61 lines
2.3 KiB
Text
61 lines
2.3 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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Authors: Floris van Doorn
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Quotients. This is a quotient without truncation for an arbitrary type-valued binary relation.
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See also .set_quotient
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-/
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/-
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The hit quotient is primitive, declared in init.hit.
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The constructors are, given {A : Type} (R : A → A → Type),
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* class_of : A → quotient R (A implicit, R explicit)
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* eq_of_rel : Π{a a' : A}, R a a' → class_of a = class_of a' (R explicit)
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-/
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open eq equiv sigma.ops
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namespace quotient
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variables {A : Type} {R : A → A → Type}
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protected definition elim {P : Type} (Pc : A → P) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a')
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(x : quotient R) : P :=
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quotient.rec Pc (λa a' H, pathover_of_eq (Pp H)) x
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protected definition elim_on [reducible] {P : Type} (x : quotient R)
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(Pc : A → P) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') : P :=
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quotient.elim Pc Pp x
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theorem elim_eq_of_rel {P : Type} (Pc : A → P)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') {a a' : A} (H : R a a')
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: ap (quotient.elim Pc Pp) (eq_of_rel R H) = Pp H :=
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begin
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apply eq_of_fn_eq_fn_inv !(pathover_constant (eq_of_rel R H)),
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rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑quotient.elim,rec_eq_of_rel],
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end
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protected definition elim_type (Pc : A → Type)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') : quotient R → Type :=
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quotient.elim Pc (λa a' H, ua (Pp H))
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protected definition elim_type_on [reducible] (x : quotient R) (Pc : A → Type)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') : Type :=
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quotient.elim_type Pc Pp x
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theorem elim_type_eq_of_rel (Pc : A → Type)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') {a a' : A} (H : R a a')
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: transport (quotient.elim_type Pc Pp) (eq_of_rel R H) = to_fun (Pp H) :=
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by rewrite [tr_eq_cast_ap_fn, ↑quotient.elim_type, elim_eq_of_rel];apply cast_ua_fn
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definition elim_type_uncurried (H : Σ(Pc : A → Type), Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a')
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: quotient R → Type :=
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quotient.elim_type H.1 H.2
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end quotient
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attribute quotient.rec [recursor]
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attribute quotient.elim [unfold 6] [recursor 6]
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attribute quotient.elim_type [unfold 5]
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attribute quotient.elim_on [unfold 4]
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attribute quotient.elim_type_on [unfold 3]
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