)
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parameters (x y z : A) (p : x = y) (q : y = z)
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theorem ap_trans_dist : ap f (p ⬝ q) = (ap f p) ⬝ (ap f q)
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:= have e1 : ap f (p ⬝ refl y) = (ap f p) ⬝ (ap f (refl y)), from refl _,
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path_rec e1 q
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theorem ap_inv_dist : ap f (p⁻¹) = (ap f p)⁻¹
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:= have e1 : ap f ((refl x)⁻¹) = (ap f (refl x))⁻¹, from refl _,
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path_rec e1 p
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theorem ap_compose : ap g (ap f p) = ap (g∘f) p
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:= have e1 : ap g (ap f (refl x)) = ap (g∘f) (refl x), from refl _,
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path_rec e1 p
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theorem ap_id : ap id p = p
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:= have e1 : ap id (refl x) = (refl x), from refl (refl x),
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path_rec e1 p
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end
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section
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parameters {A : Type} {B : A → Type} (f : Π x, B x)
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definition D [private] (x y : A) (p : x = y) := p*(f x) = f y
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definition d [private] (x : A) : D x x (refl x)
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:= refl (f x)
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theorem apd {a b : A} (p : a = b) : p*(f a) = f b
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:= path_rec (d a) p
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end
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abbreviation homotopy {A : Type} {P : A → Type} (f g : Π x, P x)
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:= Π x, f x = g x
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namespace logic
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infix `∼`:50 := homotopy
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end
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using logic
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notation `assume` binders `,` r:(scoped f, f) := r
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notation `take` binders `,` r:(scoped f, f) := r
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section
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parameters {A : Type} {B : Type}
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theorem hom_refl (f : A → B) : f ∼ f
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:= take x, refl (f x)
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theorem hom_symm {f g : A → B} : f ∼ g → g ∼ f
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:= assume h, take x, (h x)⁻¹
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theorem hom_trans {f g h : A → B} : f ∼ g → g ∼ h → f ∼ h
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:= assume h1 h2, take x, (h1 x) ⬝ (h2 x)
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theorem hom_fun {f g : A → B} {x y : A} (H : f ∼ g) (p : x = y) : (H x) ⬝ (ap g p) = (ap f p) ⬝ (H y)
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:= have e1 : (H x) ⬝ (refl (g x)) = (refl (f x)) ⬝ (H x), from
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calc (H x) ⬝ (refl (g x)) = H x : (trans_refl_right (H x))⁻¹
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... = (refl (f x)) ⬝ (H x) : trans_refl_left (H x),
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have e2 : (H x) ⬝ (ap g (refl x)) = (ap f (refl x)) ⬝ (H x), from
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calc (H x) ⬝ (ap g (refl x)) = (H x) ⬝ (refl (g x)) : {ap_refl g x}
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... = (refl (f x)) ⬝ (H x) : e1
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... = (ap f (refl x)) ⬝ (H x) : {symm (ap_refl f x)},
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path_rec e2 p
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end
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definition loop_space (A : Type) (a : A) := a = a
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notation `Ω` `(` A `,` a `)` := loop_space A a
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definition loop2d_space (A : Type) (a : A) := (refl a) = (refl a)
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notation `Ω²` `(` A `,` a `)` := loop2d_space A a
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inductive empty : Type
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theorem empty_elim (c : Type) (H : empty) : c
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:= empty_rec (λ e, c) H
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definition not (A : Type) := A → empty
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prefix `¬`:40 := not
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theorem not_intro {a : Type} (H : a → empty) : ¬ a
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:= H
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theorem not_elim {a : Type} (H1 : ¬ a) (H2 : a) : empty
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:= H1 H2
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theorem absurd {a : Type} (H1 : a) (H2 : ¬ a) : empty
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:= H2 H1
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theorem mt {a b : Type} (H1 : a → b) (H2 : ¬ b) : ¬ a
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:= assume Ha : a, absurd (H1 Ha) H2
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theorem contrapos {a b : Type} (H : a → b) : ¬ b → ¬ a
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:= assume Hnb : ¬ b, mt H Hnb
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theorem absurd_elim {a : Type} (b : Type) (H1 : a) (H2 : ¬ a) : b
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:= empty_elim b (absurd H1 H2)
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inductive unit : Type :=
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| star : unit
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notation `⋆`:max := star
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theorem absurd_not_unit (H : ¬ unit) : empty
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:= absurd star H
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theorem not_empty_trivial : ¬ empty
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:= assume H : empty, H
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theorem upun (x : unit) : x = ⋆
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:= unit_rec (refl ⋆) x
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inductive product (A : Type) (B : Type) : Type :=
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| pair : A → B → product A B
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infixr `×`:30 := product
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infixr `∧`:30 := product
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notation `(` h `,` t:(foldl `,` (e r, pair r e) h) `)` := t
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definition pr1 {A : Type} {B : Type} (p : A × B) : A
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:= product_rec (λ a b, a) p
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definition pr2 {A : Type} {B : Type} (p : A × B) : B
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:= product_rec (λ a b, b) p
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theorem uppt {A : Type} {B : Type} (p : A × B) : (pr1 p, pr2 p) = p
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:= product_rec (λ x y, refl (x, y)) p
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inductive sum (A : Type) (B : Type) : Type :=
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| inl : A → sum A B
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| inr : B → sum A B
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namespace logic
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infixr `+`:25 := sum
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end
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using logic
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infixr `∨`:25 := sum
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theorem sum_elim {a : Type} {b : Type} {c : Type} (H1 : a + b) (H2 : a → c) (H3 : b → c) : c
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:= sum_rec H2 H3 H1
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theorem resolve_right {a : Type} {b : Type} (H1 : a + b) (H2 : ¬ a) : b
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:= sum_elim H1 (assume Ha, absurd_elim b Ha H2) (assume Hb, Hb)
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theorem resolve_left {a : Type} {b : Type} (H1 : a + b) (H2 : ¬ b) : a
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:= sum_elim H1 (assume Ha, Ha) (assume Hb, absurd_elim a Hb H2)
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theorem sum_flip {a : Type} {b : Type} (H : a + b) : b + a
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:= sum_elim H (assume Ha, inr b Ha) (assume Hb, inl a Hb)
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inductive bool : Type :=
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| true : bool
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| false : bool
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theorem bool_cases (p : bool) : p = true ∨ p = false
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:= bool_rec (inl _ (refl true)) (inr _ (refl false)) p
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inductive Sigma {A : Type} (B : A → Type) : Type :=
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| sigma_intro : Π a, B a → Sigma B
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notation `Σ` binders `,` r:(scoped P, Sigma P) := r
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definition dpr1 {A : Type} {B : A → Type} (p : Σ x, B x) : A
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:= Sigma_rec (λ a b, a) p
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definition dpr2 {A : Type} {B : A → Type} (p : Σ x, B x) : B (dpr1 p)
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:= Sigma_rec (λ a b, b) p
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