215 lines
8 KiB
Text
215 lines
8 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Floris van Doorn
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Theorems about sums/coproducts/disjoint unions
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-/
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open lift eq is_equiv equiv equiv.ops prod prod.ops is_trunc sigma bool
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namespace sum
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universe variables u v u' v'
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variables {A : Type.{u}} {B : Type.{v}} (z z' : A + B) {P : A → Type.{u'}} {Q : A → Type.{v'}}
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protected definition eta : sum.rec inl inr z = z :=
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by induction z; all_goals reflexivity
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protected definition code [unfold 3 4] : A + B → A + B → Type.{max u v}
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| code (inl a) (inl a') := lift.{u v} (a = a')
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| code (inr b) (inr b') := lift.{v u} (b = b')
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| code _ _ := lift empty
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protected definition decode [unfold 3 4] : Π(z z' : A + B), sum.code z z' → z = z'
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| decode (inl a) (inl a') := λc, ap inl (down c)
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| decode (inl a) (inr b') := λc, empty.elim (down c) _
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| decode (inr b) (inl a') := λc, empty.elim (down c) _
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| decode (inr b) (inr b') := λc, ap inr (down c)
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variables {z z'}
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protected definition encode [unfold 3 4 5] (p : z = z') : sum.code z z' :=
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by induction p; induction z; all_goals exact up idp
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variables (z z')
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definition sum_eq_equiv [constructor] : (z = z') ≃ sum.code z z' :=
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equiv.MK sum.encode
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!sum.decode
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abstract begin
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intro c, induction z with a b, all_goals induction z' with a' b',
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all_goals (esimp at *; induction c with c),
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all_goals induction c, -- c either has type empty or a path
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all_goals reflexivity
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end end
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abstract begin
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intro p, induction p, induction z, all_goals reflexivity
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end end
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section
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variables {a a' : A} {b b' : B}
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definition eq_of_inl_eq_inl [unfold 5] (p : inl a = inl a' :> A + B) : a = a' :=
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down (sum.encode p)
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definition eq_of_inr_eq_inr [unfold 5] (p : inr b = inr b' :> A + B) : b = b' :=
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down (sum.encode p)
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definition empty_of_inl_eq_inr (p : inl a = inr b) : empty := down (sum.encode p)
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definition empty_of_inr_eq_inl (p : inr b = inl a) : empty := down (sum.encode p)
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/- Transport -/
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definition sum_transport (p : a = a') (z : P a + Q a)
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: p ▸ z = sum.rec (λa, inl (p ▸ a)) (λb, inr (p ▸ b)) z :=
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by induction p; induction z; all_goals reflexivity
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/- Pathovers -/
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definition etao (p : a = a') (z : P a + Q a)
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: z =[p] sum.rec (λa, inl (p ▸ a)) (λb, inr (p ▸ b)) z :=
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by induction p; induction z; all_goals constructor
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protected definition codeo (p : a = a') : P a + Q a → P a' + Q a' → Type.{max u' v'}
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| codeo (inl x) (inl x') := lift.{u' v'} (x =[p] x')
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| codeo (inr y) (inr y') := lift.{v' u'} (y =[p] y')
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| codeo _ _ := lift empty
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protected definition decodeo (p : a = a') : Π(z : P a + Q a) (z' : P a' + Q a'),
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sum.codeo p z z' → z =[p] z'
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| decodeo (inl x) (inl x') := λc, apo (λa, inl) p (down c)
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| decodeo (inl x) (inr y') := λc, empty.elim (down c) _
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| decodeo (inr y) (inl x') := λc, empty.elim (down c) _
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| decodeo (inr y) (inr y') := λc, apo (λa, inr) p (down c)
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variables {z z'}
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protected definition encodeo {p : a = a'} {z : P a + Q a} {z' : P a' + Q a'} (q : z =[p] z')
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: sum.codeo p z z' :=
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by induction q; induction z; all_goals exact up idpo
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variables (z z')
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definition sum_pathover_equiv [constructor] (p : a = a') (z : P a + Q a) (z' : P a' + Q a')
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: (z =[p] z') ≃ sum.codeo p z z' :=
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equiv.MK sum.encodeo
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!sum.decodeo
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abstract begin
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intro c, induction z with a b, all_goals induction z' with a' b',
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all_goals (esimp at *; induction c with c),
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all_goals induction c, -- c either has type empty or a pathover
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all_goals reflexivity
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end end
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abstract begin
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intro q, induction q, induction z, all_goals reflexivity
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end end
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end
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/- Functorial action -/
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variables {A' B' : Type} (f : A → A') (g : B → B')
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definition sum_functor [unfold 7] : A + B → A' + B'
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| sum_functor (inl a) := inl (f a)
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| sum_functor (inr b) := inr (g b)
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/- Equivalences -/
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definition is_equiv_sum_functor [constructor] [Hf : is_equiv f] [Hg : is_equiv g]
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: is_equiv (sum_functor f g) :=
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adjointify (sum_functor f g)
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(sum_functor f⁻¹ g⁻¹)
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abstract begin
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intro z, induction z,
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all_goals (esimp; (apply ap inl | apply ap inr); apply right_inv)
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end end
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abstract begin
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intro z, induction z,
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all_goals (esimp; (apply ap inl | apply ap inr); apply right_inv)
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end end
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definition sum_equiv_sum_of_is_equiv [constructor] [Hf : is_equiv f] [Hg : is_equiv g]
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: A + B ≃ A' + B' :=
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equiv.mk _ (is_equiv_sum_functor f g)
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definition sum_equiv_sum [constructor] (f : A ≃ A') (g : B ≃ B') : A + B ≃ A' + B' :=
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equiv.mk _ (is_equiv_sum_functor f g)
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definition sum_equiv_sum_left [constructor] (g : B ≃ B') : A + B ≃ A + B' :=
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sum_equiv_sum equiv.refl g
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definition sum_equiv_sum_right [constructor] (f : A ≃ A') : A + B ≃ A' + B :=
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sum_equiv_sum f equiv.refl
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definition flip [unfold 3] : A + B → B + A
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| flip (inl a) := inr a
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| flip (inr b) := inl b
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definition sum_comm_equiv [constructor] (A B : Type) : A + B ≃ B + A :=
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begin
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fapply equiv.MK,
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exact flip,
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exact flip,
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all_goals (intro z; induction z; all_goals reflexivity)
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end
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definition sum_assoc_equiv (A B C : Type) : A + (B + C) ≃ (A + B) + C :=
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begin
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fapply equiv.MK,
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all_goals try (intro z; induction z with u v;
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all_goals try induction u; all_goals try induction v),
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all_goals try (repeat append (append (apply inl) (apply inr)) assumption; now),
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all_goals reflexivity
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end
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definition sum_empty_equiv [constructor] (A : Type) : A + empty ≃ A :=
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begin
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fapply equiv.MK,
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intro z, induction z, assumption, contradiction,
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exact inl,
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intro a, reflexivity,
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intro z, induction z, reflexivity, contradiction
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end
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definition sum_rec_unc {P : A + B → Type} (fg : (Πa, P (inl a)) × (Πb, P (inr b))) : Πz, P z :=
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sum.rec fg.1 fg.2
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definition is_equiv_sum_rec [constructor] (P : A + B → Type)
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: is_equiv (sum_rec_unc : (Πa, P (inl a)) × (Πb, P (inr b)) → Πz, P z) :=
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begin
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apply adjointify sum_rec_unc (λf, (λa, f (inl a), λb, f (inr b))),
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intro f, apply eq_of_homotopy, intro z, focus (induction z; all_goals reflexivity),
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intro h, induction h with f g, reflexivity
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end
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definition equiv_sum_rec [constructor] (P : A + B → Type)
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: (Πa, P (inl a)) × (Πb, P (inr b)) ≃ Πz, P z :=
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equiv.mk _ !is_equiv_sum_rec
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definition imp_prod_imp_equiv_sum_imp [constructor] (A B C : Type)
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: (A → C) × (B → C) ≃ (A + B → C) :=
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!equiv_sum_rec
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definition is_trunc_sum (n : trunc_index) [HA : is_trunc (n.+2) A] [HB : is_trunc (n.+2) B]
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: is_trunc (n.+2) (A + B) :=
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begin
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apply is_trunc_succ_intro, intro z z',
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apply is_trunc_equiv_closed_rev, apply sum_eq_equiv,
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induction z with a b, all_goals induction z' with a' b', all_goals esimp,
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all_goals exact _,
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end
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/-
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Sums are equivalent to dependent sigmas where the first component is a bool.
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The current construction only works for A and B in the same universe.
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If we need it for A and B in different universes, we need to insert some lifts.
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-/
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definition sum_of_sigma_bool {A B : Type.{u}} (v : Σ(b : bool), bool.rec A B b) : A + B :=
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by induction v with b x; induction b; exact inl x; exact inr x
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definition sigma_bool_of_sum {A B : Type.{u}} (z : A + B) : Σ(b : bool), bool.rec A B b :=
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by induction z with a b; exact ⟨ff, a⟩; exact ⟨tt, b⟩
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definition sum_equiv_sigma_bool [constructor] (A B : Type.{u})
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: A + B ≃ Σ(b : bool), bool.rec A B b :=
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equiv.MK sigma_bool_of_sum
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sum_of_sigma_bool
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begin intro v, induction v with b x, induction b, all_goals reflexivity end
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begin intro z, induction z with a b, all_goals reflexivity end
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end sum
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