76 lines
2.7 KiB
Text
76 lines
2.7 KiB
Text
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/-
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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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Declaration of set-quotients, i.e. quotient of a mere relation which is then set-truncated.
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-/
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import .quotient .trunc
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open eq is_trunc trunc quotient equiv
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namespace set_quotient
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section
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parameters {A : Type} (R : A → A → hprop)
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-- set-quotients are just truncations of (type) quotients
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definition set_quotient : Type := trunc 0 (quotient (λa a', trunctype.carrier (R a a')))
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definition class_of (a : A) : set_quotient :=
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tr (class_of _ a)
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definition eq_of_rel {a a' : A} (H : R a a') : class_of a = class_of a' :=
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ap tr (eq_of_rel _ H)
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theorem is_hset_set_quotient : is_hset set_quotient :=
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begin unfold set_quotient, exact _ end
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protected definition rec {P : set_quotient → Type} [Pt : Πaa, is_hset (P aa)]
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(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a')
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(x : set_quotient) : P x :=
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begin
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apply (@trunc.rec_on _ _ P x),
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{ intro x', apply Pt},
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{ intro y, fapply (quotient.rec_on y),
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{ exact Pc},
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{ intros, apply equiv.to_inv !(pathover_compose _ tr), apply Pp}}
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end
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protected definition rec_on [reducible] {P : set_quotient → Type} (x : set_quotient)
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[Pt : Πaa, is_hset (P aa)] (Pc : Π(a : A), P (class_of a))
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a') : P x :=
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rec Pc Pp x
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theorem rec_eq_of_rel {P : set_quotient → Type} [Pt : Πaa, is_hset (P aa)]
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(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a')
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{a a' : A} (H : R a a') : apdo (rec Pc Pp) (eq_of_rel H) = Pp H :=
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!is_hset.elimo
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protected definition elim {P : Type} [Pt : is_hset P] (Pc : A → P)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') (x : set_quotient) : P :=
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rec Pc (λa a' H, pathover_of_eq (Pp H)) x
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protected definition elim_on [reducible] {P : Type} (x : set_quotient) [Pt : is_hset P]
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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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elim Pc Pp x
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theorem elim_eq_of_rel {P : Type} [Pt : is_hset P] (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 (elim Pc Pp) (eq_of_rel H) = Pp H :=
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begin
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apply eq_of_fn_eq_fn_inv !(pathover_constant (eq_of_rel H)),
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rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_eq_of_rel],
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end
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/-
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there are no theorems to eliminate to the universe here,
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because the universe is generally not a set
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-/
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end
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end set_quotient
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attribute set_quotient.class_of [constructor]
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attribute set_quotient.rec set_quotient.elim [unfold-c 7] [recursor 7]
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attribute set_quotient.rec_on set_quotient.elim_on [unfold-c 4]
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