59 lines
2 KiB
Text
59 lines
2 KiB
Text
-- Copyright (c) 2014 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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import algebra.relation logic.nonempty data.subtype
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import logic.axioms.classical logic.axioms.hilbert logic.axioms.funext
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import .basic
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namespace quotient
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open relation nonempty subtype
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-- abstract quotient
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-- -----------------
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definition prelim_map {A : Type} (R : A → A → Prop) (a : A) :=
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-- TODO: it is interesting how the elaborator fails here
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-- epsilon (fun b, R a b)
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@epsilon _ (nonempty.intro a) (fun b, R a b)
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-- TODO: only needed R reflexive (or weaker: R a a)
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theorem prelim_map_rel {A : Type} {R : A → A → Prop} (H : is_equivalence R) (a : A)
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: R a (prelim_map R a) :=
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have reflR : reflexive R, from is_reflexive.infer R,
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epsilon_spec (exists_intro a (reflR a))
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-- TODO: only needed: R PER
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theorem prelim_map_congr {A : Type} {R : A → A → Prop} (H1 : is_equivalence R) {a b : A}
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(H2 : R a b) : prelim_map R a = prelim_map R b :=
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have symmR : symmetric R, from is_symmetric.infer R,
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have transR : transitive R, from is_transitive.infer R,
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have H3 : ∀c, R a c ↔ R b c, from
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take c,
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iff.intro
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(assume H4 : R a c, transR (symmR H2) H4)
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(assume H4 : R b c, transR H2 H4),
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have H4 : (fun c, R a c) = (fun c, R b c), from
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funext (take c, iff_to_eq (H3 c)),
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have H5 [visible] : nonempty A, from
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nonempty.intro a,
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show epsilon (λc, R a c) = epsilon (λc, R b c), from
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congr_arg _ H4
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definition quotient {A : Type} (R : A → A → Prop) : Type := image (prelim_map R)
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definition quotient_abs {A : Type} (R : A → A → Prop) : A → quotient R :=
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fun_image (prelim_map R)
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definition quotient_elt_of {A : Type} (R : A → A → Prop) : quotient R → A := elt_of
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-- TODO: I had to make is_quotient transparent -- change this?
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theorem quotient_is_quotient {A : Type} (R : A → A → Prop) (H : is_equivalence R)
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: is_quotient R (quotient_abs R) (quotient_elt_of R) :=
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representative_map_to_quotient_equiv
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H
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(prelim_map_rel H)
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(@prelim_map_congr _ _ H)
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end quotient
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